Convert linear regression model to incremental learner
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 pair 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 performance metrics.
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 an linear regression model to the entire data set.
TTMdl = fitrlinear(X,Y)
TTMdl = RegressionLinear ResponseName: 'Y' ResponseTransform: 'none' Beta: [312×1 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: [1×2 table] ResponseTransform: 'none' Beta: [312×1 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 information TTMdl
extracted from the training data.
IncrementalMdl
is warm (IsWarm
is 1
), which means that incremental learning functions can start tracking performance metrics.
The incrementalLearner
function trains the model using the adaptive scaleinvariant solver, whereas fitrlinear
trained TTMdl
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.
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 of the standard SGD or ASGD solvers 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 will be used to train a model traditionally, and the second set will be used 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.
Use fit
to fit the incremental model to the rest of the data. At each iteration:
Simulate a data stream by processing ten observations at a time.
Overwrite the previous incremental model with a new one fitted to the incoming observation.
Store the learning rate and ${\beta}_{1}$ to see the evolution of the coefficients and learning rate.
% 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 that has experienced all the data in the stream.
To see how the learning rate and ${\beta}_{1}$ evolved during training, plot them on separate subplots.
subplot(2,1,1) plot(beta1) hold on ylabel('\beta_1') xline(IncrementalMdl.EstimationPeriod/numObsPerChunk,'r.'); subplot(2,1,2) plot(learnrate) ylabel('Learning Rate') xline(IncrementalMdl.EstimationPeriod/numObsPerChunk,'r.'); xlabel('Iteration')
The learning rate jumps to its autotuned value after the estimation period.
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 ten observations.
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 all of 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 observation.
Store ${\beta}_{1}$, 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); % 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 observation, and then fits the model to that observation.
To see how the performance metrics and ${\beta}_{1}$ evolved during training, plot them on separate subplots.
figure; subplot(2,2,1) plot(beta1) ylabel('\beta_1') xlim([0 nchunk]); xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r.'); xlabel('Iteration') subplot(2,2,2) h = plot(ei.Variables); xlim([0 nchunk]); ylabel('Epsilon Insensitive Loss') xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r.'); legend(h,ei.Properties.VariableNames) xlabel('Iteration') subplot(2,2,3) h = plot(mse.Variables); xlim([0 nchunk]); ylabel('MSE') xline(IncrementalMdl.MetricsWarmupPeriod/numObsPerChunk,'r.'); legend(h,mse.Properties.VariableNames) xlabel('Iteration') subplot(2,2,4) 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 ${\beta}_{1}$ during all incremental learning iterations.
Compute performance metrics after the metrics warmup period only.
Compute the cumulative metrics during each iteration.
Compute the window metrics after processing 500 observations.
Mdl
— Traditionally trained linear regression modelRegressionLinear
model objectTraditionally 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 using selectModels
.
Incremental learning functions support only numeric input predictor data. If Mdl
was fit to categorical data, use dummyvar
to convert each categorical variable to a numeric matrix of dummy variables, and concatenate all dummy variable matrices and any other numeric predictors. For more details, see Dummy Variables.
Specify optional
commaseparated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
'Solver','scaleinvariant','MetricsWindowSize',100
specifies the adaptive scaleinvariant solver for objective optimization, and specifies processing 100 observations before updating the performance metrics.'Solver'
— Objective function minimization technique'scaleinvariant'
(default)  '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]  This algorithm is parameter free and can adapt to differences in predictor scales. Try this algorithm before using SGD or ASGD. 
'sgd'  Stochastic gradient descent (SGD) [3][2]  To train effectively with SGD, standardize the data and specify adequate values for hyperparameters using options listed in SGD and ASGD Solver Options. 
'asgd'  Average stochastic gradient descent (ASGD) [4]  To train effectively with ASGD, standardize the data and specify adequate values for hyperparameters using options listed in SGD and ASGD Solver Options. 
Example: 'Solver','sgd'
Data Types: char
 string
'EstimationPeriod'
— Number of observations processed to estimate hyperparametersNumber 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
forces 'EstimationPeriod'
to 0
.
If Mdl
is not prepared for incremental learning, incrementalLearner
sets 'EstimationPeriod'
to 1000
.
For more details, see Estimation Period.
Example: 'EstimationPeriod',100
Data Types: single
 double
'Standardize'
— Flag to standardize predictor datafalse
(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 sizeMinibatch size, specified as the commaseparated pair consisting of 'BatchSize'
and a positive integer. At each iteration during training, incrementalLearner
uses min(BatchSize,numObs)
observations to compute the subgradient, where numObs
is the number of observations in the training data passed to fit
or updateMetricsAndFit
.
If Mdl.ModelParameters.Solver
is 'sgd'
or 'asgd'
, you cannot set 'BatchSize'
. Instead, incrementalLearner
sets 'BatchSize'
to Mdl.ModelParameters.BatchSize
.
Otherwise, BatchSize
is 10
.
Example: 'BatchSize',1
Data Types: single
 double
'Lambda'
— Ridge (L2) regularization term strengthRidge (L2) regularization term strength, specified as the commaseparated pair consisting of 'Lambda'
and a nonnegative scalar.
When Mdl.Regularization
is 'ridge (L2)'
:
If Mdl.ModelParameters.Solver
is 'sgd'
or 'asgd'
, you cannot set 'Lambda'
. Instead, incrementalLearner
sets 'Lambda'
to Mdl.Lambda
.
