incrementalLearner
Convert binary classification support vector machine (SVM) model to incremental learner
Since R2020b
Description
returns a binary classification linear model for incremental learning,
IncrementalMdl
= incrementalLearner(Mdl
)IncrementalMdl
, using the traditionally trained linear SVM model
object or SVM model template object in Mdl
.
If you specify a traditionally trained model, then its property values reflect the
knowledge gained from Mdl
(parameters and hyperparameters of the
model). Therefore, IncrementalMdl
can predict labels given new
observations, and it is warm, meaning that its predictive performance
is tracked.
uses additional options specified by one or more namevalue
arguments. Some options require you to train IncrementalMdl
= incrementalLearner(Mdl
,Name,Value
)IncrementalMdl
before its
predictive performance is tracked. For example,
'MetricsWarmupPeriod',50,'MetricsWindowSize',100
specifies a preliminary
incremental training period of 50 observations before performance metrics are tracked, and
specifies processing 100 observations before updating the window performance metrics.
Examples
Convert Traditionally Trained Model to Incremental Learner
Train an SVM model by using fitcsvm
, and then convert it to an incremental learner.
Load and Preprocess Data
Load the human activity data set.
load humanactivity
For details on the data set, enter Description
at the command line.
Responses can be one of five classes: Sitting
, Standing
, Walking
, Running
, or Dancing
. Dichotomize the response by identifying whether the subject is moving (actid
> 2).
Y = actid > 2;
Train SVM Model
Fit an SVM model to the entire data set. Discard the support vectors (Alpha
) from the model so that the software uses the linear coefficients (Beta
) for prediction.
TTMdl = fitcsvm(feat,Y); TTMdl = discardSupportVectors(TTMdl)
TTMdl = ClassificationSVM ResponseName: 'Y' CategoricalPredictors: [] ClassNames: [0 1] ScoreTransform: 'none' NumObservations: 24075 Beta: [60x1 double] Bias: 6.4280 KernelParameters: [1x1 struct] BoxConstraints: [24075x1 double] ConvergenceInfo: [1x1 struct] IsSupportVector: [24075x1 logical] Solver: 'SMO'
TTMdl
is a ClassificationSVM
model object representing a traditionally trained SVM model.
Convert Trained Model
Convert the traditionally trained SVM model to a binary classification linear model for incremental learning.
IncrementalMdl = incrementalLearner(TTMdl)
IncrementalMdl = incrementalClassificationLinear IsWarm: 1 Metrics: [1x2 table] ClassNames: [0 1] ScoreTransform: 'none' Beta: [60x1 double] Bias: 6.4280 Learner: 'svm'
IncrementalMdl
is an incrementalClassificationLinear
model object prepared for incremental learning using SVM.
The
incrementalLearner
function Initializes the incremental learner by passing learned coefficients to it, along with other informationTTMdl
extracted from the training data.IncrementalMdl
is warm (IsWarm
is1
), which means that incremental learning functions can start tracking performance metrics.The
incrementalLearner
function specifies to train the model using the adaptive scaleinvariant solver, whereasfitcsvm
trainedTTMdl
using theSMO
solver.
Predict Responses
An incremental learner created from converting a traditionally trained model can generate predictions without further processing.
Predict classification scores for all observations using both models.
[~,ttscores] = predict(TTMdl,feat); [~,ilcores] = predict(IncrementalMdl,feat); compareScores = norm(ttscores(:,1)  ilcores(:,1))
compareScores = 0
The difference between the scores generated by the models is 0.
Specify SGD Solver and Standardize Predictor Data
The default solver is the adaptive scaleinvariant solver. If you specify this solver, you do not need to tune any parameters for training. However, if you specify either the standard SGD or ASGD solver instead, you can also specify an estimation period, during which the incremental fitting functions tune the learning rate.
