incrementalRegressionKernel

Kernel regression model for incremental learning

Since R2022a

Description

The incrementalRegressionKernel function creates an incrementalRegressionKernel model object, which represents a binary Gaussian kernel regression model for incremental learning. The kernel model maps data in a low-dimensional space into a high-dimensional space, then fits a linear model in the high-dimensional space. Supported linear models include support vector machine (SVM) and least-squares regression.

Unlike other Statistics and Machine Learning Toolbox™ model objects, incrementalRegressionKernel can be called directly. Also, you can specify learning options, such as performance metrics configurations and the objective solver, before fitting the model to data. After you create an incrementalRegressionKernel object, it is prepared for incremental learning.

incrementalRegressionKernel is best suited for incremental learning. For a traditional approach to training a kernel regression model (such as creating a model by fitting it to data, performing cross-validation, tuning hyperparameters, and so on), see fitrkernel.

Creation

You can create an incrementalRegressionKernel model object in several ways:

• Call the function directly — Configure incremental learning options, or specify learner-specific options, by calling incrementalRegressionKernel directly. This approach is best when you do not have data yet or you want to start incremental learning immediately.

• Convert a traditionally trained model — To initialize a model for incremental learning using the model parameters and hyperparameters of a trained model object (RegressionKernel), you can convert the traditionally trained model to an incrementalRegressionKernel model object by passing it to the incrementalLearner function.

• Call an incremental learning function — fit, updateMetrics, and updateMetricsAndFit accept a configured incrementalRegressionKernel model object and data as input, and return an incrementalRegressionKernel model object updated with information learned from the input model and data.

Syntax

Mdl = incrementalRegressionKernel()

Mdl = incrementalRegressionKernel(Name=Value)

Description

Mdl = incrementalRegressionKernel() returns a default incremental learning model object for binary Gaussian kernel regression, Mdl. Properties of a default model contain placeholders for unknown model parameters. You must train a default model before you can track its performance or generate predictions from it.

Mdl = incrementalRegressionKernel(Name=Value) sets properties and additional options using name-value arguments. For example, incrementalRegressionKernel(Solver="sgd",LearnRateSchedule="constant") specifies to use the stochastic gradient descent (SGD) solver with a constant learning rate.

Input Arguments

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: Metrics="mse",MetricsWarmupPeriod=100 sets the model performance metric to the weighted mean squared error and the metrics warm-up period to 100.

Regression Options

RandomStream — Random number stream

global stream (default) | random stream object

Random number stream for reproducibility of data transformation, specified as a random stream object. For details, see Random Feature Expansion.

Use RandomStream to reproduce the random basis functions used by incrementalRegressionKernel to transform the predictor data to a high-dimensional space. For details, see Managing the Global Stream Using RandStream and Creating and Controlling a Random Number Stream.

Example: RandomStream=RandStream("mlfg6331_64")

Standardize — Flag to standardize predictor data

"auto" (default) | false | true

Since R2023b

Flag to standardize the predictor data, specified as a value in this table.

Value	Description
"auto"	incrementalRegressionKernel determines whether the predictor variables need to be standardized. See Standardize Data.
true	The software standardizes the predictor data. For more details, see Standardize Data.
false	The software does not standardize the predictor data.

Example: Standardize=true

Data Types: logical | char | string

SGD and ASGD (Average SGD) Solver Options

BatchSize — Mini-batch size

10 (default) | positive integer

Mini-batch size, specified as a positive integer. At each learning cycle during training, incrementalRegressionKernel 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, 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.

When Mdl is warm (see IsWarm), updateMetrics and updateMetricsAndFit track performance metrics in the Metrics property of Mdl.

The following table lists the built-in loss function names and which learners, specified in 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 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 Learner is "svm".

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

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

Data Types: char | string | struct | cell | function_handle

Properties

You can set most properties by using name-value argument syntax when you call incrementalRegressionKernel directly. You can set some properties when you call incrementalLearner to convert a traditionally trained model. You cannot set the properties FittedLoss, NumTrainingObservations, SolverOptions, and IsWarm.

Regression Model Parameters

Epsilon — Half of the width of epsilon insensitive band

"auto" | nonnegative scalar

This property is read-only.

Half of the width of the epsilon insensitive band, specified as "auto" or a nonnegative scalar. incrementalRegressionKernel stores the Epsilon value as a numeric scalar.

