incrementalLearner

Convert support vector machine (SVM) regression model to incremental learner

Syntax

IncrementalMdl = incrementalLearner(Mdl)

IncrementalMdl = incrementalLearner(Mdl,Name,Value)

Description

IncrementalMdl = incrementalLearner(Mdl) returns a linear regression model for incremental learning, IncrementalMdl, using the hyperparameters and coefficients of the traditionally trained linear SVM model for regression, Mdl. Because its property values reflect the knowledge gained from Mdl, IncrementalMdl can predict labels given new observations, and it is warm, meaning that its predictive performance is tracked.

example

IncrementalMdl = incrementalLearner(Mdl,Name,Value) uses additional options specified by one or more name-value pair arguments. Some options require you to train 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 performance metrics.

Examples

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Convert Traditionally Trained Model to Incremental Learner

Open Live Script

Train an SVM regression model by using fitrsvm, and then convert it to an incremental learner.

Load and Preprocess Data

Load the 2015 NYC housing data set. For more details on the data, see NYC Open Data.

load NYCHousing2015

Extract the response variable SALEPRICE from the table. For numerical stability, scale SALEPRICE by 1e6.

Y = NYCHousing2015.SALEPRICE/1e6;
NYCHousing2015.SALEPRICE = [];

Create dummy variable matrices from the categorical predictors.

catvars = ["BOROUGH" "BUILDINGCLASSCATEGORY" "NEIGHBORHOOD"];
dumvarstbl = varfun(@(x)dummyvar(categorical(x)),NYCHousing2015,...
    'InputVariables',catvars);
dumvarmat = table2array(dumvarstbl);
NYCHousing2015(:,catvars) = [];

Treat all other numeric variables in the table as linear predictors of sales price. Concatenate the matrix of dummy variables to the rest of the predictor data.

idxnum = varfun(@isnumeric,NYCHousing2015,'OutputFormat','uniform');
X = [dumvarmat NYCHousing2015{:,idxnum}];

Train SVM Regression Model

Fit an SVM regression model to 5000 randomly drawn observations from the data set. Discard the support vectors (Alpha) from the model so that the software uses linear coefficients (Beta) for prediction.

N = numel(Y);
n = 5000;
rng(1); % For reproducibility
idx = randsample(N,n);

TTMdl = fitrsvm(X(idx,:),Y(idx));
TTMdl = discardSupportVectors(TTMdl)

TTMdl = 
  RegressionSVM
             ResponseName: 'Y'
    CategoricalPredictors: []
        ResponseTransform: 'none'
                     Beta: [312×1 double]
                     Bias: -3.2469e+11
         KernelParameters: [1×1 struct]
          NumObservations: 5000
           BoxConstraints: [5000×1 double]
          ConvergenceInfo: [1×1 struct]
          IsSupportVector: [5000×1 logical]
                   Solver: 'SMO'


  Properties, Methods

TTMdl is a RegressionSVM model object representing a traditionally trained SVM regression model.

Convert Trained Model

Convert the traditionally trained SVM regression model to a linear regression model for incremental learning.

IncrementalMdl = incrementalLearner(TTMdl)

IncrementalMdl = 
  incrementalRegressionLinear

               IsWarm: 1
              Metrics: [1×2 table]
    ResponseTransform: 'none'
                 Beta: [312×1 double]
                 Bias: -3.2469e+11
              Learner: 'svm'


  Properties, Methods

IncrementalMdl is an incrementalRegressionLinear model object prepared for incremental learning using SVM.

The incrementalLearner function Initializes the incremental learner by passing learned coefficients to it, along with other information TTMdl extracted from the training data.
IncrementalMdl is warm (IsWarm is 1), which means that incremental learning functions can start tracking performance metrics.
The incrementalLearner function trains the model using the adaptive scale-invariant solver, whereas fitrsvm trained TTMdl using the SMO solver.

Predict Responses

An incremental learner created from converting a traditionally trained model can generate predictions without further processing.

Predict sales prices for all observations using both models.

ttyfit = predict(TTMdl,X);
ilyfit = predict(IncrementalMdl,X);
compareyfit = norm(ttyfit - ilyfit)

compareyfit = 0

The difference between the fitted values generated by the models is 0.