Load the human activity data set.
load humanactivity
For details on the data set, enter Description
at the command line.
Responses can be one of five classes: Sitting
, Standing
, Walking
, Running
, and Dancing
. Dichotomize the response by identifying whether the subject is moving (actid
> 2).
Y = actid > 2;
Randomly split the data in half: the first half for training a model traditionally, and the second half for incremental learning.
n = numel(Y); rng(1) % For reproducibility cvp = cvpartition(n,'Holdout',0.5); idxtt = training(cvp); idxil = test(cvp); % First half of data Xtt = feat(idxtt,:); Ytt = Y(idxtt); % Second half of data Xil = feat(idxil,:); Yil = Y(idxil);
Fit an SVM model to the first half of the data. Standardize the predictor data by setting 'Standardize',true
.
TTMdl = fitcsvm(Xtt,Ytt,'Standardize',true);
The Mu
and Sigma
properties of TTMdl
contain the predictor data sample means and standard deviations, respectively.
Suppose that the distribution of the predictors is not expected to change in the future. Convert the traditionally trained SVM model to a binary classification linear model for incremental learning. Specify the standard SGD solver and an estimation period of 2000
observations (the default is 1000
when a learning rate is required).
IncrementalMdl = incrementalLearner(TTMdl,'Solver','sgd','EstimationPeriod',2000);
IncrementalMdl
is an incrementalClassificationLinear
model object. Because the predictor data of TTMdl
is standardized (TTMdl.Mu
and TTMdl.Sigma
are nonempty), incrementalLearner
prepares incremental learning functions to standardize supplied predictor data by using the previously learned moments (stored in IncrementalMdl.Mu
and IncrementalMdl.Sigma
).
Fit the incremental model to the second half of the data by using the fit
function. At each iteration:
Simulate a data stream by processing 10 observations at a time.
Overwrite the previous incremental model with a new one fitted to the incoming observations.
Store the initial learning rate and ${\beta}_{1}$ to see how the coefficients and rate evolve during training.
% Preallocation nil = numel(Yil); numObsPerChunk = 10; nchunk = floor(nil/numObsPerChunk); learnrate = [IncrementalMdl.LearnRate; zeros(nchunk,1)]; beta1 = [IncrementalMdl.Beta(1); zeros(nchunk,1)]; % Incremental fitting for j = 1:nchunk ibegin = min(nil,numObsPerChunk*(j1) + 1); iend = min(nil,numObsPerChunk*j); idx = ibegin:iend; IncrementalMdl = fit(IncrementalMdl,Xil(idx,:),Yil(idx)); beta1(j + 1) = IncrementalMdl.Beta(1); learnrate(j + 1) = IncrementalMdl.LearnRate; end
IncrementalMdl
is an incrementalClassificationLinear
model object trained on all the data in the stream.
To see how the initial learning rate and ${\beta}_{1}$ evolve during training, plot them on separate tiles.
t = tiledlayout(2,1); nexttile plot(beta1) ylabel('\beta_1') xline(IncrementalMdl.EstimationPeriod/numObsPerChunk,'r.') nexttile plot(learnrate) ylabel('Initial Learning Rate') xline(IncrementalMdl.EstimationPeriod/numObsPerChunk,'r.') xlabel(t,'Iteration')
The initial learning rate jumps from 0.7
to its autotuned value after the estimation period. During training, the software uses a learning rate that gradually decays from the initial value specified in the LearnRateSchedule property of IncrementalMdl
.
Because fit
does not fit the model to the streaming data during the estimation period, ${\beta}_{1}$ is constant for the first 200 iterations (2000 observations). Then, ${\beta}_{1}$ changes during incremental fitting.
Configure Performance Metric Options
Use a trained SVM model to initialize an incremental learner. Prepare the incremental learner by specifying a metrics warmup period, during which the updateMetricsAndFit
function only fits the model. Specify a metrics window size of 500 observations.
Load the human activity data set.
load humanactivity
For details on the data set, enter Description
