TrainingOptionsLBFGS

Training options for limited-memory BFGS (L-BFGS) optimizer

Since R2023b

Description

Use a TrainingOptionsLBFGS object to set training options for the limited-memory BFGS (L-BFGS) optimizer, including line search method and gradient and step tolerances.

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

Creation

Create a TrainingOptionsLBFGS object by using the trainingOptions function and specifying "lbfgs" as the first input argument.

Properties

L-BFGS

MaxIterations — Maximum number of iterations

1000 (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.

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

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.

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

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.

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

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.

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

GradientTolerance — Relative gradient tolerance

1e-5 (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.

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

StepTolerance — Step size tolerance

1e-5 (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.

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

InitialStepSize — Initial step size

[] (default) | "auto" | real finite scalar

Since R2024b

Initial step size, specified as one of these values:

• [] — Do not use an initial step size to determine the initial Hessian approximation.

• "auto" — Determine the initial step size automatically. The software uses an initial step size of $$\Vert s_0{\Vert }_{\infty }=\frac{1}{2}\Vert W_0{\Vert }_{\infty }+0.1$$, where W₀ are the initial learnable parameters of the network.

• Positive real scalar — Use the specified value as the initial step size $$\Vert s_0{\Vert }_{\infty }$$.

If InitialStepSize is "auto" or a positive real scalar, then the software approximates the initial inverse Hessian using $${\lambda }_0=\frac{\Vert s_0{\Vert }_{\infty }}{\Vert \nabla J(W_0){\Vert }_{\infty }}$$, where λ₀ is the initial inverse Hessian factor and $$\nabla J(W_0)$$ denotes the gradients of the loss with respect to the initial learnable parameters. For more information, see Limited-Memory BFGS.

Data Formats

InputDataFormats — Description of input data dimensions

"auto" (default) | string array | cell array of character vectors | character vector

Description of the input data dimensions, specified as a string array, character vector, or cell array of character vectors.

If InputDataFormats is "auto", then the software uses the formats expected by the network input. Otherwise, the software uses the specified formats for the corresponding network input.

A data format is a string of characters, where each character describes the type of the corresponding data dimension.

The characters are:

• "S" — Spatial

• "C" — Channel

• "B" — Batch

• "T" — Time

• "U" — Unspecified

For example, consider an array containing a batch of sequences where the first, second, and third dimensions correspond to channels, observations, and time steps, respectively. You can specify that this array has the format "CBT" (channel, batch, time).

You can specify multiple dimensions labeled "S" or "U". You can use the labels "C", "B", and "T" once each, at most. The software ignores singleton trailing "U" dimensions after the second dimension.

For a neural networks with multiple inputs net, specify an array of input data formats, where InputDataFormats(i) corresponds to the input net.InputNames(i).

For more information, see Deep Learning Data Formats.

Data Types: char | string | cell

TargetDataFormats — Description of target data dimensions

"auto" (default) | string array | cell array of character vectors | character vector

Description of the target data dimensions, specified as one of these values:

• "auto" — If the target data has the same number of dimensions as the input data, then the trainnet function uses the format specified by InputDataFormats. If the target data has a different number of dimensions to the input data, then the trainnet function uses the format expected by the loss function.

• String array, character vector, or cell array of character vectors — The trainnet function uses the data formats you specify.

A data format is a string of characters, where each character describes the type of the corresponding data dimension.

The characters are:

• "S" — Spatial

• "C" — Channel

• "B" — Batch

• "T" — Time

• "U" — Unspecified

For example, consider an array containing a batch of sequences where the first, second, and third dimensions correspond to channels, observations, and time steps, respectively. You can specify that this array has the format "CBT" (channel, batch, time).

You can specify multiple dimensions labeled "S" or "U". You can use the labels "C", "B", and "T" once each, at most. The software ignores singleton trailing "U" dimensions after the second dimension.

For more information, see Deep Learning Data Formats.

Data Types: char | string | cell

Monitoring

Plots — Plots to display during neural network training

"none" (default) | "training-progress"

Plots to display during neural network training, specified as one of these values:

• "none" — Do not display plots during training.

• "training-progress" — Plot training progress.

The plot shows the training and validation loss, training and validation metrics specified by the Metrics property, and additional information about the training progress.

To programmatically open and close the training progress plot after training, use the show and close functions with the second output of the trainnet function. You can use the show function to view the training progress even if the Plots training option is specified as "none".