Specify SGD Solver

Open Live Script

The default solver is the adaptive scale-invariant 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 and shuffle the 2015 NYC housing data set. For more details on the data, see NYC Open Data.

load NYCHousing2015

rng(1) % For reproducibility
n = size(NYCHousing2015,1);
shuffidx = randsample(n,n);
NYCHousing2015 = NYCHousing2015(shuffidx,:);

Extract the response variable SALEPRICE from the table. For numerical stability, scale SALEPRICE by 1e6.

Y = NYCHousing2015.SALEPRICE/1e6;
NYCHousing2015.SALEPRICE = [];

Create dummy variable matrices from the categorical predictors.

catvars = ["BOROUGH" "BUILDINGCLASSCATEGORY" "NEIGHBORHOOD"];
dumvarstbl = varfun(@(x)dummyvar(categorical(x)),NYCHousing2015,...
    'InputVariables',catvars);
dumvarmat = table2array(dumvarstbl);
NYCHousing2015(:,catvars) = [];

Treat all other numeric variables in the table as linear predictors of sales price. Concatenate the matrix of dummy variables to the rest of the predictor data.

idxnum = varfun(@isnumeric,NYCHousing2015,'OutputFormat','uniform');
X = [dumvarmat NYCHousing2015{:,idxnum}];

Randomly partition the data into 5% and 95% sets: the first set for training a model traditionally, and the second set for incremental learning.

cvp = cvpartition(n,'Holdout',0.95);
idxtt = training(cvp);
idxil = test(cvp);

% 5% set for traditional training 
Xtt = X(idxtt,:);
Ytt = Y(idxtt);

% 95% set for incremental learning
Xil = X(idxil,:);
Yil = Y(idxil);

Fit an SVM regression model to 5% of the data.

TTMdl = fitrsvm(Xtt,Ytt);

Convert the traditionally trained SVM regression model to a linear regression model for incremental learning. Specify the standard SGD solver and an estimation period of 2e4 observations (the default is 1000 when a learning rate is required).

IncrementalMdl = incrementalLearner(TTMdl,'Solver','sgd','EstimationPeriod',2e4);

IncrementalMdl is an incrementalRegressionLinear model object.

Fit the incremental model to the rest 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 observation.
Store the learning rate and $β_{1}$ to see how the coefficients and learning 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*(j-1) + 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 incrementalRegressionLinear model object trained on all the data in the stream.

To see how the learning rate and $β_{1}$ evolved during training, plot them on separate subplots.

subplot(2,1,1)
plot(beta1)
hold on
ylabel('\beta_1')
xline(IncrementalMdl.EstimationPeriod/numObsPerChunk,'r-.');
subplot(2,1,2)
plot(learnrate)
ylabel('Learning Rate')
xline(IncrementalMdl.EstimationPeriod/numObsPerChunk,'r-.');
xlabel('Iteration')

The learning rate jumps to its auto-tuned value after the estimation period.

Because fit does not fit the model to the streaming data during the estimation period, $β_{1}$ is constant for the first 2000 iterations (20,000 observations). Then, $β_{1}$ changes slightly as fit fits the model to each new chunk of 10 observations.

Configure Performance Metric Options

Open Live Script

Use a trained SVM regression model to initialize an incremental learner. Prepare the incremental learner by specifying a metrics warm-up period, during which the updateMetricsAndFit function only fits the model. Specify a metrics window size of 500 observations.

Load the robot arm data set.

load robotarm

For details on the data set, enter Description at the command line.

Randomly partition the data into 5% and 95% sets: the first set for training a model traditionally, and the second set for incremental learning.

n = numel(ytrain);

rng(1) % For reproducibility
cvp = cvpartition(n,'Holdout',0.95);
idxtt = training(cvp);
idxil = test(cvp);

% 5% set for traditional training
Xtt = Xtrain(idxtt,:);
Ytt = ytrain(idxtt);

% 95% set for incremental learning
Xil = Xtrain(idxil,:);
Yil = ytrain(idxil);

Fit an SVM regression model to the first set.

