templateLinear
Linear classification learner template
Description
templateLinear
creates a template suitable for fitting a linear classification model to highdimensional data for multiclass problems.
The template specifies the binary learner model, regularization type and strength, and solver, among other things. After creating the template, train the model by passing the template and data to fitcecoc
.
returns a linear classification learner template.t
= templateLinear()
If you specify a default template, then the software uses default values for all input arguments during training.
returns a template with additional options specified by one or more namevalue pair arguments. For example, you can specify to implement logistic regression, specify the regularization type or strength, or specify the solver to use for objectivefunction minimization.t
= templateLinear(Name,Value
)
If you display t
in the Command Window, then all options appear empty ([]
) except options that you specify using namevalue pair arguments. During training, the software uses default values for empty options.
Examples
Train Multiclass Linear Classification Model
Train an ECOC model composed of multiple binary, linear classification models.
Load the NLP data set.
load nlpdata
X
is a sparse matrix of predictor data, and Y
is a categorical vector of class labels. There are more than two classes in the data.
Create a default linearclassificationmodel template.
t = templateLinear();
To adjust the default values, see the NameValue Pair Arguments on templateLinear
page.
Train an ECOC model composed of multiple binary, linear classification models that can identify the product given the frequency distribution of words on a documentation web page. For faster training time, transpose the predictor data, and specify that observations correspond to columns.
X = X'; rng(1); % For reproducibility Mdl = fitcecoc(X,Y,'Learners',t,'ObservationsIn','columns')
Mdl = CompactClassificationECOC ResponseName: 'Y' ClassNames: [comm dsp ecoder fixedpoint ... ] ScoreTransform: 'none' BinaryLearners: {78x1 cell} CodingMatrix: [13x78 double] Properties, Methods
Alternatively, you can train an ECOC model composed of default linear classification models using 'Learners','Linear'
.
To conserve memory, fitcecoc
returns trained ECOC models composed of linear classification learners in CompactClassificationECOC
model objects.
Input Arguments
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: 'Learner','logistic','Regularization','lasso','CrossVal','on'
specifies to implement logistic regression with a lasso penalty, and to implement 10fold crossvalidation.
Lambda
— Regularization term strength
'auto'
(default)  nonnegative scalar  vector of nonnegative values
Regularization term strength, specified as the commaseparated pair consisting of 'Lambda'
and 'auto'
, a nonnegative scalar, or a vector of nonnegative values.
For
'auto'
,Lambda
= 1/n.If you specify a crossvalidation, namevalue pair argument (e.g.,
CrossVal
), then n is the number of infold observations.Otherwise, n is the training sample size.
For a vector of nonnegative values,
templateLinear
sequentially optimizes the objective function for each distinct value inLambda
in ascending order.If
Solver
is'sgd'
or'asgd'
andRegularization
is'lasso'
,templateLinear
does not use the previous coefficient estimates as a warm start for the next optimization iteration. Otherwise,templateLinear
uses warm starts.If
Regularization
is'lasso'
, then any coefficient estimate of 0 retains its value whentemplateLinear
optimizes using subsequent values inLambda
.templateLinear
returns coefficient estimates for each specified regularization strength.
Example: 'Lambda',10.^((10:2:2))
Data Types: char
 string
 double
 single
Learner
— Linear classification model type
'svm'
(default)  'logistic'
Linear classification model type, specified as the commaseparated
pair consisting of 'Learner'
and 'svm'
or 'logistic'
.
In this table, $$f\left(x\right)=x\beta +b.$$
β is a vector of p coefficients.
x is an observation from p predictor variables.
b is the scalar bias.
Value  Algorithm  Response Range  Loss Function 

'svm'  Support vector machine  y ∊ {–1,1}; 1 for the positive class and –1 otherwise  Hinge: $$\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,1yf\left(x\right)\right]$$ 
'logistic'  Logistic regression  Same as 'svm'  Deviance (logistic): $$\ell \left[y,f\left(x\right)\right]=\mathrm{log}\left\{1+\mathrm{exp}\left[yf\left(x\right)\right]\right\}$$ 
Example: 'Learner','logistic'
Regularization
— Complexity penalty type
'lasso'
 'ridge'
Complexity penalty type, specified as the commaseparated pair
consisting of 'Regularization'
and 'lasso'
or 'ridge'
.
The software composes the objective function for minimization
from the sum of the average loss function (see Learner
)
and the regularization term in this table.
