incrementalLearner

Convert kernel regression model to incremental learner

Since R2022a

Syntax

IncrementalMdl = incrementalLearner(Mdl)

IncrementalMdl = incrementalLearner(Mdl,Name=Value)

Description

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

IncrementalMdl = incrementalLearner(Mdl,Name=Value) uses additional options specified by one or more name-value 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 window performance metrics.

Examples

Convert Traditionally Trained Model to Incremental Learner

Train a kernel regression model by using fitrkernel, 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 = [];

To reduce computational cost for this example, remove the NEIGHBORHOOD column, which contains a categorical variable with 254 categories.

NYCHousing2015.NEIGHBORHOOD = [];

Create dummy variable matrices from the other categorical predictors.

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

Treat all other numeric variables in the table as 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 Kernel Regression Model

Fit a kernel regression model to the entire data set.

Mdl = fitrkernel(X,Y)

Mdl = 
  RegressionKernel
              ResponseName: 'Y'
                   Learner: 'svm'
    NumExpansionDimensions: 2048
               KernelScale: 1
                    Lambda: 1.0935e-05
             BoxConstraint: 1
                   Epsilon: 0.0549


  Properties, Methods

Mdl is a RegressionKernel model object representing a traditionally trained kernel regression model.

Convert Trained Model

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

IncrementalMdl = incrementalLearner(Mdl)

IncrementalMdl = 
  incrementalRegressionKernel

                    IsWarm: 1
                   Metrics: [1×2 table]
         ResponseTransform: 'none'
    NumExpansionDimensions: 2048
               KernelScale: 1


  Properties, Methods

IncrementalMdl is an incrementalRegressionKernel model object prepared for incremental learning.

• The incrementalLearner function initializes the incremental learner by passing model parameters to it, along with other information Mdl extracted from the training data.

• IncrementalMdl is warm (IsWarm is 1), which means that incremental learning functions can start tracking performance metrics.

• incrementalRegressionKernel trains the model using the adaptive scale-invariant solver, whereas fitrkernel trained Mdl using the Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) 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(Mdl,X);
ilyfit = predict(IncrementalMdl,X);
compareyfit = norm(ttyfit - ilyfit)

compareyfit = 
0

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

Configure Performance Metric Options

Use a trained kernel regression model to initialize an incremental learner. Prepare the incremental learner by specifying a metrics warm-up period and a metrics window size.

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 a kernel regression model to the first set.

TTMdl = fitrkernel(Xtt,Ytt);

Convert the traditionally trained kernel regression model to a 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 observations.

• Store the cumulative metrics, window metrics, and number of training observations 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"]);
numtrainobs = [IncrementalMdl.NumTrainingObservations; zeros(nchunk+1,1)];

% 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",:};
    numtrainobs(j+1) = IncrementalMdl.NumTrainingObservations;
end

IncrementalMdl is an incrementalRegressionKernel 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.

Plot a trace plot of the number of training observations and the performance metrics on separate tiles.

t = tiledlayout(4,1);
nexttile
plot(numtrainobs)
xlim([0 nchunk])
xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,"--")
ylabel(["Number of Training","Observations"])
nexttile
plot(ei.Variables)
xlim([0 nchunk])
ylabel(["Epsilon Insensitive","Loss"])
xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,"--")
legend(ei.Properties.VariableNames,Location="best")
nexttile
plot(mse.Variables)
xlim([0 nchunk])
ylabel("MSE")
xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,"--")
nexttile
plot(mae.Variables)
xlim([0 nchunk])
ylabel("MAE")
xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,"--")
xlabel(t,"Iteration")

The plot suggests that updateMetricsAndFit does the following:

• Fit the model during all incremental learning iterations.

• Compute the performance metrics after the metrics warm-up period only.

• Compute the cumulative metrics during each iteration.

• Compute the window metrics after processing 500 observations.

Specify SGD Solver

The default solver for incrementalRegressionKernel is the adaptive scale-invariant solver, which does not require hyperparameter tuning before you fit a model. However, if you specify either the standard stochastic gradient descent (SGD) or average SGD (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 = [];

To reduce computational cost for this example, remove the NEIGHBORHOOD column, which contains a categorical variable with 254 categories.