If you specify "auto" when you call incrementalRegressionKernel, incremental fitting functions estimate Epsilon during the estimation period, specified by EstimationPeriod, using this procedure:

• If iqr(Y) ≠ 0, Epsilon is iqr(Y)/13.49, where Y is the estimation period response data.

• If iqr(Y) = 0 or before you fit Mdl to data, Epsilon is 0.1.

The default Epsilon value depends on how you create the model:

• If you convert a traditionally trained model whose Learner property is 'svm', Epsilon is specified by the corresponding property of the traditionally trained model.

• Otherwise, the default value is "auto".

If Learner is "leastsquares", you cannot set Epsilon and its value is NaN.

Data Types: single | double

FittedLoss — Loss function used to fit linear model

'epsiloninsensitive' | 'mse'

This property is read-only.

Loss function used to fit the linear model, specified as 'epsiloninsensitive' or 'mse'.

Value	Algorithm	Loss Function	Learner Value
'epsiloninsensitive'	Support vector machine regression	Epsilon insensitive: $$\ell \left[y,f\left(x\right)\right]=\max \left[0,\left|y-f\left(x\right)\right|-\epsilon \right]$$	'svm'
'mse'	Linear regression through ordinary least squares	Mean squared error (MSE): $$\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[y-f\left(x\right)\right]}^2$$	'leastsquares'

KernelScale — Kernel scale parameter

"auto" | positive scalar

This property is read-only.

Kernel scale parameter, specified as "auto" or a positive scalar. incrementalRegressionKernel stores the KernelScale value as a numeric scalar. The software obtains a random basis for feature expansion by using the kernel scale parameter. For details, see Random Feature Expansion.

If you specify "auto" when creating the model object, the software selects an appropriate kernel scale parameter using a heuristic procedure. This procedure uses subsampling, so estimates might vary from one call to another. Therefore, to reproduce results, set a random number seed by using rng before training.

The default KernelScale value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, KernelScale is specified by the corresponding property of the traditionally trained model.

• Otherwise, the default value is 1.

Data Types: char | string | single | double

Learner — Linear regression model type

"svm" | "leastsquares"

This property is read-only.

Linear regression model type, specified as "svm" or "leastsquares". incrementalRegressionKernel stores the Learner value as a character vector.

In the following table, $$f\left(x\right)=T(x)\beta +b.$$

• x is an observation (row vector) from p predictor variables.

• $$T(·)$$ is a transformation of an observation (row vector) for feature expansion. T(x) maps x in $${ℝ}^p$$ to a high-dimensional space ($${ℝ}^m$$).

• β is a vector of coefficients.

• b is the scalar bias.

Value	Algorithm	Loss Function	FittedLoss Value
"svm"	Support vector machine regression	Epsilon insensitive: $$\ell \left[y,f\left(x\right)\right]=\max \left[0,\left|y-f\left(x\right)\right|-\epsilon \right]$$	'epsiloninsensitive'
"leastsquares"	Linear regression through ordinary least squares	Mean squared error (MSE): $$\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[y-f\left(x\right)\right]}^2$$	'mse'

The default Learner value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, Learner is specified by the corresponding property of the traditionally trained model.

• Otherwise, the default value is "svm".

NumExpansionDimensions — Number of dimensions of expanded space

"auto" | positive integer

This property is read-only.

Number of dimensions of the expanded space, specified as "auto" or a positive integer. incrementalRegressionKernel stores the NumExpansionDimensions value as a numeric scalar.

For "auto", the software selects the number of dimensions using 2.^ceil(min(log2(p)+5,15)), where p is the number of predictors. For details, see Random Feature Expansion.

The default NumExpansionDimensions value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, NumExpansionDimensions is specified by the corresponding property of the traditionally trained model.

• Otherwise, the default value is "auto".

Data Types: char | string | single | double

NumPredictors — Number of predictor variables

nonnegative numeric scalar

This property is read-only.

Number of predictor variables, specified as a nonnegative numeric scalar.

The default NumPredictors value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, NumPredictors is specified by the corresponding property of the traditionally trained model.

• If you create Mdl by calling incrementalRegressionKernel directly, you can specify NumPredictors by using name-value argument syntax. If you do not specify the value, then the default value is 0, and incremental fitting functions infer NumPredictors from the predictor data during training.

