# TrainingOptionsLM

## 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

### 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.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`InitialDampingFactor`

— Initial damping factor

`0.001`

(default) | positive scalar

Initial damping factor, specified as a positive scalar.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`MaxDampingFactor`

— Maximum damping factor

`1e10`

(default) | positive scalar

Maximum damping factor, specified as a positive scalar.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`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.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`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.

**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`

### 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 | 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 F_{1}-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.

**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

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"`

.

**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.

### 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.

**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.

### 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"`

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 LM Optimizer

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

### Levenberg–Marquardt

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.

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

$${W}_{k+1}={W}_{k}+\Delta {W}_{k},$$

where *ΔW _{k}* the change of the weights at iteration

*k*given by

$$\Delta {W}_{k}=-{\left({H}_{k}\right)}^{-1}\nabla {E}_{k}.$$

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

$${H}_{k}={J}_{k}^{\top}{J}_{k}+{\mu}_{k}I,$$

where *J _{k}* is the Jacobian matrix at iteration

*k*,

*μ*is the damping factor at iteration

_{k}*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**

## See Also

`trainingOptions`

| `trainnet`

| `dlnetwork`

| `analyzeNetwork`

| Deep Network Designer

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