NYCHousing2015.NEIGHBORHOOD = [];

Create dummy variable matrices from the categorical predictors.

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

Treat all other numeric variables in the table as 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 a kernel regression model to 5% of the data.

Mdl = fitrkernel(Xtt,Ytt);

Convert the traditionally trained kernel regression model to a 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(Mdl,Solver="sgd",EstimationPeriod=2e4);

IncrementalMdl is an incrementalRegressionKernel model object configured for incremental learning.

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 observations.

• Store the initial learning rate and number of training observations to see how they evolve during training.

% Preallocation
nil = numel(Yil);
numObsPerChunk = 10;
nchunk = floor(nil/numObsPerChunk);
learnrate = [zeros(nchunk,1)];
numtrainobs = [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));
    learnrate(j) = IncrementalMdl.SolverOptions.LearnRate;
    numtrainobs(j) = IncrementalMdl.NumTrainingObservations;
end

IncrementalMdl is an incrementalRegressionKernel model object trained on all the data in the stream.

Plot a trace plot of the number of training observations and the initial learning rate on separate tiles.

t = tiledlayout(2,1);
nexttile
plot(numtrainobs)
xlim([0 nchunk])
xline(IncrementalMdl.EstimationPeriod/numObsPerChunk,"-.");
ylabel("Number of Training Observations")
nexttile
plot(learnrate)
xlim([0 nchunk])
ylabel("Initial Learning Rate")
xline(IncrementalMdl.EstimationPeriod/numObsPerChunk,"-.");
xlabel(t,"Iteration")

The plot suggests that fit does not fit the model to the streaming data during the estimation period. 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.

Input Arguments

Mdl — Traditionally trained Gaussian kernel regression model

RegressionKernel model object

Traditionally trained Gaussian kernel regression model, specified as a RegressionKernel model object returned by fitrkernel.

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. Use dummyvar 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.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: Solver="sgd",MetricsWindowSize=100 specifies the stochastic gradient descent solver for objective optimization, and specifies processing 100 observations before updating the window performance metrics.

General Options

Solver — Objective function minimization technique

"scale-invariant" (default) | "sgd" | "asgd"

Objective function minimization technique, specified as 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. To shuffle an incoming chunk of data before the fit function fits the model, set Shuffle to true.
"sgd"	Stochastic gradient descent (SGD) [2][3]	To train effectively with SGD, standardize the data and specify adequate values for hyperparameters using options listed in SGD and ASGD Solver Options. The fit function always shuffles an incoming chunk of data before fitting the model.
"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. The fit function always shuffles an incoming chunk of data before fitting the model.

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 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

SGD and ASGD Solver Options

BatchSize — Mini-batch size

10 (default) | positive integer

Mini-batch size, specified as a positive integer. At each learning cycle during training, incrementalLearner uses BatchSize observations to compute the subgradient.

The number of observations in the last mini-batch (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 5 observations for the last learning cycle.

Example: BatchSize=5

Data Types: single | double

Lambda — Ridge (L2) regularization term strength

1e-5 (default) | nonnegative scalar

Ridge (L2) regularization term strength, specified as a nonnegative scalar.

Example: Lambda=0.01

Data Types: single | double

LearnRate — Initial learning rate

"auto" (default) | positive scalar

Initial learning rate, specified as "auto" or a positive scalar.

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 and updateMetricsAndFit change the rate to 1/sqrt(1+max(sum(X.^2,2))) at the end of EstimationPeriod.

Example: LearnRate=0.001

Data Types: single | double | char | string

LearnRateSchedule — Learning rate schedule

"decaying" (default) | "constant"

Learning rate schedule, specified as 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.
"decaying"	The learning rate at learning cycle t is $${\gamma }_t=\frac{{\gamma }_0}{{\left(1+\lambda {\gamma }_{0}t\right)}^c}.$$ λ is the value of Lambda. If Solver is "sgd", c = 1. If Solver is "asgd": c = 2/3 if Learner is "leastsquares". c = 3/4 if Learner is "svm" [4].