To switch the y-axis scale to logarithmic, use the axes toolbar.

For more information about the plot, see Monitor Deep Learning Training Progress.

Metrics — Metrics to monitor

[] (default) | character vector | string array | function handle | deep.DifferentiableFunction object (since R2024a) | cell array | metric object

Metrics to monitor, specified as one of these values:

• Built-in metric or loss function name — Specify metrics as a string scalar, character vector, or a cell array or string array of one or more of these names:

  • Metrics:

    • "accuracy" — Accuracy (also known as top-1 accuracy)

    • "auc" — Area under ROC curve (AUC)

    • "fscore" — F-score (also known as F₁-score)

    • "precision" — Precision

    • "recall" — Recall

    • "rmse" — Root mean squared error

    • "mape" — Mean absolute percentage error (MAPE) (since R2024b)

  • Loss functions:

    • "crossentropy" — Cross-entropy loss for classification tasks. (since R2024b)

    • "indexcrossentropy" — Index cross-entropy loss for classification tasks. (since R2024b)

    • "binary-crossentropy" — Binary cross-entropy loss for binary and multilabel classification tasks. (since R2024b)

    • "mae" / "mean-absolute-error" / "l1loss" — Mean absolute error for regression tasks. (since R2024b)

    • "mse" / "mean-squared-error" / "l2loss" — Mean squared error for regression tasks. (since R2024b)

    • "huber" — Huber loss for regression tasks (since R2024b)

  Note that setting the loss function as "crossentropy" and specifying "index-crossentropy" as a metric or setting the loss function as "index-crossentropy" and specifying "crossentropy" as a metric is not supported.

• Built-in metric object — If you need more flexibility, you can use built-in metric objects. The software supports these built-in metric objects:

  • AccuracyMetric

  • AUCMetric

  • FScoreMetric

  • PrecisionMetric

  • RecallMetric

  • RMSEMetric

  When you create a built-in metric object, you can specify additional options such as the averaging type and whether the task is single-label or multilabel.

• Custom metric function handle — If the metric you need is not a built-in metric, then you can specify custom metrics using a function handle. The function must have the syntax metric = metricFunction(Y,T), where Y corresponds to the network predictions and T corresponds to the target responses. For networks with multiple outputs, the syntax must be metric = metricFunction(Y1,…,YN,T1,…TM), where N is the number of outputs and M is the number of targets. For more information, see Define Custom Metric Function.

• deep.DifferentiableFunction object (since R2024a) — Function object with custom backward function. For more information, see Define Custom Deep Learning Operations.

• Custom metric object — If you need greater customization, then you can define your own custom metric object. For an example that shows how to create a custom metric, see Define Custom Metric Object. For general information about creating custom metrics, see Define Custom Deep Learning Metric Object. Specify your custom metric as the Metrics option of the trainingOptions function.

If you specify a metric as a function handle, a deep.DifferentiableFunction object, or a custom metric object and train the neural network using the trainnet function, then the layout of the targets that the software passes to the metric depends on the data type of the targets, and the loss function that you specify in the trainnet function and the other metrics that you specify:

• If the targets are numeric arrays, then the software passes the targets to the metric directly.

• If the loss function is "index-crossentropy" and the targets are categorical arrays, then the software automatically converts the targets to numeric class indices and passes them to the metric.

• For other loss functions, if the targets are categorical arrays, then the software automatically converts the targets to one-hot encoded vectors and then passes them to the metric.

Example: Metrics=["accuracy","fscore"]

Example: Metrics=["accuracy",@myFunction,precisionObj]

ObjectiveMetricName — Name of objective metric

"loss" (default) | string scalar | character vector

Since R2024a

Name of objective metric to use for early stopping and returning the best network, specified as a string scalar or character vector.

The metric name must be "loss" or match the name of a metric specified by the Metrics argument. Metrics specified using function handles are not supported. To specify the ObjectiveMetricName value as the name of a custom metric, the value of the Maximize property of the custom metric object must be nonempty. For more information, see Define Custom Deep Learning Metric Object.

For more information about specifying the objective metric for early stopping, see ValidationPatience. For more information about returning the best network using the objective metric, see OutputNetwork.

Data Types: char | string

Verbose — Flag to display training progress information

1 (true) (default) | 0 (false)

Flag to display training progress information in the command window, specified as 1 (true) or 0 (false).

