incrementalLearner
Description
returns a Gaussian kernel regression model for incremental learning,
IncrementalMdl
= incrementalLearner(Mdl
)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.
uses additional options specified by one or more namevalue
arguments. Some options require you to train IncrementalMdl
= incrementalLearner(Mdl
,Name=Value
)IncrementalMdl
before its
predictive performance is tracked. For example,
MetricsWarmupPeriod=50,MetricsWindowSize=100
specifies a preliminary
incremental training period of 50 observations before performance metrics are tracked, and
specifies processing 100 observations before updating the window performance metrics.
Examples
Convert Traditionally Trained Model to Incremental Learner
Train 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: 1.0935e05 BoxConstraint: 1 Epsilon: 0.0549
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 prepared for incremental learning.
The
incrementalLearner
function initializes the incremental learner by passing model parameters to it, along with other informationMdl
extracted from the training data.IncrementalMdl
is warm (IsWarm
is1
), which means that incremental learning functions can start tracking performance metrics.incrementalRegressionKernel
trains the model using the adaptive scaleinvariant solver, whereasfitrkernel
trainedMdl
using the Limitedmemory BroydenFletcherGoldfarbShanno (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 warmup 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 warmup 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*(j1) + 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 warmup 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 scaleinvariant 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*(j1) + 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.
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue arguments must appear after other arguments, but the order of the
pairs does not matter.
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.
Solver
— Objective function minimization technique
"scaleinvariant"
(default)  "sgd"
 "asgd"
Objective function minimization technique, specified as a value in this table.
Value  Description  Notes 

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

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

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

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
forcesEstimationPeriod
to0
.If
Mdl
is not prepared for incremental learning,incrementalLearner
setsEstimationPeriod
to1000
.
For more details, see Estimation Period.
Example: EstimationPeriod=100
Data Types: single
 double
BatchSize
— Minibatch size
10
(default)  positive integer
Minibatch size, specified as a positive integer. At each learning cycle during
training, incrementalLearner
uses BatchSize
observations to compute the subgradient.
The number of observations in the last minibatch (last learning cycle in each
function call of fit
or
updateMetricsAndFit
) can be smaller than BatchSize
.
For example, if you supply 25 observations to fit
or
updateMetricsAndFit
, the function uses 10 observations for the
first two learning cycles and 5 observations for the last learning cycle.
Example: BatchSize=5
Data Types: single
 double
Lambda
— Ridge (L2) regularization term strength
1e5
(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
andupdateMetricsAndFit
change the rate to1/sqrt(1+max(sum(X.^2,2)))
at the end ofEstimationPeriod
.
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 ɣ_{0}.
Value  Description 

"constant"  The learning rate is ɣ_{0} for all learning cycles. 
"decaying"  The learning rate at learning cycle t is $${\gamma}_{t}=\frac{{\gamma}_{0}}{{\left(1+\lambda {\gamma}_{0}t\right)}^{c}}.$$

Example: LearnRateSchedule="constant"
Data Types: char
 string
Shuffle
— Flag for shuffling observations
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
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 builtin loss function name, string
vector of names, function handle (@metricName
), structure array of
function handles, or cell vector of names, function handles, or structure
arrays.
The following table lists the builtin loss function names 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 builtin 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 nby1 numeric vector, where each element is the loss of the corresponding observation in the data processed by the incremental learning functions during a learning cycle.You specify the function name (
customMetric
).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 builtin 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 builtin 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_ , where
is metric
in
Metrics  Row name for @(Y,YFit)customMetric(Y,YFit)... is
CustomMetric_1 
By default:
Metrics
is"epsiloninsensitive"
ifMdl.Learner
is'svm'
.Metrics
is"mse"
ifMdl.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 crossvalidation to tune hyperparameters, and infer the predictor distribution.
Given incoming observations, an incremental learning model processes data in any of the following ways, but usually in this order:
Predict labels.
Measure the predictive performance.
Check for structural breaks or drift in the model.
Fit the model to the incoming observations.
For more details, see Incremental Learning Overview.
Adaptive ScaleInvariant Solver for Incremental Learning
The adaptive scaleinvariant solver for incremental learning, introduced in [1], is a gradientdescentbased objective solver for training linear predictive models. The solver is hyperparameter free, insensitive to differences in predictor variable scales, and does not require prior knowledge of the distribution of the predictor variables. These characteristics make it well suited to incremental learning.
The standard SGD and ASGD solvers are sensitive to differing scales among the predictor variables, resulting in models that can perform poorly. To achieve better accuracy using SGD and ASGD, you can standardize the predictor data, and tune the regularization and learning rate parameters. For traditional machine learning, enough data is available to enable hyperparameter tuning by crossvalidation and predictor standardization. However, for incremental learning, enough data might not be available (for example, observations might be available only one at a time) and the distribution of the predictors might be unknown. These characteristics make parameter tuning and predictor standardization difficult or impossible to do during incremental learning.
The incremental fitting functions 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 highdimensional 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 x_{1}x_{2}′ 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 highdimensional space (called the “kernel trick”). However, evaluating G(x_{1},x_{2}), 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 $${\mathbb{R}}^{p}$$ to a highdimensional space ($${\mathbb{R}}^{m}$$). The Random Kitchen Sinks [1] scheme uses the random transformation
$$T(x)={m}^{1/2}\mathrm{exp}\left(iZx\text{'}\right)\text{'},$$
where $$Z\in {\mathbb{R}}^{m\times p}$$ 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(mlog
p) 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
namevalue
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 
 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:

