# MGL Manual

## 2 Introduction

### 2.1 Overview

MGL is a Common Lisp machine learning library by Gábor Melis with some parts originally contributed by Ravenpack International. It mainly concentrates on various forms of neural networks (boltzmann machines, feed-forward and recurrent backprop nets). Most of MGL is built on top of MGL-MAT so it has BLAS and CUDA support.

In general, the focus is on power and performance not on ease of use. Perhaps one day there will be a cookie cutter interface with restricted functionality if a reasonable compromise is found between power and utility.

Here is the official repository and the HTML documentation for the latest version.

### 2.3 Dependencies

MGL used to rely on LLA to interface to BLAS and LAPACK. That's mostly history by now, but configuration of foreign libraries is still done via LLA. See the README in LLA on how to set things up. Note that these days OpenBLAS is easier to set up and just as fast as ATLAS.

CL-CUDA and MGL-MAT are the two main dependencies and also the ones not yet in quicklisp, so just drop them into quicklisp/local-projects/. If there is no suitable GPU on the system or the CUDA SDK is not installed, MGL will simply fall back on using BLAS and Lisp code. Wrapping code in MGL-MAT:WITH-CUDA* is basically all that's needed to run on the GPU, and with MGL-MAT:CUDA-AVAILABLE-P one can check whether the GPU is really being used.

### 2.4 Code Organization

MGL consists of several packages dedicated to different tasks. For example, package MGL-RESAMPLE is about Resampling and MGL-GD is about Gradient Descent and so on. On one hand, having many packages makes it easier to cleanly separate API and implementation and also to explore into a specific task. At other times, they can be a hassle, so the MGL package itself reexports every external symbol found in all the other packages that make up MGL and MGL-MAT (see MAT Manual) on which it heavily relies.

One exception to this rule is the bundled, but independent MGL-GNUPLOT library.

The built in tests can be run with:

(ASDF:OOS 'ASDF:TEST-OP '#:MGL)


Note, that most of the tests are rather stochastic and can fail once in a while.

### 2.5 Glossary

Ultimately machine learning is about creating models of some domain. The observations in the modelled domain are called instances (also known as examples or samples). Sets of instances are called datasets. Datasets are used when fitting a model or when making predictions. Sometimes the word predictions is too specific, and the results obtained from applying a model to some instances are simply called results.

## 3 Datasets

###### [in package MGL-DATASET]

An instance can often be any kind of object of the user's choice. It is typically represented by a set of numbers which is called a feature vector or by a structure holding the feature vector, the label, etc. A dataset is a SEQUENCE of such instances or a Samplers object that produces instances.

• FN DATASET

Call FN with each instance in DATASET. This is basically equivalent to iterating over the elements of a sequence or a sampler (see Samplers).

• FN DATASETS &KEY (IMPUTE NIL IMPUTEP)

Call FN with a list of instances, one from each dataset in DATASETS. Return nothing. If IMPUTE is specified then iterate until the largest dataset is consumed imputing IMPUTE for missing values. If IMPUTE is not specified then iterate until the smallest dataset runs out.

(map-datasets #'prin1 '((0 1 2) (:a :b)))
.. (0 :A)(1 :B)

(map-datasets #'prin1 '((0 1 2) (:a :b)) :impute nil)
.. (0 :A)(1 :B)(2 NIL)


It is of course allowed to mix sequences with samplers:

(map-datasets #'prin1
(list '(0 1 2)
(make-sequence-sampler '(:a :b) :max-n-samples 2)))
.. (0 :A)(1 :B)


### 3.1 Samplers

Some algorithms do not need random access to the entire dataset and can work with a stream observations. Samplers are simple generators providing two functions: SAMPLE and FINISHEDP.

• SAMPLER

See if SAMPLER has run out of examples.

• SAMPLER MAX-SIZE

Return a list of samples of length at most MAX-SIZE or less if SAMPLER runs out.

• SEQ &KEY MAX-N-SAMPLES

Create a sampler that returns elements of SEQ in their original order. If MAX-N-SAMPLES is non-nil, then at most MAX-N-SAMPLES are sampled.

• SEQ &KEY MAX-N-SAMPLES (REORDER #'MGL-RESAMPLE:SHUFFLE)

Create a sampler that returns elements of SEQ in random order. If MAX-N-SAMPLES is non-nil, then at most MAX-N-SAMPLES are sampled. The first pass over a shuffled copy of SEQ, and this copy is reshuffled whenever the sampler reaches the end of it. Shuffling is performed by calling the REORDER function.

• #<FUNCTION-SAMPLER "infinitely empty" >

This is the default dataset for MGL-OPT:MINIMIZE. It's an infinite stream of NILs.

#### 3.1.1 Function Sampler

• FUNCTION-SAMPLER (:GENERATOR)

A generator function of no arguments that returns the next sample.

• FUNCTION-SAMPLER (:MAX-N-SAMPLES = NIL)

• FUNCTION-SAMPLER (:NAME = NIL)

An arbitrary object naming the sampler. Only used for printing the sampler object.

• FUNCTION-SAMPLER (:N-SAMPLES = 0)

## 4 Resampling

###### [in package MGL-RESAMPLE]

The focus of this package is on resampling methods such as cross-validation and bagging which can be used for model evaluation, model selection, and also as a simple form of ensembling. Data partitioning and sampling functions are also provided because they tend to be used together with resampling.

### 4.1 Partitions

The following functions partition a dataset (currently only SEQUENCEs are supported) into a number of partitions. For each element in the original dataset there is exactly one partition that contains it.

• FRACTIONS SEQ &KEY WEIGHT

Partition SEQ into a number of subsequences. FRACTIONS is either a positive integer or a list of non-negative real numbers. WEIGHT is NIL or a function that returns a non-negative real number when called with an element from SEQ. If FRACTIONS is a positive integer then return a list of that many subsequences with equal sum of weights bar rounding errors, else partition SEQ into subsequences, where the sum of weights of subsequence I is proportional to element I of FRACTIONS. If WEIGHT is NIL, then it's element is assumed to have the same weight.

To split into 5 sequences:

(fracture 5 '(0 1 2 3 4 5 6 7 8 9))
=> ((0 1) (2 3) (4 5) (6 7) (8 9))


To split into two sequences whose lengths are proportional to 2 and 3:

(fracture '(2 3) '(0 1 2 3 4 5 6 7 8 9))
=> ((0 1 2 3) (4 5 6 7 8 9))


• SEQ &KEY (KEY #'IDENTITY) (TEST #'EQL)

Return the list of strata of SEQ. SEQ is a sequence of elements for which the function KEY returns the class they belong to. Such classes are opaque objects compared for equality with TEST. A stratum is a sequence of elements with the same (under TEST) KEY.

(stratify '(0 1 2 3 4 5 6 7 8 9) :key #'evenp)
=> ((0 2 4 6 8) (1 3 5 7 9))


• FRACTIONS SEQ &KEY (KEY #'IDENTITY) (TEST #'EQL) WEIGHT

Similar to FRACTURE, but also makes sure that keys are evenly distributed among the partitions (see STRATIFY). It can be useful for classification tasks to partition the data set while keeping the distribution of classes the same.

Note that the sets returned are not in random order. In fact, they are sorted internally by KEY.

For example, to make two splits with approximately the same number of even and odd numbers:

(fracture-stratified 2 '(0 1 2 3 4 5 6 7 8 9) :key #'evenp)
=> ((0 2 1 3) (4 6 8 5 7 9))


### 4.2 Cross-validation

• DATA FN &KEY (N-FOLDS 5) (FOLDS (ALEXANDRIA:IOTA N-FOLDS)) (SPLIT-FN #'SPLIT-FOLD/MOD) PASS-FOLD

Map FN over the FOLDS of DATA split with SPLIT-FN and collect the results in a list. The simplest demonstration is:

(cross-validate '(0 1 2 3 4)
(lambda (test training)
(list test training))
:n-folds 5)
=> (((0) (1 2 3 4))
((1) (0 2 3 4))
((2) (0 1 3 4))
((3) (0 1 2 4))
((4) (0 1 2 3)))


Of course, in practice one would typically train a model and return the trained model and/or its score on TEST. Also, sometimes one may want to do only some of the folds and remember which ones they were:

(cross-validate '(0 1 2 3 4)
(lambda (fold test training)
(list :fold fold test training))
:folds '(2 3)
:pass-fold t)
=> ((:fold 2 (2) (0 1 3 4))
(:fold 3 (3) (0 1 2 4)))


Finally, the way the data is split can be customized. By default SPLIT-FOLD/MOD is called with the arguments DATA, the fold (from among FOLDS) and N-FOLDS. SPLIT-FOLD/MOD returns two values which are then passed on to FN. One can use SPLIT-FOLD/CONT or SPLIT-STRATIFIED or any other function that works with these arguments. The only real constraint is that FN has to take as many arguments (plus the fold argument if PASS-FOLD) as SPLIT-FN returns.

• SEQ FOLD N-FOLDS

Partition SEQ into two sequences: one with elements of SEQ with indices whose remainder is FOLD when divided with N-FOLDS, and a second one with the rest. The second one is the larger set. The order of elements remains stable. This function is suitable as the SPLIT-FN argument of CROSS-VALIDATE.

• SEQ FOLD N-FOLDS

Imagine dividing SEQ into N-FOLDS subsequences of the same size (bar rounding). Return the subsequence of index FOLD as the first value and the all the other subsequences concatenated into one as the second value. The order of elements remains stable. This function is suitable as the SPLIT-FN argument of CROSS-VALIDATE.

• SEQ FOLD N-FOLDS &KEY (KEY #'IDENTITY) (TEST #'EQL) WEIGHT

Split SEQ into N-FOLDS partitions (as in FRACTURE-STRATIFIED). Return the partition of index FOLD as the first value, and the concatenation of the rest as the second value. This function is suitable as the SPLIT-FN argument of CROSS-VALIDATE (mostly likely as a closure with KEY, TEST, WEIGHT bound).

### 4.3 Bagging

• SEQ FN &KEY (RATIO 1) N WEIGHT (REPLACEMENT T) KEY (TEST #'EQL) (RANDOM-STATE *RANDOM-STATE*)

Sample from SEQ with SAMPLE-FROM (passing RATIO, WEIGHT, REPLACEMENT), or SAMPLE-STRATIFIED if KEY is not NIL. Call FN with the sample. If N is NIL then keep repeating this until FN performs a non-local exit. Else N must be a non-negative integer, N iterations will be performed, the primary values returned by FN collected into a list and returned. See SAMPLE-FROM and SAMPLE-STRATIFIED for examples.

• RATIO SEQ &KEY WEIGHT REPLACEMENT (RANDOM-STATE *RANDOM-STATE*)

Return a sequence constructed by sampling with or without REPLACEMENT from SEQ. The sum of weights in the result sequence will approximately be the sum of weights of SEQ times RATIO. If WEIGHT is NIL then elements are assumed to have equal weights, else WEIGHT should return a non-negative real number when called with an element of SEQ.

To randomly select half of the elements:

(sample-from 1/2 '(0 1 2 3 4 5))
=> (5 3 2)

To randomly select some elements such that the sum of their weights constitute about half of the sum of weights across the whole sequence:

(sample-from 1/2 '(0 1 2 3 4 5 6 7 8 9) :weight #'identity)
=> ;; sums to 28 that's near 45/2
(9 4 1 6 8)

To sample with replacement (that is, allowing the element to be sampled multiple times):

(sample-from 1 '(0 1 2 3 4 5) :replacement t)
=> (1 1 5 1 4 4)

• RATIO SEQ &KEY WEIGHT REPLACEMENT (KEY #'IDENTITY) (TEST #'EQL) (RANDOM-STATE *RANDOM-STATE*)

Like SAMPLE-FROM but makes sure that the weighted proportion of classes in the result is approximately the same as the proportion in SEQ. See STRATIFY for the description of KEY and TEST.

### 4.4 CV Bagging

• DATA FN &KEY N (N-FOLDS 5) (FOLDS (ALEXANDRIA:IOTA N-FOLDS)) (SPLIT-FN #'SPLIT-FOLD/MOD) PASS-FOLD (RANDOM-STATE *RANDOM-STATE*)

Perform cross-validation on different shuffles of DATA N times and collect the results. Since CROSS-VALIDATE collects the return values of FN, the return value of this function is a list of lists of FN results. If N is NIL, don't collect anything just keep doing repeated CVs until FN performs a non-local exit.

The following example simply collects the test and training sets for 2-fold CV repeated 3 times with shuffled data:

;;; This is non-deterministic.
(bag-cv '(0 1 2 3 4) #'list :n 3 :n-folds 2)
=> ((((2 3 4) (1 0))
((1 0) (2 3 4)))
(((2 1 0) (4 3))
((4 3) (2 1 0)))
(((1 0 3) (2 4))
((2 4) (1 0 3))))

CV bagging is useful when a single CV is not producing stable results. As an ensemble method, CV bagging has the advantage over bagging that each example will occur the same number of times and after the first CV is complete there is a complete but less reliable estimate for each example which gets refined by further CVs.

### 4.5 Miscellaneous Operations

• SEQ &KEY (KEY #'IDENTITY) (TEST #'EQL)

Return a sequence that's a reordering of SEQ such that elements belonging to different strata (under KEY and TEST, see STRATIFY) are distributed evenly. The order of elements belonging to the same stratum is unchanged.

For example, to make sure that even and odd numbers are distributed evenly:

(spread-strata '(0 2 4 6 8 1 3 5 7 9) :key #'evenp)
=> (0 1 2 3 4 5 6 7 8 9)


Same thing with unbalanced classes:

(spread-strata (vector 0 2 3 5 6 1 4)
:key (lambda (x)
(if (member x '(1 4))
t
nil)))
=> #(0 1 2 3 4 5 6)


• SEQS &KEY RESULT-TYPE

Make a single sequence out of the sequences in SEQS so that in the returned sequence indices of elements belonging to the same source sequence are spread evenly across the whole range. The result is a list is RESULT-TYPE is LIST, it's a vector if RESULT-TYPE is VECTOR. If RESULT-TYPE is NIL, then it's determined by the type of the first sequence in SEQS.

(zip-evenly '((0 2 4) (1 3)))
=> (0 1 2 3 4)


## 5 Core

### 5.1 Persistence

• FILENAME OBJECT

Load weights of OBJECT from FILENAME. Return OBJECT.

• FILENAME OBJECT &KEY (IF-EXISTS :ERROR) (ENSURE T)

Save weights of OBJECT to FILENAME. If ENSURE, then ENSURE-DIRECTORIES-EXIST is called on FILENAME. IF-EXISTS is passed on to OPEN. Return OBJECT.

• OBJECT STREAM

Read the weights of OBJECT from the bivalent STREAM where weights mean the learnt parameters. There is currently no sanity checking of data which will most certainly change in the future together with the serialization format. Return OBJECT.

• OBJECT STREAM

Write weight of OBJECT to the bivalent STREAM. Return OBJECT.

### 5.2 Batch Processing

Processing instances one by one during training or prediction can be slow. The models that support batch processing for greater efficiency are said to be striped.

