lbfgsState

State of limited-memory BFGS (L-BFGS) solver

Since R2023a

Description

An lbfgsState object stores information about steps in the L-BFGS algorithm.

The L-BFGS algorithm [1] is a quasi-Newton method that approximates the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. The L-BFGS algorithm is best suited for small networks and data sets that you can process in a single batch.

Use lbfgsState objects in conjunction with the lbfgsupdate function to train a neural network using the L-BFGS algorithm.

Creation

Syntax

solverState = lbfgsState

solverState = lbfgsState(Name=Value)

Description

solverState = lbfgsState creates an L-BFGS state object with a history size of 10 and an initial inverse Hessian factor of 1.

solverState = lbfgsState(Name=Value) sets the HistorySize and InitialInverseHessianFactor properties using one or more name-value arguments.

Properties

L-BFGS State

HistorySize — Number of state updates to store

10 (default) | positive integer

Number of state updates to store, specified as a positive integer. Values between 3 and 20 suit most tasks.

The L-BFGS algorithm uses a history of gradient calculations to approximate the Hessian matrix recursively. For more information, see Limited-Memory BFGS.

After creating the lbfgsState object, this property is read-only.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

InitialInverseHessianFactor — Initial value that characterizes approximate inverse Hessian matrix

1 (default) | positive scalar

This property is read-only.

Initial value that characterizes the approximate inverse Hessian matrix, specified as a positive scalar.

To save memory, the L-BFGS algorithm does not store and invert the dense Hessian matrix B. Instead, the algorithm uses the approximation $$B_{k-m}^{-1}\approx {\lambda }_{k}I$$, where m is the history size, the inverse Hessian factor $${\lambda }_k$$ is a scalar, and I is the identity matrix, and stores the scalar inverse Hessian factor only. The algorithm updates the inverse Hessian factor at each step.

The InitialInverseHessianFactor property is the value of $${\lambda }_0$$.

For more information, see Limited-Memory BFGS.

After creating the lbfgsState object, this property is read-only.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

InverseHessianFactor — Value that characterizes approximate inverse Hessian matrix

1 (default) | positive scalar

Value that characterizes the approximate inverse Hessian matrix, specified as a positive scalar.

To save memory, the L-BFGS algorithm does not store and invert the dense Hessian matrix B. Instead, the algorithm uses the approximation $$B_{k-m}^{-1}\approx {\lambda }_{k}I$$, where m is the history size, the inverse Hessian factor $${\lambda }_k$$ is a scalar, and I is the identity matrix, and stores the scalar inverse Hessian factor only. The algorithm updates the inverse Hessian factor at each step.

For more information, see Limited-Memory BFGS.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

StepHistory — Step history

{} (default) | cell array

Step history, specified as a cell array.

The L-BFGS algorithm uses a history of gradient calculations to approximate the Hessian matrix recursively. For more information, see Limited-Memory BFGS.

Data Types: cell

GradientsDifferenceHistory — Gradients difference history

{} (default) | cell array

Gradients difference history, specified as a cell array.

The L-BFGS algorithm uses a history of gradient calculations to approximate the Hessian matrix recursively. For more information, see Limited-Memory BFGS.

Data Types: cell

HistoryIndices — History indices

0-by-1 vector (default) | row vector

History indices, specified as a row vector.

HistoryIndices is a 1-by-HistorySize vector, where StepHistory(i) and GradientsDifferenceHistory(i) correspond to iteration HistoryIndices(i).

For more information, see Limited-Memory BFGS.

Data Types: double

Iteration Information

Loss — Loss

[] (default) | dlarray scalar | numeric scalar

This property is read-only.

Loss, specified as a dlarray scalar, a numeric scalar, or [].

If the state object is the output of the lbfgsupdate function, then Loss is the first output of the loss function that you pass to the lbfgsupdate function. Otherwise, Loss is [].

