incrementalLearner
Convert linear regression model to incremental learner
Description
returns a linear regression model for incremental learning, IncrementalMdl
= incrementalLearner(Mdl
)IncrementalMdl
, using the hyperparameters and coefficients of the traditionally trained linear regression model Mdl
. Because its property values reflect the knowledge gained from Mdl
, 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 linear regression model by using fitrlinear
, 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 = [];
Create dummy variable matrices from the categorical predictors.
catvars = ["BOROUGH" "BUILDINGCLASSCATEGORY" "NEIGHBORHOOD"]; dumvarstbl = varfun(@(x)dummyvar(categorical(x)),NYCHousing2015,... 'InputVariables',catvars); dumvarmat = table2array(dumvarstbl); NYCHousing2015(:,catvars) = [];
Treat all other numeric variables in the table as linear 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 Linear Regression Model
Fit a linear regression model to the entire data set.
TTMdl = fitrlinear(X,Y)
TTMdl = RegressionLinear ResponseName: 'Y' ResponseTransform: 'none' Beta: [312x1 double] Bias: 0.0956 Lambda: 1.0935e05 Learner: 'svm' Properties, Methods
TTMdl
is a RegressionLinear
model object representing a traditionally trained linear regression model.
Convert Trained Model
Convert the traditionally trained linear regression model to a linear regression model for incremental learning.
IncrementalMdl = incrementalLearner(TTMdl)
IncrementalMdl = incrementalRegressionLinear IsWarm: 1 Metrics: [1x2 table] ResponseTransform: 'none' Beta: [312x1 double] Bias: 0.0956 Learner: 'svm' Properties, Methods
IncrementalMdl
is an incrementalRegressionLinear
model object prepared for incremental learning using SVM.
The
incrementalLearner
function Initializes the incremental learner by passing learned coefficients to it, along with other informationTTMdl
extracted from the training data.IncrementalMdl
is warm (IsWarm
is1
), which means that incremental learning functions can start tracking performance metrics.incrementalRegressionLinear
trains the model using the adaptive scaleinvariant solver, whereasfitrlinear
trainedTTMdl
using the dual SGD 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(TTMdl,X); ilyfit = predict(IncrementalMdl,X); compareyfit = norm(ttyfit  ilyfit)
compareyfit = 0
The difference between the fitted values generated by the models is 0.
Specify SGD Solver
The default solver is the adaptive scaleinvariant solver. If you specify this solver, you do not need to tune any parameters for training. However, if you specify either the standard SGD or 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 = [];
Create dummy variable matrices from the categorical predictors.
catvars = ["BOROUGH" "BUILDINGCLASSCATEGORY" "NEIGHBORHOOD"]; dumvarstbl = varfun(@(x)dummyvar(categorical(x)),NYCHousing2015,... 'InputVariables',catvars); dumvarmat = table2array(dumvarstbl); NYCHousing2015(:,catvars) = [];
Treat all other numeric variables in the table as linear 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 linear regression model to 5% of the data.
TTMdl = fitrlinear(Xtt,Ytt);
Convert the traditionally trained linear regression model to a linear regression 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(TTMdl,'Solver','sgd','EstimationPeriod',2e4);
IncrementalMdl
is an incrementalRegressionLinear
model object.
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 ${\beta}_{1}$ to see how the coefficients and rate evolve during training.
% Preallocation nil = numel(Yil); numObsPerChunk = 10; nchunk = floor(nil/numObsPerChunk); learnrate = [IncrementalMdl.LearnRate; zeros(nchunk,1)]; beta1 = [IncrementalMdl.Beta(1); 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)); beta1(j + 1) = IncrementalMdl.Beta(1); learnrate(j + 1) = IncrementalMdl.LearnRate; end
IncrementalMdl
is an incrementalRegressionLinear
model object trained on all the data in the stream.
To see how the initial learning rate and ${\beta}_{1}$ evolve during training, plot them on separate tiles.
t = tiledlayout(2,1); nexttile plot(beta1) hold on ylabel('\beta_1') xline(IncrementalMdl.EstimationPeriod/numObsPerChunk,'r.') nexttile plot(learnrate) ylabel('Initial Learning Rate') xline(IncrementalMdl.EstimationPeriod/numObsPerChunk,'r.') xlabel(t,'Iteration')
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
.
Because fit
does not fit the model to the streaming data during the estimation period, ${\beta}_{1}$ is constant for the first 2000 iterations (20,000 observations). Then, ${\beta}_{1}$ changes slightly as fit
fits the model to each new chunk of 10 observations.
Configure Performance Metric Options
Use a trained linear regression model to initialize an incremental learner. Prepare the incremental learner by specifying a metrics warmup period, during which the updateMetricsAndFit
function only fits the model. Specify a metrics window size of 500 observations.
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 linear regression model to the first set.
TTMdl = fitrlinear(Xtt,Ytt);
Convert the traditionally trained linear regression model to a linear regression 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 ${\beta}_{10}$, the cumulative metrics, and the window metrics 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"]); beta1 = zeros(nchunk+1,1); beta1(1) = IncrementalMdl.Beta(10); % 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",:}; beta1(j + 1) = IncrementalMdl.Beta(10); end
IncrementalMdl
is an incrementalRegressionLinear
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.
To see how the performance metrics and ${\beta}_{10}$ evolve during training, plot them on separate tiles.
tiledlayout(2,2) nexttile plot(beta1) ylabel('\beta_{10}') xlim([0 nchunk]) xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r.') xlabel('Iteration') nexttile h = plot(ei.Variables); xlim([0 nchunk]) ylabel('Epsilon Insensitive Loss') xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r.') legend(h,ei.Properties.VariableNames) xlabel('Iteration') nexttile h = plot(mse.Variables); xlim([0 nchunk]); ylabel('MSE') xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r.') legend(h,mse.Properties.VariableNames) xlabel('Iteration') nexttile h = plot(mae.Variables); xlim([0 nchunk]); ylabel('MAE') xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r.') legend(h,mae.Properties.VariableNames) xlabel('Iteration')
The plot suggests that updateMetricsAndFit
does the following:
Fit ${\beta}_{10}$ 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.
Input Arguments
Mdl
— Traditionally trained linear regression model
RegressionLinear
model object
Traditionally trained linear regression model, specified as a RegressionLinear
model object returned by fitrlinear
.
Note
If
Mdl.Lambda
is a numeric vector, you must select the model corresponding to one regularization strength in the regularization path by usingselectModels
.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. Usedummyvar
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.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'Solver','scaleinvariant','MetricsWindowSize',100
specifies
the adaptive scaleinvariant solver for objective optimization, and specifies processing 100
observations before updating the window performance metrics.
Solver
— Objective function minimization technique
'scaleinvariant'
 'sgd'
 'asgd'
Objective function minimization technique, specified as the commaseparated pair consisting of 'Solver'
and a value in this table.
Value  Description  Notes 