Value  Description 

'lasso'  Lasso (L1) penalty: $$\lambda {\displaystyle \sum _{j=1}^{p}\left{\beta}_{j}\right}$$ 
'ridge'  Ridge (L2) penalty: $$\frac{\lambda}{2}{\displaystyle \sum _{j=1}^{p}{\beta}_{j}^{2}}$$ 
To specify the regularization term strength, which is λ in
the expressions, use Lambda
.
The software excludes the bias term (β_{0}) from the regularization penalty.
If Solver
is 'sparsa'
,
then the default value of Regularization
is 'lasso'
.
Otherwise, the default is 'ridge'
.
Tip
For predictor variable selection, specify
'lasso'
. For more on variable selection, see Introduction to Feature Selection.For optimization accuracy, specify
'ridge'
.
Example: 'Regularization','lasso'
Solver
— Objective function minimization technique
'sgd'
 'asgd'
 'dual'
 'bfgs'
 'lbfgs'
 'sparsa'
 string array  cell array of character vectors
Objective function minimization technique, specified as the commaseparated pair consisting of 'Solver'
and a character vector or string scalar, a string array, or a cell array of character vectors with values from this table.
Value  Description  Restrictions 

'sgd'  Stochastic gradient descent (SGD) [4][2]  
'asgd'  Average stochastic gradient descent (ASGD) [7]  
'dual'  Dual SGD for SVM [1][6]  Regularization must be 'ridge' and Learner must be 'svm' . 
'bfgs'  BroydenFletcherGoldfarbShanno quasiNewton algorithm (BFGS) [3]  Inefficient if X is very highdimensional. 
'lbfgs'  Limitedmemory BFGS (LBFGS) [3]  Regularization must be 'ridge' . 
'sparsa'  Sparse Reconstruction by Separable Approximation (SpaRSA) [5]  Regularization must be 'lasso' . 
If you specify:
A ridge penalty (see
Regularization
) and the predictor data set contains 100 or fewer predictor variables, then the default solver is'bfgs'
.An SVM model (see
Learner
), a ridge penalty, and the predictor data set contains more than 100 predictor variables, then the default solver is'dual'
.A lasso penalty and the predictor data set contains 100 or fewer predictor variables, then the default solver is
'sparsa'
.
Otherwise, the default solver is
'sgd'
. Note that the default solver can change when
you perform hyperparameter optimization. For more information, see Regularization method determines the linear learner solver used during hyperparameter optimization.
If you specify a string array or cell array of solver names, then, for
each value in Lambda
, the software uses the
solutions of solver j as a warm start for solver
j + 1.
Example: {'sgd' 'lbfgs'}
applies SGD to solve the
objective, and uses the solution as a warm start for
LBFGS.
Tip
SGD and ASGD can solve the objective function more quickly than other solvers, whereas LBFGS and SpaRSA can yield more accurate solutions than other solvers. Solver combinations like
{'sgd' 'lbfgs'}
and{'sgd' 'sparsa'}
can balance optimization speed and accuracy.When choosing between SGD and ASGD, consider that:
SGD takes less time per iteration, but requires more iterations to converge.
ASGD requires fewer iterations to converge, but takes more time per iteration.
If the predictor data is high dimensional and
Regularization
is'ridge'
, setSolver
to any of these combinations:'sgd'
'asgd'
'dual'
ifLearner
is'svm'
'lbfgs'
{'sgd','lbfgs'}
{'asgd','lbfgs'}
{'dual','lbfgs'}
ifLearner
is'svm'
Although you can set other combinations, they often lead to solutions with poor accuracy.
If the predictor data is moderate through low dimensional and
Regularization
is'ridge'
, setSolver
to'bfgs'
.If
Regularization
is'lasso'
, setSolver
to any of these combinations:'sgd'
'asgd'
'sparsa'
{'sgd','sparsa'}
{'asgd','sparsa'}
Example: 'Solver',{'sgd','lbfgs'}
Beta
— Initial linear coefficient estimates
zeros(p
,1)
(default)  numeric vector  numeric matrix
p
,1)Initial linear coefficient estimates (β),
specified as the commaseparated pair consisting of 'Beta'
and
a pdimensional numeric vector or a pbyL numeric
matrix. p is the number of predictor variables
in X
and L is the number of
regularizationstrength values (for more details, see Lambda
).