Gradients — Gradients

[] (default) | dlarray object | numeric array | cell array | structure | table

This property is read-only.

Gradients, specified as a dlarray object, a numeric array, a cell array, a structure, a table, or [].

If the state object is the output of the lbfgsupdate function, then Gradients is the second output of the loss function that you pass to the lbfgsupdate function. Otherwise, Gradients is [].

AdditionalLossFunctionOutputs — Additional loss function outputs

1-by-0 cell array (default) | cell array

This property is read-only.

Additional loss function outputs, specified as a cell array.

If the state object is the output of the lbfgsupdate function, then AdditionalLossFunctionOutputs is a cell array containing additional outputs of the loss function that you pass to the lbfgsupdate function. Otherwise, AdditionalLossFunctionOutputs is a 1-by-0 cell array.

Data Types: cell

StepNorm — Norm of step

[] (default) | dlarray scalar | numeric scalar

This property is read-only.

Norm of the step, specified as a dlarray scalar, numeric scalar, or [].

If the state object is the output of the lbfgsupdate function, then StepNorm is the norm of the step that the lbfgsupdate function calculates. Otherwise, StepNorm is [].

GradientsNorm — Norm of gradients

[] (default) | dlarray scalar | numeric scalar

This property is read-only.

Norm of the gradients, specified as a dlarray scalar, a numeric scalar, or [].

If the state object is the output of the lbfgsupdate function, then GradientsNorm is the norm of the second output of the loss function that you pass to the lbfgsupdate function. Otherwise, GradientsNorm is [].

LineSearchStatus — Status of line search algorithm

"" (default) | "completed" | "failed"

This property is read-only.

Status of the line search algorithm, specified as "", "completed", or "failed".

If the state object is the output of the lbfgsupdate function, then LineSearchStatus is one of these values:

• "completed" — The algorithm finds a learning rate that satisfies the LineSearchMethod and MaxNumLineSearchIterations options that the lbfgsupdate function uses.

• "failed" — The algorithm fails to find a learning rate that satisfies the LineSearchMethod and MaxNumLineSearchIterations options that the lbfgsupdate function uses.

Otherwise, LineSearchStatus is "".

LineSearchMethod — Method solver uses to find suitable learning rate

"" (default) | "weak-wolfe" | "strong-wolfe" | "backtracking"

This property is read-only.

Method solver uses to find a suitable learning rate, specified as "weak-wolfe", "strong-wolfe", "backtracking", or [].

If the state object is the output of the lbfgsupdate function, then LineSearchMethod is the line search method that the lbfgsupdate function uses. Otherwise, LineSearchMethod is "".

MaxNumLineSearchIterations — Maximum number of line search iterations

0 (default) | nonnegative integer

This property is read-only.

Maximum number of line search iterations, specified as a nonnegative integer.

If the state object is the output of the lbfgsupdate function, then MaxNumLineSearchIterations is the maximum number of line search iterations that the lbfgsupdate function uses. Otherwise, MaxNumLineSearchIterations is 0.

Data Types: double

Examples

Create L-BFGS Solver State Object

Create an L-BFGS solver state object.

solverState = lbfgsState

solverState = 
  LBFGSState with properties:

             InverseHessianFactor: 1
                      StepHistory: {}
       GradientsDifferenceHistory: {}
                   HistoryIndices: [1x0 double]

   Iteration Information
                             Loss: []
                        Gradients: []
    AdditionalLossFunctionOutputs: {1x0 cell}
                    GradientsNorm: []
                         StepNorm: []
                 LineSearchStatus: ""

  Show all properties

Update Learnable Parameters in Neural Network

Load the iris flower dataset.

[XTrain, TTrain] = iris_dataset;

Convert the predictors to a dlarray object with format "CB" (channel, batch).