TTMdl = fitrsvm(Xtt,Ytt);

Convert the traditionally trained SVM regression model to a linear regression model for incremental learning. Specify the following:

A performance metrics warm-up period of 2000 observations.
A metrics window size of 500 observations.
Use of epsilon insensitive loss, MSE, and mean absolute error (MAE) to measure the performance of the model. The software supports epsilon insensitive loss and MSE. Create an anonymous function that measures the absolute error of each new observation. Create a structure array containing the name MeanAbsoluteError and its corresponding function.

maefcn = @(z,zfit)abs(z - zfit);
maemetric = struct("MeanAbsoluteError",maefcn);
IncrementalMdl = incrementalLearner(TTMdl,'MetricsWarmupPeriod',2000,'MetricsWindowSize',500,...
    'Metrics',{'epsiloninsensitive' 'mse' maemetric});

Fit the incremental model to the rest of the data by using the updateMetricsAndfit function. At each iteration:

Simulate a data stream by processing 50 observations at a time.
Overwrite the previous incremental model with a new one fitted to the incoming observation.
Store the estimated coefficient $β_{10}$ , the cumulative metrics, and the window metrics to see how they evolve during incremental learning.

% Preallocation
nil = numel(Yil);
numObsPerChunk = 50;
nchunk = floor(nil/numObsPerChunk);
ei = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]);
mse = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]);
mae = array2table(zeros(nchunk,2),'VariableNames',["Cumulative" "Window"]);
beta1 = zeros(nchunk,1);    

% Incremental fitting
for j = 1:nchunk
    ibegin = min(nil,numObsPerChunk*(j-1) + 1);
    iend   = min(nil,numObsPerChunk*j);
    idx = ibegin:iend;    
    IncrementalMdl = updateMetricsAndFit(IncrementalMdl,Xil(idx,:),Yil(idx));
    ei{j,:} = IncrementalMdl.Metrics{"EpsilonInsensitiveLoss",:};
    mse{j,:} = IncrementalMdl.Metrics{"MeanSquaredError",:};
    mae{j,:} = IncrementalMdl.Metrics{"MeanAbsoluteError",:};
    beta1(j + 1) = IncrementalMdl.Beta(10);
end

IncrementalMdl is an incrementalRegressionLinear 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 observation, and then fits the model to that observation.

To see how the performance metrics and $β_{10}$ evolved during training, plot them on separate subplots.

figure;
subplot(2,2,1)
plot(beta1)
ylabel('\beta_{10}')
xlim([0 nchunk]);
xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r-.');
xlabel('Iteration')
subplot(2,2,2)
h = plot(ei.Variables);
xlim([0 nchunk]);
ylabel('Epsilon Insensitive Loss')
xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r-.');
legend(h,ei.Properties.VariableNames)
xlabel('Iteration')
subplot(2,2,3)
h = plot(mse.Variables);
xlim([0 nchunk]);
ylabel('MSE')
xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r-.');
legend(h,mse.Properties.VariableNames)
xlabel('Iteration')
subplot(2,2,4)
h = plot(mae.Variables);
xlim([0 nchunk]);
ylabel('MAE')
xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r-.');
legend(h,mae.Properties.VariableNames)
xlabel('Iteration')

The plot suggests that updateMetricsAndFit does the following:

Fit $β_{10}$ during all incremental learning iterations.
Compute performance metrics after the metrics warm-up period only.
Compute the cumulative metrics during each iteration.
Compute the window metrics after processing 500 observations.

Input Arguments

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`Mdl` — Traditionally trained linear SVM model for regression
`RegressionSVM` model object | `CompactRegressionSVM` model object

Traditionally trained linear SVM model for regression, specified as a model object returned by its training or processing function.

Model Object	Training or Processing Function
`RegressionSVM`	`fitrsvm`
`CompactRegressionSVM`	`fitrsvm` or `compact`

Note

Incremental learning functions support only numeric input predictor data. If Mdl was fit to categorical data, use dummyvar to convert each categorical variable to a numeric matrix of dummy variables, and concatenate all dummy variable matrices and any other numeric predictors. For more details, see Dummy Variables.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Solver','scale-invariant','MetricsWindowSize',100 specifies the adaptive scale-invariant solver for objective optimization, and specifies processing 100 observations before updating the performance metrics.

General Options

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`'Solver'` — Objective function minimization technique
`'scale-invariant'` (default) | `'sgd'` | `'asgd'`

Objective function minimization technique, specified as the comma-separated pair consisting of 'Solver' and a value in this table.