If you specify a pdimensional vector, then the software optimizes the objective function L times using this process.
The software optimizes using
Beta
as the initial value and the minimum value ofLambda
as the regularization strength.The software optimizes again using the resulting estimate from the previous optimization as a warm start, and the next smallest value in
Lambda
as the regularization strength.The software implements step 2 until it exhausts all values in
Lambda
.
If you specify a pbyL matrix, then the software optimizes the objective function L times. At iteration
j
, the software usesBeta(:,
as the initial value and, after it sortsj
)Lambda
in ascending order, usesLambda(
as the regularization strength.j
)
If you set 'Solver','dual'
, then the software
ignores Beta
.
Data Types: single
 double
Bias
— Initial intercept estimate
numeric scalar  numeric vector
Initial intercept estimate (b), specified
as the commaseparated pair consisting of 'Bias'
and
a numeric scalar or an Ldimensional numeric vector. L is
the number of regularizationstrength values (for more details, see Lambda
).
If you specify a scalar, then the software optimizes the objective function L times using this process.
The software optimizes using
Bias
as the initial value and the minimum value ofLambda
as the regularization strength.The uses the resulting estimate as a warm start to the next optimization iteration, and uses the next smallest value in
Lambda
as the regularization strength.The software implements step 2 until it exhausts all values in
Lambda
.
If you specify an Ldimensional vector, then the software optimizes the objective function L times. At iteration
j
, the software usesBias(
as the initial value and, after it sortsj
)Lambda
in ascending order, usesLambda(
as the regularization strength.j
)By default:
If
Learner
is'logistic'
, then let g_{j} be 1 ifY(
is the positive class, and 1 otherwise.j
)Bias
is the weighted average of the g for training or, for crossvalidation, infold observations.If
Learner
is'svm'
, thenBias
is 0.
Data Types: single
 double
FitBias
— Linear model intercept inclusion flag
true
(default)  false
Linear model intercept inclusion flag, specified as the commaseparated
pair consisting of 'FitBias'
and true
or false
.
Value  Description 

true  The software includes the bias term b in the linear model, and then estimates it. 
false  The software sets b = 0 during estimation. 
Example: 'FitBias',false
Data Types: logical
PostFitBias
— Flag to fit linear model intercept after optimization
false
(default)  true
Flag to fit the linear model intercept after optimization, specified
as the commaseparated pair consisting of 'PostFitBias'
and true
or false
.
Value  Description 

false  The software estimates the bias term b and the coefficients β during optimization. 
true 
To estimate b, the software:

If you specify true
, then FitBias
must
be true.
Example: 'PostFitBias',true
Data Types: logical
Verbose
— Verbosity level
0
(default)  1
Verbosity level, specified as the commaseparated pair consisting of 'Verbose'
and either 0
or 1
. Verbose
controls the display of diagnostic information at the command line.
Value  Description 

0  templateLinear does not display diagnostic information. 
1  templateLinear periodically displays the value of the objective function, gradient magnitude, and other diagnostic information. 
Example: 'Verbose',1
Data Types: single
 double
BatchSize
— Minibatch size
positive integer
Minibatch size, specified as the commaseparated pair consisting of 'BatchSize'
and a positive integer. At each iteration, the software estimates the gradient using BatchSize
observations from the training data.
If the predictor data is a numeric matrix, then the default value is
10
.If the predictor data is a sparse matrix, then the default value is
max([10,ceil(sqrt(ff))])
, whereff = numel(X)/nnz(X)
, that is, the fullness factor ofX
.
Example: 'BatchSize',100
Data Types: single
 double
LearnRate
— Learning rate
positive scalar
Learning rate, specified as the commaseparated pair consisting of 'LearnRate'
and a positive scalar. LearnRate
controls the optimization step size by scaling the subgradient.
If
Regularization
is'ridge'
, thenLearnRate
specifies the initial learning rate γ_{0}.templateLinear
determines the learning rate for iteration t, γ_{t}, using$${\gamma}_{t}=\frac{{\gamma}_{0}}{{\left(1+\lambda {\gamma}_{0}t\right)}^{c}}.$$
If
Regularization
is'lasso'
, then, for all iterations,LearnRate
is constant.
By default, LearnRate
is 1/sqrt(1+max((sum(X.^2,obsDim))))
, where obsDim
is 1
if the observations compose the columns of the predictor data X
, and 2
otherwise.