Typically, during or after creating a model, one sets MAX-N-STRIPES on it a positive integer. When a batch of instances is to be fed to the model it is first broken into subbatches of length that's at most MAX-N-STRIPES. For each subbatch, SET-INPUT (FIXDOC) is called and a before method takes care of setting N-STRIPES to the actual number of instances in the subbatch. When MAX-N-STRIPES is set internal data structures may be resized which is an expensive operation. Setting N-STRIPES is a comparatively cheap operation, often implemented as matrix reshaping.

Note that for models made of different parts (for example, MGL-BP:BPN consists of MGL-BP:LUMPs) , setting these values affects the constituent parts, but one should never change the number stripes of the parts directly because that would lead to an internal inconsistency in the model.

• The number of stripes with which the OBJECT is capable of dealing simultaneously.

• MAX-N-STRIPES OBJECT

Allocate the necessary stuff to allow for MAX-N-STRIPES number of stripes to be worked with simultaneously in OBJECT. This is called when MAX-N-STRIPES is SETF'ed.

• N-STRIPES OBJECT

Set the number of stripes (out of MAX-N-STRIPES) that are in use in OBJECT. This is called when N-STRIPES is SETF'ed.

• SPECS &BODY BODY

Bind start and optionally end indices belonging to stripes in striped objects.

(WITH-STRIPES ((STRIPE1 OBJECT1 START1 END1)
(STRIPE2 OBJECT2 START2)
...)
...)


This is how one's supposed to find the index range corresponding to the Nth input in an input lump of a bpn:

 (with-stripes ((n input-lump start end))
(loop for i upfrom start below end
do (setf (mref (nodes input-lump) i) 0d0)))


Note how the input lump is striped, but the matrix into which we are indexing (NODES) is not known to WITH-STRIPES. In fact, for lumps the same stripe indices work with NODES and MGL-BP:DERIVATIVES.

• STRIPE OBJECT

Return the start index of STRIPE in some array or matrix of OBJECT.

• STRIPE OBJECT

Return the end index (exclusive) of STRIPE in some array or matrix of OBJECT.

• INSTANCES MODEL

Set INSTANCES as inputs in MODEL. SAMPLES is always a SEQUENCE of instances even for models not capable of batch operation. It sets N-STRIPES to (LENGTH INSTANCES) in a :BEFORE method.

• FN DATASET MODEL

Call FN with batches of instances from DATASET suitable for MODEL. The number of instances in a batch is MAX-N-STRIPES of MODEL or less if there are no more instances left.

### 5.3 Executors

• FN INSTANCES PROTOTYPE-EXECUTOR

Divide INSTANCES between executors that perform the same function as PROTOTYPE-EXECUTOR and call FN with the instances and the executor for which the instances are.

Some objects conflate function and call: the forward pass of a MGL-BP:BPN computes output from inputs so it is like a function but it also doubles as a function call in the sense that the bpn (function) object changes state during the computation of the output. Hence not even the forward pass of a bpn is thread safe. There is also the restriction that all inputs must be of the same size.

For example, if we have a function that builds bpn a for an input of a certain size, then we can create a factory that creates bpns for a particular call. The factory probably wants to keep the weights the same though. In Parameterized Executor Cache, MAKE-EXECUTOR-WITH-PARAMETERS is this factory.

Parallelization of execution is another possibility MAP-OVER-EXECUTORS allows, but there is no prebuilt solution for it, yet.

The default implementation simply calls FN with INSTANCES and PROTOTYPE-EXECUTOR.

#### 5.3.1 Parameterized Executor Cache

• PARAMETERS CACHE

Create a new executor for PARAMETERS. CACHE is a PARAMETERIZED-EXECUTOR-CACHE-MIXIN. In the BPN gaussian process example, PARAMETERS would be a list of input dimensions.

## 6 Monitoring

###### [in package MGL-CORE]

When training or applying a model, one often wants to track various statistics. For example, in the case of training a neural network with cross-entropy loss, these statistics could be the average cross-entropy loss itself, classification accuracy, or even the entire confusion matrix and sparsity levels in hidden layers. Also, there is the question of what to do with the measured values (log and forget, add to some counter or a list).

So there may be several phases of operation when we want to keep an eye on. Let's call these events. There can also be many fairly independent things to do in response to an event. Let's call these monitors. Some monitors are a composition of two operations: one that extracts some measurements and another that aggregates those measurements. Let's call these two measurers and counters, respectively.

For example, consider training a backpropagation neural network. We want to look at the state of of network just after the backward pass. MGL-BP:BP-LEARNER has a [MONITORS] event hook corresponding to the moment after backpropagating the gradients. Suppose we are interested in how the training cost evolves:

(push (make-instance 'monitor
:measurer (lambda (instances bpn)
(declare (ignore instances))
(mgl-bp:cost bpn))
:counter (make-instance 'basic-counter))
(monitors learner))


During training, this monitor will track the cost of training examples behind the scenes. If we want to print and reset this monitor periodically we can put another monitor on MGL-OPT:ITERATIVE-OPTIMIZER's MGL-OPT:ON-N-INSTANCES-CHANGED accessor:

(push (lambda (optimizer gradient-source n-instances)
(declare (ignore optimizer))
(when (zerop (mod n-instances 1000))
(format t "n-instances: ~S~%" n-instances)
(when (counter monitor)
(format t "~A~%" (counter monitor))
(reset-counter (counter monitor)))))
(mgl-opt:on-n-instances-changed optimizer))


Note that the monitor we push can be anything as long as APPLY-MONITOR is implemented on it with the appropriate signature. Also note that the ZEROP + MOD logic is fragile, so you will likely want to use MGL-OPT:MONITOR-OPTIMIZATION-PERIODICALLY instead of doing the above.

So that's the general idea. Concrete events are documented where they are signalled. Often there are task specific utilities that create a reasonable set of default monitors (see Classification Monitors).

• MONITORS &REST ARGUMENTS

Call APPLY-MONITOR on each monitor in MONITORS and ARGUMENTS. This is how an event is fired.

• MONITOR &REST ARGUMENTS

Apply MONITOR to ARGUMENTS. This sound fairly generic, because it is. MONITOR can be anything, even a simple function or symbol, in which case this is just CL:APPLY. See Monitors for more.

• MONITOR

Return an object representing the state of MONITOR or NIL, if it doesn't have any (say because it's a simple logging function). Most monitors have counters into which they accumulate results until they are printed and reset. See Counters for more.

• FN DATASET MODEL MONITORS

Call FN with batches of instances from DATASET until it runs out (as in DO-BATCHES-FOR-MODEL). FN is supposed to apply MODEL to the batch and return some kind of result (for neural networks, the result is the model state itself). Apply MONITORS to each batch and the result returned by FN for that batch. Finally, return the list of counters of MONITORS.

The purpose of this function is to collect various results and statistics (such as error measures) efficiently by applying the model only once, leaving extraction of quantities of interest from the model's results to MONITORS.

See the model specific versions of this functions such as MGL-BP:MONITOR-BPN-RESULTS.

• OBJECT

Return monitors associated with OBJECT. See various methods such as [MONITORS] for more documentation.

### 6.1 Monitors

• A monitor that has another monitor called MEASURER embedded in it. When this monitor is applied, it applies the measurer and passes the returned values to ADD-TO-COUNTER called on its COUNTER slot. One may further specialize APPLY-MONITOR to change that.

This class is useful when the same event monitor is applied repeatedly over a period and its results must be aggregated such as when training statistics are being tracked or when predictions are begin made. Note that the monitor must be compatible with the event it handles. That is, the embedded MEASURER must be prepared to take the arguments that are documented to come with the event.

• MONITOR (:COUNTER)

The COUNTER of a monitor carries out the aggregation of results returned by MEASURER. The See Counters for a library of counters.

### 6.2 Measurers

MEASURER is a part of MONITOR objects, an embedded monitor that computes a specific quantity (e.g. classification accuracy) from the arguments of event it is applied to (e.g. the model results). Measurers are often implemented by combining some kind of model specific extractor with a generic measurer function.

All generic measurer functions return their results as multiple values matching the arguments of ADD-TO-COUNTER for a counter of a certain type (see Counters) so as to make them easily used in a MONITOR:

(multiple-value-call #'add-to-counter <some-counter>
<call-to-some-measurer>)


The counter class compatible with the measurer this way is noted for each function.

For a list of measurer functions see Classification Measurers.

### 6.3 Counters

• COUNTER &REST ARGS

Add ARGS to COUNTER in some way. See specialized methods for type specific documentation. The kind of arguments to be supported is the what the measurer functions (see Measurers) intended to be paired with the counter return as multiple values.

• COUNTER

Return any number of values representing the state of COUNTER. See specialized methods for type specific documentation.

• Return any number of values representing the state of COUNTER in such a way that passing the returned values as arguments ADD-TO-COUNTER on a fresh instance of the same type recreates the original state.

• COUNTER

Restore state of COUNTER to what it was just after creation.

#### 6.3.1 Attributes

• This is a utility class that all counters subclass. The ATTRIBUTES plist can hold basically anything. Currently the attributes are only used when printing and they can be specified by the user. The monitor maker functions such as those in Classification Monitors also add attributes of their own to the counters they create.

With the :PREPEND-ATTRIBUTES initarg when can easily add new attributes without clobbering the those in the :INITFORM, (:TYPE "rmse") in this case.

(princ (make-instance 'rmse-counter
:prepend-attributes '(:event "pred."
:dataset "test")))
;; pred. test rmse: 0.000e+0 (0)
=> #<RMSE-COUNTER pred. test rmse: 0.000e+0 (0)>


• ATTRIBUTED (:ATTRIBUTES = NIL)

A plist of attribute keys and values.

• (ATTRIBUTEDS) &BODY BODY

Note the width of values for each attribute key which is the number of characters in the value's PRINC-TO-STRING'ed representation. In BODY, if attributes with they same key are printed they are forced to be at least this wide. This allows for nice, table-like output:

(let ((attributeds
(list (make-instance 'basic-counter
:attributes '(:a 1 :b 23 :c 456))
(make-instance 'basic-counter
:attributes '(:a 123 :b 45 :c 6)))))
(map nil (lambda (attributed)
(format t "~A~%" attributed))
attributeds)))
;; 1   23 456: 0.000e+0 (0)
;; 123 45 6  : 0.000e+0 (0)


• ATTRIBUTEDS

Log (see LOG-MSG) ATTRIBUTEDS non-escaped (as in PRINC or ~A) with the output being as table-like as possible.

#### 6.3.2 Counter classes

In addition to the really basic ones here, also see Classification Counters.

• A simple counter whose ADD-TO-COUNTER takes two additional parameters: an increment to the internal sums of called the NUMERATOR and DENOMINATOR. COUNTER-VALUES returns two values:

• NUMERATOR divided by DENOMINATOR (or 0 if DENOMINATOR is 0) and

• DENOMINATOR

Here is an example the compute the mean of 5 things received in two batches:

 (let ((counter (make-instance 'basic-counter)))
counter)
=> #<BASIC-COUNTER 2.00000e+0 (5)>


• A counter that simply concatenates sequences.

cl-transcript (let ((counter (make-instance 'concat-counter))) (add-to-counter counter '(1 2 3) #(4 5)) (add-to-counter counter '(6 7)) (counter-values counter)) => (1 2 3 4 5 6 7) 

• CONCAT-COUNTER (:CONCATENATION-TYPE = 'LIST)

A type designator suitable as the RESULT-TYPE argument to CONCATENATE.

## 7 Classification

###### [in package MGL-CORE]

To be able to measure classification related quantities, we need to define what the label of an instance is. Customization is possible by implementing a method for a specific type of instance, but these functions only ever appear as defaults that can be overridden.

• INSTANCE

Return the label of INSTANCE as a non-negative integer.

• Return a one dimensional array of probabilities representing the distribution of labels. The probability of the label with LABEL-INDEX I is element at index I of the returned arrray.

The following two functions are basically the same as the previous two, but in batch mode: they return a sequence of label indices or distributions. These are called on results produced by models. Implement these for a model and the monitor maker functions below will automatically work. See FIXDOC: for bpn and boltzmann.

• RESULTS

Return a sequence of label indices for RESULTS produced by some model for a batch of instances. This is akin to LABEL-INDEX.

### 7.1 Classification Monitors

The following functions return a list monitors. The monitors are for events of signature (INSTANCES MODEL) such as those produced by MONITOR-MODEL-RESULTS and its various model specific variations. They are model-agnostic functions, extensible to new classifier types.

The monitor makers above can be extended to support new classifier types via the following generic functions.

• MODEL OPERATION-MODE LABEL-INDEX-FN ATTRIBUTES

Identical to MAKE-CLASSIFICATION-ACCURACY-MONITORS bar the keywords arguments. Specialize this to add to support for new model types. The default implementation also allows for some extensibility: if LABEL-INDICES is defined on MODEL, then it will be used to extract label indices from model results.

• MODEL OPERATION-MODE LABEL-INDEX-DISTRIBUTION-FN ATTRIBUTES

Identical to MAKE-CROSS-ENTROPY-MONITORS bar the keywords arguments. Specialize this to add to support for new model types. The default implementation also allows for some extensibility: if LABEL-INDEX-DISTRIBUTIONS is defined on MODEL, then it will be used to extract label distributions from model results.

### 7.2 Classification Measurers

The functions here compare some known good solution (also known as ground truth or target) to a prediction or approximation and return some measure of their [dis][]similarity. They are model independent, hence one has to extract the ground truths and predictions first. Rarely used directly, they are mostly hidden behind Classification Monitors.

• TRUTHS PREDICTIONS &KEY (TEST #'EQL) TRUTH-KEY PREDICTION-KEY WEIGHT

Return the number of correct classifications and as the second value the number of instances (equal to length of TRUTHS in the non-weighted case). TRUTHS (keyed by TRUTH-KEY) is a sequence of opaque class labels compared with TEST to another sequence of classes labels in PREDICTIONS (keyed by PREDICTION-KEY). If WEIGHT is non-nil, then it is a function that returns the weight of an element of TRUTHS. Weighted cases add their weight to both counts (returned as the first and second values) instead of 1 as in the non-weighted case.

Note how the returned values are suitable for MULTIPLE-VALUE-CALL with #'ADD-TO-COUNTER and a CLASSIFICATION-ACCURACY-COUNTER.

• TRUTHS PREDICTIONS &KEY TRUTH-KEY PREDICTION-KEY (MIN-PREDICTION-PR 1.0d-15)

Return the sum of the cross-entropy between pairs of elements with the same index of TRUTHS and PREDICTIONS. TRUTH-KEY is a function that's when applied to an element of TRUTHS returns a sequence representing some kind of discrete target distribution (P in the definition below). TRUTH-KEY may be NIL which is equivalent to the IDENTITY function. PREDICTION-KEY is the same kind of key for PREDICTIONS, but the sequence it returns represents a distribution that approximates (Q below) the true one.

Cross-entropy of the true and approximating distributions is defined as:

cross-entropy(p,q) = - sum_i p(i) * log(q(i))


of which this function returns the sum over the pairs of elements of TRUTHS and PREDICTIONS keyed by TRUTH-KEY and PREDICTION-KEY.

Due to the logarithm, if q(i) is close to zero, we run into numerical problems. To prevent this, all q(i) that are less than MIN-PREDICTION-PR are treated as if they were MIN-PREDICTION-PR.

The second value returned is the sum of p(i) over all TRUTHS and all I. This is normally equal to (LENGTH TRUTHS), since elements of TRUTHS represent a probability distribution, but this is not enforced which allows relative importance of elements to be controlled.

