TrainingOptionsLM

Training options for Levenberg–Marquardt (LM) optimizer

Since R2024b

Description

Use a TrainingOptionsLM object to set training options for the Levenberg–Marquardt (LM) optimizer.

The LM algorithm [1] interpolates between gradient descent and Gauss-Newton methods, and can be more robust for small neural networks. It approximates second order derivatives using a Jacobian outer product. Use the LM algorithm for regression networks with small numbers of learnable parameters, where you can process the data set in a single batch.

Creation

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

Properties

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LM

`MaxIterations` — Maximum number of iterations
`1000` (default) | positive integer

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

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

`InitialDampingFactor` — Initial damping factor
`0.001` (default) | positive scalar

Initial damping factor, specified as a positive scalar.

`MaxDampingFactor` — Maximum damping factor
`1e10` (default) | positive scalar

Maximum damping factor, specified as a positive scalar.

`DampingIncreaseFactor` — Factor for increasing damping factor
`10` (default) | positive scalar greater than 1

Factor for increasing damping factor, specified as a positive scalar greater than 1.

`DampingDecreaseFactor` — Factor for decreasing damping factor
`0.1` (default) | positive scalar less than 1

Factor for decreasing damping factor, specified as a positive scalar less than 1.

`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.

`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 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 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. Training plot axes toolbar with log scale enabled and the tooltip "Log scale y-axis".

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 | 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)
- Loss functions:
  - "crossentropy" — Cross-entropy loss for classification tasks.
  - "indexcrossentropy" — Index cross-entropy loss for classification tasks.
  - "binary-crossentropy" — Binary cross-entropy loss for binary and multilabel classification tasks.
  - "mae" / "mean-absolute-error" / "l1loss" — Mean absolute error for regression tasks.
  - "mse" / "mean-squared-error" / "l2loss" — Mean squared error for regression tasks.
  - "huber" — Huber loss for regression tasks
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:
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 — 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

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.

`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).

`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

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.

`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".

`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.

Normalization

`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.

`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

$\begin{array}{l} μ^{*} = λ_{μ} \hat{μ} + (1 - λ_{μ}) μ \\ σ^{2}^{*} = λ_{σ^{2}} \hat{σ^{2}} + (1- λ_{σ^{2}}) σ^{2} \end{array}$
where $μ^{*}$ and $σ^{2}^{*}$ denote the updated mean and variance, respectively, $λ_{μ}$ and $λ_{σ^{2}}$ denote the mean and variance decay values, respectively, $\hat{μ}$ and $\hat{σ^{2}}$ denote the mean and variance of the layer input, respectively, and $μ$ and $σ^{2}$ 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.

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.

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"`

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.

Examples

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Create Training Options for LM Optimizer

Open Live Script

Create a set of options for training a neural network using the LM optimizer.

Use an initial damping factor of 0.002.
Use a maximum damping factor of $1 0^{- 9}$ .
Increase the damping using a factor of 12.
Decrease the damping using a factor of 0.2.

options = trainingOptions("lm", ...
    InitialDampingFactor=0.002, ...
    MaxDampingFactor=1e9, ...
    DampingIncreaseFactor=12, ...
    DampingDecreaseFactor=0.2)

options = 
  TrainingOptionsLM with properties:

                   MaxIterations: 1000
            InitialDampingFactor: 0.0020
                MaxDampingFactor: 1.0000e+09
           DampingDecreaseFactor: 0.2000
           DampingIncreaseFactor: 12
               GradientTolerance: 1.0000e-05
                   StepTolerance: 1.0000e-05
                  SequenceLength: 'longest'
             CheckpointFrequency: 30
                         Verbose: 1
                VerboseFrequency: 50
                  ValidationData: []
             ValidationFrequency: 50
              ValidationPatience: Inf
             ObjectiveMetricName: 'loss'
                  CheckpointPath: ''
            ExecutionEnvironment: 'auto'
                       OutputFcn: []
                         Metrics: []
                           Plots: 'none'
            SequencePaddingValue: 0
        SequencePaddingDirection: 'right'
                InputDataFormats: "auto"
               TargetDataFormats: "auto"
         ResetInputNormalization: 1
    BatchNormalizationStatistics: 'auto'
                   OutputNetwork: 'auto'
                    Acceleration: "auto"

Algorithms

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Levenberg–Marquardt

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

$W_{k + 1} = W_{k} + Δ W_{k},$

where ΔW_k the change of the weights at iteration k given by

$Δ W_{k} = - {(H_{k})}^{- 1} \nabla E_{k} .$

Here, H_k is the approximated Hessian at iteration k and $\nabla E_{k}$ is the gradient of the loss at iteration k with respect to the learnable parameters. The algorithm approximates the Hessian using

$H_{k} = J_{k}^{⊤} J_{k} + μ_{k} I,$

where J_k is the Jacobian matrix at iteration k, μ_k is the damping factor at iteration k, and I is the identity matrix.

The solver uses the damping factor to adjust the step size taken each iteration and adaptively updates it each iteration. It increases and decreases the damping factor when iterations increase and decrease the loss, respectively. These adjustments make the optimizer take larger and smaller steps when the loss is increasing and decreasing, respectively.

When the loss increases or decreases, the solver adaptively increases or decreases the damping factor by multiplying it by DampingIncreaseFactor and DampingDecreaseFactor, respectively.

References

[1] Marquardt, Donald W. “An Algorithm for Least-Squares Estimation of Nonlinear Parameters.” Journal of the Society for Industrial and Applied Mathematics 11, no. 2 (June 1963): 431–41. https://doi.org/10.1137/0111030.

Version History

Introduced in R2024b

TrainingOptionsLM

Description

Creation

Properties

LM

MaxIterations — Maximum number of iterations 1000 (default) | positive integer

InitialDampingFactor — Initial damping factor 0.001 (default) | positive scalar

MaxDampingFactor — Maximum damping factor 1e10 (default) | positive scalar

DampingIncreaseFactor — Factor for increasing damping factor 10 (default) | positive scalar greater than 1

DampingDecreaseFactor — Factor for decreasing damping factor 0.1 (default) | positive scalar less than 1

GradientTolerance — Relative gradient tolerance 1e-5 (default) | positive scalar

StepTolerance — Step size tolerance 1e-5 (default) | positive scalar

Data Formats

InputDataFormats — Description of input data dimensions "auto" (default) | string array | cell array of character vectors | character vector

TargetDataFormats — Description of target data dimensions "auto" (default) | string array | cell array of character vectors | character vector

Monitoring

Plots — Plots to display during neural network training "none" (default) | "training-progress"

Metrics — Metrics to monitor [] (default) | character vector | string array | function handle | deep.DifferentiableFunction object | cell array | metric object

ObjectiveMetricName — Name of objective metric "loss" (default) | string scalar | character vector

Verbose — Flag to display training progress information 1 (true) (default) | 0 (false)

VerboseFrequency — Frequency of verbose printing 50 (default) | positive integer

OutputFcn — Output functions function handle | cell array of function handles