incrementalLearner
Convert kernel model for binary classification to incremental learner
Since R2022a
Description
returns a binary Gaussian kernel classification model for incremental learning,
IncrementalMdl
= incrementalLearner(Mdl
)IncrementalMdl
, using the traditionally trained kernel model object
or kernel model template object in Mdl
.
If you specify a traditionally trained model, then its property values reflect the
knowledge gained from Mdl
(parameters and hyperparameters of the
model). Therefore, IncrementalMdl
can predict labels 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 classification model for binary learning by using fitckernel
, and then convert it to an incremental learner.
Load and Preprocess Data
Load the human activity data set.
load humanactivity
For details on the data set, enter Description
at the command line.
Responses can be one of five classes: Sitting
, Standing
, Walking
, Running
, or Dancing
. Dichotomize the response by identifying whether the subject is moving (actid
> 2).
Y = actid > 2;
Train Kernel Classification Model
Fit a kernel classification model to the entire data set.
Mdl = fitckernel(feat,Y)
Mdl = ClassificationKernel ResponseName: 'Y' ClassNames: [0 1] Learner: 'svm' NumExpansionDimensions: 2048 KernelScale: 1 Lambda: 4.1537e05 BoxConstraint: 1
Mdl
is a ClassificationKernel
model object representing a traditionally trained kernel classification model.
Convert Trained Model
Convert the traditionally trained kernel classification model to a model for incremental learning.
IncrementalMdl = incrementalLearner(Mdl,Solver="sgd",LearnRate=1)
IncrementalMdl = incrementalClassificationKernel IsWarm: 1 Metrics: [1x2 table] ClassNames: [0 1] ScoreTransform: 'none' NumExpansionDimensions: 2048 KernelScale: 1
IncrementalMdl
is an incrementalClassificationKernel
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
is 1), which means that incremental learning functions can start tracking performance metrics.incrementalClassificationKernel
trains the model using the adaptive scaleinvariant solver, whereasfitckernel
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 classification scores for all observations using both models.
[~,ttscores] = predict(Mdl,feat); [~,ilscores] = predict(IncrementalMdl,feat); compareScores = norm(ttscores(:,1)  ilscores(:,1))
compareScores = 0
The difference between the scores generated by the models is 0.
Configure Performance Metric Options
Use a trained kernel classification model to initialize an incremental learner. Prepare the incremental learner by specifying a metrics warmup period and a metrics window size.
Load the human activity data set.
load humanactivity
For details on the data set, enter Description
at the command line.
Responses can be one of five classes: Sitting, Standing, Walking, Running, and Dancing. Dichotomize the response by identifying whether the subject is moving (actid
> 2).
Y = actid > 2;
Because the data set is grouped by activity, shuffle it for simplicity. Then, randomly split the data in half: the first half for training a model traditionally, and the second half for incremental learning.
n = numel(Y); rng(1) % For reproducibility cvp = cvpartition(n,Holdout=0.5); idxtt = training(cvp); idxil = test(cvp); shuffidx = randperm(n); X = feat(shuffidx,:); Y = Y(shuffidx); % First half of data Xtt = X(idxtt,:); Ytt = Y(idxtt); % Second half of data Xil = X(idxil,:); Yil = Y(idxil);
Fit a kernel classification model to the first half of the data.
Mdl = fitckernel(Xtt,Ytt);
Convert the traditionally trained kernel classification 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 classification error and hinge loss to measure the performance of the model
IncrementalMdl = incrementalLearner(Mdl, ... MetricsWarmupPeriod=2000,MetricsWindowSize=500, ... Metrics=["classiferror","hinge"]);
Fit the incremental model to the second half of the data by using the updateMetricsAndFit
function. At each iteration:
Simulate a data stream by processing 20 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 = 20; nchunk = ceil(nil/numObsPerChunk); ce = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); hinge = array2table(zeros(nchunk,2),VariableNames=["Cumulative","Window"]); 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 = updateMetricsAndFit(IncrementalMdl,Xil(idx,:),Yil(idx)); ce{j,:} = IncrementalMdl.Metrics{"ClassificationError",:}; hinge{j,:} = IncrementalMdl.Metrics{"HingeLoss",:}; numtrainobs(j) = IncrementalMdl.NumTrainingObservations; end
IncrementalMdl
is an incrementalClassificationKernel
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(3,1); nexttile plot(numtrainobs) xlim([0 nchunk]) ylabel(["Number of","Training Observations"]) xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,"") nexttile plot(ce.Variables) xlim([0 nchunk]) ylabel("Classification Error") xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,"") legend(ce.Properties.VariableNames,Location="best") nexttile plot(hinge.Variables) xlim([0 nchunk]) ylabel("Hinge Loss") 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 (25 iterations).
Specify SGD Solver
The default solver for incrementalClassificationKernel
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 the human activity data set.
load humanactivity
For details on the data set, enter Description
at the command line.
Responses can be one of five classes: Sitting
, Standing
, Walking
, Running
, and Dancing
. Dichotomize the response by identifying whether the subject is moving (actid
> 2).
Y = actid > 2;
Randomly split the data in half: the first half for training a model traditionally, and the second half for incremental learning.
n = numel(Y); rng(1) % For reproducibility cvp = cvpartition(n,Holdout=0.5); idxtt = training(cvp); idxil = test(cvp); % First half of data Xtt = feat(idxtt,:); Ytt = Y(idxtt); % Second half of data Xil = feat(idxil,:); Yil = Y(idxil);
Fit a kernel classification model to the first half of the data.
TTMdl = fitckernel(Xtt,Ytt);
Convert the traditionally trained kernel classification model to a model for incremental learning. Specify the standard SGD solver and an estimation period of 2000
observations (the default is 1000
when a learning rate is required).
IncrementalMdl = incrementalLearner(TTMdl,Solver="sgd",EstimationPeriod=2000);
IncrementalMdl
is an incrementalClassificationKernel
model object configured for incremental learning.
Fit the incremental model to the second half 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 incrementalClassificationKernel
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 the fit
function 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 model or model template
ClassificationKernel
model object  kernel model template
Traditionally trained Gaussian kernel model or kernel model template, specified as a
ClassificationKernel
model object returned by fitckernel
or
a template object returned by templateKernel
, respectively.
If Mdl
is a kernel model template object,
incrementalLearner
determines whether to standardize the
predictor variables based on the Standardize
property of the model
template object. For more information, see Standardize Data
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
"classiferror"
(default)  string vector  function handle  cell vector  structure array  "binodeviance"
 "exponential"
 "hinge"
 "logit"
 "quadratic"
Model performance metrics to track during incremental learning with the updateMetrics
or updateMetricsAndFit
function, 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. You can specify more than one by using a string vector.
Name  Description 