The third value returned is a plist that maps each index occurring in the distribution sequences to a list of two elements:

 sum_j p_j(i) * log(q_j(i))


and

sum_j p_j(i)


where J indexes into TRUTHS and PREDICTIONS.

(measure-cross-entropy '((0 1 0)) '((0.1 0.7 0.2)))
=> 0.35667497
1
(2 (0.0 0)
1 (0.35667497 1)
0 (0.0 0))


Note how the returned values are suitable for MULTIPLE-VALUE-CALL with #'ADD-TO-COUNTER and a CROSS-ENTROPY-COUNTER.

• PREDICTIONS PRED &KEY (KEY #'IDENTITY) WEIGHT

Return the area under the ROC curve for PREDICTIONS representing predictions for a binary classification problem. PRED is a predicate function for deciding whether a prediction belongs to the so called positive class. KEY returns a number for each element which is the predictor's idea of how much that element is likely to belong to the class, although it's not necessarily a probability.

If WEIGHT is NIL, then all elements of PREDICTIONS count as 1 towards the unnormalized sum within AUC. Else WEIGHT must be a function like KEY, but it should return the importance (a positive real number) of elements. If the weight of an prediction is 2 then it's as if there were another identical copy of that prediction in PREDICTIONS.

The algorithm is based on algorithm 2 in the paper 'An introduction to ROC analysis' by Tom Fawcett.

ROC AUC is equal to the probability of a randomly chosen positive having higher KEY (score) than a randomly chosen negative element. With equal scores in mind, a more precise version is: AUC is the expectation of the above probability over all possible sequences sorted by scores.

• TRUTHS PREDICTIONS &KEY (TEST #'EQL) TRUTH-KEY PREDICTION-KEY WEIGHT

Create a CONFUSION-MATRIX from TRUTHS and PREDICTIONS. TRUTHS (keyed by TRUTH-KEY) is a sequence of class labels compared with TEST to another sequence of class labels in PREDICTIONS (keyed by PREDICTION-KEY). If WEIGHT is non-nil, then it is a function that returns the weight of an element of TRUTHS. Weighted cases add their weight to both counts (returned as the first and second values).

Note how the returned confusion matrix can be added to another with ADD-TO-COUNTER.

### 7.3 Classification Counters

• A BASIC-COUNTER with "acc." as its :TYPE attribute and a PRINT-OBJECT method that prints percentages.

#### 7.3.1 Confusion Matrices

• A confusion matrix keeps count of classification results. The correct class is called target' and the output of the classifier is calledprediction'.

• &KEY (TEST #'EQL)

Classes are compared with TEST.

• MATRIX CLASSES

Return a list of CLASSES sorted for presentation purposes.

• MATRIX CLASS

Name of CLASS for presentation purposes.

• MATRIX TARGET PREDICTION

• FN MATRIX

Call FN with TARGET, PREDICTION, COUNT paramaters for each cell in the confusion matrix. Cells with a zero count may be ommitted.

• A list of all classes. The default is to collect classes from the counts. This can be overridden if, for instance, some classes are not present in the results.

• MATRIX &KEY FILTER

Return the overall accuracy of the results in MATRIX. It's computed as the number of correctly classified cases (hits) divided by the name of cases. Return the number of hits and the number of cases as the second and third value. If FILTER function is given, then call it with the target and the prediction of the cell. Disregard cell for which FILTER returns NIL.

Precision and recall can be easily computed by giving the right filter, although those are provided in separate convenience functions.

• MATRIX PREDICTION

Return the accuracy over the cases when the classifier said PREDICTION.

• MATRIX TARGET

Return the accuracy over the cases when the correct class is TARGET.

• MATRIX RESULT-MATRIX

Add MATRIX into RESULT-MATRIX.

## 8 Features

### 8.1 Feature Selection

The following scoring functions all return an EQUAL hash table that maps features to scores.

• DOCUMENTS MAPPER &KEY (KEY #'IDENTITY)

Return scored features as an EQUAL hash table whose keys are features of DOCUMENTS and values are counts of occurrences of features. MAPPER takes a function and a document and calls function with features of the document.

(sort (alexandria:hash-table-alist
(count-features '(("hello" "world")
("this" "is" "our" "world"))
(lambda (fn document)
(map nil fn document))))
#'string< :key #'car)
=> (("hello" . 1) ("is" . 1) ("our" . 1) ("this" . 1) ("world" . 2))


• DOCUMENTS MAPPER CLASS-FN &KEY (CLASSES (ALL-DOCUMENT-CLASSES DOCUMENTS CLASS-FN))

Return scored features as an EQUAL hash table whose keys are features of DOCUMENTS and values are their log likelihood ratios. MAPPER takes a function and a document and calls function with features of the document.

(sort (alexandria:hash-table-alist
(feature-llrs '((:a "hello" "world")
(:b "this" "is" "our" "world"))
(lambda (fn document)
(map nil fn (rest document)))
#'first))
#'string< :key #'car)
=> (("hello" . 2.6032386) ("is" . 2.6032386) ("our" . 2.6032386)
("this" . 2.6032386) ("world" . 4.8428774e-8))


• DOCUMENTS MAPPER CLASS-FN &KEY (CLASSES (ALL-DOCUMENT-CLASSES DOCUMENTS CLASS-FN))

Return scored features as an EQUAL hash table whose keys are features of DOCUMENTS and values are their disambiguities. MAPPER takes a function and a document and calls function with features of the document.

From the paper 'Using Ambiguity Measure Feature Selection Algorithm for Support Vector Machine Classifier'.

### 8.2 Feature Encoding

Features can rarely be fed directly to algorithms as is, they need to be transformed in some way. Suppose we have a simple language model that takes a single word as input and predicts the next word. However, both input and output is to be encoded as float vectors of length 1000. What we do is find the top 1000 words by some measure (see Feature Selection) and associate these words with the integers in [0..999][] (this is ENCODEing). By using for example one-hot encoding, we translate a word into a float vector when passing in the input. When the model outputs the probability distribution of the next word, we find the index of the max and find the word associated with it (this is DECODEing)

• ENCODER DECODED

Encode DECODED with ENCODER. This interface is generic enough to be almost meaningless. See ENCODER/DECODER for a simple, MGL-NLP:BAG-OF-WORDS-ENCODER for a slightly more involved example.

If ENCODER is a function designator, then it's simply FUNCALLed with DECODED.

• DECODER ENCODED

Decode ENCODED with ENCODER. For an DECODER / ENCODER pair, (DECODE DECODER (ENCODE ENCODER OBJECT)) must be equal in some sense to OBJECT.

If DECODER is a function designator, then it's simply FUNCALLed with ENCODED.

• Implements O(1) ENCODE and DECODE by having an internal decoded-to-encoded and an encoded-to-decoded EQUAL hash table. ENCODER/DECODER objects can be saved and loaded (see Persistence) as long as the elements in the hash tables have read/write consitency.

(let ((indexer
(make-indexer
(alexandria:alist-hash-table '(("I" . 3) ("me" . 2) ("mine" . 1)))
2)))
(values (encode indexer "I")
(encode indexer "me")
(encode indexer "mine")
(decode indexer 0)
(decode indexer 1)
(decode indexer 2)))
=> 0
=> 1
=> NIL
=> "I"
=> "me"
=> NIL


Also see Bag of Words.

###### [in package MGL-OPT]

We have a real valued, differentiable function F and the task is to find the parameters that minimize its value. Optimization starts from a single point in the parameter space of F, and this single point is updated iteratively based on the gradient and value of F at or around the current point.

Note that while the stated problem is that of global optimization, for non-convex functions, most algorithms will tend to converge to a local optimum.

Currently, there are two optimization algorithms: Gradient Descent (with several variants) and Conjugate Gradient both of which are first order methods (they do not need second order gradients) but more can be added with the Extension API.

• OPTIMIZER GRADIENT-SOURCE &KEY (WEIGHTS (LIST-SEGMENTS GRADIENT-SOURCE)) (DATASET *INFINITELY-EMPTY-DATASET*)

Minimize the value of the real valued function represented by GRADIENT-SOURCE by updating some of its parameters in WEIGHTS (a MAT or a sequence of MATs). Return WEIGHTS. DATASET (see Datasets) is a set of unoptimized parameters of the same function. For example, WEIGHTS may be the weights of a neural network while DATASET is the training set consisting of inputs suitable for SET-INPUT. The default DATASET, (*INFINITELY-EMPTY-DATASET*) is suitable for when all parameters are optimized, so there is nothing left to come from the environment.

Optimization terminates if DATASET is a sampler and it runs out or when some other condition met (see TERMINATION, for example). If DATASET is a SEQUENCE, then it is reused over and over again.

Examples for various optimizers are provided in Gradient Descent and Conjugate Gradient.

### 9.1 Iterative Optimizer

• ITERATIVE-OPTIMIZER (:N-INSTANCES = 0)

The number of instances this optimizer has seen so far. Incremented automatically during optimization.

• ITERATIVE-OPTIMIZER (:TERMINATION = NIL)

If a number, it's the number of instances to train on in the sense of N-INSTANCES. If N-INSTANCES is equal or greater than this value optimization stops. If TERMINATION is NIL, then optimization will continue. If it is T, then optimization will stop. If it is a function of no arguments, then its return value is processed as if it was returned by TERMINATION.

• ITERATIVE-OPTIMIZER (:ON-OPTIMIZATION-STARTED = NIL)

An event hook with parameters (OPTIMIZER GRADIENT-SOURCE N-INSTANCES). Called after initializations are performed (INITIALIZE-OPTIMIZER, INITIALIZE-GRADIENT-SOURCE) but before optimization is started.

• ITERATIVE-OPTIMIZER (:ON-OPTIMIZATION-FINISHED = NIL)

An event hook with parameters (OPTIMIZER GRADIENT-SOURCE N-INSTANCES). Called when optimization has finished.

• ITERATIVE-OPTIMIZER (:ON-N-INSTANCES-CHANGED = NIL)

An event hook with parameters (OPTIMIZER GRADIENT-SOURCE N-INSTANCES). Called when optimization of a batch of instances is done and N-INSTANCES is incremented.

Now let's discuss a few handy utilities.

• OPTIMIZER PERIODIC-FNS

For each periodic function in the list of PERIODIC-FNS, add a monitor to OPTIMIZER's ON-OPTIMIZATION-STARTED, ON-OPTIMIZATION-FINISHED and ON-N-INSTANCES-CHANGED hooks. The monitors are simple functions that just call each periodic function with the event parameters (OPTIMIZER GRADIENT-SOURCE N-INSTANCES). Return OPTIMIZER.

To log and reset the monitors of the gradient source after every 1000 instances seen by OPTIMIZER:

(monitor-optimization-periodically optimizer
'((:fn log-my-test-error
:period 2000)
(:fn reset-optimization-monitors
:period 1000
:last-eval 0)))


Note how we don't pass it's allowed to just pass the initargs for a PERIODIC-FN instead of PERIODIC-FN itself. The :LAST-EVAL 0 bit prevents RESET-OPTIMIZATION-MONITORS from being called at the start of the optimization when the monitors are empty anyway.

Log the counters of the monitors of OPTIMIZER and GRADIENT-SOURCE and reset them.

A utility that's often called at the start of optimization (from ON-OPTIMIZATION-STARTED). The default implementation logs the description of GRADIENT-SOURCE (as in DESCRIBE) and OPTIMIZER and calls LOG-MAT-ROOM.

### 9.2 Cost Function

The function being minimized is often called the cost or the loss function.

• MODEL

Return the value of the cost function being minimized. Calling this only makes sense in the context of an ongoing optimization (see MINIMIZE). The cost is that of a batch of instances.

• MODEL OPERATION-MODE ATTRIBUTES

Identical to MAKE-COST-MONITORS bar the keywords arguments. Specialize this to add to support for new model types.

###### [in package MGL-GD]

Gradient descent is a first-order optimization algorithm. Relying completely on first derivatives, it does not even evaluate the function to be minimized. Let's see how to minimize a numerical lisp function with respect to some of its parameters.

(cl:defpackage :mgl-example-sgd
(:use #:common-lisp #:mgl))

(in-package :mgl-example-sgd)

;;; Create an object representing the sine function.
(defparameter *diff-fn-1*
(make-instance 'mgl-diffun:diffun
:fn #'sin
;; We are going to optimize its only parameter.
:weight-indices '(0)))

;;; Minimize SIN. Note that there is no dataset involved because all
;;; parameters are being optimized.
(minimize (make-instance 'sgd-optimizer :termination 1000)
*diff-fn-1*
:weights (make-mat 1))
;;; => A MAT with a single value of about -pi/2.

;;; Create a differentiable function for f(x,y)=(x-y)^2. X is a
;;; parameter whose values come from the DATASET argument passed to
;;; MINIMIZE. Y is a parameter to be optimized (a 'weight').
(defparameter *diff-fn-2*
(make-instance 'mgl-diffun:diffun
:fn (lambda (x y)
(expt (- x y) 2))
:parameter-indices '(0)
:weight-indices '(1)))

;;; Find the Y that minimizes the distance from the instances
;;; generated by the sampler.
(minimize (make-instance 'sgd-optimizer :batch-size 10)
*diff-fn-2*
:weights (make-mat 1)
:dataset (make-instance 'function-sampler
:generator (lambda ()
(list (+ 10
(gaussian-random-1))))
:max-n-samples 1000))
;;; => A MAT with a single value of about 10, the expected value of
;;; the instances in the dataset.

;;; The dataset can be a SEQUENCE in which case we'd better set
;;; TERMINATION else optimization would never finish.
(minimize (make-instance 'sgd-optimizer :termination 1000)
*diff-fn-2*
:weights (make-mat 1)
:dataset '((0) (1) (2) (3) (4) (5)))
;;; => A MAT with a single value of about 2.5.

We are going to see a number of accessors for optimizer paramaters. In general, it's allowed to SETF real slot accessors (as opposed to readers and writers) at any time during optimization and so is defining a method on an optimizer subclass that computes the value in any way. For example, to decay the learning rate on a per mini-batch basis:

(defmethod learning-rate ((optimizer my-sgd-optimizer))
(* (slot-value optimizer 'learning-rate)
(expt 0.998
(/ (n-instances optimizer) 60000))))

#### 9.3.1 Batch Based Optimizers

First let's see everything common to all batch based optimizers, then discuss SGD Optimizer, Adam Optimizer and Normalized Batch Optimizer. All batch based optimizers are ITERATIVE-OPTIMIZERs, so see Iterative Optimizer too.

• GD-OPTIMIZER (:BATCH-SIZE = 1)

After having gone through BATCH-SIZE number of inputs, weights are updated. With BATCH-SIZE 1, one gets Stochastics Gradient Descent. With BATCH-SIZE equal to the number of instances in the dataset, one gets standard, 'batch' gradient descent. With BATCH-SIZE between these two extremes, one gets the most practical 'mini-batch' compromise.

• GD-OPTIMIZER (:LEARNING-RATE = 0.1)

This is the step size along the gradient. Decrease it if optimization diverges, increase it if it doesn't make progress.

• GD-OPTIMIZER (:MOMENTUM = 0)

A value in the [0, 1) interval. MOMENTUM times the previous weight change is added to the gradient. 0 means no momentum.