When this property is 1 (true), the software displays this information:

Variable	Description
Iteration	Iteration number.
TimeElapsed	Time elapsed in hours, minutes, and seconds.
TrainingLoss	Training loss.
ValidationLoss	Validation loss. If you do not specify validation data, then the software does not display this information.
GradientNorm	Norm of the gradients.
StepNorm	Norm of the steps.

If you specify additional metrics in the training options, then they also appear in the verbose output. For example, if you set the Metrics training option to "accuracy", then the information includes the TrainingAccuracy and ValidationAccuracy variables.

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

VerboseFrequency — Frequency of verbose printing

50 (default) | positive integer

Frequency of verbose printing, which is the number of iterations between printing to the Command Window, specified as a positive integer.

If you validate the neural network during training, then the software also prints to the command window every time validation occurs.

To enable this property, set the Verbose training option to 1 (true).

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

OutputFcn — Output functions

function handle | cell array of function handles

Output functions to call during training, specified as a function handle or cell array of function handles. The software calls the functions once before the start of training, after each iteration, and once when training is complete.

The functions must have the syntax stopFlag = f(info), where info is a structure containing information about the training progress, and stopFlag is a scalar that indicates to stop training early. If stopFlag is 1 (true), then the software stops training. Otherwise, the software continues training.

The trainnet function passes the output function the structure info that contains these fields:

Field	Description
Iteration	Iteration number
TimeElapsed	Time elapsed in hours, minutes, and seconds
TrainingLoss	Training loss
ValidationLoss	Validation loss. If you do not specify validation data, then the software does not display this information.
GradientNorm	Norm of the gradients
StepNorm	Norm of the steps
State	Iteration training state, specified as "start", "iteration", or "done".

If you specify additional metrics in the training options, then they also appear in the training information. For example, if you set the Metrics training option to "accuracy", then the information includes the TrainingAccuracy and ValidationAccuracy fields.

If a field is not calculated or relevant for a certain call to the output functions, then that field contains an empty array.

For an example showing how to use output functions, see Custom Stopping Criteria for Deep Learning Training.

Data Types: function_handle | cell

Validation

ValidationData — Data to use for validation during training

[] (default) | datastore | cell array | minibatchqueue object (since R2024a)

Data to use for validation during training, specified as [], a datastore, a table, a cell array, or a minibatchqueue object that contains the validation predictors and targets.

During training, the software uses the validation data to calculate the validation loss and metric values. To specify the validation frequency, use the ValidationFrequency training option. You can also use the validation data to stop training automatically when the validation objective metric stops improving. By default, the objective metric is set to the loss. To turn on automatic validation stopping, use the ValidationPatience training option.

If ValidationData is [], then the software does not validate the neural network during training.

If your neural network has layers that behave differently during prediction than during training (for example, dropout layers), then the validation loss can be lower than the training loss.

If ValidationData is [], then the software does not validate the neural network during training.

Specify the validation data as a datastore, minibatchqueue object, or the cell array {predictors,targets}, where predictors contains the validation predictors and targets contains the validation targets. Specify the validation predictors and targets using any of the formats supported by the trainnet function.

For more information, see the input arguments of the trainnet function.

ValidationFrequency — Frequency of neural network validation

50 (default) | positive integer

Frequency of neural network validation in number of iterations, specified as a positive integer.

The ValidationFrequency value is the number of iterations between evaluations of validation metrics. To specify validation data, use the ValidationData training option.

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

ValidationPatience — Patience of validation stopping

Inf (default) | positive integer

Patience of validation stopping of neural network training, specified as a positive integer or Inf.

ValidationPatience specifies the number of times that the objective metric on the validation set can be worse than or equal to the previous best value before neural network training stops. If ValidationPatience is Inf, then the values of the validation metric do not cause training to stop early. The software aims to maximize or minimize the metric, as specified by the Maximize property of the metric. When the objective metric is "loss", the software aims to minimize the loss value.

The returned neural network depends on the OutputNetwork training option. To return the neural network with the best validation metric value, set the OutputNetwork training option to "best-validation".

Before R2024a: The software computes the validation patience using the validation loss value.

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

OutputNetwork — Neural network to return when training completes

"auto" (default) | "last-iteration" | "best-validation"

Neural network to return when training completes, specified as one of the following:

• "auto" – Use "best-validation" if ValidationData is specified. Otherwise, use "last-iteration".

• "best-validation" – Return the neural network corresponding to the training iteration with the best validation metric value, where the metric to optimize is specified by the ObjectiveMetricName option. To use this option, you must specify the ValidationData training option.