Data Types: double

NumTrainingObservations — Number of observations fit to incremental model

0 (default) | nonnegative numeric scalar

This property is read-only.

Number of observations fit to the incremental model Mdl, specified as a nonnegative numeric scalar. NumTrainingObservations increases when you pass Mdl and training data to fit or updateMetricsAndFit.

Note

If you convert a traditionally trained model to create Mdl, incrementalRegressionKernel does not add the number of observations fit to the traditionally trained model to NumTrainingObservations.

Data Types: double

ResponseTransform — Response transformation function

"none" | function handle

This property is read-only.

Response transformation function, specified as "none" or a function handle. incrementalRegressionKernel stores the ResponseTransform value as a character vector or function handle.

ResponseTransform describes how incremental learning functions transform raw response values.

For a MATLAB^® function or a function that you define, enter its function handle; for example, ResponseTransform=@function, where function accepts an n-by-1 vector (the original responses) and returns a vector of the same length (the transformed responses).

The default ResponseTransform value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, ResponseTransform is specified by the corresponding property of the traditionally trained model.

• Otherwise, the default value is "none".

Data Types: char | string | function_handle

Training Parameters

EstimationPeriod — Number of observations processed to estimate hyperparameters

nonnegative integer

This property is read-only.

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), incrementalRegressionKernel forces EstimationPeriod to 0.

• If Mdl is not prepared for incremental learning, incrementalRegressionKernel sets EstimationPeriod to 1000.

For more details, see Estimation Period.

Data Types: single | double

Mu — Predictor means

numeric vector | []

Since R2023b

This property is read-only.

Predictor means, specified as a numeric vector.

If Mu is an empty array [] and you specify Standardize=true, incremental fitting functions set Mu to the predictor variable means estimated during the estimation period specified by EstimationPeriod.

You cannot specify Mu directly.

Data Types: single | double

Sigma — Predictor standard deviations

numeric vector | []

Since R2023b

This property is read-only.

Predictor standard deviations, specified as a numeric vector.

If Sigma is an empty array [] and you specify Standardize=true, incremental fitting functions set Sigma to the predictor variable standard deviations estimated during the estimation period specified by EstimationPeriod.

You cannot specify Sigma directly.

Data Types: single | double

Solver — Objective function minimization technique

"scale-invariant" | "sgd" | "asgd"

This property is read-only.

Objective function minimization technique, specified as "scale-invariant", "sgd", or "asgd". incrementalRegressionKernel stores the Solver value as a character vector.

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 (Average SGD) 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 (Average SGD) Solver Options. The fit function always shuffles an incoming chunk of data before fitting the model.

The default Solver value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, the Solver name-value argument of the incrementalLearner function sets this property. The default value of the argument is "scale-invariant".

• Otherwise, the default value is "scale-invariant".

Data Types: char | string

SolverOptions — Objective solver configurations

structure array

This property is read-only.

Objective solver configurations, specified as a structure array. The fields of SolverOptions depend on Solver.

• For the SGD and ASGD solvers, the structure array includes the Solver, BatchSize, Lambda, LearnRate, and LearnRateSchedule fields.

• For the adaptive scale-invariant solver, the structure array includes the Solver and Shuffle fields.

You can specify the field values using the corresponding name-value arguments when you create the model object by calling incrementalRegressionKernel directly, or when you convert a traditionally trained model using the incrementalLearner function.

Data Types: struct

Performance Metrics Parameters

IsWarm — Flag indicating whether model tracks performance metrics

false or 0 | true or 1

This property is read-only.

Flag indicating whether the incremental model tracks performance metrics, specified as logical 0 (false) or 1 (true).

The incremental model Mdl is warm (IsWarm becomes true) after incremental fitting functions fit (EstimationPeriod + MetricsWarmupPeriod) observations to the incremental model.

Value	Description
true or 1	The incremental model Mdl is warm. Consequently, updateMetrics and updateMetricsAndFit track performance metrics in the Metrics property of Mdl.
false or 0	updateMetrics and updateMetricsAndFit do not track performance metrics.

Data Types: logical

Metrics — Model performance metrics

table

This property is read-only.

Model performance metrics updated during incremental learning by updateMetrics and updateMetricsAndFit, specified as a table with two columns and m rows, where m is the number of metrics specified by the Metrics name-value argument.