Value	Description	Notes
`'scale-invariant'`	Adaptive scale-invariant solver for incremental learning [1]	This algorithm is parameter free and can adapt to differences in predictor scales. Try this algorithm before using SGD or ASGD.
`'sgd'`	Stochastic gradient descent (SGD) [3][2]	To train effectively with SGD, standardize the data and specify adequate values for hyperparameters using options listed in SGD and ASGD Solver Options.
`'asgd'`	Average stochastic gradient descent (ASGD) [4]	To train effectively with ASGD, standardize the data and specify adequate values for hyperparameters using options listed in SGD and ASGD Solver Options.

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 comma-separated pair consisting of 'EstimationPeriod' and a nonnegative integer.

Note

If Mdl is prepared for incremental learning (all hyperparameters required for training are specified), incrementalLearner forces 'EstimationPeriod' to 0.
If Mdl is not prepared for incremental learning, incrementalLearner sets 'EstimationPeriod' to 1000.

For more details, see Estimation Period.

Example: 'EstimationPeriod',100

Data Types: single | double

`'Standardize'` — Flag to standardize predictor data
`'auto'` (default) | `false` | `true`

Flag to standardize the predictor data, specified as the comma-separated pair consisting of 'Standardize' and a value in this table.

Value	Description
`'auto'`	`incrementalLearner` determines whether the predictor variables need to be standardized. See Standardize Data.
`true`	The software standardizes the predictor data.
`false`	The software does not standardize the predictor data.

Under some conditions, incrementalLearner can override your specification. For more details, see Standardize Data.

Example: 'Standardize',true

Data Types: logical | char | string

SGD and ASGD Solver Options

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`'BatchSize'` — Mini-batch size
`10` (default) | positive integer

Mini-batch size, specified as the comma-separated pair consisting of 'BatchSize' and a positive integer. At each iteration during training, incrementalLearner uses min(BatchSize,numObs) observations to compute the subgradient, where numObs is the number of observations in the training data passed to fit or updateMetricsAndFit.

Example: 'BatchSize',1

Data Types: single | double

`'Lambda'` — Ridge (L²) regularization term strength
`1e-5` (default) | nonnegative scalar

Ridge (L²) regularization term strength, specified as the comma-separated pair consisting of 'Lambda' and a nonnegative scalar.

Example: 'Lambda',0.01

Data Types: single | double

`'LearnRate'` — Learning rate
`'auto'` (default) | positive scalar

Learning rate, specified as the comma-separated pair consisting of 'LearnRate' and 'auto' or a positive scalar. LearnRate controls the optimization step size by scaling the objective subgradient.

For 'auto':

If EstimationPeriod is 0, the initial learning rate is 0.7.
If EstimationPeriod > 0, the initial learning rate is 1/sqrt(1+max(sum(X.^2,obsDim))), where obsDim is 1 if the observations compose the columns of the predictor data, and 2 otherwise. fit and updateMetricsAndFit set the value when you pass the model and training data to either function.

The name-value pair argument 'LearnRateSchedule' determines the learning rate for subsequent learning cycles.

Example: 'LearnRate',0.001

Data Types: single | double | char | string

`'LearnRateSchedule'` — Learning rate schedule
`'decaying'` (default) | `'constant'`

Learning rate schedule, specified as the comma-separated pair consisting of 'LearnRateSchedule' and a value in this table, where LearnRate specifies the initial learning rate ɣ₀.

Value Description

'constant' The learning rate is ɣ₀ for all learning cycles.

Value	Description
`'constant'`	The learning rate is ɣ₀ for all learning cycles.
`'decaying'`	The learning rate at learning cycle t is $γ_{t} = \frac{γ_{0}}{{(1 + λ γ_{0} t)}^{c}} .$ λ is the value of `Lambda`. If `Solver` is `'sgd'`, then c = 1. If `Solver` is `'asgd'`, then c is 0.75 [4].

'decaying'

The learning rate at learning cycle t is

$γ_{t} = \frac{γ_{0}}{{(1 + λ γ_{0} t)}^{c}} .$

λ is the value of Lambda.
If Solver is 'sgd', then c = 1.
If Solver is 'asgd', then c is 0.75 [4].

Example: 'LearnRateSchedule','constant'

Data Types: char | string

Adaptive Scale-Invariant Solver Options

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`'Shuffle'` — Flag for shuffling observations in batch
`true` (default) | `false`

Flag for shuffling the observations in the batch at each iteration, specified as the comma-separated pair consisting of 'Shuffle' and a value in this table.