• GD-OPTIMIZER (:MOMENTUM-TYPE = :NORMAL)

One of :NORMAL, :NESTEROV or :NONE. For pure optimization Nesterov's momentum may be better, but it may also increases chances of overfitting. Using :NONE is equivalent to 0 momentum, but it also uses less memory. Note that with :NONE, MOMENTUM is ignored even it it is non-zero.

• GD-OPTIMIZER (:WEIGHT-DECAY = 0)

An L2 penalty. It discourages large weights, much like a zero mean gaussian prior. WEIGHT-DECAY * WEIGHT is added to the gradient to penalize large weights. It's as if the function whose minimum is sought had WEIGHT-DECAY*sum_i{0.5 * WEIGHT_i^2} added to it.

• GD-OPTIMIZER (:WEIGHT-PENALTY = 0)

An L1 penalty. It encourages sparsity. SIGN(WEIGHT) * WEIGHT-PENALTY is added to the gradient pushing the weight towards negative infinity. It's as if the function whose minima is sought had WEIGHT-PENALTY*sum_i{abs(WEIGHT_i)} added to it. Putting it on feature biases consitutes a sparsity constraint on the features.

• GD-OPTIMIZER (:USE-SEGMENT-DERIVATIVES-P = NIL)

Save memory if both the gradient source (the model being optimized) and the optimizer support this feature. It works like this: the accumulator into which the gradient source is asked to place the derivatives of a segment will be SEGMENT-DERIVATIVES of the segment. This allows the optimizer not to allocate an accumulator matrix into which the derivatives are summed.

• GD-OPTIMIZER (:AFTER-UPDATE-HOOK = NIL)

A list of functions with no arguments called after each weight update.

• BATCH-GD-OPTIMIZER (:BEFORE-UPDATE-HOOK = NIL)

A list of functions of no parameters. Each function is called just before a weight update takes place (after accumulated gradients have been divided the length of the batch). Convenient to hang some additional gradient accumulating code on.

##### SGD Optimizer

• With BATCH-SIZE(0 1 2) 1 this is Stochastic Gradient Descent. With higher batch sizes, one gets mini-batch and Batch Gradient Descent.

Assuming that ACCUMULATOR has the sum of gradients for a mini-batch, the weight update looks like this:

$$\Delta_w^{t+1} = momentum \Delta_w^t + \frac{accumulator}{batchsize} + l_2 w + l_1 sign(w)$$

$$w^{t+1} = w^{t} - learningrate \Delta_w$$

which is the same as the more traditional formulation:

$$\Delta_w^{t+1} = momentum * \Delta_w^{t} + learningrate \left(\frac{\frac{df}{dw}}{batchsize} + l_2 w + l_1 sign(w)\right)$$

$$w^{t+1} = w^{t} - \Delta_w$$

but the former works better when batch size, momentum or learning rate change during the course of optimization. The above is with normal momentum, Nesterov's momentum (see MOMENTUM-TYPE) momentum is also available.

See Batch Based Optimizers for the description of the various options common to all batch based optimizers.

• Adam is a first-order stochasistic gradient descent optimizer. It maintains an internal estimation for the mean and raw variance of each derivative as exponential moving averages. The step it takes is basically M/(sqrt(V)+E) where M is the estimated mean, V is the estimated variance, and E is a small adjustment factor to prevent the gradient from blowing up. See version 5 of the paper for more.

Note that using momentum is not supported with Adam. In fact, an error is signalled if it's not :NONE.

See Batch Based Optimizers for the description of the various options common to all batch based optimizers.

A number between 0 and 1 that determines how fast the estimated mean of derivatives is updated. 0 basically gives you RMSPROP (if VARIANCE-DECAY is not too large) or AdaGrad (if VARIANCE-DECAY is close to 1 and the learning rate is annealed. This is $\beta_1$ in the paper.

• ADAM-OPTIMIZER (:MEAN-DECAY-DECAY = (- 1 1.0d-7))

A value that should be close to 1. MEAN-DECAY is multiplied by this value after each update. This is $\lambda$ in the paper.

A number between 0 and 1 that determines how fast the estimated variance of derivatives is updated. This is $\beta_2$ in the paper.

Within the bowels of adam, the estimated mean is divided by the square root of the estimated variance (per weight) which can lead to numerical problems if the denominator is near zero. To avoid this, VARIANCE-ADJUSTMENT, which should be a small positive number, is added to the denominator. This is epsilon in the paper.

##### Normalized Batch Optimizer

• Like BATCH-GD-OPTIMIZER but keeps count of how many times each weight was used in the batch and divides the accumulated gradient by this count instead of dividing by N-INSTANCES-IN-BATCH. This only makes a difference if there are missing values in the learner that's being trained. The main feature that distuinguishes this class from PER-WEIGHT-BATCH-GD-OPTIMIZER is that batches end at same time for all weights.

• NORMALIZED-BATCH-GD-OPTIMIZER

Number of uses of the weight in its current batch.

#### 9.3.2 Segmented GD Optimizer

• BASE-GD-OPTIMIZER

An optimizer that delegates training of segments to other optimizers. Useful to delegate training of different segments to different optimizers (capable of working with segmentables) or simply to not train all segments.

• SEGMENTED-GD-OPTIMIZER (:SEGMENTER)

When this optimizer is initialized it loops over the segment of the learner with MAP-SEGMENTS. SEGMENTER is a function that is called with each segment and returns an optimizer or NIL. Several segments may be mapped to the same optimizer. After the segment->optimizer mappings are collected, each optimizer is initialized by INITIALIZE-OPTIMIZER with the list of segments mapped to it.

• SEGMENTED-GD-OPTIMIZER

SEGMENTED-GD-OPTIMIZER inherits from ITERATIVE-OPTIMIZER, so see Iterative Optimizer too.

#### 9.3.3 Per-weight Optimization

• GD-OPTIMIZER

This is much like Batch Based Optimizers but it is more clever about when to update weights. Basically every weight has its own batch independent from the batches of others. This has desirable properties. One can for example put two neural networks together without adding any connections between them and the learning will produce results equivalent to the separated case. Also, adding inputs with only missing values does not change anything.

Due to its very non-batch nature, there is no CUDA implementation of this optimizer.

• PER-WEIGHT-BATCH-GD-OPTIMIZER

Number of uses of the weight in its current batch.

#### 9.3.4 Utilities

• MATS L2-UPPER-BOUND &KEY CALLBACK

Scale MATS so that their $L_2$ norm does not exceed L2-UPPER-BOUND.

Compute the norm of of MATS as if they were a single vector. If the norm is greater than L2-UPPER-BOUND, then scale each matrix destructively by the norm divided by L2-UPPER-BOUND and if non-NIL call the function CALLBACK with the scaling factor.

• BATCH-GD-OPTIMIZER L2-UPPER-BOUND &KEY CALLBACK

Make it so that the norm of the batch normalized gradients accumulated by BATCH-GD-OPTIMIZER is clipped to L2-UPPER-BOUND before every update. See CLIP-L2-NORM.

###### [in package MGL-CG]

Conjugate gradient is a first-order optimization algorithm. It's more advanced than gradient descent as it does line searches which unfortunately also makes it unsuitable for non-deterministic functions. Let's see how to minimize a numerical lisp function with respect to some of its parameters.

;;; Create an object representing the sine function.
(defparameter *diff-fn-1*
(make-instance 'mgl-diffun:diffun
:fn #'sin
;; We are going to optimize its only parameter.
:weight-indices '(0)))

;;; Minimize SIN. Note that there is no dataset involved because all
;;; parameters are being optimized.
(minimize (make-instance 'cg-optimizer
:batch-size 1
:termination 1)
*diff-fn-1*
:weights (make-mat 1))
;;; => A MAT with a single value of about -pi/2.

;;; Create a differentiable function for f(x,y)=(x-y)^2. X is a
;;; parameter whose values come from the DATASET argument passed to
;;; MINIMIZE. Y is a parameter to be optimized (a 'weight').
(defparameter *diff-fn-2*
(make-instance 'mgl-diffun:diffun
:fn (lambda (x y)
(expt (- x y) 2))
:parameter-indices '(0)
:weight-indices '(1)))

;;; Find the Y that minimizes the distance from the instances
;;; generated by the sampler.
(minimize (make-instance 'cg-optimizer :batch-size 10)
*diff-fn-2*
:weights (make-mat 1)
:dataset (make-instance 'function-sampler
:generator (lambda ()
(list (+ 10
(gaussian-random-1))))
:max-n-samples 1000))
;;; => A MAT with a single value of about 10, the expected value of
;;; the instances in the dataset.

;;; The dataset can be a SEQUENCE in which case we'd better set
;;; TERMINATION else optimization would never finish. Note how a
;;; single epoch suffices.
(minimize (make-instance 'cg-optimizer :termination 6)
*diff-fn-2*
:weights (make-mat 1)
:dataset '((0) (1) (2) (3) (4) (5)))
;;; => A MAT with a single value of about 2.5.

• FN W &KEY (MAX-N-LINE-SEARCHES *DEFAULT-MAX-N-LINE-SEARCHES*) (MAX-N-EVALUATIONS-PER-LINE-SEARCH *DEFAULT-MAX-N-EVALUATIONS-PER-LINE-SEARCH*) (MAX-N-EVALUATIONS *DEFAULT-MAX-N-EVALUATIONS*) (SIG *DEFAULT-SIG*) (RHO *DEFAULT-RHO*) (INT *DEFAULT-INT*) (EXT *DEFAULT-EXT*) (RATIO *DEFAULT-RATIO*) SPARE-VECTORS

CG-OPTIMIZER passes each batch of data to this function with its CG-ARGS passed on.

Minimize a differentiable multivariate function with conjugate gradient. The Polak-Ribiere flavour of conjugate gradients is used to compute search directions, and a line search using quadratic and cubic polynomial approximations and the Wolfe-Powell stopping criteria is used together with the slope ratio method for guessing initial step sizes. Additionally a bunch of checks are made to make sure that exploration is taking place and that extrapolation will not be unboundedly large.

FN is a function of two parameters: WEIGHTS(0 1) and DERIVATIVES. WEIGHTS(0 1) is a MAT of the same size as W that is where the search start from. DERIVATIVES is also a MAT of that size and it is where FN shall place the partial derivatives. FN returns the value of the function that is being minimized.

CG performs a number of line searches and invokes FN at each step. A line search invokes FN at most MAX-N-EVALUATIONS-PER-LINE-SEARCH number of times and can succeed in improving the minimum by the sufficient margin or it can fail. Note, the even a failed line search may improve further and hence change the weights it's just that the improvement was deemed too small. CG stops when either:

• two line searches fail in a row

• MAX-N-LINE-SEARCHES is reached

• MAX-N-EVALUATIONS is reached

CG returns a MAT that contains the best weights, the minimum, the number of line searches performed, the number of succesful line searches and the number of evaluations.

When using MAX-N-EVALUATIONS remember that there is an extra evaluation of FN before the first line search.

SPARE-VECTORS is a list of preallocated MATs of the same size as W. Passing 6 of them covers the current need of the algorithm and it will not cons up vectors of size W at all.

NOTE: If the function terminates within a few iterations, it could be an indication that the function values and derivatives are not consistent (ie, there may be a bug in the implementation of FN function).

SIG and RHO are the constants controlling the Wolfe-Powell conditions. SIG is the maximum allowed absolute ratio between previous and new slopes (derivatives in the search direction), thus setting SIG to low (positive) values forces higher precision in the line-searches. RHO is the minimum allowed fraction of the expected (from the slope at the initial point in the linesearch). Constants must satisfy 0 < RHO < SIG < 1. Tuning of SIG (depending on the nature of the function to be optimized) may speed up the minimization; it is probably not worth playing much with RHO.

• Don't reevaluate within INT of the limit of the current bracket.

• Extrapolate maximum EXT times the current step-size.

• SIG and RHO are the constants controlling the Wolfe-Powell conditions. SIG is the maximum allowed absolute ratio between previous and new slopes (derivatives in the search direction), thus setting SIG to low (positive) values forces higher precision in the line-searches.

• RHO is the minimum allowed fraction of the expected (from the slope at the initial point in the linesearch). Constants must satisfy 0 < RHO < SIG < 1.

• Maximum allowed slope ratio.

• CG-OPTIMIZER (:BATCH-SIZE)

After having gone through BATCH-SIZE number of instances, weights are updated. Normally, CG operates on all available data, but it may be useful to introduce some noise into the optimization to reduce overfitting by using smaller batch sizes. If BATCH-SIZE is not set, it is initialized to the size of the dataset at the start of optimization.

• CG-OPTIMIZER (:CG-ARGS = 'NIL)

• CG-OPTIMIZER (:ON-CG-BATCH-DONE = NIL)

An event hook called when processing a conjugate gradient batch is done. The handlers on the hook are called with 8 arguments:

(optimizer gradient-source instances
best-w best-f n-line-searches
n-succesful-line-searches n-evaluations)


The latter 5 of which are the return values of the CG function.

• OPTIMIZER GRADIENT-SOURCE INSTANCES BEST-W BEST-F N-LINE-SEARCHES N-SUCCESFUL-LINE-SEARCHES N-EVALUATIONS

This is a function can be added to ON-CG-BATCH-DONE. The default implementation simply logs the event arguments.

• CG-OPTIMIZER (:SEGMENT-FILTER = (CONSTANTLY T))

A predicate function on segments that filters out uninteresting segments. Called from INITIALIZE-OPTIMIZER*.

### 9.5 Extension API

#### 9.5.1 Implementing Optimizers

The following generic functions must be specialized for new optimizer types.

Called automatically before training starts, this function sets up OPTIMIZER to be suitable for optimizing GRADIENT-SOURCE. It typically creates appropriately sized accumulators for the gradients.

• OPTIMIZER

Several weight matrices known as segments can be optimized by a single optimizer. This function returns them as a list.

The rest are just useful for utilities for implementing optimizers.

• This is a utility class for optimizers that have a list of SEGMENTS and (the weights being optimized) is able to copy back and forth between those segments and a single MAT (the accumulator).

• SEGMENT-SET (:SEGMENTS)

A list of weight matrices.

• (SEGMENT &OPTIONAL START) SEGMENT-SET &BODY BODY

Iterate over SEGMENTS in SEGMENT-SET. If START is specified, the it is bound to the start index of SEGMENT within SEGMENT-SET. The start index is the sum of the sizes of previous segments.

• SEGMENT-SET MAT

Copy the values of MAT to the weight matrices of SEGMENT-SET as if they were concatenated into a single MAT.

• SEGMENT-SET MAT

Copy the values of SEGMENT-SET to MAT as if they were concatenated into a single MAT.

Weights can be stored in a multitude of ways. Optimizers need to update weights, so it is assumed that weights are stored in any number of MAT objects called segments.

The generic functions in this section must all be specialized for new gradient sources except where noted.

Apply FN to each segment of GRADIENT-SOURCE.

• FN SEGMENT

Call FN with start and end of intervals of consecutive indices that are not missing in SEGMENT. Called by optimizers that support partial updates. The default implementation assumes that all weights are present. This only needs to be specialized if one plans to use an optimizer that knows how to deal unused/missing weights such as MGL-GD:NORMALIZED-BATCH-GD-OPTIMIZER and OPTIMIZER MGL-GD:PER-WEIGHT-BATCH-GD-OPTIMIZER.

