# l2loss

## Syntax

## Description

The L_{2} loss operation computes the
L_{2} loss (based on the squared L_{2} norm) given
network predictions and target values. When the `Reduction`

option is
`"sum"`

and the `NormalizationFactor`

option is
`"batch-size"`

, the computed value is known as the mean squared error
(MSE).

The `l2loss`

function calculates the L_{2} loss
using `dlarray`

data.
Using `dlarray`

objects makes working with high
dimensional data easier by allowing you to label the dimensions. For example, you can label
which dimensions correspond to spatial, time, channel, and batch dimensions using the
`"S"`

, `"T"`

, `"C"`

, and
`"B"`

labels, respectively. For unspecified and other dimensions, use the
`"U"`

label. For `dlarray`

object functions that operate
over particular dimensions, you can specify the dimension labels by formatting the
`dlarray`

object directly, or by using the `DataFormat`

option.

specifies additional options using one or more name-value arguments. For example,
`loss`

= l2loss(___,`Name=Value`

)`l2loss(Y,targets,Reduction="none")`

computes the
L_{2} loss without reducing the output to a scalar.

## Examples

### Mean Squared Error Loss

Create an array of predictions for 12 observations over 10 responses.

```
numResponses = 10;
numObservations = 12;
Y = rand(numResponses,numObservations);
dlY = dlarray(Y,'CB');
```

View the size and format of the predictions.

size(dlY)

`ans = `*1×2*
10 12

dims(dlY)

ans = 'CB'

Create an array of random targets.

targets = rand(numResponses,numObservations);

View the size of the targets.

size(targets)

`ans = `*1×2*
10 12

Compute the mean squared error (MSE) loss between the predictions and the targets using the `l2loss`

function.

loss = l2loss(dlY,targets)

loss = 1x1 dlarray 1.4748

### Masked Mean Squared Error for Padded Sequences

Create arrays of predictions and targets for 12 sequences of varying lengths over 10 responses.

numResponses = 10; numObservations = 12; maxSequenceLength = 15; sequenceLengths = randi(maxSequenceLength,[1 numObservations]); Y = cell(numObservations,1); targets = cell(numObservations,1); for i = 1:numObservations Y{i} = rand(numResponses,sequenceLengths(i)); targets{i} = rand(numResponses,sequenceLengths(i)); end

View the cell arrays of predictions and targets.

Y

`Y=`*12×1 cell array*
{10x13 double}
{10x14 double}
{10x2 double}
{10x14 double}
{10x10 double}
{10x2 double}
{10x5 double}
{10x9 double}
{10x15 double}
{10x15 double}
{10x3 double}
{10x15 double}

targets

`targets=`*12×1 cell array*
{10x13 double}
{10x14 double}
{10x2 double}
{10x14 double}
{10x10 double}
{10x2 double}
{10x5 double}
{10x9 double}
{10x15 double}
{10x15 double}
{10x3 double}
{10x15 double}

Pad the prediction and target sequences in the second dimension using the `padsequences`

function and also return the corresponding mask.

[Y,mask] = padsequences(Y,2); targets = padsequences(targets,2);

Convert the padded sequences to `dlarray`

with the format `"CTB"`

(channel, time, batch). Because formatted `dlarray`

objects automatically permute the dimensions of the underlying data, keep the order consistent by also converting the targets and mask to formatted `dlarray`

objects with the format `"CTB"`

(channel, batch, time).

dlY = dlarray(Y,"CTB"); targets = dlarray(targets,"CTB"); mask = dlarray(mask,"CTB");

View the sizes of the prediction scores, targets, and mask.

size(dlY)

`ans = `*1×3*
10 12 15

size(targets)

`ans = `*1×3*
10 12 15

size(mask)

`ans = `*1×3*
10 12 15

Compute the mean squared error (MSE) between the predictions and the targets. To prevent the loss values calculated from padding from contributing to the loss, set the `Mask`

option to the mask returned by the `padsequences`

function.

loss = l2loss(dlY,targets,Mask=mask)

loss = 1x1 dlarray 16.3668

## Input Arguments

`Y`

— Predictions

`dlarray`

object | numeric array

Predictions, specified as a formatted or unformatted `dlarray`

object,
or a numeric array. When `Y`

is not a formatted
`dlarray`

, you must specify the dimension format using the
`DataFormat`

argument.

If `Y`

is a numeric array, `targets`

must be a
`dlarray`

object.