Value	Description
`true`	The software shuffles observations in each incoming batch of data before processing the set. 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

Performance Metrics Options

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`'Metrics'` — Model performance metrics to track during incremental learning
`"epsiloninsensitive"` (default) | string vector | function handle | cell vector | structure array | `"mse"` | ...

Model performance metrics to track during incremental learning with updateMetrics and updateMetricsAndFit, specified as the comma-separated pair consisting of 'Metrics' and a built-in 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 built-in loss function names. You can specify more than one by using a string vector.

Name	Description
`"epsiloninsensitive"`	Epsilon insensitive loss
`"mse"`	Weighted mean squared error

For more details on the built-in loss functions, see loss.

Example: 'Metrics',["epsiloninsensitive" "mse"]

To specify a custom function that returns a performance metric, use function handle notation. The function must have this form:

metric = customMetric(Y,YFit)

The output argument metric is an n-by-1 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 select the function name (customMetric).
Y is a length n numeric vector of observed responses, where n is the sample size.
YFit is a length n numeric vector of corresponding predicted responses.

To specify multiple custom metrics and assign a custom name to each, use a structure array. To specify a combination of built-in and custom metrics, use a cell vector.

Example: 'Metrics',struct('Metric1',@customMetric1,'Metric2',@customMetric2)

Example: 'Metrics',{@customMetric1 @customeMetric2 'mse' 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 built-in metric	Row name for `"epsiloninsensitive"` is `"EpsilonInsensitiveLoss"`
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_j`, where `j` is metric `j` in `Metrics`	Row name for `@(Y,YFit)customMetric(Y,YFit)...` 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 the comma-separated pair consisting of 'MetricsWarmupPeriod' and a nonnegative integer. The incremental model is warm after incremental fitting functions fit MetricsWarmupPeriod observations to the incremental model (EstimationPeriod + MetricsWarmupPeriod observations).

For more details on performance metrics options, see Performance Metrics.

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 the comma-separated pair consisting of 'MetricsWindowSize' and a positive integer.

For more details on performance metrics options, see Performance Metrics.

Data Types: single | double

Output Arguments

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`IncrementalMdl` — Linear regression model for incremental learning
`incrementalRegressionLinear` model object

Linear regression model for incremental learning, returned as an incrementalRegressionLinear model object. IncrementalMdl is also configured to generate predictions given new data (see predict).

To initialize IncrementalMdl for incremental learning, incrementalLearner passes the values of the Mdl properties in this table to congruent properties of IncrementalMdl.

Property	Description
`Beta`	Linear model coefficients, a numeric vector
`Bias`	Model intercept, a numeric scalar
`Epsilon`	Half the width of the epsilon insensitive band, a nonnegative scalar
`Mu`	Predictor variable means, a numeric vector
`Sigma`	Predictor variable standard deviations, a numeric vector

More About

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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 cross-validation 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.

Adaptive Scale-Invariant Solver for Incremental Learning

The adaptive scale-invariant solver for incremental learning, introduced in [1], is a gradient-descent-based 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 can require tuning. For traditional machine learning, enough data is available to enable hyperparameter tuning by cross-validation 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 regression fit and updateMetricsAndFit use the more conservative ScInOL1 version of the algorithm.

Algorithms

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Estimation Period

During the estimation period, incremental fitting functions fit and updateMetricsAndFit use the first incoming EstimationPeriod observations to estimate (tune) hyperparameters required for incremental training. This table describes the hyperparameters and when they are estimated or tuned.

Hyperparameter Model Property Use Hyperparameters Estimated

Predictor means and standard deviations

Hyperparameter	Model Property	Use	Hyperparameters Estimated
Predictor means and standard deviations	`Mu` and `Sigma`	Standardize predictor data	When both these conditions apply: When you set `'Standardize',true` (see Standardize Data) `IncrementalMdl.Mu` and `IncrementalMdl.Sigma` are empty arrays `[]`.
Learning rate	`LearnRate`	Adjust solver step size	When both of these conditions apply: You change the solver of `Mdl` to SGD or ASGD (see `Solver`). You do not set the `'LearnRate'` name-value pair argument.

Mu and Sigma

Standardize predictor data

When both these conditions apply:

When you set 'Standardize',true (see Standardize Data)
IncrementalMdl.Mu and IncrementalMdl.Sigma are empty arrays [].

Learning rate

LearnRate

Adjust solver step size

When both of these conditions apply:

You change the solver of Mdl to SGD or ASGD (see Solver).
You do not set the 'LearnRate' name-value pair argument.