at the command line
Responses can be one of five classes: Sitting
, Standing
, Walking
, Running
, and Dancing
. Dichotomize the response by identifying whether the subject is moving (actid
> 2).
Y = actid > 2;
Because the data set is grouped by activity, shuffle it to reduce bias. Then, randomly split the data in half: the first half for training a model traditionally, and the second half for incremental learning.
n = numel(Y); rng(1) % For reproducibility cvp = cvpartition(n,'Holdout',0.5); idxtt = training(cvp); idxil = test(cvp); shuffidx = randperm(n); X = feat(shuffidx,:); Y = Y(shuffidx); % First half of data Xtt = X(idxtt,:); Ytt = Y(idxtt); % Second half of data Xil = X(idxil,:); Yil = Y(idxil);
Fit an SVM model to the first half of the data.
TTMdl = fitcsvm(Xtt,Ytt);
Convert the traditionally trained SVM model to a binary classification linear model for incremental learning. Specify the following:
A performance metrics warmup period of 2000 observations
A metrics window size of 500 observations
Use of classification error and hinge loss to measure the performance of the model
IncrementalMdl = incrementalLearner(TTMdl,'MetricsWarmupPeriod',2000,'MetricsWindowSize',500,... 'Metrics',["classiferror" "hinge"]);
Fit the incremental model to the second half of the data by using the updateMetricsAndFit
function. At each iteration:
Simulate a data stream by processing 20 observations at a time.
Overwrite the previous incremental model with a new one fitted to the incoming observations.
Store ${\beta}_{1}$, the cumulative metrics, and the window metrics to see how they evolve during incremental learning.
% Preallocation nil = numel(Yil); numObsPerChunk = 20; nchunk = ceil(nil/numObsPerChunk); ce = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); hinge = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]); beta1 = [IncrementalMdl.Beta(1); zeros(nchunk,1)]; % Incremental fitting for j = 1:nchunk ibegin = min(nil,numObsPerChunk*(j1) + 1); iend = min(nil,numObsPerChunk*j); idx = ibegin:iend; IncrementalMdl = updateMetricsAndFit(IncrementalMdl,Xil(idx,:),Yil(idx)); ce{j,:} = IncrementalMdl.Metrics{"ClassificationError",:}; hinge{j,:} = IncrementalMdl.Metrics{"HingeLoss",:}; beta1(j + 1) = IncrementalMdl.Beta(1); end
IncrementalMdl
is an incrementalClassificationLinear
model object trained on all the data in the stream. During incremental learning and after the model is warmed up, updateMetricsAndFit
checks the performance of the model on the incoming observations, and then fits the model to those observations.
To see how the performance metrics and ${\beta}_{1}$ evolve during training, plot them on separate tiles.
t = tiledlayout(3,1); nexttile plot(beta1) ylabel('\beta_1') xlim([0 nchunk]); xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r.'); nexttile h = plot(ce.Variables); xlim([0 nchunk]); ylabel('Classification Error') xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r.'); legend(h,ce.Properties.VariableNames,'Location','northwest') nexttile h = plot(hinge.Variables); xlim([0 nchunk]); ylabel('Hinge Loss') xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r.'); legend(h,hinge.Properties.VariableNames,'Location','northwest') xlabel(t,'Iteration')
The plot suggests that updateMetricsAndFit
does the following:
Fit ${\beta}_{1}$ during all incremental learning iterations.
Compute the performance metrics after the metrics warmup period only.
Compute the cumulative metrics during each iteration.
Compute the window metrics after processing 500 observations (25 iterations).
Input Arguments
Mdl
— Traditionally trained model or model template
ClassificationSVM
model object  CompactClassificationSVM
model object  SVM model template
Traditionally trained linear SVM model or SVM model template, specified as a model object returned by its training or processing function.
Model Object or Template Object  Training or Processing Function 