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

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

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

If
Mdl.Regularization
is'ridge (L2)'
andMdl.ModelParameters.Solver
is'sgd'
or'asgd'
, the defaultSolver
value isMdl.ModelParameters.Solver
.Otherwise, the default
Solver
value is'scaleinvariant'
.
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 the commaseparated pair consisting of 'EstimationPeriod'
and 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
Standardize
— Flag to standardize predictor data
false
(default)  true
Flag to standardize the predictor data, specified as the commaseparated pair consisting of 'Standardize'
and a value in this table.
Value  Description 

true  The software standardizes the predictor data. For more details, see Standardize Data. 
false  The software does not standardize the predictor data. 
Example: 'Standardize',true
Data Types: logical
BatchSize
— Minibatch size
positive integer
Minibatch size, specified as the commaseparated pair consisting of
'BatchSize'
and a positive integer. At each learning cycle during
training, incrementalLearner
uses BatchSize
observations to
compute the subgradient.
The number of observations for 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 uses 5 observations for the last
learning cycle.
If
Mdl.Regularization
is'ridge (L2)'
andMdl.ModelParameters.Solver
is'sgd'
or'asgd'
, you cannot setBatchSize
. Instead,incrementalLearner
setsBatchSize
toMdl.ModelParameters.BatchSize
.Otherwise,
BatchSize
is10
.
Example: 'BatchSize',1
Data Types: single
 double
Lambda
— Ridge (L2) regularization term strength
nonnegative scalar
Ridge (L2) regularization term strength, specified as the commaseparated pair consisting of 'Lambda'
and a nonnegative scalar.
If
Mdl.Regularization
is'ridge (L2)'
andMdl.ModelParameters.Solver
is'sgd'
or'asgd'
, you cannot setLambda
. Instead,incrementalLearner
setsLambda
toMdl.Lambda
.Otherwise,
Lambda
is1e5
.
Note
incrementalLearner
does not support lasso regularization. If
Mdl.Regularization
is 'lasso (L1)'
,
incrementalLearner
uses ridge regularization instead, and sets the
Solver
namevalue pair argument to
'scaleinvariant'
by default.
Example: 'Lambda',0.01
Data Types: single
 double
LearnRate
— Initial learning rate
'auto'
 positive scalar
Initial learning rate, specified as the commaseparated pair consisting of
'LearnRate'
and '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,obsDim)))
at the end ofEstimationPeriod
. TheobsDim
value is1
if the observations compose the columns of the predictor data; otherwise, the value is2
.
By default:
If
Mdl.Regularization
is'ridge (L2)'
andMdl.ModelParameters.Solver
is'sgd'
or'asgd'
, you cannot setLearnRate
. Instead,incrementalLearner
setsLearnRate
toMdl.ModelParameters.LearnRate
.Otherwise,
LearnRate
is'auto'
.
Example: 'LearnRate',0.001
Data Types: single
 double
 char
 string
LearnRateSchedule
— Learning rate schedule
'decaying'
(default)  'constant'
Learning rate schedule, specified as the commaseparated pair consisting of 'LearnRateSchedule'
and 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 $${\gamma}_{t}=\frac{{\gamma}_{0}}{{\left(1+\lambda {\gamma}_{0}t\right)}^{c}}.$$