The functions fit only the last estimation period observation to the incremental model, and they do not use any of the observations to track the performance of the model. 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 standardized the predictor data when you trained the input model Mdl by using fitrsvm, the following conditions apply:
- incrementalLearner passes the means in Mdl.Mu and standard deviations in Mdl.Sigma to the congruent incremental learning model properties.
- Incremental learning functions always standardize the predictor data, regardless of the value of the 'Standardize' name-value pair argument.
When you set 'Standardize',true, and IncrementalMdl.Mu and IncrementalMdl.Sigma are empty, the following conditions apply:
- If the estimation period is positive (see the EstimationPeriod property of IncrementalMdl), incremental fitting functions estimate means and standard deviations using the estimation period observations.
- If the estimation period is 0, incrementalLearner forces the estimation period to 1000. Consequently, incremental fitting functions estimate new predictor variable means and standard deviations during the forced estimation period.
If you set 'Standardize','auto' (the default), the following conditions apply.
- If IncrementalMdl.Mu and IncrementalMdl.Sigma are empty, incremental learning functions do not standardize predictor variables.
- Otherwise, incremental learning functions standardize the predictor variables using their means and standard deviations in IncrementalMdl.Mu and IncrementalMdl.Sigma, respectively. Incremental fitting functions do not estimate new means and standard deviations regardless of the length of the 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

$x_{j}^{*} = \frac{x_{j} - μ_{j}^{*}}{σ_{j}^{*}} .$
- x_j is predictor j, and x_jk is observation k of predictor j in the estimation period.
- $μ_{j}^{*} = \frac{1}{\sum_{k} w_{k}} \sum_{k} w_{k} x_{j k} .$
- ${(σ_{j}^{*})}^{2} = \frac{1}{\sum_{k} w_{k}} \sum_{k} w_{k} {(x_{j k} - μ_{j}^{*})}^{2} .$
- w_j is observation weight j.

Performance Metrics

The updateMetrics and updateMetricsAndFit 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 is warm after fit or updateMetricsAndFit fit the incremental model to 'MetricsWarmupPeriod' observations, which is the metrics warm-up period.
If 'EstimationPeriod' > 0, the functions estimate hyperparameters before fitting the model to data. Therefore, the functions must process an additional EstimationPeriod observations before the model starts the metrics warm-up period.
The Metrics property of the incremental model stores two forms of each performance metric as variables (columns) of a table, Cumulative and Window, with individual metrics in rows. When the incremental model is warm, updateMetrics and updateMetricsAndFit 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' name-value pair argument. 'MetricsWindowSize' also determines the frequency at which the software updates Window metrics. For example, if MetricsWindowSize is 20, the functions compute metrics based on the last 20 observations in the supplied data (X((end – 20 + 1):end,:) and Y((end – 20 + 1):end)).
  Incremental functions that track performance metrics within a window use the following process:
  1. For each specified metric, store a buffer of length MetricsWindowSize and a buffer of observation weights.
  2. Populate elements of the metrics buffer with the model performance based on batches of incoming observations, and store corresponding observations weights in the weights buffer.
  3. 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 incoming MetricsWindowSize observations enter the buffer, and the earliest observations are removed from the buffer. For example, suppose MetricsWindowSize 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.

References

[1] Kempka, Michał, Wojciech Kotłowski, and Manfred K. Warmuth. "Adaptive Scale-Invariant Online Algorithms for Learning Linear Models." CoRR (February 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] Shalev-Shwartz, S., Y. Singer, and N. Srebro. “Pegasos: Primal Estimated Sub-Gradient 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.

Documentation

incrementalLearner

Syntax

Description

Examples

Convert Traditionally Trained Model to Incremental Learner

Specify SGD Solver

Configure Performance Metric Options

Input Arguments

`Mdl` — Traditionally trained linear SVM model for regression
`RegressionSVM` model object | `CompactRegressionSVM` model object

Name-Value Pair Arguments

`'Solver'` — Objective function minimization technique
`'scale-invariant'` (default) | `'sgd'` | `'asgd'`

`'EstimationPeriod'` — Number of observations processed to estimate hyperparameters
nonnegative integer

`'Standardize'` — Flag to standardize predictor data
`'auto'` (default) | `false` | `true`

`'BatchSize'` — Mini-batch size
`10` (default) | positive integer

`'Lambda'` — Ridge (L²) regularization term strength
`1e-5` (default) | nonnegative scalar

`'LearnRate'` — Learning rate
`'auto'` (default) | positive scalar

`'LearnRateSchedule'` — Learning rate schedule
`'decaying'` (default) | `'constant'`

`'Shuffle'` — Flag for shuffling observations in batch
`true` (default) | `false`

`'Metrics'` — Model performance metrics to track during incremental learning
`"epsiloninsensitive"` (default) | string vector | function handle | cell vector | structure array | `"mse"` | ...