ClassificationSVM model
object  fitcsvm 
CompactClassificationSVM model
object  fitcsvm or compact 
SVM model template object  templateSVM 
Note
Incremental learning functions support only numeric input predictor data. If
Mdl
was trained on categorical data, you must prepare an encoded version of the categorical data to use incremental learning functions. Usedummyvar
to convert each categorical variable to a numeric matrix of dummy variables. Then, concatenate all dummy variable matrices and any other numeric predictors, in the same way that the training function encodes categorical data. For more details, see Dummy Variables.If
Mdl
is a SVM model template object,incrementalLearner
determines whether to standardize the predictor variables based on theStandardize
property of the model template object. For more information, see Standardize Data.
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'Solver','scaleinvariant','MetricsWindowSize',100
specifies
the adaptive scaleinvariant solver for objective optimization, and specifies processing 100
observations before updating the window performance metrics.
Solver
— Objective function minimization technique
'scaleinvariant'
(default)  'sgd'
 'asgd'
Objective function minimization technique, specified as a value in this table.
Value  Description  Notes 

'scaleinvariant'  Adaptive scaleinvariant solver for incremental learning [1] 

'sgd'  Stochastic gradient descent (SGD) [3][2] 

'asgd'  Average stochastic gradient descent (ASGD) [4] 

The linear model for incremental learning (IncrementalMdl
) does
not support the solver used to train the traditionally trained linear SVM model
Mdl
or the solver specified in the SVM model template object
Mdl
. By default, the incrementalLearner
function
sets IncrementalMdl
to use the adaptive scaleinvariant solver
('scaleinvariant'
).
Example: 'Solver','sgd'
Data Types: char
 string
EstimationPeriod
— Number of observations processed to estimate hyperparameters
nonnegative integer
Number of observations processed by the incremental model to estimate hyperparameters before training or tracking performance metrics, specified as the commaseparated pair consisting of 'EstimationPeriod'
and a nonnegative integer.
Note
If
Mdl
is prepared for incremental learning (all hyperparameters required for training are specified),incrementalLearner
forcesEstimationPeriod
to0
.If
Mdl
is not prepared for incremental learning,incrementalLearner
setsEstimationPeriod
to1000
.
For more details, see Estimation Period.
Example: 'EstimationPeriod',100
Data Types: single
 double
BatchSize
— Minibatch size
10
(default)  positive integer
Minibatch size, specified as the commaseparated pair consisting of
'BatchSize'
and a positive integer. At each learning cycle during
training, incrementalLearner
uses BatchSize
observations to
compute the subgradient.
The number of observations for the last minibatch (last learning cycle in each function
call of fit
or updateMetricsAndFit
) can be
smaller than BatchSize
. For example, if you supply 25 observations to
fit
or updateMetricsAndFit
, the function uses
10 observations for the first two learning cycles and uses 5 observations for the last
learning cycle.
Example: 'BatchSize',1
Data Types: single
 double
Lambda
— Ridge (L^{2}) regularization term strength
1e5
(default)  nonnegative scalar
Ridge (L^{2}) regularization term strength, specified as the commaseparated pair consisting of 'Lambda'
and a nonnegative scalar.
Example: 'Lambda',0.01
Data Types: single
 double
LearnRate
— Initial learning rate
'auto'
(default)  positive scalar
Initial learning rate, specified as the commaseparated pair consisting of
'LearnRate'
and 'auto'
or a positive scalar.
LearnRate
controls the optimization step size by scaling the
objective subgradient.
The learning rate controls the optimization step size by scaling the objective
subgradient. LearnRate
specifies an initial value for the learning
rate, and LearnRateSchedule
determines
the learning rate for subsequent learning cycles.
When you specify 'auto'
:
The initial learning rate is
0.7
.If
EstimationPeriod
>0
,fit
andupdateMetricsAndFit
change the rate to1/sqrt(1+max(sum(X.^2,obsDim)))
at the end ofEstimationPeriod
. When the observations are the columns of the predictor dataX
collected during the estimation period, theobsDim
value is1
; otherwise, the value is2
.
Example: 'LearnRate',0.001
Data Types: single
 double
 char
 string
LearnRateSchedule
— Learning rate schedule
'decaying'
(default)  'constant'
Learning rate schedule, specified as the commaseparated pair consisting of 'LearnRateSchedule'
and a value in this table, where LearnRate
specifies the initial learning rate ɣ_{0}.
Value  Description 

'constant'  The learning rate is ɣ_{0} for all learning cycles. 
'decaying'  The learning rate at learning cycle t is $${\gamma}_{t}=\frac{{\gamma}_{0}}{{\left(1+\lambda {\gamma}_{0}t\right)}^{c}}.$$