Example: 'LearnRate',0.01
Data Types: single
 double
OptimizeLearnRate
— Flag to decrease learning rate
true
(default)  false
Flag to decrease the learning rate when the software detects
divergence (that is, overstepping the minimum), specified as the
commaseparated pair consisting of 'OptimizeLearnRate'
and true
or false
.
If OptimizeLearnRate
is 'true'
,
then:
For the few optimization iterations, the software starts optimization using
LearnRate
as the learning rate.If the value of the objective function increases, then the software restarts and uses half of the current value of the learning rate.
The software iterates step 2 until the objective function decreases.
Example: 'OptimizeLearnRate',true
Data Types: logical
TruncationPeriod
— Number of minibatches between lasso truncation runs
10
(default)  positive integer
Number of minibatches between lasso truncation runs, specified
as the commaseparated pair consisting of 'TruncationPeriod'
and
a positive integer.
After a truncation run, the software applies a soft threshold
to the linear coefficients. That is, after processing k = TruncationPeriod
minibatches,
the software truncates the estimated coefficient j using
$${\widehat{\beta}}_{j}^{\ast}=\{\begin{array}{ll}{\widehat{\beta}}_{j}{u}_{t}\hfill & \text{if}\text{\hspace{0.17em}}{\widehat{\beta}}_{j}>{u}_{t},\hfill \\ 0\hfill & \text{if}\text{\hspace{0.17em}}\left{\widehat{\beta}}_{j}\right\le {u}_{t},\hfill \\ {\widehat{\beta}}_{j}+{u}_{t}\hfill & \text{if}\text{\hspace{0.17em}}{\widehat{\beta}}_{j}<{u}_{t}.\hfill \end{array}\begin{array}{r}\hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$$
For SGD, $${\widehat{\beta}}_{j}$$ is the estimate of coefficient j after processing k minibatches. $${u}_{t}=k{\gamma}_{t}\lambda .$$ γ_{t} is the learning rate at iteration t. λ is the value of
Lambda
.For ASGD, $${\widehat{\beta}}_{j}$$ is the averaged estimate coefficient j after processing k minibatches, $${u}_{t}=k\lambda .$$
If Regularization
is 'ridge'
,
then the software ignores TruncationPeriod
.
Example: 'TruncationPeriod',100
Data Types: single
 double
BatchLimit
— Maximal number of batches
positive integer
Maximal number of batches to process, specified as the commaseparated
pair consisting of 'BatchLimit'
and a positive
integer. When the software processes BatchLimit
batches,
it terminates optimization.
By default:
If you specify
BatchLimit
, thentemplateLinear
uses the argument that results in processing the fewest observations, eitherBatchLimit
orPassLimit
.
Example: 'BatchLimit',100
Data Types: single
 double
BetaTolerance
— Relative tolerance on linear coefficients and bias term
1e4
(default)  nonnegative scalar
Relative tolerance on the linear coefficients and the bias term (intercept), specified
as the commaseparated pair consisting of 'BetaTolerance'
and a
nonnegative scalar.
Let $${B}_{t}=\left[{\beta}_{t}{}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{t}\right]$$, that is, the vector of the coefficients and the bias term at optimization iteration t. If $${\Vert \frac{{B}_{t}{B}_{t1}}{{B}_{t}}\Vert}_{2}<\text{BetaTolerance}$$, then optimization terminates.
If the software converges for the last solver specified in
Solver
, then optimization terminates. Otherwise, the software uses
the next solver specified in Solver
.
Example: 'BetaTolerance',1e6
Data Types: single
 double
NumCheckConvergence
— Number of batches to process before next convergence check
positive integer
Number of batches to process before next convergence check, specified as the
commaseparated pair consisting of 'NumCheckConvergence'
and a
positive integer.
To specify the batch size, see BatchSize
.
The software checks for convergence about 10 times per pass through the entire data set by default.
Example: 'NumCheckConvergence',100
Data Types: single
 double
PassLimit
— Maximal number of passes
1
(default)  positive integer
Maximal number of passes through the data, specified as the commaseparated pair consisting of 'PassLimit'
and a positive integer.
The software processes all observations when it completes one pass through the data.
When the software passes through the data PassLimit
times, it terminates optimization.
If you specify BatchLimit
, then
templateLinear
uses the argument that results in
processing the fewest observations, either
BatchLimit
or
PassLimit
.