• "last-iteration" – Return the neural network corresponding to the last training iteration.

Regularization and Normalization

L2Regularization — Factor for L₂ regularization

0.0001 (default) | nonnegative scalar

Factor for L₂ regularization (weight decay), specified as a nonnegative scalar. For more information, see L2 Regularization.

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

ResetInputNormalization — Option to reset input layer normalization

1 (true) (default) | 0 (false)

Option to reset input layer normalization, specified as one of the following:

• 1 (true) — Reset the input layer normalization statistics and recalculate them at training time.

• 0 (false) — Calculate normalization statistics at training time when they are empty.

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

BatchNormalizationStatistics — Mode to evaluate statistics in batch normalization layers

"auto" (default) | "population" | "moving"

Mode to evaluate the statistics in batch normalization layers, specified as one of the following:

• "population" — Use the population statistics. After training, the software finalizes the statistics by passing through the training data once more and uses the resulting mean and variance.

• "moving" — Approximate the statistics during training using a running estimate given by update steps

  $$\begin{array}{l}{\mu }^{*}={\lambda }_{\mu }\widehat{\mu }+(1-{\lambda }_{\mu })\mu \\ {\sigma }^2{}^{*}={\lambda }_{{\sigma }^2}\widehat{{\sigma }^2}\text{}\text{+}\text{}\text{(1-}{\lambda }_{{\sigma }^2})\text{}{\sigma }^2\end{array}$$

  where $${\mu }^{*}$$ and $${\sigma }^2{}^{*}$$ denote the updated mean and variance, respectively, $${\lambda }_{\mu }$$ and $${\lambda }_{{\sigma }^2}$$ denote the mean and variance decay values, respectively, $$\widehat{\mu }$$ and $$\widehat{{\sigma }^2}$$ denote the mean and variance of the layer input, respectively, and $$\mu$$ and $${\sigma }^2$$ denote the latest values of the moving mean and variance values, respectively. After training, the software uses the most recent value of the moving mean and variance statistics. This option supports CPU and single GPU training only.

• "auto" — Use the "moving" option.

Gradient Clipping

GradientThreshold — Gradient threshold

Inf (default) | positive scalar

Gradient threshold, specified as Inf or a positive scalar. If the gradient exceeds the value of GradientThreshold, then the gradient is clipped according to the GradientThresholdMethod training option.

For more information, see Gradient Clipping.

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

GradientThresholdMethod — Gradient threshold method

"l2norm" (default) | "global-l2norm" | "absolute-value"

Gradient threshold method used to clip gradient values that exceed the gradient threshold, specified as one of the following:

• "l2norm" — If the L₂ norm of the gradient of a learnable parameter is larger than GradientThreshold, then scale the gradient so that the L₂ norm equals GradientThreshold.

• "global-l2norm" — If the global L₂ norm, L, is larger than GradientThreshold, then scale all gradients by a factor of GradientThreshold/L. The global L₂ norm considers all learnable parameters.

• "absolute-value" — If the absolute value of an individual partial derivative in the gradient of a learnable parameter is larger than GradientThreshold, then scale the partial derivative to have magnitude equal to GradientThreshold and retain the sign of the partial derivative.

For more information, see Gradient Clipping.

Sequence

SequenceLength — Option to pad or truncate input sequences

"longest" (default) | "shortest"

Option to pad or truncate the input sequences, specified as one of these options:

• "longest" — Pad sequences to have the same length as the longest sequence. This option does not discard any data, though padding can introduce noise to the neural network.

• "shortest" — Truncate sequences to have the same length as the shortest sequence. This option ensures that the function does not add padding, at the cost of discarding data.

To learn more about the effects of padding and truncating the input sequences, see Sequence Padding and Truncation.

SequencePaddingDirection — Direction of padding or truncation

"right" (default) | "left"

Direction of padding or truncation, specified as one of these options:

• "right" — Pad or truncate sequences on the right. The sequences start at the same time step and the software truncates or adds padding to the end of each sequence.

• "left" — Pad or truncate sequences on the left. The software truncates or adds padding to the start of each sequence so that the sequences end at the same time step.

Because recurrent layers process sequence data one time step at a time, when the recurrent layer OutputMode property is "last", any padding in the final time steps can negatively influence the layer output. To pad or truncate sequence data on the left, set the SequencePaddingDirection argument to "left".