Example: 'LearnRateSchedule','constant'
Data Types: char
 string
Shuffle
— Flag for shuffling observations in batch
true
(default)  false
Flag for shuffling the observations in the batch at each iteration, specified as the commaseparated pair consisting of 'Shuffle'
and a value in this table.
Value  Description 

true  The software shuffles an incoming chunk of data before the
fit function fits the model. This action
reduces bias induced by the sampling scheme. 
false  The software processes the data in the order received. 
Example: 'Shuffle',false
Data Types: logical
Metrics
— Model performance metrics to track during incremental learning
"classiferror"
(default)  string vector  function handle  cell vector  structure array  "binodeviance"
 "exponential"
 "hinge"
 "logit"
 "quadratic"
Model performance metrics to track during incremental learning with the updateMetrics
or updateMetricsAndFit
function, specified as a builtin loss function name, string vector of names, function handle (@metricName
), structure array of function handles, or cell vector of names, function handles, or structure arrays.
The following table lists the builtin loss function names. You can specify more than one by using a string vector.
Name  Description 

"binodeviance"  Binomial deviance 
"classiferror"  Classification error 
"exponential"  Exponential loss 
"hinge"  Hinge loss 
"logit"  Logistic loss 
"quadratic"  Quadratic loss 
For more details on the builtin loss functions, see loss
.
Example: 'Metrics',["classiferror" "hinge"]
To specify a custom function that returns a performance metric, use function handle notation. The function must have this form:
metric = customMetric(C,S)
The output argument
metric
is an nby1 numeric vector, where each element is the loss of the corresponding observation in the data processed by the incremental learning functions during a learning cycle.You specify the function name (
customMetric
).C
is an nby2 logical matrix with rows indicating the class to which the corresponding observation belongs. The column order corresponds to the class order in the model for incremental learning. CreateC
by settingC(
=p
,q
)1
, if observation
is in classp
, for each observation in the specified data. Set the other element in rowq
top
0
.S
is an nby2 numeric matrix of predicted classification scores.S
is similar to thescore
output ofpredict
, where rows correspond to observations in the data, and the column order corresponds to the class order in the model for incremental learning.S(
is the classification score of observationp
,q
)
being classified in classp
.q
To specify multiple custom metrics and assign a custom name to each, use a structure array. To specify a combination of builtin and custom metrics, use a cell vector.
Example: 'Metrics',struct('Metric1',@customMetric1,'Metric2',@customMetric2)
Example: 'Metrics',{@customMetric1 @customMetric2 'logit' struct('Metric3',@customMetric3)}
updateMetrics
and updateMetricsAndFit
store specified metrics in a table in the property IncrementalMdl.Metrics
. The data type of Metrics
determines the row names of the table.
'Metrics' Value Data Type  Description of Metrics Property Row Name  Example 