Example: 'PassLimit',5
Data Types: single
 double
BetaTolerance
— Relative tolerance on linear coefficients and bias term
1e4
(default)  nonnegative scalar
Relative tolerance on the linear coefficients and the bias term (intercept), specified
as the commaseparated pair consisting of 'BetaTolerance'
and a
nonnegative scalar.
Let $${B}_{t}=\left[{\beta}_{t}{}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{t}\right]$$, that is, the vector of the coefficients and the bias term at optimization iteration t. If $${\Vert \frac{{B}_{t}{B}_{t1}}{{B}_{t}}\Vert}_{2}<\text{BetaTolerance}$$, then optimization terminates.
If you also specify DeltaGradientTolerance
, then optimization
terminates when the software satisfies either stopping criterion.
If the software converges for the last solver specified in
Solver
, then optimization terminates. Otherwise, the software uses
the next solver specified in Solver
.
Example: 'BetaTolerance',1e6
Data Types: single
 double
DeltaGradientTolerance
— Gradientdifference tolerance
1
(default)  nonnegative scalar
Gradientdifference tolerance between upper and lower pool KarushKuhnTucker
(KKT) complementarity conditions violators, specified as the
commaseparated pair consisting of 'DeltaGradientTolerance'
and
a nonnegative scalar.
If the magnitude of the KKT violators is less than
DeltaGradientTolerance
, then the software terminates optimization.If the software converges for the last solver specified in
Solver
, then optimization terminates. Otherwise, the software uses the next solver specified inSolver
.
Example: 'DeltaGapTolerance',1e2
Data Types: double
 single
NumCheckConvergence
— Number of passes through entire data set to process before next convergence check
5
(default)  positive integer
Number of passes through entire data set to process before next convergence check,
specified as the commaseparated pair consisting of
'NumCheckConvergence'
and a positive integer.
Example: 'NumCheckConvergence',100
Data Types: single
 double
PassLimit
— Maximal number of passes
10
(default)  positive integer
Maximal number of passes through the data, specified as the
commaseparated pair consisting of 'PassLimit'
and
a positive integer.
When the software completes one pass through the data, it has processed all observations.
When the software passes through the data PassLimit
times,
it terminates optimization.
Example: 'PassLimit',5
Data Types: single
 double
BetaTolerance
— Relative tolerance on linear coefficients and bias term
1e4
(default)  nonnegative scalar
Relative tolerance on the linear coefficients and the bias term (intercept), specified as the commaseparated pair consisting of 'BetaTolerance'
and a nonnegative scalar.
Let $${B}_{t}=\left[{\beta}_{t}{}^{\prime}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{t}\right]$$, that is, the vector of the coefficients and the bias term at optimization iteration t. If $${\Vert \frac{{B}_{t}{B}_{t1}}{{B}_{t}}\Vert}_{2}<\text{BetaTolerance}$$, then optimization terminates.
If you also specify GradientTolerance
, then optimization terminates when the software satisfies either stopping criterion.
If the software converges for the last solver specified in
Solver
, then optimization terminates. Otherwise, the software uses
the next solver specified in Solver
.
Example: 'BetaTolerance',1e6
Data Types: single
 double
GradientTolerance
— Absolute gradient tolerance
1e6
(default)  nonnegative scalar
Absolute gradient tolerance, specified as the commaseparated pair consisting of 'GradientTolerance'
and a nonnegative scalar.
Let $$\nabla {\mathcal{L}}_{t}$$ be the gradient vector of the objective function with respect to the coefficients and bias term at optimization iteration t. If $${\Vert \nabla {\mathcal{L}}_{t}\Vert}_{\infty}=\mathrm{max}\left\nabla {\mathcal{L}}_{t}\right<\text{GradientTolerance}$$, then optimization terminates.
If you also specify BetaTolerance
, then optimization terminates when the
software satisfies either stopping criterion.
If the software converges for the last solver specified in the
software, then optimization terminates. Otherwise, the software uses
the next solver specified in Solver
.
Example: 'GradientTolerance',1e5
Data Types: single
 double
HessianHistorySize
— Size of history buffer for Hessian approximation
15
(default)  positive integer
Size of history buffer for Hessian approximation, specified
as the commaseparated pair consisting of 'HessianHistorySize'
and
a positive integer. That is, at each iteration, the software composes
the Hessian using statistics from the latest HessianHistorySize
iterations.
The software does not support 'HessianHistorySize'
for
SpaRSA.