XTrain = dlarray(XTrain,"CB");

Define the network architecture.

numInputFeatures = size(XTrain,1);
numClasses = size(TTrain,1);
numHiddenUnits = 32;

layers = [
    featureInputLayer(numInputFeatures)
    fullyConnectedLayer(numHiddenUnits)
    reluLayer
    fullyConnectedLayer(numHiddenUnits)
    reluLayer
    fullyConnectedLayer(numClasses)
    softmaxLayer];

net = dlnetwork(layers);

Define the modelLoss function, listed in the Model Loss Function section of the example. This function takes as input a neural network, input data, and targets. The function returns the loss and the gradients of the loss with respect to the network learnable parameters.

The lbfgsupdate function requires a loss function with the syntax [loss,gradients] = f(net). Create a variable that parameterizes the evaluated modelLoss function to take a single input argument.

lossFcn = @(net) dlfeval(@modelLoss,net,XTrain,TTrain);

Initialize an L-BFGS solver state object with a maximum history size of 3 and an initial inverse Hessian approximation factor of 1.1.

solverState = lbfgsState( ...
    HistorySize=3, ...
    InitialInverseHessianFactor=1.1);

Train the network for 10 epochs.

numEpochs = 10;
for i = 1:numEpochs
    [net, solverState] = lbfgsupdate(net,lossFcn,solverState);
end

Model Loss Function

The modelLoss function takes as input a neural network net, input data X, and targets T. The function returns the loss and the gradients of the loss with respect to the network learnable parameters.

function [loss, gradients] = modelLoss(net, X, T)

Y = forward(net,X);
loss = crossentropy(Y,T);
gradients = dlgradient(loss,net.Learnables);

end

Algorithms

Limited-Memory BFGS

The L-BFGS algorithm [1] is a quasi-Newton method that approximates the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. The L-BFGS algorithm is best suited for small networks and data sets that you can process in a single batch.

The algorithm updates learnable parameters W at iteration k+1 using the update step given by

$$W_{k+1}=W_k-{\eta }_k{B}_k^{-1}\nabla J(W_k),$$

where W_k denotes the weights at iteration k, $${\eta }_k$$ is the learning rate at iteration k, B_k is an approximation of the Hessian matrix at iteration k, and $$\nabla J(W_k)$$ denotes the gradients of the loss with respect to the learnable parameters at iteration k.

The L-BFGS algorithm computes the matrix-vector product $$B_k^{-1}\nabla J(W_k)$$ directly. The algorithm does not require computing the inverse of B_k.

To save memory, the L-BFGS algorithm does not store and invert the dense Hessian matrix B. Instead, the algorithm uses the approximation $$B_{k-m}^{-1}\approx {\lambda }_{k}I$$, where m is the history size, the inverse Hessian factor $${\lambda }_k$$ is a scalar, and I is the identity matrix, and stores the scalar inverse Hessian factor only. The algorithm updates the inverse Hessian factor at each step.

To compute the matrix-vector product $$B_k^{-1}\nabla J(W_k)$$ directly, the L-BFGS algorithm uses this recursive algorithm:

1. Set $$r=B_{k-m}^{-1}\nabla J(W_k)$$, where m is the history size.

2. For $$i=m,\dots ,1$$:

   1. Let $$\beta =\frac{1}{s_{k-i}^{\top }{y}_{k-i}}{y}_{k-i}^{\top }r$$, where $$s_{k-i}$$ and $$y_{k-i}$$ are the step and gradient differences for iteration $$k-i$$, respectively.

   2. Set $$r=r+\text{}{s}_{k-i}\text{}\left(a_{k-i}-\beta \right)$$, where $$a$$ is derived from $$s$$, $$y$$, and the gradients of the loss with respect to the loss function. For more information, see [1].

3. Return $$B_k^{-1}\nabla J(W_k)=r$$.

References

[1] Liu, Dong C., and Jorge Nocedal. "On the limited memory BFGS method for large scale optimization." Mathematical programming 45, no. 1 (August 1989): 503-528. https://doi.org/10.1007/BF01589116.

Version History

Introduced in R2023a