For sequence-to-sequence neural networks (when the OutputMode property is "sequence" for each recurrent layer), any padding in the first time steps can negatively influence the predictions for the earlier time steps. To pad or truncate sequence data on the right, set the SequencePaddingDirection option to "right".

To learn more about the effects of padding and truncating sequences, see Sequence Padding and Truncation.

SequencePaddingValue — Value by which to pad input sequences

0 (default) | scalar

Value by which to pad the input sequences, specified as a scalar.

Do not pad sequences with NaN, because doing so can propagate errors through the neural network.

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

Hardware and Acceleration

ExecutionEnvironment — Hardware resource

"auto" (default) | "gpu" | "cpu"

Hardware resource, specified as one of these values:

• "auto" — Use a GPU if one is available. Otherwise, use the CPU.

• "gpu" — Use the GPU. Using a GPU requires a Parallel Computing Toolbox™ license and a supported GPU device. For information about supported devices, see GPU Computing Requirements (Parallel Computing Toolbox). If Parallel Computing Toolbox or a suitable GPU is not available, then the software returns an error.

• "cpu" — Use the CPU.

Acceleration — Performance optimization

"auto" (default) | "none"

Since R2024a

Performance optimization, specified as one of these values:

• "auto" – Automatically apply a number of optimizations suitable for the input network and hardware resources.

• "none" – Disable all optimizations.

Checkpoints

CheckpointPath — Path for saving checkpoint neural networks

"" (default) | string scalar | character vector

Path for saving the checkpoint neural networks, specified as a string scalar or character vector.

• If you do not specify a path (that is, you use the default ""), then the software does not save any checkpoint neural networks.

• If you specify a path, then the software saves checkpoint neural networks to this path and assigns a unique name to each neural network. You can then load any checkpoint neural network and resume training from that neural network.

  If the folder does not exist, then you must first create it before specifying the path for saving the checkpoint neural networks. If the path you specify does not exist, then the software throws an error.

Data Types: char | string

CheckpointFrequency — Frequency of saving checkpoint neural networks

30 (default) | positive integer

Frequency of saving checkpoint neural networks in iterations, specified as a positive integer.

This option only has an effect when CheckpointPath is nonempty.

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

Examples

Create Training Options for the L-BFGS Optimizer

Create a set of options for training a neural network using the L-BFGS optimizer:

• Determine the learn rate using the "strong-wolfe" line search method.

• Stop training when the relative gradient is less than or equal to 1e-5.

• Turn on the training progress plot.

options = trainingOptions("lbfgs", ...
    LineSearchMethod="strong-wolfe", ...
    GradientTolerance=1e-5, ...
    Plots="training-progress")

options = 
  TrainingOptionsLBFGS with properties:

                   MaxIterations: 1000
                     HistorySize: 10
     InitialInverseHessianFactor: 1
                 InitialStepSize: []
                LineSearchMethod: 'strong-wolfe'
      MaxNumLineSearchIterations: 20
               GradientTolerance: 1.0000e-05
                   StepTolerance: 1.0000e-05
                  SequenceLength: 'longest'
             CheckpointFrequency: 30
                L2Regularization: 1.0000e-04
         GradientThresholdMethod: 'l2norm'
               GradientThreshold: Inf
                         Verbose: 1
                VerboseFrequency: 50
                  ValidationData: []
             ValidationFrequency: 50
              ValidationPatience: Inf
             ObjectiveMetricName: 'loss'
                  CheckpointPath: ''
            ExecutionEnvironment: 'auto'
                       OutputFcn: []
                         Metrics: []
                           Plots: 'training-progress'
            SequencePaddingValue: 0
        SequencePaddingDirection: 'right'
                InputDataFormats: "auto"
               TargetDataFormats: "auto"
         ResetInputNormalization: 1
    BatchNormalizationStatistics: 'auto'
                   OutputNetwork: 'auto'
                    Acceleration: "auto"

Algorithms

Limited-Memory BFGS

The L-BFGS algorithm [1] is a quasi-Newton method that approximates the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. Use the L-BFGS algorithm 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. The algorithm then 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 $$B_k^{-1}\nabla J(W_k)=r$$.

Gradient Clipping

If the gradients increase in magnitude exponentially, then the training is unstable and can diverge within a few iterations. This "gradient explosion" is indicated by a training loss that goes to NaN or Inf. Gradient clipping helps prevent gradient explosion by stabilizing the training at higher learning rates and in the presence of outliers [2]. Gradient clipping enables networks to be trained faster, and does not usually impact the accuracy of the learned task.

