incrementalRegressionKernel
Kernel regression model for incremental learning
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 lowdimensional
space into a highdimensional space, then fits a linear model in the highdimensional space.
Supported linear models include support vector machine (SVM) and leastsquares
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 crossvalidation, 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 learnerspecific 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 anincrementalRegressionKernel
model object by passing it to theincrementalLearner
function.Call an incremental learning function —
fit
,updateMetrics
, andupdateMetricsAndFit
accept a configuredincrementalRegressionKernel
model object and data as input, and return anincrementalRegressionKernel
model object updated with information learned from the input model and data.
Description
returns
a default incremental learning model object for binary Gaussian kernel regression,
Mdl
= incrementalRegressionKernel()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.
sets properties and additional
options using namevalue arguments. For example,
Mdl
= incrementalRegressionKernel(Name=Value
)incrementalRegressionKernel(Solver="sgd",LearnRateSchedule="constant")
specifies to use the stochastic gradient descent (SGD) solver with a constant learning
rate.
Input 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: Metrics="mse",MetricsWarmupPeriod=100
sets the model
performance metric to the weighted mean squared error and the metrics warmup period to
100
.
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
highdimensional space. For details, see Managing the Global Stream Using RandStream
and Creating and Controlling a Random Number Stream.
Example: RandomStream=RandStream("mlfg6331_64")
BatchSize
— Minibatch size
10
(default)  positive integer
Minibatch 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 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, 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.
When Mdl
is warm (see IsWarm
), updateMetrics
and updateMetricsAndFit
track performance metrics in the Metrics
property of
Mdl
.
The following table lists the builtin 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 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 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"
ifLearner
is"svm"
.Metrics
is"mse"
ifLearner
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 namevalue 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 readonly.
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
isiqr(Y)/13.49
, whereY
is the estimation period response data.If
iqr(Y)
= 0 or before you fitMdl
to data,Epsilon
is0.1
.
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 readonly.
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: $$\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,\leftyf\left(x\right)\right\epsilon \right]$$  'svm' 
'mse'  Linear regression through ordinary least squares  Mean squared error (MSE): $$\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[yf\left(x\right)\right]}^{2}$$  'leastsquares' 
KernelScale
— Kernel scale parameter
"auto"
 positive scalar
This property is readonly.
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 can 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 readonly.
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.
$$T(\xb7)$$ is a transformation of an observation (row vector) for feature expansion. T(x) maps x in $${\mathbb{R}}^{p}$$ to a highdimensional space ($${\mathbb{R}}^{m}$$).
β is a vector of coefficients.
b is the scalar bias.
Value  Algorithm  Loss Function  FittedLoss Value 

"svm"  Support vector machine regression  Epsilon insensitive: $$\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,\leftyf\left(x\right)\right\epsilon \right]$$  'epsiloninsensitive' 
"leastsquares"  Linear regression through ordinary least squares  Mean squared error (MSE): $$\ell \left[y,f\left(x\right)\right]=\frac{1}{2}{\left[yf\left(x\right)\right]}^{2}$$  '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 readonly.
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 readonly.
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 callingincrementalRegressionKernel
directly, you can specifyNumPredictors
by using namevalue argument syntax. If you do not specify the value, then the default value is0
, and incremental fitting functions inferNumPredictors
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 readonly.
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 readonly.
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
nby1 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 readonly.
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
forcesEstimationPeriod
to0
.If
Mdl
is not prepared for incremental learning,incrementalRegressionKernel
setsEstimationPeriod
to1000
.
For more details, see Estimation Period.
Data Types: single
 double
Solver
— Objective function minimization technique
"scaleinvariant"
 "sgd"
 "asgd"
This property is readonly.
Objective function minimization technique, specified as "scaleinvariant"
, "sgd"
, or "asgd"
. incrementalRegressionKernel
stores the Solver
value as a character vector.
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] 

The default Solver
value depends on how you create the model:
If you convert a traditionally trained model to create
Mdl
, theSolver
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument is"scaleinvariant"
.Otherwise, the default value is
"scaleinvariant"
.
Data Types: char
 string
SolverOptions
— Objective solver configurations
structure array
This property is readonly.
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
, andLearnRateSchedule
fields.For the adaptive scaleinvariant solver, the structure array includes the
Solver
andShuffle
fields.
You can specify the field values using the corresponding namevalue 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 readonly.
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 readonly.
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
namevalue
argument.
The columns of Metrics
are labeled Cumulative
and Window
.
Cumulative
: Elementj
is the model performance, as measured by metricj
, from the time the model became warm (IsWarm
is1
).Window
: Elementj
is the model performance, as measured by metricj
, evaluated over all observations within the window specified by theMetricsWindowSize
property. The software updatesWindow
after it processesMetricsWindowSize
observations.
Rows are labeled by the specified metrics. For details, see the
Metrics
namevalue argument of
incrementalLearner
or incrementalRegressionKernel
.
Data Types: table
MetricsWarmupPeriod
— Number of observations fit before tracking performance metrics
nonnegative integer
This property is readonly.
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
, theMetricsWarmupPeriod
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument is0
.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 readonly.
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
, theMetricsWindowSize
namevalue argument of theincrementalLearner
function sets this property. The default value of the argument is200
.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 Properties, Methods
Mdl.EstimationPeriod
ans = 1000
Mdl
is an incrementalRegressionKernel
model object. All its properties are readonly.
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 floatingpoint scalar by using the Epsilon
namevalue 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*(j1) + 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 warmup 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 warmup 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 warmup 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 Properties, Methods
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*(j1) + 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 warmup 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: 2.1977e05 BoxConstraint: 1 Epsilon: 0.0547 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: [1x2 table] ResponseTransform: 'none' NumExpansionDimensions: 2048 KernelScale: 1 Properties, Methods
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:
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 theMetrics
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.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.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*(j1) + 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 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 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 

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

Half the width of the epsilon insensitive band  Epsilon
 Control number of support vectors  The hyperparameter is estimated when all of these conditions apply:

During the estimation period, fit
does not fit the model, and updateMetricsAndFit
does not fit the model or update the performance metrics. At the end of the estimation period, the functions update the properties that store the hyperparameters.
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.
Version History
Introduced in R2022a
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