The columns of Metrics are labeled Cumulative and Window.

• Cumulative: Element j is the model performance, as measured by metric j, from the time the model became warm (IsWarm is 1).

• Window: Element j is the model performance, as measured by metric j, evaluated over all observations within the window specified by the MetricsWindowSize property. The software updates Window after it processes MetricsWindowSize observations.

Rows are labeled by the specified metrics. For details, see the Metrics name-value argument of incrementalLearner or incrementalRegressionKernel.

Data Types: table

MetricsWarmupPeriod — Number of observations fit before tracking performance metrics

nonnegative integer

This property is read-only.

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 default MetricsWarmupPeriod value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, the MetricsWarmupPeriod name-value argument of the incrementalLearner function sets this property. The default value of the argument is 0.

• Otherwise, the default value is 1000.

For more details, see Performance Metrics.

Data Types: single | double

MetricsWindowSize — Number of observations to use to compute window performance metrics

positive integer

This property is read-only.

Number of observations to use to compute window performance metrics, specified as a positive integer.

The default MetricsWindowSize value depends on how you create the model:

• If you convert a traditionally trained model to create Mdl, the MetricsWindowSize name-value argument of the incrementalLearner function sets this property. The default value of the argument is 200.

• Otherwise, the default value is 200.

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

Data Types: single | double

Object Functions

fit	Train kernel model for incremental learning
updateMetrics	Update performance metrics in kernel incremental learning model given new data
updateMetricsAndFit	Update performance metrics in kernel incremental learning model given new data and train model
loss	Loss of kernel incremental learning model on batch of data
predict	Predict responses for new observations from kernel incremental learning model
perObservationLoss	Per observation regression error of model for incremental learning
reset	Reset incremental regression model

Examples

Create Incremental Learner Without Any Prior Information

Create an incremental kernel model without any prior information. Track the model performance on streaming data, and fit the model to the data.

Create a default incremental kernel SVM model for regression.

Mdl = incrementalRegressionKernel()

Mdl = 
  incrementalRegressionKernel

                    IsWarm: 0
                   Metrics: [1x2 table]
         ResponseTransform: 'none'
    NumExpansionDimensions: 0
               KernelScale: 1

Mdl.EstimationPeriod

ans = 
1000

Mdl is an incrementalRegressionKernel model object. All its properties are read-only.

Mdl must be fit to data before you can use it to perform any other operations. The software sets the estimation period to 1000 because half the width of the epsilon insensitive band Epsilon is unknown. You can set Epsilon to a positive floating-point scalar by using the Epsilon name-value argument. This action results in a default estimation period of 0.

Load the robot arm data set.

load robotarm

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

Fit the incremental model to the training data by using the updateMetricsAndFit function. To simulate a data stream, fit the model in chunks of 50 observations at a time. At each iteration:

• Process 50 observations.

• 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
n = numel(ytrain);
numObsPerChunk = 50;
nchunk = floor(n/numObsPerChunk);
ei = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); 
numtrainobs = zeros(nchunk+1,1);

% Incremental fitting
for j = 1:nchunk
    ibegin = min(n,numObsPerChunk*(j-1) + 1);
    iend   = min(n,numObsPerChunk*j);
    idx = ibegin:iend;    
    Mdl = updateMetricsAndFit(Mdl,Xtrain(idx,:),ytrain(idx));
    ei{j,:} = Mdl.Metrics{"EpsilonInsensitiveLoss",:};
    numtrainobs(j+1) = Mdl.NumTrainingObservations;
end

Mdl is an incrementalRegressionKernel model object trained on all the data in the stream. While updateMetricsAndFit processes the first 1000 observations, it stores the response values to estimate Epsilon; the function does not fit the model until after this estimation period. 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(2,1);
nexttile
plot(numtrainobs)
xlim([0 nchunk])
ylabel("Number of Training Observations")
xline(Mdl.EstimationPeriod/numObsPerChunk,"-.")
xline((Mdl.EstimationPeriod + Mdl.MetricsWarmupPeriod)/numObsPerChunk,"--")
nexttile
plot(ei.Variables)
xlim([0 nchunk])
ylabel("Epsilon Insensitive Loss")
xline(Mdl.EstimationPeriod/numObsPerChunk,"-.")
xline((Mdl.EstimationPeriod + Mdl.MetricsWarmupPeriod)/numObsPerChunk,"--")
legend(ei.Properties.VariableNames,Location="best")
xlabel(t,"Iteration")

The plot suggests that updateMetricsAndFit does the following:

• After the estimation period (first 20 iterations), 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 200 observations (4 iterations).