Example: LearnRateSchedule="constant"

Data Types: char | string

Adaptive Scale-Invariant Solver Options

Shuffle — Flag for shuffling observations

true or 1 (default) | false or 0

Flag for shuffling the observations at each iteration, specified as logical 1 (true) or 0 (false).

Value	Description
logical 1 (true)	The software shuffles the observations in an incoming chunk of data before the fit function fits the model. This action reduces bias induced by the sampling scheme.
logical 0 (false)	The software processes the data in the order received.

Example: Shuffle=false

Data Types: logical

Performance Metrics Options

Metrics — Model performance metrics to track during incremental learning

"epsiloninsensitive" | "mse" | string vector | function handle | cell vector | structure array

Model performance metrics to track during incremental learning with updateMetrics or updateMetricsAndFit, specified as 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 and which learners, specified in Mdl.Learner, support them. You can specify more than one loss function by using a string vector.

Name	Description	Learner Supporting Metric
"epsiloninsensitive"	Epsilon insensitive loss	'svm'
"mse"	Weighted mean squared error	'svm' and 'leastsquares'

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 specify 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,@customMetric2,"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

By default:

• Metrics is "epsiloninsensitive" if Mdl.Learner is 'svm'.

• Metrics is "mse" if Mdl.Learner is 'leastsquares'.

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=250

Data Types: single | double

Output Arguments

IncrementalMdl — Gaussian kernel regression model for incremental learning

incrementalRegressionKernel model object

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

The incrementalLearner function initializes IncrementalMdl for incremental learning using the model information in Mdl. The following table shows the Mdl properties that incrementalLearner passes to corresponding properties of IncrementalMdl. The function also passes other model information required to initialize IncrementalMdl, such as learned model coefficients and the random number stream.

Property	Description
Epsilon	Half the width of the epsilon insensitive band, a nonnegative scalar. incrementalLearner passes this value only when Mdl.Learner is 'svm'.
KernelScale	Kernel scale parameter, a positive scalar
Learner	Linear regression model type, a character vector
Mu	Predictor variable means, a numeric vector
NumExpansionDimensions	Number of dimensions of expanded space, a positive integer
NumPredictors	Number of predictors, a positive integer
ResponseTransform	Response transformation function, a function name or function handle
Sigma	Predictor variable standard deviations, a numeric vector

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 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.

For more details, see Incremental Learning Overview.

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. 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 fit and updateMetricsAndFit use the more aggressive ScInOL2 version of the algorithm.

Random Feature Expansion

Random feature expansion, such as Random Kitchen Sinks [1] or Fastfood [2], is a scheme to approximate Gaussian kernels of the kernel regression algorithm for big data in a computationally efficient way. Random feature expansion is more practical for big data applications that have large training sets, but can also be applied to smaller data sets that fit in memory.

After mapping the predictor data into a high-dimensional space, the kernel regression algorithm searches for an optimal function that deviates from each response data point (y_i) by values no greater than the epsilon margin (ε).

Some regression problems cannot be described adequately using a linear model. In such cases, obtain a nonlinear regression model by replacing the dot product x₁x₂′ with a nonlinear kernel function $$G(x_1,x_2)=\langle \phi (x_1),\phi (x_2)\rangle$$, where x_i is the ith observation (row vector) and φ(x_i) is a transformation that maps x_i to a high-dimensional space (called the “kernel trick”). However, evaluating G(x₁,x₂), the Gram matrix, for each pair of observations is computationally expensive for a large data set (large n).