Example: 'HessianHistorySize',10
Data Types: single
 double
IterationLimit
— Maximal number of optimization iterations
1000
(default)  positive integer
Maximal number of optimization iterations, specified as the
commaseparated pair consisting of 'IterationLimit'
and
a positive integer. IterationLimit
applies to these
values of Solver
: 'bfgs'
, 'lbfgs'
,
and 'sparsa'
.
Example: 'IterationLimit',500
Data Types: single
 double
Output Arguments
t
— Linear classification model learner template
template object
Linear classification model learner template, returned as a template object. To train a linear classification model using highdimensional data for multiclass problems, pass t
to fitcecoc
.
If you display t
to the Command Window, then all, unspecified options appear empty ([]
). However, the software replaces empty options with their corresponding default values during training.
More About
Warm Start
A warm start is initial estimates of the beta coefficients and bias term supplied to an optimization routine for quicker convergence.
Tips
It is a best practice to orient your predictor matrix so that observations correspond to columns and to specify
'ObservationsIn','columns'
. As a result, you can experience a significant reduction in optimizationexecution time.If the predictor data has few observations, but many predictor variables, then:
Specify
'PostFitBias',true
.For SGD or ASGD solvers, set
PassLimit
to a positive integer that is greater than 1, for example, 5 or 10. This setting often results in better accuracy.
For SGD and ASGD solvers,
BatchSize
affects the rate of convergence.If
BatchSize
is too small, then the software achieves the minimum in many iterations, but computes the gradient per iteration quickly.If
BatchSize
is too large, then the software achieves the minimum in fewer iterations, but computes the gradient per iteration slowly.
Large learning rate (see
LearnRate
) speedup convergence to the minimum, but can lead to divergence (that is, overstepping the minimum). Small learning rates ensure convergence to the minimum, but can lead to slow termination.If
Regularization
is'lasso'
, then experiment with various values ofTruncationPeriod
. For example, setTruncationPeriod
to1
,10
, and then100
.For efficiency, the software does not standardize predictor data. To standardize the predictor data (
X
), enterX = bsxfun(@rdivide,bsxfun(@minus,X,mean(X,2)),std(X,0,2));
The code requires that you orient the predictors and observations as the rows and columns of
X
, respectively. Also, for memoryusage economy, the code replaces the original predictor data the standardized data.
References
[1] Hsieh, C. J., K. W. Chang, C. J. Lin, S. S. Keerthi, and S. Sundararajan. “A Dual Coordinate Descent Method for LargeScale Linear SVM.” Proceedings of the 25th International Conference on Machine Learning, ICML ’08, 2001, pp. 408–415.
[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] Nocedal, J. and S. J. Wright. Numerical Optimization, 2nd ed., New York: Springer, 2006.
[4] 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.
[5] Wright, S. J., R. D. Nowak, and M. A. T. Figueiredo. “Sparse Reconstruction by Separable Approximation.” Trans. Sig. Proc., Vol. 57, No 7, 2009, pp. 2479–2493.
[6] Xiao, Lin. “Dual Averaging Methods for Regularized Stochastic Learning and Online Optimization.” J. Mach. Learn. Res., Vol. 11, 2010, pp. 2543–2596.
[7] Xu, Wei. “Towards Optimal One Pass Large Scale Learning with Averaged Stochastic Gradient Descent.” CoRR, abs/1107.2490, 2011.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
Usage notes and limitations when you train a model by passing a linear model template and tall arrays to fitcecoc
:
The default values for these namevalue pair arguments are different when you work with tall arrays.
'Lambda'
— Can be'auto'
(default) or a scalar'Regularization'
— Supports only'ridge'
'Solver'
— Supports only'lbfgs'
'FitBias'
— Supports onlytrue
'Verbose'
— Default value is1
'BetaTolerance'
— Default value is relaxed to1e–3
'GradientTolerance'
— Default value is relaxed to1e–3
'IterationLimit'
— Default value is relaxed to20
When
fitcecoc
uses atemplateLinear
object with tall arrays, the only available solver is LBFGS. The software implements LBFGS by distributing the calculation of the loss and gradient among different parts of the tall array at each iteration. If you do not specify initial values forBeta
andBias
, the software refines the initial estimates of the parameters by fitting the model locally to parts of the data and combining the coefficients by averaging.
For more information, see Tall Arrays.
Version History
See Also
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