• (MAT MAT)

When the segment is really a MAT, then just return it.

A utility function that returns the list of segments from MAP-SEGMENTS on GRADIENT-SOURCE.

Called automatically before MINIMIZE* is called, this function may be specialized if GRADIENT-SOURCE needs some kind of setup.

The default method does nothing.

• GRADIENT-SOURCE SINK BATCH MULTIPLIER VALUEP

Add MULTIPLIER times the sum of first-order gradients to accumulators of SINK (normally accessed with DO-GRADIENT-SINK) and if VALUEP, return the sum of values of the function being optimized for a BATCH of instances. GRADIENT-SOURCE is the object representing the function being optimized, SINK is gradient sink.

Note the number of instances in BATCH may be larger than what GRADIENT-SOURCE process in one go (in the sense of say, MAX-N-STRIPES), so DO-BATCHES-FOR-MODEL or something like (GROUP BATCH MAX-N-STRIPES) can be handy.

Optimizers call ACCUMULATE-GRADIENTS* on gradient sources. One parameter of ACCUMULATE-GRADIENTS* is the SINK. A gradient sink knows what accumulator matrix (if any) belongs to a segment. Sinks are defined entirely by MAP-GRADIENT-SINK.

• Call FN of lambda list (SEGMENT ACCUMULATOR) on each segment and their corresponding accumulator MAT in SINK.

## 10 Differentiable Functions

###### [in package MGL-DIFFUN]

• DIFFUN (:FN)

A real valued lisp function. It may have any number of parameters.

• DIFFUN (:PARAMETER-INDICES = NIL)

The list of indices of parameters that we don't optimize. Values for these will come from the DATASET argument of MINIMIZE.

• DIFFUN (:WEIGHT-INDICES = NIL)

The list of indices of parameters to be optimized, the values of which will come from the [WEIGHTS] argument of MINIMIZE.

## 11 Backpropagation Neural Networks

### 11.1 Backprop Overview

Backpropagation Neural Networks are just functions with lots of parameters called weights and a layered structure when presented as a computational graph. The network is trained to MINIMIZE some kind of loss function whose value the network computes.

In this implementation, a BPN is assembled from several LUMPs (roughly corresponding to layers). Both feed-forward and recurrent neural nets are supported (FNN and RNN, respectively). BPNs can contain not only LUMPs but other BPNs, too. As we see, networks are composite objects and the abstract base class for composite and simple parts is called CLUMP.

• A CLUMP is a LUMP or a BPN. It represents a differentiable function. Arguments of clumps are given during instantiation. Some arguments are clumps themselves so they get permenantly wired together like this:

(->v*m (->input :size 10 :name 'input)
(->weight :dimensions '(10 20) :name 'weight)
:name 'activation)

The above creates three clumps: the vector-matrix multiplication clumps called ACTIVATION which has a reference to its operands: INPUT and WEIGHT. Note that the example just defines a function, no actual computation has taken place, yet.

This wiring of CLUMPs is how one builds feed-forward nets (FNN) or recurrent neural networks (RNN) that are CLUMPs themselves so one can build nets in a hiearchical style if desired. Non-composite CLUMPs are called LUMP (note the loss of C that stands for composite). The various LUMP subtypes correspond to different layer types (->SIGMOID, ->DROPOUT, ->RELU, ->TANH, etc).

At this point, you may want to jump ahead to get a feel for how things work by reading the FNN Tutorial.

### 11.2 Clump API

These are mostly for extension purposes. About the only thing needed from here for normal operation is NODES when clamping inputs or extracting predictions.

• For efficiency, forward and backprop phases do their stuff in batch mode: passing a number of instances through the network in batches. Thus clumps must be able to store values of and gradients for each of these instances. However, some clumps produce the same result for each instance in a batch. These clumps are the weights, the parameters of the network. STRIPEDP returns true iff CLUMP does not represent weights (i.e. it's not a ->WEIGHT).

For striped clumps, their NODES and DERIVATIVES are MAT objects with a leading dimension (number of rows in the 2d case) equal to the number of instances in the batch. Non-striped clumps have no restriction on their shape apart from what their usage dictates.

• OBJECT

Returns a MAT object representing the state or result of OBJECT. The first dimension of the returned matrix is equal to the number of stripes.

CLUMPs' NODES holds the result computed by the most recent FORWARD. For ->INPUT lumps, this is where input values shall be placed (see SET-INPUT). Currently, the matrix is always two dimensional but this restriction may go away in the future.

• Compute the values of the function represented by CLUMP for all stripes and place the results into NODES of CLUMP.

• Compute the partial derivatives of the function represented by CLUMP and add them to DERIVATIVES of the corresponding argument clumps. The DERIVATIVES of CLUMP contains the sum of partial derivatives of all clumps by the corresponding output. This function is intended to be called after a FORWARD pass.

Take the ->SIGMOID clump for example when the network is being applied to a batch of two instances x1 and x2. x1 and x2 are set in the ->INPUT lump X. The sigmoid computes 1/(1+exp(-x)) where X is its only argument clump.

f(x) = 1/(1+exp(-x))


When BACKWARD is called on the sigmoid lump, its DERIVATIVES is a 2x1 MAT object that contains the partial derivatives of the loss function:

dL(x1)/df
dL(x2)/df


Now the BACKWARD method of the sigmoid needs to add dL(x1)/dx1 and dL(x2)/dx2 to DERIVATIVES of X. Now, dL(x1)/dx1 = dL(x1)/df * df(x1)/dx1 and the first term is what we have in DERIVATIVES of the sigmoid so it only needs to calculate the second term.

In addition to the above, clumps also have to support SIZE(0 1), N-STRIPES, MAX-N-STRIPES (and the SETF methods of the latter two) which can be accomplished just by inheriting from BPN, FNN, RNN, or a LUMP.

### 11.3 BPNs

• BPN (:N-STRIPES = 1)

The current number of instances the network has. This is automatically set to the number of instances passed to SET-INPUT, so it rarely has to be manipulated directly although it can be set. When set N-STRIPES of all CLUMPS get set to the same value.

• BPN (:MAX-N-STRIPES = NIL)

The maximum number of instances the network can operate on in parallel. Within BUILD-FNN or BUILD-RNN, it defaults to MAX-N-STRIPES of that parent network, else it defaults to 1. When set MAX-N-STRIPES of all CLUMPS get set to the same value.

• NAME BPN &KEY (ERRORP T)

Find the clump with NAME among CLUMPS of BPN. As always, names are compared with EQUAL. If not found, then return NIL or signal and error depending on ERRORP.

• CLUMP BPN

Add CLUMP to BPN. MAX-N-STRIPES of CLUMP gets set to that of BPN. It is an error to add a clump with a name already used by one of the CLUMPS of BPN.

#### 11.3.1 Training

BPNs are trained to minimize the loss function they compute. Before a BPN is passed to MINIMIZE (as its GRADIENT-SOURCE argument), it must be wrapped in a BP-LEARNER object. BP-LEARNER has [MONITORS] slot which is used for example by RESET-OPTIMIZATION-MONITORS.

Without the bells an whistles, the basic shape of training is this:

(minimize optimizer (make-instance 'bp-learner :bpn bpn)
:dataset dataset)

#### 11.3.2 Monitoring

• RNN &KEY (COUNTER-VALUES-FN #'COUNTER-RAW-VALUES) (MAKE-COUNTER #'MAKE-STEP-MONITOR-MONITOR-COUNTER)

Return a list of monitors, one for every monitor in STEP-MONITORS of RNN. These monitors extract the results from their warp counterpairs with COUNTER-VALUES-FN and add them to their own counter that's created by MAKE-COUNTER. Wow. Ew. The idea is that one does something like this do monitor warped prediction:

(let ((*warp-time* t))
(setf (step-monitors rnn)
(make-cost-monitors rnn :attributes '(:event "warped pred.")))
(monitor-bpn-results dataset rnn
;; Just collect and reset the warp
;; monitors after each batch of
;; instances.
(make-step-monitor-monitors rnn)))

• STEP-COUNTER

In an RNN, STEP-COUNTER aggregates results of all the time steps during the processing of instances in the current batch. Return a new counter into which results from STEP-COUNTER can be accumulated when the processing of the batch is finished. The default implementation creates a copy of STEP-COUNTER.

#### 11.3.3 Feed-Forward Nets

FNN and RNN have a lot in common (see their common superclass, BPN). There is very limited functionality that's specific to FNNs so let's get them out of they way before we study a full example.

• (&KEY FNN (CLASS ''FNN) INITARGS MAX-N-STRIPES NAME) &BODY CLUMPS

Syntactic sugar to assemble FNNs from CLUMPs. Like LET*, it is a sequence of bindings (of symbols to CLUMPs). The names of the clumps created default to the symbol of the binding. In case a clump is not bound to a symbol (because it was created in a nested expression), the local function CLUMP can be used to find the clump with the given name in the fnn being built. Example:

(build-fnn ()
(features (->input :size n-features))
(biases (->weight :size n-features))
(weights (->weight :size (* n-hiddens n-features)))
(activations0 (->v*m :weights weights :x (clump 'features)))
(activations (->+ :args (list biases activations0)))
(output (->sigmoid :x activations)))


##### FNN Tutorial

Hopefully this example from example/digit-fnn.lisp illustrates the concepts involved. If it's too dense despite the comments, then read up on Datasets, Gradient Based Optimization and come back.

(cl:defpackage :mgl-example-digit-fnn
(:use #:common-lisp #:mgl))

(in-package :mgl-example-digit-fnn)

;;; There are 10 possible digits used as inputs ...
(defparameter *n-inputs* 10)
;;; and we want to learn the rule that maps the input digit D to (MOD
;;; (1+ D) 3).
(defparameter *n-outputs* 3)

;;; We define a feed-forward net to be able to specialize how inputs
;;; are translated by adding a SET-INPUT method later.
(defclass digit-fnn (fnn)
())

;;; Build a DIGIT-FNN with a single hidden layer of rectified linear
;;; units and a softmax output.
(defun make-digit-fnn (&key (n-hiddens 5))
(build-fnn (:class 'digit-fnn)
(input (->input :size *n-inputs*))
(hidden-activation (->activation input :size n-hiddens))
(hidden (->relu hidden-activation))
(output-activation (->activation hidden :size *n-outputs*))
(output (->softmax-xe-loss output-activation))))

;;; This method is called with batches of 'instances' (input digits in
;;; this case) by MINIMIZE and also by MONITOR-BPN-RESULTS before
;;; performing a forward pass (i.e. computing the value of the
;;; function represented by the network). Its job is to encode the
;;; inputs by populating rows of the NODES matrix of the INPUT clump.
;;;
;;; Each input is encoded as a row of zeros with a single 1 at index
;;; determined by the input digit. This is called one-hot encoding.
;;; The TARGET could be encoded the same way, but instead we use the
;;; sparse option supported by TARGET of ->SOFTMAX-XE-LOSS.
(defmethod set-input (digits (fnn digit-fnn))
(let* ((input (nodes (find-clump 'input fnn)))
(output-lump (find-clump 'output fnn)))
(fill! 0 input)
(loop for i upfrom 0
for digit in digits
do (setf (mref input i digit) 1))
(setf (target output-lump)
(mapcar (lambda (digit)
(mod (1+ digit) *n-outputs*))
digits))))

;;; Train the network by minimizing the loss (cross-entropy here) with
(defun train-digit-fnn ()
(let ((optimizer
;; First create the optimizer for MINIMIZE.
(make-instance 'segmented-gd-optimizer
:segmenter
;; We train each weight lump with the same
;; parameters and, in fact, the same
;; optimizer. But it need not be so, in
;; general.
(constantly
(make-instance 'sgd-optimizer
:learning-rate 1
:momentum 0.9
:batch-size 100))))
(fnn (make-digit-fnn)))
;; The number of instances the FNN can work with in parallel. It's
;; usually equal to the batch size or is a its divisor.
(setf (max-n-stripes fnn) 50)
;; Initialize all weights randomly.
(map-segments (lambda (weights)
(gaussian-random! (nodes weights) :stddev 0.01))
fnn)
;; Arrange for training and test error to be logged.
(monitor-optimization-periodically
optimizer '((:fn log-test-error :period 10000)
(:fn reset-optimization-monitors :period 1000)))
;; Finally, start the optimization.
(minimize optimizer
;; Dress FNN in a BP-LEARNER and attach monitors for the
;; cost to it. These monitors are going to be logged and
;; reset after every 100 training instance by
;; RESET-OPTIMIZATION-MONITORS above.
(make-instance 'bp-learner
:bpn fnn
:monitors (make-cost-monitors
fnn :attributes (:event "train")))
;; Training stops when the sampler runs out (after 10000
;; instances).
:dataset (make-sampler 10000))))

;;; Return a sampler object that produces MAX-N-SAMPLES number of
;;; random inputs (numbers between 0 and 9).
(defun make-sampler (max-n-samples)
(make-instance 'function-sampler :max-n-samples max-n-samples
:generator (lambda () (random *n-inputs*))))

;;; Log the test error. Also, describe the optimizer and the bpn at
;;; the beginning of training. Called periodically during training
;;; (see above).
(defun log-test-error (optimizer learner)
(when (zerop (n-instances optimizer))
(describe optimizer)
(describe (bpn learner)))
(monitor-bpn-results (make-sampler 1000) (bpn learner)
(make-cost-monitors
(bpn learner) :attributes (:event "pred.")))))