Half the width of the epsilon insensitive band  Epsilon
 Control the number of support vectors  The software does not estimate If you create an

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 usingfitrkernel
, the following conditions apply:incrementalLearner
passes the means inMdl.Mu
and standard deviations inMdl.Sigma
to the corresponding incremental learning model properties.Incremental learning functions always standardize the predictor data.
When 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
andupdateMetricsAndFit
functions are incremental learning functions that track model performance metrics (Metrics
) from new data only when the incremental model is warm (IsWarm
property istrue
). An incremental model becomes warm afterfit
orupdateMetricsAndFit
fits the incremental model toMetricsWarmupPeriod
observations, which is the metrics warmup period.If
EstimationPeriod
> 0, thefit
andupdateMetricsAndFit
functions estimate hyperparameters before fitting the model to data. Therefore, the functions must process an additionalEstimationPeriod
observations before the model starts the metrics warmup period.The
Metrics
property of the incremental model stores two forms of each performance metric as variables (columns) of a table,Cumulative
andWindow
, with individual metrics in rows. When the incremental model is warm,updateMetrics
andupdateMetricsAndFit
update the metrics at the following frequencies:Cumulative
— The functions compute cumulative metrics since the start of model performance tracking. The functions update metrics every time you call the functions and base the calculation on the entire supplied data set.Window
— The functions compute metrics based on all observations within a window determined byMetricsWindowSize
, which also determines the frequency at which the software updatesWindow
metrics. For example, ifMetricsWindowSize
is 20, the functions compute metrics based on the last 20 observations in the supplied data (X((end – 20 + 1):end,:)
andY((end – 20 + 1):end)
).Incremental functions that track performance metrics within a window use the following process:
Store a buffer of length
MetricsWindowSize
for each specified metric, and store a buffer of observation weights.Populate elements of the metrics buffer with the model performance based on batches of incoming observations, and store corresponding observation weights in the weights buffer.
When the buffer is full, overwrite the
Window
field of theMetrics
property with the weighted average performance in the metrics window. If the buffer overfills when the function processes a batch of observations, the latest incomingMetricsWindowSize
observations enter the buffer, and the earliest observations are removed from the buffer. For example, supposeMetricsWindowSize
is 20, the metrics buffer has 10 values from a previously processed batch, and 15 values are incoming. To compose the length 20 window, the functions use the measurements from the 15 incoming observations and the latest 5 measurements from the previous batch.
The software omits an observation with a
NaN
prediction when computing theCumulative
andWindow
performance metric values.
References
[1] Kempka, Michał, Wojciech Kotłowski, and Manfred K. Warmuth. "Adaptive ScaleInvariant Online Algorithms for Learning Linear Models." Preprint, submitted February 10, 2019. https://arxiv.org/abs/1902.07528.
[2] Langford, J., L. Li, and T. Zhang. “Sparse Online Learning Via Truncated Gradient.” J. Mach. Learn. Res., Vol. 10, 2009, pp. 777–801.
[3] ShalevShwartz, S., Y. Singer, and N. Srebro. “Pegasos: Primal Estimated SubGradient Solver for SVM.” Proceedings of the 24th International Conference on Machine Learning, ICML ’07, 2007, pp. 807–814.
[4] Xu, Wei. “Towards Optimal One Pass Large Scale Learning with Averaged Stochastic Gradient Descent.” CoRR, abs/1107.2490, 2011.
[5] Rahimi, A., and B. Recht. “Random Features for LargeScale 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.
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