There are two types of gradient clipping.

• Norm-based gradient clipping rescales the gradient based on a threshold, and does not change the direction of the gradient. The "l2norm" and "global-l2norm" values of GradientThresholdMethod are norm-based gradient clipping methods.

• Value-based gradient clipping clips any partial derivative greater than the threshold, which can result in the gradient arbitrarily changing direction. Value-based gradient clipping can have unpredictable behavior, but sufficiently small changes do not cause the network to diverge. The "absolute-value" value of GradientThresholdMethod is a value-based gradient clipping method.

L₂ Regularization

Adding a regularization term for the weights to the loss function $$E\left(\theta \right)$$ is one way to reduce overfitting [3], [4]. The regularization term is also called weight decay. The loss function with the regularization term takes the form

$$E_R\left(\theta \right)=E\left(\theta \right)+\lambda \Omega \left(w\right),$$

where $$w$$ is the weight vector, $$\lambda$$ is the regularization factor (coefficient), and the regularization function $$\Omega \left(w\right)$$ is

$$\Omega \left(w\right)=\frac{1}{2}{w}^{T}w.$$

Note that the biases are not regularized [4]. You can specify the regularization factor $$\lambda$$ by using the L2Regularization training option. You can also specify different regularization factors for different layers and parameters.

The loss function that the software uses for network training includes the regularization term. However, the loss value displayed in the command window and training progress plot during training is the loss on the data only and does not include the regularization term.

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.

[2] Pascanu, R., T. Mikolov, and Y. Bengio. "On the difficulty of training recurrent neural networks". Proceedings of the 30th International Conference on Machine Learning. Vol. 28(3), 2013, pp. 1310–1318.

[3] Bishop, C. M. Pattern Recognition and Machine Learning. Springer, New York, NY, 2006.

[4] Murphy, K. P. Machine Learning: A Probabilistic Perspective. The MIT Press, Cambridge, Massachusetts, 2012.

Version History

Introduced in R2023b

R2024b: Monitor and plot more metrics during training

Use new and updated metric objects during training and testing.

• MAPEMetric — Mean absolute percentage error (MAPE)

• AccuracyMetric with new NumTopKClasses option — Top-k accuracy

• FScoreMetric with new Beta option — F_β-score

You can also directly specify these new built-in metric and loss names:

• "mape"— Mean absolute percentage error (MAPE)

• "crossentropy" — Cross-entropy loss

• "index-crossentropy" — Index cross-entropy loss

• "binary-crossentropy" — Binary cross-entropy loss

• "mse" / "mean-squared-error" / "l2loss" — Mean squared error

• "mae" / "mean-absolute-error" / "l1loss" — Mean absolute error

• "huber" — Huber loss

R2024b: Specify initial step size for L-BFGS solver

Specify the initial step size for the L-BFGS solver using the InitialStepSize argument.

R2024a: Specify validation data using minibatchqueue object

Specify validation data as a minibatchqueue object using the ValidationData argument.

R2024a: Automatic performance optimization

Accelerate training with automatic performance optimization. When you train a network using the trainnet function, automatic performance optimization is enabled by default. You can disable performance optimization by setting the Acceleration option to "none" using the trainingOptions function.

R2024a: Specify metrics as deep.DifferentiableFunction object

Specify the metrics as deep.DifferentiableFunction object.

R2024a: OutputNetwork default is "auto"

Starting in R2024a, the OutputNetwork training option default value is "auto". If you have specified validation data, then the software returns the network corresponding to the best validation metric value. If you have not specified validation data, then the software returns the network corresponding to the last training iteration. If you have validation data and want to replicate the previous default, then set OutputNetwork to "last-iteration".

This change applies when using the training options with trainnet only. If you are using the training options with the trainNetwork function, then there is no behavior change and by default the software returns the network corresponding to the last training iteration.

R2024a: OutputNetwork value "best-validation-loss" is not recommended

Specifying OutputNetwork as "best-validation-loss" is not recommended. If you have code that set OutputNetwork to "best-validation-loss", then use "best-validation" instead. The software returns the network corresponding to the best validation metric value as specified by the ObjectiveMetricName option. By default, the ObjectiveMetricName value is set to "loss". This behavior applies when using the training options with the trainnet function only.

When using the training options with the trainNetwork function, if you specify OutputNetwork as "best-validation", then software always returns the network with the best validation loss value.