#|

;;; Transcript follows:
(repeatably ()
(let ((*log-time* nil))
(train-digit-fnn)))
.. training at n-instances: 0
.. train cost: 0.000e+0 (0)
.. #<SEGMENTED-GD-OPTIMIZER {100E112E93}>
..  SEGMENTED-GD-OPTIMIZER description:
..    N-INSTANCES = 0
..    OPTIMIZERS = (#<SGD-OPTIMIZER
..                    #<SEGMENT-SET
..                      (#<->WEIGHT # :SIZE 15 1/1 :NORM 0.04473>
..                       #<->WEIGHT # :SIZE 3 1/1 :NORM 0.01850>
..                       #<->WEIGHT # :SIZE 50 1/1 :NORM 0.07159>
..                       #<->WEIGHT # :SIZE 5 1/1 :NORM 0.03056>)
..                      {100E335B73}>
..                    {100E06DF83}>)
..    SEGMENTS = (#<->WEIGHT (HIDDEN OUTPUT-ACTIVATION) :SIZE
..                  15 1/1 :NORM 0.04473>
..                #<->WEIGHT (:BIAS OUTPUT-ACTIVATION) :SIZE
..                  3 1/1 :NORM 0.01850>
..                #<->WEIGHT (INPUT HIDDEN-ACTIVATION) :SIZE
..                  50 1/1 :NORM 0.07159>
..                #<->WEIGHT (:BIAS HIDDEN-ACTIVATION) :SIZE
..                  5 1/1 :NORM 0.03056>)
..
.. #<SGD-OPTIMIZER {100E06DF83}>
..  GD-OPTIMIZER description:
..    N-INSTANCES = 0
..    SEGMENT-SET = #<SEGMENT-SET
..                    (#<->WEIGHT (HIDDEN OUTPUT-ACTIVATION) :SIZE
..                       15 1/1 :NORM 0.04473>
..                     #<->WEIGHT (:BIAS OUTPUT-ACTIVATION) :SIZE
..                       3 1/1 :NORM 0.01850>
..                     #<->WEIGHT (INPUT HIDDEN-ACTIVATION) :SIZE
..                       50 1/1 :NORM 0.07159>
..                     #<->WEIGHT (:BIAS HIDDEN-ACTIVATION) :SIZE
..                       5 1/1 :NORM 0.03056>)
..                    {100E335B73}>
..    LEARNING-RATE = 1.00000e+0
..    MOMENTUM = 9.00000e-1
..    MOMENTUM-TYPE = :NORMAL
..    WEIGHT-DECAY = 0.00000e+0
..    WEIGHT-PENALTY = 0.00000e+0
..    N-AFTER-UPATE-HOOK = 0
..    BATCH-SIZE = 100
..
..  BATCH-GD-OPTIMIZER description:
..    N-BEFORE-UPATE-HOOK = 0
..  #<DIGIT-FNN {100E11A423}>
..   BPN description:
..     CLUMPS = #(#<->INPUT INPUT :SIZE 10 1/50 :NORM 0.00000>
..                #<->ACTIVATION
..                  (HIDDEN-ACTIVATION :ACTIVATION) :STRIPES 1/50
..                  :CLUMPS 4>
..                #<->RELU HIDDEN :SIZE 5 1/50 :NORM 0.00000>
..                #<->ACTIVATION
..                  (OUTPUT-ACTIVATION :ACTIVATION) :STRIPES 1/50
..                  :CLUMPS 4>
..                #<->SOFTMAX-XE-LOSS OUTPUT :SIZE 3 1/50 :NORM 0.00000>)
..     N-STRIPES = 1
..     MAX-N-STRIPES = 50
..   pred. cost: 1.100d+0 (1000.00)
.. training at n-instances: 1000
.. train cost: 1.093d+0 (1000.00)
.. training at n-instances: 2000
.. train cost: 5.886d-1 (1000.00)
.. training at n-instances: 3000
.. train cost: 3.574d-3 (1000.00)
.. training at n-instances: 4000
.. train cost: 1.601d-7 (1000.00)
.. training at n-instances: 5000
.. train cost: 1.973d-9 (1000.00)
.. training at n-instances: 6000
.. train cost: 4.882d-10 (1000.00)
.. training at n-instances: 7000
.. train cost: 2.771d-10 (1000.00)
.. training at n-instances: 8000
.. train cost: 2.283d-10 (1000.00)
.. training at n-instances: 9000
.. train cost: 2.123d-10 (1000.00)
.. training at n-instances: 10000
.. train cost: 2.263d-10 (1000.00)
.. pred. cost: 2.210d-10 (1000.00)
..
==> (#<->WEIGHT (:BIAS HIDDEN-ACTIVATION) :SIZE 5 1/1 :NORM 2.94294>
-->  #<->WEIGHT (INPUT HIDDEN-ACTIVATION) :SIZE 50 1/1 :NORM 11.48995>
-->  #<->WEIGHT (:BIAS OUTPUT-ACTIVATION) :SIZE 3 1/1 :NORM 3.39103>
-->  #<->WEIGHT (HIDDEN OUTPUT-ACTIVATION) :SIZE 15 1/1 :NORM 11.39339>)

|#

#### 11.3.4 Recurrent Neural Nets

##### RNN Tutorial

Hopefully this example from example/sum-sign-fnn.lisp illustrates the concepts involved. Make sure you are comfortable with FNN Tutorial before reading this.

(cl:defpackage :mgl-example-sum-sign-rnn
(:use #:common-lisp #:mgl))

(in-package :mgl-example-sum-sign-rnn)

;;; There is a single input at each time step...
(defparameter *n-inputs* 1)
;;; and we want to learn the rule that outputs the sign of the sum of
;;; inputs so far in the sequence.
(defparameter *n-outputs* 3)

;;; Generate a training example that's a sequence of random length
;;; between 1 and LENGTH. Elements of the sequence are lists of two
;;; elements:
;;;
;;; 1. The input for the network (a single random number).
;;;
;;; 2. The sign of the sum of inputs so far encoded as 0, 1, 2 (for
;;;    negative, zero and positive values). To add a twist, the sum is
;;;    reset whenever a negative input is seen.
(defun make-sum-sign-instance (&key (length 10))
(let ((length (max 1 (random length)))
(sum 0))
(loop for i below length
collect (let ((x (1- (* 2 (random 2)))))
(incf sum x)
(when (< x 0)
(setq sum x))
(list x (cond ((minusp sum) 0)
((zerop sum) 1)
(t 2)))))))

;;; Build an RNN with a single lstm hidden layer and softmax output.
;;; For each time step, a SUM-SIGN-FNN will be instantiated.
(defun make-sum-sign-rnn (&key (n-hiddens 1))
(build-rnn ()
(build-fnn (:class 'sum-sign-fnn)
(input (->input :size 1))
(h (->lstm input :name 'h :size n-hiddens))
(prediction (->softmax-xe-loss (->activation h :name 'prediction
:size *n-outputs*))))))

;;; We define this class to be able to specialize how inputs are
;;; translated by adding a SET-INPUT method later.
(defclass sum-sign-fnn (fnn)
())

;;; We have a batch of instances from MAKE-SUM-SIGN-INSTANCE for the
;;; RNN. This function is invoked with elements of these instances
;;; belonging to the same time step (i.e. at the same index) and sets
;;; the input and target up.
(defmethod set-input (instances (fnn sum-sign-fnn))
(let ((input-nodes (nodes (find-clump 'input fnn))))
(setf (target (find-clump 'prediction fnn))
(loop for stripe upfrom 0
for instance in instances
collect
;; Sequences in the batch are not of equal length. The
;; RNN sends a NIL our way if a sequence has run out.
(when instance
(destructuring-bind (input target) instance
(setf (mref input-nodes stripe 0) input)
target))))))

;;; Train the network by minimizing the loss (cross-entropy here) with
(defun train-sum-sign-rnn ()
(let ((rnn (make-sum-sign-rnn)))
(setf (max-n-stripes rnn) 50)
;; Initialize the weights in the usual sqrt(1 / fan-in) style.
(map-segments (lambda (weights)
(let* ((fan-in (mat-dimension (nodes weights) 0))
(limit (sqrt (/ 6 fan-in))))
(uniform-random! (nodes weights)
:limit (* 2 limit))
(.+! (- limit) (nodes weights))))
rnn)
(minimize (monitor-optimization-periodically
:learning-rate 0.2
:mean-decay 0.9
:mean-decay-decay 0.9
:variance-decay 0.9
:batch-size 100)
'((:fn log-test-error :period 30000)
(:fn reset-optimization-monitors :period 3000)))
(make-instance 'bp-learner
:bpn rnn
:monitors (make-cost-monitors rnn))
:dataset (make-sampler 30000))))

;;; Return a sampler object that produces MAX-N-SAMPLES number of
;;; random inputs.
(defun make-sampler (max-n-samples &key (length 10))
(make-instance 'function-sampler :max-n-samples max-n-samples
:generator (lambda ()
(make-sum-sign-instance :length length))))

;;; Log the test error. Also, describe the optimizer and the bpn at
;;; the beginning of training. Called periodically during training
;;; (see above).
(defun log-test-error (optimizer learner)
(when (zerop (n-instances optimizer))
(describe optimizer)
(describe (bpn learner)))
(let ((rnn (bpn learner)))
(append
(monitor-bpn-results (make-sampler 1000) rnn
(make-cost-monitors
rnn :attributes '(:event "pred.")))
;; Same result in a different way: monitor predictions for
;; sequences up to length 20, but don't unfold the RNN
;; unnecessarily to save memory.
(let ((*warp-time* t))
(monitor-bpn-results (make-sampler 1000 :length 20) rnn
;; Just collect and reset the warp
;; monitors after each batch of
;; instances.
(make-cost-monitors
rnn :attributes '(:event "warped pred."))))))
;; Verify that no further unfoldings took place.
(assert (<= (length (clumps rnn)) 10)))
(log-mat-room))

#|

;;; Transcript follows:
(let (;; Backprop nets do not need double float. Using single floats
;; is faster and needs less memory.
(*default-mat-ctype* :float)
;; Enable moving data in and out of GPU memory so that the RNN
;; can work with sequences so long that the unfolded network
;; wouldn't otherwise fit in the GPU.
(*cuda-window-start-time* 1)
(*log-time* nil))
;; Seed the random number generators.
(repeatably ()
;; Enable CUDA if available.
(with-cuda* ()
(train-sum-sign-rnn))))
.. training at n-instances: 0
.. cost: 0.000e+0 (0)
..  GD-OPTIMIZER description:
..    N-INSTANCES = 0
..    SEGMENT-SET = #<SEGMENT-SET
..                    (#<->WEIGHT (H #) :SIZE 1 1/1 :NORM 1.73685>
..                     #<->WEIGHT (H #) :SIZE 1 1/1 :NORM 0.31893>
..                     #<->WEIGHT (#1=# #2=# :PEEPHOLE) :SIZE
..                       1 1/1 :NORM 1.81610>
..                     #<->WEIGHT (H #2#) :SIZE 1 1/1 :NORM 0.21965>
..                     #<->WEIGHT (#1# #3=# :PEEPHOLE) :SIZE
..                       1 1/1 :NORM 1.74939>
..                     #<->WEIGHT (H #3#) :SIZE 1 1/1 :NORM 0.40377>
..                     #<->WEIGHT (H PREDICTION) :SIZE
..                       3 1/1 :NORM 2.15898>
..                     #<->WEIGHT (:BIAS PREDICTION) :SIZE
..                       3 1/1 :NORM 2.94470>
..                     #<->WEIGHT (#1# #4=# :PEEPHOLE) :SIZE
..                       1 1/1 :NORM 0.97601>
..                     #<->WEIGHT (INPUT #4#) :SIZE 1 1/1 :NORM 0.65261>
..                     #<->WEIGHT (:BIAS #4#) :SIZE 1 1/1 :NORM 0.37653>
..                     #<->WEIGHT (INPUT #1#) :SIZE 1 1/1 :NORM 0.92334>
..                     #<->WEIGHT (:BIAS #1#) :SIZE 1 1/1 :NORM 0.01609>
..                     #<->WEIGHT (INPUT #5=#) :SIZE 1 1/1 :NORM 1.09995>
..                     #<->WEIGHT (:BIAS #5#) :SIZE 1 1/1 :NORM 1.41244>
..                     #<->WEIGHT (INPUT #6=#) :SIZE 1 1/1 :NORM 0.40475>
..                     #<->WEIGHT (:BIAS #6#) :SIZE 1 1/1 :NORM 1.75358>)
..                    {1006CD8753}>
..    LEARNING-RATE = 2.00000e-1
..    MOMENTUM = NONE
..    MOMENTUM-TYPE = :NONE
..    WEIGHT-DECAY = 0.00000e+0
..    WEIGHT-PENALTY = 0.00000e+0
..    N-AFTER-UPATE-HOOK = 0
..    BATCH-SIZE = 100
..
..  BATCH-GD-OPTIMIZER description:
..    N-BEFORE-UPATE-HOOK = 0
..
..    MEAN-DECAY-RATE = 1.00000e-1
..    MEAN-DECAY-RATE-DECAY = 9.00000e-1
..    VARIANCE-DECAY-RATE = 1.00000e-1
..  #<RNN {10047C77E3}>
..   BPN description:
..     CLUMPS = #(#<SUM-SIGN-FNN :STRIPES 1/50 :CLUMPS 4>
..                #<SUM-SIGN-FNN :STRIPES 1/50 :CLUMPS 4>)
..     N-STRIPES = 1
..     MAX-N-STRIPES = 50
..
..   RNN description:
..     MAX-LAG = 1
..   pred.        cost: 1.223e+0 (4455.00)
.. warped pred. cost: 1.228e+0 (9476.00)
.. Foreign memory usage:
.. foreign arrays: 162 (used bytes: 39,600)
.. CUDA memory usage:
.. device arrays: 114 (used bytes: 220,892, pooled bytes: 19,200)
.. host arrays: 162 (used bytes: 39,600)
.. host->device copies: 6,164, device->host copies: 4,490
.. training at n-instances: 3000
.. cost: 3.323e-1 (13726.00)
.. training at n-instances: 6000
.. cost: 3.735e-2 (13890.00)
.. training at n-instances: 9000
.. cost: 1.012e-2 (13872.00)
.. training at n-instances: 12000
.. cost: 3.026e-3 (13953.00)
.. training at n-instances: 15000
.. cost: 9.267e-4 (13948.00)
.. training at n-instances: 18000
.. cost: 2.865e-4 (13849.00)
.. training at n-instances: 21000
.. cost: 8.893e-5 (13758.00)
.. training at n-instances: 24000
.. cost: 2.770e-5 (13908.00)
.. training at n-instances: 27000
.. cost: 8.514e-6 (13570.00)
.. training at n-instances: 30000
.. cost: 2.705e-6 (13721.00)
.. pred.        cost: 1.426e-6 (4593.00)
.. warped pred. cost: 1.406e-6 (9717.00)
.. Foreign memory usage:
.. foreign arrays: 216 (used bytes: 52,800)
.. CUDA memory usage:
.. device arrays: 148 (used bytes: 224,428, pooled bytes: 19,200)
.. host arrays: 216 (used bytes: 52,800)
.. host->device copies: 465,818, device->host copies: 371,990
..
==> (#<->WEIGHT (H (H :OUTPUT)) :SIZE 1 1/1 :NORM 0.10624>
-->  #<->WEIGHT (H (H :CELL)) :SIZE 1 1/1 :NORM 0.94460>
-->  #<->WEIGHT ((H :CELL) (H :FORGET) :PEEPHOLE) :SIZE 1 1/1 :NORM 0.61312>
-->  #<->WEIGHT (H (H :FORGET)) :SIZE 1 1/1 :NORM 0.38093>
-->  #<->WEIGHT ((H :CELL) (H :INPUT) :PEEPHOLE) :SIZE 1 1/1 :NORM 1.17956>
-->  #<->WEIGHT (H (H :INPUT)) :SIZE 1 1/1 :NORM 0.88011>
-->  #<->WEIGHT (H PREDICTION) :SIZE 3 1/1 :NORM 49.93808>
-->  #<->WEIGHT (:BIAS PREDICTION) :SIZE 3 1/1 :NORM 10.98112>
-->  #<->WEIGHT ((H :CELL) (H :OUTPUT) :PEEPHOLE) :SIZE 1 1/1 :NORM 0.67996>
-->  #<->WEIGHT (INPUT (H :OUTPUT)) :SIZE 1 1/1 :NORM 0.65251>
-->  #<->WEIGHT (:BIAS (H :OUTPUT)) :SIZE 1 1/1 :NORM 10.23003>
-->  #<->WEIGHT (INPUT (H :CELL)) :SIZE 1 1/1 :NORM 5.98116>
-->  #<->WEIGHT (:BIAS (H :CELL)) :SIZE 1 1/1 :NORM 0.10681>
-->  #<->WEIGHT (INPUT (H :FORGET)) :SIZE 1 1/1 :NORM 4.46301>
-->  #<->WEIGHT (:BIAS (H :FORGET)) :SIZE 1 1/1 :NORM 1.57195>
-->  #<->WEIGHT (INPUT (H :INPUT)) :SIZE 1 1/1 :NORM 0.36401>
-->  #<->WEIGHT (:BIAS (H :INPUT)) :SIZE 1 1/1 :NORM 8.63833>)

|#

• A recurrent neural net (as opposed to a feed-forward one. It is typically built with BUILD-RNN that's no more than a shallow convenience macro.