Configure Incremental Learning Options

Prepare an incremental regression learner by specifying a metrics warm-up period and a metrics window size. Train the model by using SGD, and adjust the SGD batch size, learning rate, and regularization parameter.

Load the robot arm data set.

load robotarm
n = numel(ytrain);

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

Create an incremental kernel model for regression. Configure the model as follows:

• Specify the SGD solver.

• Assume that these settings work well for the problem: a ridge regularization parameter value of 0.001, SGD batch size of 20, learning rate of 0.002, and half the width of the epsilon insensitive band for SVM of 0.05.

• Specify a metrics warm-up period of 1000 observations.

• Specify a metrics window size of 500 observations.

• Track the 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);

Mdl = incrementalRegressionKernel(Solver="sgd", ...
    Lambda=0.001,BatchSize=20,LearnRate=0.002,Epsilon=0.05, ...
    MetricsWarmupPeriod=1000,MetricsWindowSize=500, ...
    Metrics={"epsiloninsensitive","mse",maemetric})

Mdl = 
  incrementalRegressionKernel

                    IsWarm: 0
                   Metrics: [3x2 table]
         ResponseTransform: 'none'
    NumExpansionDimensions: 0
               KernelScale: 1

Mdl is an incrementalRegressionKernel model object configured for incremental learning without an estimation period.

Fit the incremental model to the data by using the updateMetricsAndFit function. At each iteration:

• Simulate a data stream by processing a chunk of 50 observations. Note that the chunk size is different from the SGD batch size.

• 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
numObsPerChunk = 50;
nchunk = floor(n/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 = zeros(nchunk,1);

% Incremental fitting
rng("default") % For reproducibility
for j = 1:nchunk
    ibegin = min(n,numObsPerChunk*(j-1) + 1);
    iend   = min(n,numObsPerChunk*j);
    idx = ibegin:iend;    
    Mdl = updateMetricsAndFit(Mdl,Xtrain(idx,:),ytrain(idx));
    ei{j,:} = Mdl.Metrics{"EpsilonInsensitiveLoss",:};
    mse{j,:} = Mdl.Metrics{"MeanSquaredError",:};
    mae{j,:} = Mdl.Metrics{"MeanAbsoluteError",:};
    numtrainobs(j) = Mdl.NumTrainingObservations;
end

Mdl 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])
ylabel(["Number of","Training Observations"])
xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--")
nexttile
plot(ei.Variables)
xlim([0 nchunk])
ylabel(["Epsilon Insensitive","Loss"])
xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--")
legend(ei.Properties.VariableNames)
nexttile
plot(mse.Variables)
xlim([0 nchunk])
ylabel("MSE")
xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--")
legend(mse.Properties.VariableNames)
nexttile
plot(mae.Variables)
xlim([0 nchunk])
ylabel("MAE")
xline(Mdl.MetricsWarmupPeriod/numObsPerChunk,"--")
legend(mae.Properties.VariableNames)
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 (10 iterations).

Convert Traditionally Trained Model to Incremental Learner

Train a kernel regression model by using fitrkernel, convert it to an incremental learner, track its performance, and fit it to streaming data. Carry over training options from traditional to incremental learning.

Load and Preprocess Data

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

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

Suppose that the data collected from Manhattan (BOROUGH = 1) was collected using a new method that doubles its quality. Create a weight variable that attributes 2 to observations collected from Manhattan, and 1 to all other observations.

NYCHousing2015.W = ones(n,1) + (NYCHousing2015.BOROUGH == 1);

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 a random sample of half the data. Specify the observation weights.

idxtt = randsample([true false],n,true);
Mdl = fitrkernel(X(idxtt,:),Y(idxtt),Weights=NYCHousing2015.W(idxtt))

Mdl = 
  RegressionKernel
              ResponseName: 'Y'
                   Learner: 'svm'
    NumExpansionDimensions: 2048
               KernelScale: 1
                    Lambda: 2.1977e-05
             BoxConstraint: 1
                   Epsilon: 0.0547

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: [1x2 table]
         ResponseTransform: 'none'
    NumExpansionDimensions: 2048
               KernelScale: 1

IncrementalMdl is an incrementalRegressionKernel model object configured for incremental learning.