"binodeviance"  Binomial deviance 
"classiferror"  Classification error 
"exponential"  Exponential loss 
"hinge"  Hinge loss 
"logit"  Logistic loss 
"quadratic"  Quadratic loss 
For more details on the builtin loss functions, see loss
.
Example: Metrics=["classiferror","hinge"]
To specify a custom function that returns a performance metric, use function handle notation. The function must have this form:
metric = customMetric(C,S)
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
).C
is an nby2 logical matrix with rows indicating the class to which the corresponding observation belongs. The column order corresponds to the class order in the model for incremental learning. CreateC
by settingC(
=p
,q
)1
, if observation
is in classp
, for each observation in the specified data. Set the other element in rowq
top
0
.S
is an nby2 numeric matrix of predicted classification scores.S
is similar to thescore
output ofpredict
, where rows correspond to observations in the data and the column order corresponds to the class order in the model for incremental learning.S(
is the classification score of observationp
,q
)
being classified in classp
.q
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,"logit",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 "classiferror" is
"ClassificationError" 
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 @(C,S)customMetric(C,S)... is
CustomMetric_1 
For more details on performance metrics options, see Performance Metrics.
Data Types: char
 string
 struct
 cell
 function_handle
MetricsWarmupPeriod
— Number of observations fit before tracking performance metrics
0
(default)  nonnegative integer
Number of observations the incremental model must be fit to before it tracks
performance metrics in its Metrics
property, specified as 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
— Binary Gaussian kernel classification model for incremental learning
incrementalClassificationKernel
model object
Binary Gaussian kernel classification model for incremental learning, returned as an
incrementalClassificationKernel
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,
regularization term strength, and the random number stream.
Input Object Mdl Type  Property  Description 