Otherwise, Lambda
is 1e5
.
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'
— Learning rate'auto'
 positive scalarLearning rate, specified as the commaseparated pair consisting of 'LearnRate'
and 'auto'
or a positive scalar. LearnRate
controls the optimization step size by scaling the objective subgradient.
If Mdl.ModelParameters.Solver
is 'sgd'
or 'asgd'
, you cannot set 'LearnRate'
. Instead, incrementalLearner
sets 'LearnRate'
to Mdl.ModelParameters.LearnRate
.
Otherwise, LearnRate
is 'auto'
.
For 'auto'
:
If EstimationPeriod
is 0
, the initial learning rate is 0.7
.
If EstimationPeriod
>
0
, the initial learning rate is 1/sqrt(1+max(sum(X.^2,obsDim)))
, where obsDim
is 1
if the observations compose the columns of the predictor data, and 2
otherwise. fit
and updateMetricsAndFit
set the value when you pass the model and training data to either function.
The namevalue pair argument 'LearnRateSchedule'
determines the learning rate for subsequent learning cycles.
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.ModelParameters.Solver
is 'sgd'
or 'asgd'
, you cannot set 'LearnRateSchedule'
.
Example: 'LearnRateSchedule','constant'
Data Types: char
 string
'Shuffle'
— Flag for shuffling observations in batchtrue
(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 observations in each incoming batch of data before processing the set. 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  Learners 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 @customeMetric2 '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"
if Mdl.Learner
is 'svm'
.
Metrics
is "mse"
if Mdl.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 metrics0
(default)  nonnegative integer  ...Number of observations the incremental model must be fit to before it tracks performance metrics in its Metrics
property, specified as the commaseparated pair consisting of 'MetricsWarmupPeriod'
and a nonnegative integer. The incremental model is warm after incremental fitting functions fit MetricsWarmupPeriod
observations to the incremental model (EstimationPeriod
+ MetricsWarmupPeriod
observations).
For more details on performance metrics options, see Performance Metrics.
Data Types: single
 double
'MetricsWindowSize'
— Number of observations to use to compute window performance metrics200
(default)  positive integer  ...Number of observations to use to compute window performance metrics, specified as the commaseparated pair consisting of 'MetricsWindowSize'
and a positive integer.
For more details on performance metrics options, see Performance Metrics.
Data Types: single
 double
IncrementalMdl
— Linear regression model for incremental learningincrementalRegressionLinear
model objectLinear 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 congruent 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 
ModelParameters.FitBias  Linear model intercept inclusion flag 
Mu  Predictor variable means, a numeric vector 
Sigma  Predictor variable standard deviations, a numeric vector 
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.
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 can require tuning. 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.
During the estimation period, incremental fitting functions fit
and updateMetricsAndFit
use the first incoming EstimationPeriod
observations to estimate (tune) hyperparameters required for incremental training. This table describes the hyperparameters and when they are estimated or tuned.
Hyperparameter  Model Property  Use  Hyperparameters Estimated 

Predictor means and standard deviations 
 Standardize predictor data  When you set 
Learning rate  LearnRate  Adjust solver step size  When both of these conditions apply:

The functions fit only the last estimation period observation to the incremental model, and they do not use any of the observations to track the performance of the model. At the end of the estimation period, the functions update the properties that store the hyperparameters.
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
, and IncrementalMdl.Mu
and IncrementalMdl.Sigma
are empty, the following conditions apply:
If the estimation period is positive (see the EstimationPeriod
property of IncrementalMdl
), incremental fitting functions estimate means and standard deviations using the estimation period observations.
If the estimation period is 0, incrementalLearner
forces the estimation period to 1000
. 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.
The updateMetrics
and updateMetricsAndFit
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 is warm after fit
or updateMetricsAndFit
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 additional EstimationPeriod
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
and Window
, with individual metrics in rows. When the incremental model is warm, updateMetrics
and updateMetricsAndFit
update the metrics at the following frequencies:
Cumulative
— The functions compute cumulative metrics since the start of model performance tracking. The functions update metrics every time you call the functions and base the calculation on the entire supplied data set.
Window
— The functions compute metrics based on all observations within a window determined by the 'MetricsWindowSize'
namevalue pair argument. 'MetricsWindowSize'
also determines the frequency at which the software updates Window
metrics. For example, if MetricsWindowSize
is 20, the functions compute metrics based on the last 20 observations in the supplied data (X((end – 20 + 1):end,:)
and Y((end – 20 + 1):end)
).
Incremental functions that track performance metrics within a window use the following process:
For each specified metric, store a buffer of length MetricsWindowSize
and a buffer of observation weights.
Populate elements of the metrics buffer with the model performance based on batches of incoming observations, and store corresponding observations 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 incoming MetricsWindowSize
observations enter the buffer, and the earliest observations are removed from the buffer. For example, suppose MetricsWindowSize
is 20, the metrics buffer has 10 values from a previously processed batch, and 15 values are incoming. To compose the length 20 window, the functions use the measurements from the 15 incoming observations and the latest 5 measurements from the previous batch.
[1] Kempka, Michał, Wojciech Kotłowski, and Manfred K. Warmuth. "Adaptive ScaleInvariant Online Algorithms for Learning Linear Models." CoRR (February 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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