An RNN takes instances as inputs that are sequences of variable length. At each time step, the next unprocessed elements of these sequences are set as input until all input sequences in the batch run out. To be able to perform backpropagation, all intermediate LUMPs must be kept around, so the recursive connections are transformed out by unfolding the network. Just how many lumps this means depends on the length of the sequences.

When an RNN is created, MAX-LAG + 1 BPNs are instantiated so that all weights are present and one can start training it.

• RNN (:MAX-LAG = 1)

The networks built by UNFOLDER may contain new weights up to time step MAX-LAG. Beyond that point, all weight lumps must be reappearances of weight lumps with the same name at previous time steps. Most recurrent networks reference only the state of lumps at the previous time step (with the function LAG), hence the default of 1. But it is possible to have connections to arbitrary time steps. The maximum connection lag must be specified when creating the RNN.

• RNN (:CUDA-WINDOW-START-TIME = *CUDA-WINDOW-START-TIME*)

Due to unfolding, the memory footprint of an RNN is almost linear in the number of time steps (i.e. the max sequence length). For prediction, this is addressed by Time Warp. For training, we cannot discard results of previous time steps because they are needed for backpropagation, but we can at least move them out of GPU memory if they are not going to be used for a while and copy them back before they are needed. Obviously, this is only relevant if CUDA is being used.

If CUDA-WINDOW-START-TIME is NIL, then this feature is turned off. Else, during training, at CUDA-WINDOW-START-TIME or later time steps, matrices belonging to non-weight lumps may be forced out of GPU memory and later brought back as neeeded.

This feature is implemented in terms of MGL-MAT:WITH-SYNCING-CUDA-FACETS that uses CUDA host memory (also known as page-locked or pinned memory) to do asynchronous copies concurrently with normal computation. The consequence of this is that it is now main memory usage that's unbounded which toghether with page-locking makes it a potent weapon to bring a machine to a halt. You were warned.

• (&KEY RNN (CLASS ''RNN) NAME INITARGS MAX-N-STRIPES (MAX-LAG 1)) &BODY BODY

Create an RNN with MAX-N-STRIPES and MAX-LAG whose UNFOLDER is BODY wrapped in a lambda. Bind symbol given as the RNN argument to the RNN object so that BODY can see it.

• &KEY (RNN *RNN*)

Return the time step RNN is currently executing or being unfolded for. It is 0 when the RNN is being unfolded for the first time.

• INSTANCES (RNN RNN)

RNNs operate on batches of instances just like FNNs. But the instances here are like datasets: sequences or samplers and they are turned into sequences of batches of instances with MAP-DATASETS :IMPUTE NIL. The batch of instances at index 2 is clamped onto the BPN at time step 2 with SET-INPUT.

When the input sequences in the batch are not of the same length, already exhausted sequences will produce NIL (due to :IMPUTE NIL) above. When such a NIL is clamped with SET-INPUT on a BPN of the RNN, SET-INPUT must set the IMPORTANCE of the ->ERROR lumps to 0 else training would operate on the noise left there by previous invocations.

##### Time Warp

The unbounded memory usage of RNNs with one BPN allocated per time step can become a problem. For training, where the gradients often have to be backpropagated from the last time step to the very beginning, this is hard to solve but with CUDA-WINDOW-START-TIME the limit is no longer GPU memory.

For prediction on the other hand, one doesn't need to keep old steps around indefinitely: they can be discarded when future time steps will never reference them again.

• Controls whether warping is enabled (see Time Warp). Don't enable it for training, as it would make backprop impossible.

• RNN (:WARP-LENGTH = 1)

An integer such that the BPN UNFOLDER creates at time step I (where (<= WARP-START I)) is identical to the BPN created at time step (+ WARP-START (MOD (- I WARP-START) WARP-LENGTH)) except for a shift in its time lagged connections.

• RNN (:STEP-MONITORS = NIL)

During training, unfolded BPNs corresponding to previous time steps may be expensive to get at because they are no longer in GPU memory. This consideration also applies to making prediction with the additional caveat that with *WARP-TIME* true, previous states are discarded so it's not possible to gather statistics after FORWARD finished.

Add monitor objects to this slot and they will be automatically applied to the RNN after each step when FORWARDing the RNN during training or prediction. To be able to easily switch between sets of monitors, in addition to a list of monitors this can be a symbol or a function, too. If it's a symbol, then its a designator for its SYMBOL-VALUE. If it's a function, then it must have no arguments and it's a designator for its return value.

### 11.4 Lumps

#### 11.4.1 Lump Base Class

• A LUMP is a simple, layerlike component of a neural network. There are many kinds of lumps, each of which performs a specific operation or just stores inputs and weights. By convention, the names of lumps start with the prefix ->. Defined as classes, they also have a function of the same name as the class to create them easily. These maker functions typically have keyword arguments corresponding to initargs of the class, with some (mainly the input lumps) turned into normal positional arguments. So instead of having to do

(make-instance '->tanh :x some-input :name 'my-tanh)


one can simply write

(->tanh some-input :name 'my-tanh)


Lumps instantiated in any way within a BUILD-FNN or BUILD-RNN are automatically added to the network being built.

A lump has its own NODES and DERIVATIVES matrices allocated for it in which the results of the forward and backward passes are stored. This is in contrast to a BPN whose NODES and DERIVATIVES are those of its last constituent CLUMP.

Since lumps almost always live within a BPN, their N-STRIPES and MAX-N-STRIPES are handled automagically behind the scenes.

• LUMP (:SIZE)

The number of values in a single stripe.

• LUMP (:DEFAULT-VALUE = 0)

Upon creation or resize the lump's nodes get filled with this value.

• Return a default for the SIZE of LUMP if one is not supplied at instantiation. The value is often computed based on the sizes of the inputs. This function is for implementing new lump types.

• LUMP (= NIL)

The values computed by the lump in the forward pass are stored here. It is an N-STRIPES * SIZE matrix that has storage allocated for MAX-N-STRIPES * SIZE elements for non-weight lumps. ->WEIGHT lumps have no stripes nor restrictions on their shape.

• LUMP

The derivatives computed in the backward pass are stored here. This matrix is very much like NODES in shape and size.

#### 11.4.2 Inputs

##### Input Lump

• A lump that has no input lumps, does not change its values in the forward pass (except when DROPOUT is non-zero), and does not compute derivatives. Clamp inputs on NODES of input lumps in SET-INPUT.

For convenience, ->INPUT can perform dropout itself although it defaults to no dropout.

(->input :size 10 :name 'some-input)
==> #<->INPUT SOME-INPUT :SIZE 10 1/1 :NORM 0.00000>


• ->INPUT (= NIL)

##### Embedding Lump

This lump is like an input and a simple activation molded together in the name of efficiency.

• ->EMBEDDING (:INPUT-ROW-INDICES)

A sequence of batch size length of row indices. To be set in SET-INPUT.

#### 11.4.3 Weight Lump

• A set of optimizable parameters of some kind. When a BPN is is trained (see Training) the NODES of weight lumps will be changed. Weight lumps perform no computation.

Weights can be created by specifying the total size or the dimensions:

(dimensions (->weight :size 10 :name 'w))
=> (1 10)
(dimensions (->weight :dimensions '(5 10) :name 'w))
=> (5 10)


• (FROM-BPN) &BODY BODY

In BODY ->WEIGHT will first look up if a weight lump of the same name exists in FROM-BPN and return that, or else create a weight lump normally. If FROM-BPN is NIL, then no weights are copied.

#### 11.4.4 Activations

##### Activation Subnet

So we have some inputs. Usually the next step is to multiply the input vector with a weight matrix and add biases. This can be done directly with ->+, ->V*M and ->WEIGHT, but it's more convenient to use activation subnets to reduce the clutter.

• Activation subnetworks are built by the function ->ACTIVATION and they have a number of lumps hidden inside them. Ultimately, this subnetwork computes a sum like sum_i x_i * W_i + sum_j y_j .* V_j + biases where x_i are input lumps, W_i are dense matrices representing connections, while V_j are peephole connection vectors that are mulitplied in an elementwise manner with their corresponding input y_j.

• INPUTS &KEY (NAME (GENSYM)) SIZE PEEPHOLES (ADD-BIAS-P T)

Create a subnetwork of class ->ACTIVATION that computes the over activation from dense connection from lumps in INPUTS, and elementwise connection from lumps in PEEPHOLES. Create new ->WEIGHT lumps as necessary. INPUTS and PEEPHOLES can be a single lump or a list of lumps. Finally, if ADD-BIAS-P, then add an elementwise bias too. SIZE must be specified explicitly, because it is not possible to determine it unless there are peephole connections.

(->activation (->input :size 10 :name 'input) :name 'h1 :size 4)
==> #<->ACTIVATION (H1 :ACTIVATION) :STRIPES 1/1 :CLUMPS 4>


This is the basic workhorse of neural networks which takes care of the linear transformation whose results and then fed to some non-linearity (->SIGMOID, ->TANH, etc).

The name of the subnetwork clump is (,NAME :ACTIVATION). The bias weight lump (if any) is named (:BIAS ,NAME). Dense connection weight lumps are named are named after the input and NAME: (,(NAME INPUT) ,NAME), while peepholes weight lumps are named (,(NAME INPUT) ,NAME :PEEPHOLE). This is useful to know if, for example, they are to be initialized differently.

##### Batch-Normalization

• This is an implementation of v3 of the Batch Normalization paper. The output of ->BATCH-NORMALIZED is its input normalized so that for all elements the mean across stripes is zero and the variance is 1. That is, the mean of the batch is subtracted from the inputs and they are rescaled by their sample stddev. Actually, after the normalization step the values are rescaled and shifted (but this time with learnt parameters) in order to keep the representational power of the model the same. The primary purpose of this lump is to speed up learning, but it also acts as a regularizer. See the paper for the details.

To normalize the output of LUMP without no additional regularizer effect:

(->batch-normalized lump :batch-size :use-population)

The above uses an exponential moving average to estimate the mean and variance of batches and these estimations are used at both training and test time. In contrast to this, the published version uses the sample mean and variance of the current batch at training time which injects noise into the process. The noise is higher for lower batch sizes and has a regularizing effect. This is the default behavior (equivalent to :BATCH-SIZE NIL):

(->batch-normalized lump)

For performance reasons one may wish to process a higher number of instances in a batch (in the sense of N-STRIPES) and get the regularization effect associated with a lower batch size. This is possible by setting :BATCH-SIZE to a divisor of the the number of stripes. Say, the number of stripes is 128, but we want as much regularization as we would get with 32:

(->batch-normalized lump :batch-size 32)

The primary input of ->BATCH-NORMALIZED is often an ->ACTIVATION(0 1) and its output is fed into an activation function (see Activation Functions).

• The primary purpose of this class is to hold the estimated mean and variance of the inputs to be normalized and allow them to be shared between multiple ->BATCH-NORMALIZED lumps that carry out the computation. These estimations are saved and loaded by SAVE-STATE and LOAD-STATE.

(->batch-normalization (->weight :name '(h1 :scale) :size 10)
(->weight :name '(h1 :shift) :size 10)
:name '(h1 :batch-normalization))

• ->BATCH-NORMALIZATION (:SCALE)

A weight lump of the same size as SHIFT. This is $\gamma$ in the paper.

• ->BATCH-NORMALIZATION (:SHIFT)

A weight lump of the same size as SCALE. This is $\beta$ in the paper.

• ->BATCH-NORMALIZATION (:BATCH-SIZE = NIL)

Normally all stripes participate in the batch. Lowering the number of stripes may increase the regularization effect, but it also makes the computation less efficient. By setting BATCH-SIZE to a divisor of N-STRIPES one can decouple the concern of efficiency from that of regularization. The default value, NIL, is equivalent to N-STRIPES. BATCH-SIZE only affects training.

With the special value :USE-POPULATION, instead of the mean and the variance of the current batch, use the population statistics for normalization. This effectively cancels the regularization effect, leaving only the faster learning.

A small positive real number that's added to the sample variance. This is $\epsilon$ in the paper.

• ->BATCH-NORMALIZATION (:POPULATION-DECAY = 0.99)

While training, an exponential moving average of batch means and standard deviances (termed population statistics) is updated. When making predictions, normalization is performed using these statistics. These population statistics are persisted by SAVE-STATE.

• INPUTS &KEY (NAME (GENSYM)) SIZE PEEPHOLES BATCH-SIZE VARIANCE-ADJUSTMENT POPULATION-DECAY

A utility functions that creates and wraps an ->ACTIVATION(0 1) in ->BATCH-NORMALIZED and with its BATCH-NORMALIZATION the two weight lumps for the scale and shift parameters. (->BATCH-NORMALIZED-ACTIVATION INPUTS :NAME 'H1 :SIZE 10) is equivalent to:

(->batch-normalized (->activation inputs :name 'h1 :size 10 :add-bias-p nil)
:name '(h1 :batch-normalized-activation))

Note how biases are turned off since normalization will cancel them anyway (but a shift is added which amounts to the same effect).

#### 11.4.5 Activation Functions

Now we are moving on to the most important non-linearities to which activations are fed.

##### Sigmoid Lump

• Applies the 1/(1 + e^{-x}) function elementwise to its inputs. This is one of the classic non-linearities for neural networks.

For convenience, ->SIGMOID can perform dropout itself although it defaults to no dropout.

(->sigmoid (->activation (->input :size 10) :size 5) :name 'this)
==> #<->SIGMOID THIS :SIZE 5 1/1 :NORM 0.00000>


The SIZE(0 1) of this lump is the size of its input which is determined automatically.

• ->SIGMOID (= NIL)

##### Tanh Lump

• Applies the TANH function to its input in an elementwise manner. The SIZE(0 1) of this lump is the size of its input which is determined automatically.

##### Scaled Tanh Lump

• Pretty much like TANH but its input and output is scaled in such a way that the variance of its output is close to 1 if the variance of its input is close to 1 which is a nice property to combat vanishing gradients. The actual function is 1.7159 * tanh(2/3 * x). The SIZE(0 1) of this lump is the size of its input which is determined automatically.

##### Relu Lump

We are somewhere around year 2007 by now.

• max(0,x) activation function. Be careful, relu units can get stuck in the off state: if they move to far to negative territory it can be very difficult to get out of it. The SIZE(0 1) of this lump is the size of its input which is determined automatically.

##### Max Lump

We are in about year 2011.

• This is basically maxout without dropout (see http://arxiv.org/abs/1302.4389). It groups its inputs by GROUP-SIZE, and outputs the maximum of each group. The SIZE(0 1) of the output is automatically calculated, it is the size of the input divided by GROUP-SIZE.

(->max (->input :size 120) :group-size 3 :name 'my-max)
==> #<->MAX MY-MAX :SIZE 40 1/1 :NORM 0.00000 :GROUP-SIZE 3>


The advantage of ->MAX over ->RELU is that flow gradient is never stopped so there is no problem of units getting stuck in off state.

• ->MAX (:GROUP-SIZE)

The number of inputs in each group.