String or character vector  Name of corresponding builtin metric  Row name for "classiferror" is "ClassificationError" 
Structure array  Field name  Row name for struct('Metric1',@customMetric1) is "Metric1" 
Function handle to function stored in a program file  Name of function  Row name for @customMetric is "customMetric" 
Anonymous function  CustomMetric_ , where is metric in Metrics  Row name for @(C,S)customMetric(C,S)... is CustomMetric_1 
For more details on performance metrics options, see Performance Metrics.
Data Types: char
 string
 struct
 cell
 function_handle
MetricsWarmupPeriod
— Number of observations fit before tracking performance metrics
0
(default)  nonnegative integer
Number of observations the incremental model must be fit to before it tracks
performance metrics in its Metrics
property, specified as a
nonnegative integer. The incremental model is warm after incremental fitting functions
fit (EstimationPeriod
+ MetricsWarmupPeriod
)
observations to the incremental model.
For more details on performance metrics options, see Performance Metrics.
Example: 'MetricsWarmupPeriod',50
Data Types: single
 double
MetricsWindowSize
— Number of observations to use to compute window performance metrics
200
(default)  positive integer
Number of observations to use to compute window performance metrics, specified as a positive integer.
For more details on performance metrics options, see Performance Metrics.
Example: 'MetricsWindowSize',100
Data Types: single
 double
Output Arguments
IncrementalMdl
— Binary classification linear model for incremental learning
incrementalClassificationLinear
model object
Binary classification linear model for incremental learning, returned as an incrementalClassificationLinear
model object. IncrementalMdl
is also configured to generate predictions given new data (see predict
).
If you specify a traditionally trained model object in
Mdl
,incrementalLearner
passes the values of theMdl
properties to corresponding properties ofIncrementalMdl
to initializeIncrementalMdl
for incremental learning.Property Description Beta
Scaled linear model coefficients, Mdl.Beta/Mdl.KernelParameters.Scale
, a numeric vectorBias
Model intercept, a numeric scalar ClassNames
Class labels for binary classification, twoelement list Mu
Predictor variable means, a numeric vector NumPredictors
Number of predictors, a positive integer Prior
Prior class label distribution, a numeric vector Sigma
Predictor variable standard deviations, a numeric vector ScoreTransform
Score transformation function, a function name or function handle Note that
incrementalLearner
does not use theCost
property of the traditionally trained model inMdl
becauseincrementalClassificationLinear
does not support this property.
More About
Incremental Learning
Incremental learning, or online learning, is a branch of machine learning concerned with processing incoming data from a data stream, possibly given little to no knowledge of the distribution of the predictor variables, aspects of the prediction or objective function (including tuning parameter values), or whether the observations are labeled. Incremental learning differs from traditional machine learning, where enough labeled data is available to fit to a model, perform crossvalidation to tune hyperparameters, and infer the predictor distribution.
Given incoming observations, an incremental learning model processes data in any of the following ways, but usually in this order:
Predict labels.
Measure the predictive performance.
Check for structural breaks or drift in the model.
Fit the model to the incoming observations.
For more details, see Incremental Learning Overview.
Adaptive ScaleInvariant Solver for Incremental Learning
The adaptive scaleinvariant solver for incremental learning, introduced in [1], is a gradientdescentbased objective solver for training linear predictive models. The solver is hyperparameter free, insensitive to differences in predictor variable scales, and does not require prior knowledge of the distribution of the predictor variables. These characteristics make it well suited to incremental learning.
The standard SGD and ASGD solvers are sensitive to differing scales among the predictor variables, resulting in models that can perform poorly. To achieve better accuracy using SGD and ASGD, you can standardize the predictor data, and tune the regularization and learning rate parameters. For traditional machine learning, enough data is available to enable hyperparameter tuning by crossvalidation and predictor standardization. However, for incremental learning, enough data might not be available (for example, observations might be available only one at a time) and the distribution of the predictors might be unknown. These characteristics make parameter tuning and predictor standardization difficult or impossible to do during incremental learning.
The incremental fitting functions for classification fit
and updateMetricsAndFit
use the more aggressive ScInOL2 version of the algorithm.
Algorithms
Estimation Period
During the estimation period, the incremental fitting functions fit
and updateMetricsAndFit
use the
first incoming EstimationPeriod
observations
to estimate (tune) hyperparameters required for incremental training. Estimation occurs only
when EstimationPeriod
is positive. This table describes the
hyperparameters and when they are estimated, or tuned.
Hyperparameter  Model Property  Usage  Conditions 

Predictor means and standard deviations 
 Standardize predictor data  The hyperparameters are estimated when both of these conditions apply:

Learning rate  LearnRate  Adjust the solver step size  The hyperparameter is estimated when both of these conditions apply:

During the estimation period, fit
does not fit the model, and updateMetricsAndFit
does not fit the model or update the performance metrics. At the end of the estimation period, the functions update the properties that store the hyperparameters.
Standardize Data
If incremental learning functions are configured to standardize predictor variables, they do so using the means and standard deviations stored in the Mu
and Sigma
properties of the incremental learning model IncrementalMdl
.
If you standardize the predictor data when you train the input model
Mdl
by usingfitcsvm
, the following conditions apply:incrementalLearner
passes the means inMdl.Mu
and standard deviations inMdl.Sigma
to the corresponding incremental learning model properties.Incremental learning functions always standardize the predictor data.
When you set
Standardize=true
by using theStandardize
namevalue argument oftemplateSVM
, and theMdl.Mu
andMdl.Sigma
properties are empty, the following conditions apply:If the estimation period is positive (see the
EstimationPeriod
property ofIncrementalMdl
), incremental fitting functions estimate the means and standard deviations using the estimation period observations.If the estimation period is 0,
incrementalLearner
forces the estimation period to1000
. Consequently, incremental fitting functions estimate new predictor variable means and standard deviations during the forced estimation period.
When incremental fitting functions estimate predictor means and standard deviations, the functions compute weighted means and weighted standard deviations using the estimation period observations. Specifically, the functions standardize predictor j (x_{j}) using
$${x}_{j}^{\ast}=\frac{{x}_{j}{\mu}_{j}^{\ast}}{{\sigma}_{j}^{\ast}}.$$
x_{j} is predictor j, and x_{jk} is observation k of predictor j in the estimation period.
$${\mu}_{j}^{\ast}=\frac{1}{{\displaystyle \sum _{k}{w}_{k}^{\ast}}}{\displaystyle \sum _{k}{w}_{k}^{\ast}{x}_{jk}}.$$
$${\left({\sigma}_{j}^{\ast}\right)}^{2}=\frac{1}{{\displaystyle \sum _{k}{w}_{k}^{\ast}}}{\displaystyle \sum _{k}{w}_{k}^{\ast}{\left({x}_{jk}{\mu}_{j}^{\ast}\right)}^{2}}.$$
$${w}_{j}^{\ast}=\frac{{w}_{j}}{{\displaystyle \sum _{\forall j\in \text{Class}k}{w}_{j}}}{p}_{k},$$
p_{k} is the prior probability of class k (
Prior
property of the incremental model).w_{j} is observation weight j.
Performance Metrics
The
updateMetrics
andupdateMetricsAndFit
functions are incremental learning functions that track model performance metrics ('Metrics'
) from new data when the incremental model is warm (IsWarm
property). An incremental model becomes warm afterfit
orupdateMetricsAndFit
fit the incremental model to'MetricsWarmupPeriod'
observations, which is the metrics warmup period.If
'EstimationPeriod'
> 0, the functions estimate hyperparameters before fitting the model to data. Therefore, the functions must process an additionalEstimationPeriod
observations before the model starts the metrics warmup period.The
Metrics
property of the incremental model stores two forms of each performance metric as variables (columns) of a table,Cumulative
andWindow
, with individual metrics in rows. When the incremental model is warm,updateMetrics
andupdateMetricsAndFit
update the metrics at the following frequencies:Cumulative
— The functions compute cumulative metrics since the start of model performance tracking. The functions update metrics every time you call the functions and base the calculation on the entire supplied data set.Window
— The functions compute metrics based on all observations within a window determined by the'MetricsWindowSize'
namevalue pair argument.'MetricsWindowSize'
also determines the frequency at which the software updatesWindow
metrics. For example, ifMetricsWindowSize
is 20, the functions compute metrics based on the last 20 observations in the supplied data (X((end – 20 + 1):end,:)
andY((end – 20 + 1):end)
).Incremental functions that track performance metrics within a window use the following process:
Store a buffer of length
MetricsWindowSize
for each specified metric, and store a buffer of observation weights.Populate elements of the metrics buffer with the model performance based on batches of incoming observations, and store corresponding observation weights in the weights buffer.
When the buffer is filled, overwrite
IncrementalMdl.Metrics.Window
with the weighted average performance in the metrics window. If the buffer is overfilled when the function processes a batch of observations, the latest incomingMetricsWindowSize
observations enter the buffer, and the earliest observations are removed from the buffer. For example, supposeMetricsWindowSize
is 20, the metrics buffer has 10 values from a previously processed batch, and 15 values are incoming. To compose the length 20 window, the functions use the measurements from the 15 incoming observations and the latest 5 measurements from the previous batch.
The software omits an observation with a
NaN
score when computing theCumulative
andWindow
performance metric values.
References
[1] Kempka, Michał, Wojciech Kotłowski, and Manfred K. Warmuth. "Adaptive ScaleInvariant Online Algorithms for Learning Linear Models." Preprint, submitted February 10, 2019. https://arxiv.org/abs/1902.07528.
[2] Langford, J., L. Li, and T. Zhang. “Sparse Online Learning Via Truncated Gradient.” J. Mach. Learn. Res., Vol. 10, 2009, pp. 777–801.
[3] ShalevShwartz, S., Y. Singer, and N. Srebro. “Pegasos: Primal Estimated SubGradient Solver for SVM.” Proceedings of the 24th International Conference on Machine Learning, ICML ’07, 2007, pp. 807–814.
[4] Xu, Wei. “Towards Optimal One Pass Large Scale Learning with Averaged Stochastic Gradient Descent.” CoRR, abs/1107.2490, 2011.
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