The random feature expansion scheme finds a random transformation so that its dot product approximates the Gaussian kernel. That is,

$$G(x_1,x_2)=\langle \phi (x_1),\phi (x_2)\rangle \approx T(x_1)T(x_2)\text{'},$$

where T(x) maps x in $${ℝ}^p$$ to a high-dimensional space ($${ℝ}^m$$). The Random Kitchen Sinks [1] scheme uses the random transformation

$$T(x)=m^{-1/2}\exp \left(iZx\text{'}\right)\text{'},$$

where $$Z\in {ℝ}^{m\times p}$$ is a sample drawn from $$N\left(0,{\sigma }^{-2}\right)$$ and σ is a kernel scale. This scheme requires O(mp) computation and storage. The Fastfood [2] scheme introduces another random basis V instead of Z using Hadamard matrices combined with Gaussian scaling matrices. This random basis reduces computation cost to O(mlogp) and reduces storage to O(m).

incrementalRegressionKernel uses the Fastfood scheme for random feature expansion, and uses linear regression to train a Gaussian kernel regression model. You can specify values for m and σ using the NumExpansionDimensions and KernelScale name-value arguments, respectively, when you create a traditionally trained model using fitrkernel or when you call incrementalRegressionKernel directly to create the model object.

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	Mu and Sigma	Standardize predictor data	The hyperparameters are not estimated.
Learning rate	LearnRate field of SolverOptions	Adjust the solver step size	The hyperparameter is estimated when both of these conditions apply: You specify the Solver name-value argument as "sgd" or "asgd". You do not specify the LearnRate name-value argument as a positive scalar.
Half the width of the epsilon insensitive band	Epsilon	Control the number of support vectors	The software does not estimate Epsilon or KernelScale if you create an incrementalRegressionKernel model object by using incrementalLearner. If you create an incrementalRegressionKernel model object by calling the incrementalRegressionKernel function, the software estimates these hyperparameters in some cases. For details, see Estimation Period on the incrementalRegressionKernel page.
Kernel scale parameter	KernelScale	Set a kernel scale parameter value for random feature expansion

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, respectively, of the incremental learning model IncrementalMdl.

• If you standardize the predictor data when you train the input model Mdl by using fitrkernel, the following conditions apply:

  • incrementalLearner passes the means in Mdl.Mu and standard deviations in Mdl.Sigma to the corresponding incremental learning model properties.

  • Incremental learning functions always standardize the predictor data.

• 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.

  • $${\mu }_j^{\ast }=\frac{1}{{\displaystyle \sum _k{w}_k}}{\displaystyle \sum _k{w}_k{x}_{jk}}.$$

  • $${\left({\sigma }_j^{\ast }\right)}^2=\frac{1}{{\displaystyle \sum _k{w}_k}}{\displaystyle \sum _k{w}_k{\left(x_{jk}-{\mu }_j^{\ast }\right)}^2}.$$

  • 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 only when the incremental model is warm (IsWarm property is true). An incremental model becomes warm after fit or updateMetricsAndFit fits the incremental model to MetricsWarmupPeriod observations, which is the metrics warm-up period.

  If EstimationPeriod > 0, the fit and updateMetricsAndFit 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 MetricsWindowSize, which 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. Store a buffer of length MetricsWindowSize for each specified metric, and store 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 observation weights in the weights buffer.

    3. When the buffer is full, overwrite the Window field of the Metrics property with the weighted average performance in the metrics window. If the buffer overfills 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.

• The software omits an observation with a NaN prediction when computing the Cumulative and Window performance metric values.

References

[1] Kempka, Michał, Wojciech Kotłowski, and Manfred K. Warmuth. "Adaptive Scale-Invariant 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] 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.

[5] Rahimi, A., and B. Recht. “Random Features for Large-Scale Kernel Machines.” Advances in Neural Information Processing Systems. Vol. 20, 2008, pp. 1177–1184.

[6] Le, Q., T. Sarlós, and A. Smola. “Fastfood — Approximating Kernel Expansions in Loglinear Time.” Proceedings of the 30th International Conference on Machine Learning. Vol. 28, No. 3, 2013, pp. 244–252.

[7] Huang, P. S., H. Avron, T. N. Sainath, V. Sindhwani, and B. Ramabhadran. “Kernel methods match Deep Neural Networks on TIMIT.” 2014 IEEE International Conference on Acoustics, Speech and Signal Processing. 2014, pp. 205–209.

Version History

Introduced in R2022a