`'MetricsWarmupPeriod'` — Number of observations fit before tracking performance metrics
`0` (default) | nonnegative integer | ...

`'MetricsWindowSize'` — Number of observations to use to compute window performance metrics
`200` (default) | positive integer | ...

Output Arguments

`IncrementalMdl` — Linear regression model for incremental learning
`incrementalRegressionLinear` model object

More About

Incremental Learning

Adaptive Scale-Invariant Solver for Incremental Learning

Algorithms

Estimation Period

Standardize Data

Performance Metrics

References

See Also

Objects

Functions

Topics

Statistics and Machine Learning Toolbox Documentation

Support

Documentation

incrementalLearner

Syntax

Description

Examples

Convert Traditionally Trained Model to Incremental Learner

Specify SGD Solver

Configure Performance Metric Options

Input Arguments

Mdl — Traditionally trained linear SVM model for regression RegressionSVM model object | CompactRegressionSVM model object

Name-Value Pair Arguments

'Solver' — Objective function minimization technique 'scale-invariant' (default) | 'sgd' | 'asgd'

'EstimationPeriod' — Number of observations processed to estimate hyperparameters nonnegative integer

'Standardize' — Flag to standardize predictor data 'auto' (default) | false | true

'BatchSize' — Mini-batch size 10 (default) | positive integer

'Lambda' — Ridge (L2) regularization term strength 1e-5 (default) | nonnegative scalar

'LearnRate' — Learning rate 'auto' (default) | positive scalar

'LearnRateSchedule' — Learning rate schedule 'decaying' (default) | 'constant'

'Shuffle' — Flag for shuffling observations in batch true (default) | false

'Metrics' — Model performance metrics to track during incremental learning "epsiloninsensitive" (default) | string vector | function handle | cell vector | structure array | "mse" | ...

'MetricsWarmupPeriod' — Number of observations fit before tracking performance metrics 0 (default) | nonnegative integer | ...

'MetricsWindowSize' — Number of observations to use to compute window performance metrics 200 (default) | positive integer | ...

Output Arguments

IncrementalMdl — Linear regression model for incremental learning incrementalRegressionLinear model object

More About

Incremental Learning

Adaptive Scale-Invariant Solver for Incremental Learning

Algorithms

Estimation Period

Standardize Data

Performance Metrics

References

See Also

Objects

Functions

Topics

Statistics and Machine Learning Toolbox Documentation

Support

`Mdl` — Traditionally trained linear SVM model for regression
`RegressionSVM` model object | `CompactRegressionSVM` model object

`'Solver'` — Objective function minimization technique
`'scale-invariant'` (default) | `'sgd'` | `'asgd'`

`'EstimationPeriod'` — Number of observations processed to estimate hyperparameters
nonnegative integer

`'Standardize'` — Flag to standardize predictor data
`'auto'` (default) | `false` | `true`

`'BatchSize'` — Mini-batch size
`10` (default) | positive integer

`'Lambda'` — Ridge (L²) regularization term strength
`1e-5` (default) | nonnegative scalar

`'LearnRate'` — Learning rate
`'auto'` (default) | positive scalar

`'LearnRateSchedule'` — Learning rate schedule
`'decaying'` (default) | `'constant'`

`'Shuffle'` — Flag for shuffling observations in batch
`true` (default) | `false`

`'Metrics'` — Model performance metrics to track during incremental learning
`"epsiloninsensitive"` (default) | string vector | function handle | cell vector | structure array | `"mse"` | ...

`'MetricsWarmupPeriod'` — Number of observations fit before tracking performance metrics
`0` (default) | nonnegative integer | ...

`'MetricsWindowSize'` — Number of observations to use to compute window performance metrics
`200` (default) | positive integer | ...

`IncrementalMdl` — Linear regression model for incremental learning
`incrementalRegressionLinear` model object