##### Min Lump

• ->MIN (:GROUP-SIZE)

The number of inputs in each group.

##### Max-Channel Lump

• Called LWTA (Local Winner Take All) or Channel-Out (see http://arxiv.org/abs/1312.1909) in the literature it is basically ->MAX, but instead of producing one output per group, it just produces zeros for all unit but the one with the maximum value in the group. This allows the next layer to get some information about the path along which information flowed. The SIZE(0 1) of this lump is the size of its input which is determined automatically.

• ->MAX-CHANNEL (:GROUP-SIZE)

The number of inputs in each group.

#### 11.4.6 Losses

Ultimately, we need to tell the network what to learn which means that the loss function to be minimized needs to be constructed as part of the network.

##### Loss Lump

• Calculate the loss for the instances in the batch. The main purpose of this lump is to provide a training signal.

An error lump is usually a leaf in the graph of lumps (i.e. there are no other lumps whose input is this one). The special thing about error lumps is that 1 (but see IMPORTANCE) is added automatically to their derivatives. Error lumps have exactly one node (per stripe) whose value is computed as the sum of nodes in their input lump.

• ->LOSS (:IMPORTANCE = NIL)

This is to support weighted instances. That is when not all training instances are equally important. If non-NIL, a 1d MAT with the importances of stripes of the batch. When IMPORTANCE is given (typically in SET-INPUT), then instead of adding 1 to the derivatives of all stripes, IMPORTANCE is added elemtwise.

##### Squared Difference Lump

In regression, the squared error loss is most common. The squared error loss can be constructed by combining ->SQUARED-DIFFERENCE with a ->LOSS.

• This lump takes two input lumps and calculates their squared difference (x - y)^2 in an elementwise manner. The SIZE(0 1) of this lump is automatically determined from the size of its inputs. This lump is often fed into ->LOSS that sums the squared differences and makes it part of the function to be minimized.

(->loss (->squared-difference (->activation (->input :size 100)
:size 10)
(->input :name 'target :size 10))
:name 'squared-error)
==> #<->LOSS SQUARED-ERROR :SIZE 1 1/1 :NORM 0.00000>


Currently this lump is not CUDAized, but it will copy data from the GPU if it needs to.

##### Softmax Cross-Entropy Loss Lump

• A specialized lump that computes the softmax of its input in the forward pass and backpropagates a cross-entropy loss. The advantage of doing these together is numerical stability. The total cross-entropy is the sum of cross-entropies per group of GROUP-SIZE elements:

$$XE(x) = - \sum_{i=1,g} t_i \ln(s_i)$$

where g is the number of classes (GROUP-SIZE), t_i are the targets (i.e. the true probabilities of the class, often all zero but one), s_i is the output of softmax calculated from input X:

$$s_i = {softmax}(x_1, x_2, ..., x_g) = \frac{e^x_i}{\sum_{j=1,g} e^x_j}$$

In other words, in the forward phase this lump takes input X, computes its elementwise EXP, normalizes each group of GROUP-SIZE elements to sum to 1 to get the softmax which is the result that goes into NODES. In the backward phase, there are two sources of gradients: the lumps that use the output of this lump as their input (currently not implemented and would result in an error) and an implicit cross-entropy loss.

One can get the cross-entropy calculated in the most recent forward pass by calling COST on this lump.

This is the most common loss function for classification. In fact, it is nearly ubiquitous. See the FNN Tutorial and the RNN Tutorial for how this loss and SET-INPUT work together.

• ->SOFTMAX-XE-LOSS (:GROUP-SIZE)

The number of elements in a softmax group. This is the number of classes for classification. Often GROUP-SIZE is equal to SIZE(0 1) (it is the default), but in general the only constraint is that SIZE(0 1) is a multiple of GROUP-SIZE.

• ->SOFTMAX-XE-LOSS (:TARGET = NIL)

Set in SET-INPUT, this is either a MAT of the same size as the input lump X or if the target is very sparse, this can also be a sequence of batch size length that contains the index value pairs of non-zero entries:

(;; first instance in batch has two non-zero targets
(;; class 10 has 30% expected probability
(10 . 0.3)
;; class 2 has 70% expected probability
(2 .  0.7))
;; second instance in batch puts 100% on class 7
7
;; more instances in the batch follow
...)


Actually, in the rare case where GROUP-SIZE is not SIZE(0 1) (i.e. there are several softmax normalization groups for every example), the length of the above target sequence is BATCH-SIZE(0 1 2) * N-GROUPS. Indices are always relative to the start of the group.

If GROUP-SIZE is large (for example, in neural language models with a huge number of words), using sparse targets can make things go much faster, because calculation of the derivative is no longer quadratic.

Giving different weights to training instances is implicitly supported. While target values in a group should sum to 1, multiplying all target values with a weight W is equivalent to training that W times on the same example.

#### 11.4.7 Stochasticity

##### Dropout Lump

• The output of this lump is identical to its input, except it randomly zeroes out some of them during training which act as a very strong regularizer. See Geoffrey Hinton's 'Improving neural networks by preventing co-adaptation of feature detectors'.

The SIZE(0 1) of this lump is the size of its input which is determined automatically.

• ->DROPOUT (:DROPOUT = 0.5)

If non-NIL, then in the forward pass zero out each node in this chunk with DROPOUT probability.

##### Gaussian Random Lump

• This lump has no input, it produces normally distributed independent random numbers with MEAN and VARIANCE (or VARIANCE-FOR-PREDICTION). This is useful building block for noise based regularization methods.

(->gaussian-random :size 10 :name 'normal :mean 1 :variance 2)
==> #<->GAUSSIAN-RANDOM NORMAL :SIZE 10 1/1 :NORM 0.00000>


• ->GAUSSIAN-RANDOM (:MEAN = 0)

The mean of the normal distribution.

• ->GAUSSIAN-RANDOM (:VARIANCE = 1)

The variance of the normal distribution.

• ->GAUSSIAN-RANDOM (:VARIANCE-FOR-PREDICTION = 0)

If not NIL, then this value overrides VARIANCE when not in training (i.e. when making predictions).

##### Binary Sampling Lump

• Treating values of its input as probabilities, sample independent binomials. Turn true into 1 and false into 0. The SIZE(0 1) of this lump is determined automatically from the size of its input.

(->sample-binary (->input :size 10) :name 'binarized-input)
==> #<->SAMPLE-BINARY BINARIZED-INPUT :SIZE 10 1/1 :NORM 0.00000>


#### 11.4.8 Arithmetic

##### Vector-Matrix Multiplication Lump

• ->V*M (:TRANSPOSE-WEIGHTS-P = NIL)

Determines whether the input is multiplied by WEIGHTS(0 1) or its transpose.

• Performs elementwise addition on its input lumps. The SIZE(0 1) of this lump is automatically determined from the size of its inputs if there is at least one. If one of the inputs is a ->WEIGHT lump, then it is added to every stripe.

(->+ (list (->input :size 10) (->weight :size 10 :name 'bias))
:name 'plus)
==> #<->+ PLUS :SIZE 10 1/1 :NORM 0.00000>


##### Elementwise Multiplication Lump

• Performs elementwise multiplication on its two input lumps. The SIZE(0 1) of this lump is automatically determined from the size of its inputs. Either input can be a ->WEIGHT lump.

(->* (->input :size 10) (->weight :size 10 :name 'scale)
:name 'mult)
==> #<->* MULT :SIZE 10 1/1 :NORM 0.00000>


#### 11.4.9 Operations for RNNs

##### LSTM Subnet

• Long-Short Term Memory subnetworks are built by the function ->LSTM and they have many lumps hidden inside them. These lumps are packaged into a subnetwork to reduce clutter.

• INPUTS &KEY NAME CELL-INIT OUTPUT-INIT SIZE (ACTIVATION-FN '->ACTIVATION(0 1)) (GATE-FN '->SIGMOID) (INPUT-FN '->TANH) (OUTPUT-FN '->TANH) (PEEPHOLES T)

Create an LSTM layer consisting of input, forget, output gates with which input, cell state and output are scaled. Lots of lumps are created, the final one representing to output of the LSTM has NAME. The rest of the lumps are named automatically based on NAME. This function returns only the output lump (m), but all created lumps are added automatically to the BPN being built.

There are many papers and tutorials on LSTMs. This version is well described in "Long Short-Term Memory Recurrent Neural Network Architectures for Large Scale Acoustic Modeling" (2014, Hasim Sak, Andrew Senior, Francoise Beaufays). Using the notation from that paper:

$$i_t = s(W_{ix} x_t + W_{im} m_{t-1} + W_{ic} \odot c_{t-1} + b_i)$$

$$f_t = s(W_{fx} x_t + W_{fm} m_{t-1} + W_{fc} \odot c_{t-1} + b_f)$$

$$c_t = f_t \odot c_{t-1} + i_t \odot g(W_{cx} x_t + W_{cm} m_{t-1} + b_c)$$

$$o_t = s(W_{ox} x_t + W_{om} m_{t-1} + W_{oc} \odot c_t + b_o)$$

$$m_t = o_t \odot h(c_t)$$

... where i, f, and o are the input, forget and output gates. c is the cell state and m is the actual output.

Weight matrices for connections from c (W_ic, W_fc and W_oc) are diagonal and represented by just the vector of diagonal values. These connections are only added if PEEPHOLES is true.

A notable difference from the paper is that in addition to being a single lump, x_t (INPUTS) can also be a list of lumps. Whenever some activation is to be calculated based on x_t, it is going to be the sum of individual activations. For example, W_ix * x_t is really sum_j W_ijx * inputs_j.

If CELL-INIT is non-NIL, then it must be a CLUMP of SIZE form which stands for the initial state of the value cell (c_{-1}). CELL-INIT being NIL is equivalent to the state of all zeros.

ACTIVATION-FN defaults to ->ACTIVATION(0 1), but it can be for example ->BATCH-NORMALIZED-ACTIVATION. In general, functions like the aforementioned two with signature like (INPUTS &KEY NAME SIZE PEEPHOLES) can be passed as ACTIVATION-FN.

##### Sequence Barrier Lump

• In an RNN, processing of stripes (instances in the batch) may require different number of time step so the final state for stripe 0 is in stripe 0 of some lump L at time step 7, while for stripe 1 it is in stripe 1 of sump lump L at time step 42.

This lump copies the per-stripe states from different lumps into a single lump so that further processing can take place (typically when the RNN is embedded in another network).

The SIZE(0 1) of this lump is automatically set to the size of the lump returned by (FUNCALL SEQ-ELT-FN 0).

• ->SEQ-BARRIER (:SEQ-ELT-FN)

A function of an [INDEX] argument that returns the lump with that index in some sequence.

• ->SEQ-BARRIER

A sequence of length batch size of indices. The element at index I is the index to be passed to SEQ-ELT-FN to find the lump whose stripe I is copied to stripe I of this this lump.

### 11.5 Utilities

• ->V*M-LUMPS L2-UPPER-BOUND

If the l2 norm of the incoming weight vector of the a unit is larger than L2-UPPER-BOUND then renormalize it to L2-UPPER-BOUND. The list of ->V*M-LUMPS is assumed to be eventually fed to the same lump.

To use it, group the activation clumps into the same GD-OPTIMIZER and hang this function on AFTER-UPDATE-HOOK, that latter of which is done for you ARRANGE-FOR-RENORMALIZING-ACTIVATIONS.

See "Improving neural networks by preventing co-adaptation of feature detectors (Hinton, 2012)", http://arxiv.org/pdf/1207.0580.pdf.

• BPN OPTIMIZER L2-UPPER-BOUND

By pushing a lambda to AFTER-UPDATE-HOOK of OPTIMIZER arrange for all weights beings trained by OPTIMIZER to be renormalized (as in RENORMALIZE-ACTIVATIONS with L2-UPPER-BOUND).

It is assumed that if the weights either belong to an activation lump or are simply added to the activations (i.e. they are biases).

## 14 Natural Language Processing

###### [in package MGL-NLP]

This in nothing more then a couple of utilities for now which may grow into a more serious toolset for NLP eventually.

• FUNCTION N

Make a function of a single argument that's suitable as the function argument to a mapper function. It calls FUNCTION with every N element.

(map nil (make-n-gram-mappee #'print 3) '(a b c d e))
..
.. (A B C)
.. (B C D)
.. (C D E)


• CANDIDATES REFERENCES &KEY CANDIDATE-KEY REFERENCE-KEY (N 4)

Compute the BLEU score for bilingual CORPUS. BLEU measures how good a translation is compared to human reference translations.

CANDIDATES (keyed by CANDIDATE-KEY) and REFERENCES (keyed by REFERENCE-KEY) are sequences of sentences. A sentence is a sequence of words. Words are compared with EQUAL, and may be any kind of object (not necessarily strings).

Currently there is no support for multiple reference translations. N determines the largest n-grams to consider.

The first return value is the BLEU score (between 0 and 1, not as a percentage). The second value is the sum of the lengths of CANDIDATES divided by the sum of the lengths of REFERENCES (or NIL, if the denominator is 0). The third is a list of n-gram precisions (also between 0 and 1 or NIL), one for each element in [1..N][].

This is basically a reimplementation of multi-bleu.perl.

(bleu '((1 2 3 4) (a b))
'((1 2 3 4) (1 2)))
=> 0.8408964
=> 1
=> (;; 1-gram precision: 4/6
2/3
;; 2-gram precision: 3/4
3/4
;; 3-gram precision: 2/2
1
;; 4-gram precision: 1/1
1)


### 14.1 Bag of Words

• ENCODE all features of a document with a sparse vector. Get the features of document from MAPPER, encode each feature with FEATURE-ENCODER. FEATURE-ENCODER may return NIL if the feature is not used. The result is a vector of encoded-feature/value conses. encoded-features are unique (under ENCODED-FEATURE-TEST) within the vector but are in no particular order.

Depending on KIND, value is calculated in various ways:

• For :FREQUENCY it is the number of times the corresponding feature was found in DOCUMENT.

• For :BINARY it is always 1.

• For :NORMALIZED-FREQUENCY and :NORMALIZED-BINARY are like the unnormalized counterparts except that as the final step values in the assembled sparse vector are normalized to sum to 1.

• Finally, :COMPACTED-BINARY is like :BINARY but the return values is not a vector of conses, but a vector of element-type ENCODED-FEATURE-TYPE.

(let* ((feature-indexer
(make-indexer
(alexandria:alist-hash-table '(("I" . 3) ("me" . 2) ("mine" . 1)))
2))
(bag-of-words-encoder
(make-instance 'bag-of-words-encoder
:feature-encoder feature-indexer
:feature-mapper (lambda (fn document)
(map nil fn document))
:kind :frequency)))
(encode bag-of-words-encoder '("All" "through" "day" "I" "me" "mine"
"I" "me" "mine" "I" "me" "mine")))
=> #((0 . 3.0d0) (1 . 3.0d0))


• BAG-OF-WORDS-ENCODER (:FEATURE-ENCODER)

• BAG-OF-WORDS-ENCODER (:FEATURE-MAPPER)

• BAG-OF-WORDS-ENCODER (:ENCODED-FEATURE-TEST = #'EQL)

• BAG-OF-WORDS-ENCODER (:ENCODED-FEATURE-TYPE = T)

• BAG-OF-WORDS-ENCODER (:KIND = :BINARY`)