nssTrainingLBFGS

L-BFGS training options object for neural state-space systems

Since R2024b

Description

L-BFGS options set object to train an idNeuralStateSpace network using nlssest.

Creation

Create an nssTrainingLBFGS object using nssTrainingOptions and specifying "lbfgs" as input argument.

Properties

UpdateMethod — Solver used to update network parameters

"LBFGS" (default)

Solver used to update network parameters, returned as a string. This property is read-only.

Use nssTrainingOptions("adam"), nssTrainingOptions("sgdm"), or nssTrainingOptions("rmsprop") to return an options set object for the Adam, SGDM, or RMSProp solvers respectively. For more information on these algorithms, see the Algorithms section of trainingOptions (Deep Learning Toolbox).

MaxIterations — Maximum number of iterations

100 (default) | positive integer

Maximum number of iterations to use for training, specified as a positive integer.

The L-BFGS solver is a full-batch solver, which means that it processes the entire training set in a single iteration.

LineSearchMethod — Method to find suitable learning rate

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

Method to find suitable learning rate, specified as one of these values:

• "weak-wolfe" — Search for a learning rate that satisfies the weak Wolfe conditions. This method maintains a positive definite approximation of the inverse Hessian matrix.

• "strong-wolfe" — Search for a learning rate that satisfies the strong Wolfe conditions. This method maintains a positive definite approximation of the inverse Hessian matrix.

• "backtracking" — Search for a learning rate that satisfies sufficient decrease conditions. This method does not maintain a positive definite approximation of the inverse Hessian matrix.

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 (Deep Learning Toolbox).

InitialInverseHessianFactor — Initial value that characterizes approximate inverse Hessian matrix

1 (default) | positive scalar

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. The algorithm then stores the scalar inverse Hessian factor only. The algorithm updates the inverse Hessian factor at each step.

The initial inverse hessian factor is the value of $${\lambda }_0$$.

For more information, see Limited-Memory BFGS (Deep Learning Toolbox).

MaxNumLineSearchIterations — Maximum number of line search iterations

20 (default) | positive integer

Maximum number of line search iterations to determine the learning rate, specified as a positive integer.

GradientTolerance — Relative gradient tolerance

1e-6 (default) | positive scalar

Relative gradient tolerance, specified as a positive scalar.

The software stops training when the relative gradient is less than or equal to GradientTolerance.

StepTolerance — Step size tolerance

1e-6 (default) | positive scalar

Step size tolerance, specified as a positive scalar.

The software stops training when the step that the algorithm takes is less than or equal to StepTolerance.

LossFcn — Type of function used to calculate loss

"MeanAbsoluteError" (default) | "MeanSquaredError"

Type of function used to calculate loss, specified as one of the following:

• "MeanAbsoluteError" — uses the mean value of the absolute error.

• "MeanSquaredError" — uses the mean value of the squared error.

PlotLossFcn — Option to plot the value of the loss function during training

true (default) | false

Option to plot the value of the loss function during training, specified as one of the following:

• true — plots the value of the loss function during training.

• false — does not plot the value of the loss function during training.

Lambda — Loss function regularization constant

0 (default) | positive scalar

Constant coefficient applied to the regularization term added to the loss function, specified as a positive scalar.

The loss function with the regularization term is given by:

${\widehat{V}}_N\left(\theta \right)=\frac{1}{N}{\displaystyle \sum _{t=1}^N{\epsilon }^2\left(t,\theta \right)+ \frac{1}{N}\lambda {\Vert \theta \Vert }^2}$

where t is the time variable, N is the size of the batch, ε is the sum of the reconstruction loss and autoencoder loss, θ is a concatenated vector of weights and biases of the neural network, and λ is the regularization constant that you can tune.

For more information, see Regularized Estimates of Model Parameters.

Beta — Coefficient applied to tune the reconstruction loss of an autoencoder

0 (default) | nonnegative scalar

Coefficient applied to tune the reconstruction loss of an autoencoder, specified as a nonnegative scalar.

Reconstruction loss measures the difference between the original input (x) and its reconstruction (xr) after encoding and decoding. You calculate this loss as the L2 norm of (x - xr) divided by the batch size (N).

WindowSize — Size of data frames

intmax (default) | positive integer

Number of samples in each frame or batch when segmenting data for model training, specified as a positive integer.

Overlap — Size of overlap

0 (default) | integer

Number of samples in the overlap between successive frames when segmenting data for model training, specified as an integer. A negative integer indicates that certain data samples are skipped when creating the data frames.

ODESolverOptions — ODE solver options for continuous-time systems

nssDLODE45 (default)

ODE solver options to integrate continuous-time neural state-space systems, specified as an nssDLODE45 object.

Use dot notation to access properties such as the following:

• Solver — Solver type, set as "dlode45". This is a read-only property.

• InitialStepSize — Initial step size, specified as a positive scalar. If you do not specify an initial step size, then the solver bases the initial step size on the slope of the solution at the initial time point.

• MaxStepSize — Maximum step size, specified as a positive scalar. It is an upper bound on the size of any step taken by the solver. The default is one tenth of the difference between final and initial time.

• AbsoluteTolerance — Absolute tolerance, specified as a positive scalar. It is the largest allowable absolute error. Intuitively, when the solution approaches 0, AbsoluteTolerance is the threshold below which you do not worry about the accuracy of the solution since it is effectively 0.

• RelativeTolerance — Relative tolerance, specified as a positive scalar. This tolerance measures the error relative to the magnitude of each solution component. Intuitively, it controls the number of significant digits in a solution, (except when it is smaller than the absolute tolerance).

For more information, see odeset.

InputInterSample — Input interpolation method

'foh' (default) | 'zoh' | 'spline' | 'cubic' | 'makima' | 'pchip'

Input interpolation method, specified as one of the following:

• 'zoh' — uses zero-order hold interpolation method.

• 'foh' — uses first-order hold interpolation method.

• 'cubic' — uses cubic interpolation method.

• 'makima' — uses modified Akima interpolation method.

• 'pchip' — uses shape-preserving piecewise cubic interpolation method.

• 'spline' — uses spline interpolation method.

This is the interpolation method used to interpolate the input when integrating continuous-time neural state-space systems. For more information, see interpolation methods in interp1.

Object Functions

Examples

Create L-BFGS Option Set to Train a Neural State-Space System

Use nssTrainingOptions to return an options set object to train an idNeuralStateSpace system.

lbfgsOpts = nssTrainingOptions("lbfgs")

lbfgsOpts = 
  nssTrainingLBFGS with properties:

                   UpdateMethod: "LBFGS"
               LineSearchMethod: "weak-wolfe"
                  MaxIterations: 100
     MaxNumLineSearchIterations: 20
                    HistorySize: 10
    InitialInverseHessianFactor: 1
              GradientTolerance: 1.0000e-06
                  StepTolerance: 1.0000e-06
                         Lambda: 0
                           Beta: 0
                        LossFcn: "MeanAbsoluteError"
                    PlotLossFcn: 1
               ODESolverOptions: [1x1 idoptions.nssDLODE45]
               InputInterSample: 'foh'
                     WindowSize: 2.1475e+09
                        Overlap: 0

Use dot notation to access the object properties.

lbfgsOpts.PlotLossFcn = false;

You can use lbfgsOpts as an input argument to nlssest to specify the training options for the state or the non-trivial output network of an idNeuralStateSpace object.

Version History

Introduced in R2024b