ClassificationKernel model object or kernel model template
object  KernelScale  Kernel scale parameter, a positive scalar 
Learner  Linear classification model type, a character vector  
NumExpansionDimensions  Number of dimensions of expanded space, a positive integer  
ClassificationKernel model object  ClassNames  Class labels for binary classification, a twoelement list 
Mu  Predictor variable means, a numeric vector  
NumPredictors  Number of predictors, a positive integer  
Prior  Prior class label distribution, a numeric vector  
ScoreTransform  Score transformation function, a function name or function handle  
Sigma  Predictor variable standard deviations, a numeric vector 
Note that incrementalLearner
does not use the
Cost
property of the traditionally trained model in
Mdl
because incrementalClassificationKernel
does
not support this property.
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 [5] or Fastfood [6], is a scheme to approximate Gaussian kernels of the kernel classification algorithm to use 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.
The kernel classification algorithm searches for an optimal hyperplane that separates the data into two classes after mapping features into a highdimensional space. Nonlinear features that are not linearly separable in a lowdimensional space can be separable in the expanded highdimensional space. All the calculations for hyperplane classification use only dot products. You can obtain a nonlinear classification model by replacing the dot product x_{1}x_{2}' with the 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}) (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 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 scheme introduces another random
basis V instead of Z using Hadamard matrices combined
with Gaussian scaling matrices. This random basis reduces the computation cost to O(mlog
p) and reduces storage to O(m).
incrementalClassificationKernel
uses the
Fastfood scheme for random feature expansion, and uses linear classification to train a Gaussian
kernel classification 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 fitckernel
or when
you callincrementalClassificationKernel
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 estimated when both of these conditions apply:

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

Kernel scale parameter  KernelScale  Set a kernel scale parameter value for random feature expansion  The software does not estimate If you create an

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 usingfitckernel
, 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 you set
Standardize=true
by using theStandardize
namevalue argument oftemplateKernel
, and theMdl.Mu
andMdl.Sigma
properties are empty, the following conditions apply:If the estimation period is positive (see the
EstimationPeriod
property ofIncrementalMdl
), incremental fitting functions estimate the means and standard deviations using the estimation period observations.If the estimation period is 0,
incrementalLearner
forces the estimation period to1000
. Consequently, incremental fitting functions estimate new predictor variable means and standard deviations during the forced estimation period.
When incremental fitting functions estimate predictor means and standard deviations, the functions compute weighted means and weighted standard deviations using the estimation period observations. Specifically, the functions standardize predictor j (x_{j}) using
$${x}_{j}^{\ast}=\frac{{x}_{j}{\mu}_{j}^{\ast}}{{\sigma}_{j}^{\ast}}.$$
x_{j} is predictor j, and x_{jk} is observation k of predictor j in the estimation period.
$${\mu}_{j}^{\ast}=\frac{1}{{\displaystyle \sum _{k}{w}_{k}^{\ast}}}{\displaystyle \sum _{k}{w}_{k}^{\ast}{x}_{jk}}.$$
$${\left({\sigma}_{j}^{\ast}\right)}^{2}=\frac{1}{{\displaystyle \sum _{k}{w}_{k}^{\ast}}}{\displaystyle \sum _{k}{w}_{k}^{\ast}{\left({x}_{jk}{\mu}_{j}^{\ast}\right)}^{2}}.$$
$${w}_{j}^{\ast}=\frac{{w}_{j}}{{\displaystyle \sum _{\forall j\in \text{Class}k}{w}_{j}}}{p}_{k},$$
p_{k} is the prior probability of class k (
Prior
property of the incremental model).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
score 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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