If Mdl.Regularization
is 'ridge
(L2)'
and Mdl.ModelParameters.Solver
is
'sgd'
or 'asgd'
, you cannot set
LearnRateSchedule
. Instead, incrementalLearner
sets
LearnRateSchedule
to 'decaying'
.
Example: 'LearnRateSchedule','constant'
Data Types: char
 string
Shuffle
— Flag for shuffling observations in batch
true
(default)  false
Flag for shuffling the observations in the batch at each iteration, specified as the commaseparated pair consisting of 'Shuffle'
and a value in this table.
Value  Description 

true  The software shuffles an incoming chunk of data before the
fit function fits the model. This action
reduces bias induced by the sampling scheme. 
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
and updateMetricsAndFit
, specified as the commaseparated pair consisting of 'Metrics'
and 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',100
Data Types: single
 double
Output Arguments
IncrementalMdl
— Linear regression model for incremental learning
incrementalRegressionLinear
model object
Linear regression model for incremental learning, returned as an incrementalRegressionLinear
model object. IncrementalMdl
is also configured to generate predictions given new data (see predict
).
To initialize IncrementalMdl
for incremental learning, incrementalLearner
passes the values of the Mdl
properties in this table to corresponding properties of IncrementalMdl
.
Property  Description 

Beta  Linear model coefficients, a numeric vector 
Bias  Model intercept, a numeric scalar 
Epsilon  Half the width of the epsilon insensitive band, a nonnegative scalar 
Learner  Linear regression model type, a character vector 
ModelParameters.FitBias  Linear model intercept inclusion flag 
NumPredictors  Number of predictors, a positive integer 
ResponseTransform  Response transformation function, a function name or function handle 
If Mdl.Regularization
is 'ridge (L2)'
and
Mdl.ModelParameters.Solver
is 'sgd'
or
'asgd'
, incrementalLearner
also passes the values of
Mdl
properties in this table.
Property  Description 

Lambda  Ridge (L2) regularization term strength, a nonnegative scalar 
ModelParameters.LearnRate  Learning rate, a positive scalar 
ModelParameters.BatchSize  Minibatch size, a positive integer 
ModelParameters.Solver  Objective function minimization technique, a character 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 for regression fit
and updateMetricsAndFit
use the more conservative ScInOL1 version of the algorithm.
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 you set

Learning rate  LearnRate  Adjust solver step size  The hyperparameter is estimated when both 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.
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 of the incremental learning model IncrementalMdl
.
When you set
'Standardize',true
, andIncrementalMdl.Mu
andIncrementalMdl.Sigma
are empty, the following conditions apply:If the estimation period is positive (see the
EstimationPeriod
property ofIncrementalMdl
), incremental fitting functions estimate 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
$${x}_{j}^{\ast}=\frac{{x}_{j}{\mu}_{j}^{\ast}}{{\sigma}_{j}^{\ast}}.$$
where
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 when the incremental model is warm (IsWarm
property). An incremental model becomes warm afterfit
orupdateMetricsAndFit
fit the incremental model to'MetricsWarmupPeriod'
observations, which is the metrics warmup period.If
'EstimationPeriod'
> 0, the 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 by the'MetricsWindowSize'
namevalue pair argument.'MetricsWindowSize'
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 filled, overwrite
IncrementalMdl.Metrics.Window
with the weighted average performance in the metrics window. If the buffer is overfilled 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.
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.
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