Separately Track Performance Metrics and Fit Model

Perform incremental learning on the rest of the data by using the updateMetrics and fit functions. Simulate a data stream by processing 500 observations at a time. At each iteration:

1. Call updateMetrics to update the cumulative and window epsilon insensitive loss of the model given the incoming chunk of observations. Overwrite the previous incremental model to update the Metrics property. Note that the function does not fit the model to the chunk of data—the chunk is "new" data for the model. Specify the observation weights.

2. Call fit to fit the model to the incoming chunk of observations. Overwrite the previous incremental model to update the model parameters. Specify the observation weights.

3. Store the losses and number of training observations.

% Preallocation
idxil = ~idxtt;
nil = sum(idxil);
numObsPerChunk = 500;
nchunk = floor(nil/numObsPerChunk);
ei = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]);
numtrainobs = zeros(nchunk,1);
Xil = X(idxil,:);
Yil = Y(idxil);
Wil = NYCHousing2015.W(idxil);

% Incremental fitting
for j = 1:nchunk
    ibegin = min(nil,numObsPerChunk*(j-1) + 1);
    iend   = min(nil,numObsPerChunk*j);
    idx = ibegin:iend;
    IncrementalMdl = updateMetrics(IncrementalMdl,Xil(idx,:),Yil(idx), ...
        Weights=Wil(idx));
    ei{j,:} = IncrementalMdl.Metrics{"EpsilonInsensitiveLoss",:};
    IncrementalMdl = fit(IncrementalMdl,Xil(idx,:),Yil(idx), ...
        Weights=Wil(idx));
    numtrainobs(j) = IncrementalMdl.NumTrainingObservations;
end

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

Alternatively, you can use updateMetricsAndFit to update performance metrics of the model given a new chunk of data, and then fit the model to the data.

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

t = tiledlayout(2,1);
nexttile
plot(numtrainobs)
xlim([0 nchunk])
ylabel("Number of Training Observations")
nexttile
plot(ei.Variables)
xlim([0 nchunk])
ylabel("Epsilon Insensitive Loss")
legend(ei.Properties.VariableNames)
xlabel(t,"Iteration")

The cumulative loss gradually changes with each iteration (chunk of 500 observations), whereas the window loss jumps. Because the metrics window is 200 by default, updateMetrics measures the performance based on the latest 200 observations in each 500 observation chunk.

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 estimated when both of these conditions apply: Incremental fitting functions are configured to standardize predictor data (see Standardize Data). Mdl.Mu and Mdl.Sigma are empty arrays [].
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 hyperparameter is estimated when all of these conditions apply: You create an incrementalRegressionKernel model object by calling the incrementalRegressionKernel function directly. The learner is SVM (see Learner). You do not specify the Epsilon name-value argument as a nonnegative scalar.
Kernel scale parameter	KernelScale	Set a kernel scale parameter value for random feature expansion	The hyperparameter is estimated when you set the KernelScale name-value argument to "auto".

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 Mdl.

• When you set Standardize=true and specify a positive estimation period (see EstimationPeriod), and Mdl.Mu and Mdl.Sigma are empty, incremental fitting functions estimate means and standard deviations using the estimation period observations.

• When you set Standardize="auto" (the default), the following conditions apply:

  • If you create incrementalRegressionKernel by converting a traditionally trained kernel regression model (RegressionKernel), and the Mu and Sigma properties of the model being converted are empty arrays [], incremental learning functions do not standardize predictor variables. If the Mu and Sigma properties of the model being converted are nonempty, incremental learning functions standardize the predictor variables using the specified means and standard deviations. Incremental fitting functions do not estimate new means and standard deviations, regardless of the length of the estimation period.

  • If you do not convert a traditionally trained model, incremental learning functions standardize the predictor data only when you specify an SGD solver (see Solver) and a positive estimation period (see EstimationPeriod).

• 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

R2023b: Kernel models support standardization of predictors

incrementalRegressionKernel supports the standardization of numeric predictors. That is, you can specify the Standardize value as true to center and scale each numeric predictor variable by the corresponding column mean and standard deviation. The software does not standardize the categorical predictors.