`targets`

— Target responses

`dlarray`

| numeric array

Target responses, specified as a formatted or unformatted `dlarray`

or a
numeric array.

The size of each dimension of `targets`

must match the size of the
corresponding dimension of `Y`

.

If `targets`

is a formatted `dlarray`

, then its format must
be the same as the format of `Y`

, or the same as
`DataFormat`

if `Y`

is
unformatted.

If `targets`

is an unformatted `dlarray`

or a numeric array,
then the function applies the format of `Y`

or the value of
`DataFormat`

to `targets`

.

**Tip**

Formatted `dlarray`

objects automatically permute the dimensions of the
underlying data to have the order `"S"`

(spatial), `"C"`

(channel), `"B"`

(batch), `"T"`

(time), then
`"U"`

(unspecified). To ensure that the dimensions of
`Y`

and `targets`

are consistent, when
`Y`

is a formatted `dlarray`

, also specify
`targets`

as a formatted `dlarray`

.

`weights`

— Weights

`dlarray`

| numeric array

Weights, specified as a formatted or unformatted `dlarray`

or a numeric array.

If `weights`

is a vector and `Y`

has two or more
nonsingleton dimensions, then `weights`

must be a formatted
`dlarray`

, where the dimension label of the nonsingleton dimension is
either `"C"`

(channel) or `"B"`

(batch) and has a size
that matches the size of the corresponding dimension in `Y`

.

If `weights`

is a formatted `dlarray`

with two or more
nonsingleton dimensions, then its format must match the format of
`Y`

.

If `weights`

is not a formatted `dlarray`

and has two or
more nonsingleton dimensions, then its size must match the size of
`Y`

and the function uses the same format as
`Y`

. Alternatively, to specify the weights format, use the
`WeightsFormat`

option.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

**Example: **`loss = l2loss(Y,targets,Reduction="none")`

specifies to
compute the L_{2} loss without reducing the output to a
scalar

`Mask`

— Mask indicating which elements to include for loss computation

`dlarray`

| logical array | numeric array

Mask indicating which elements to include for loss computation, specified as a
`dlarray`

object, a logical array, or a numeric array with the same
size as `Y`

.

The function includes and excludes elements of the input data for loss computation when the corresponding value in the mask is 1 and 0, respectively.

If `Mask`

is a formatted `dlarray`

object, then its
format must match that of `Y`

. If `Mask`

is not a
formatted `dlarray`

object, then the function uses the same format as
`Y`

.

If you specify the `DataFormat`

argument, then the function also
uses the specified format for the mask.

The size of each dimension of `Mask`

must match the size of the
corresponding dimension in `Y`

. The default value is a logical array
of ones.

**Tip**

Formatted `dlarray`

objects automatically permute the dimensions of the
underlying data to have this order: `"S"`

(spatial), `"C"`

(channel), `"B"`

(batch), `"T"`

(time), and
`"U"`

(unspecified). For example, `dlarray`

objects
automatically permute the dimensions of data with format `"TSCSBS"`

to have
format `"SSSCBT"`

.

To ensure that the dimensions of `Y`

and the mask are consistent, when
`Y`

is a formatted `dlarray`

, also specify the mask as
a formatted `dlarray`

.

`Reduction`

— Loss value array reduction mode

`"sum"`

(default) | `"none"`

Loss value array reduction mode, specified as `"sum"`

or
`"none"`

.

If the `Reduction`

argument is `"sum"`

, then the function
sums all elements in the array of loss values. In this case, the output
`loss`

is a scalar.

If the `Reduction`

argument is `"none"`

, then the
function does not reduce the array of loss values. In this case, the output
`loss`

is an unformatted `dlarray`

object
of the same size as `Y`

.

`NormalizationFactor`

— Divisor for normalizing reduced loss

`"batch-size"`

(default) | `"all-elements"`

| `"mask-included"`

| `"none"`

Divisor for normalizing the reduced loss when `Reduction`

is
`"sum"`

, specified as one of the following:

`"batch-size"`

— Normalize the loss by dividing it by the number of observations in`Y`

.`"all-elements"`

— Normalize the loss by dividing it by the number of elements of`Y`

.`"mask-included"`

— Normalize the loss by dividing the loss values by the product of the number of observations and the number of included elements specified by the mask for each observation independently. To use this option, you must specify a mask using the`Mask`

option.`"none"`

— Do not normalize the loss.

`DataFormat`

— Description of data dimensions

character vector | string scalar

Description of the data dimensions, specified as a character vector or string scalar.

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.

If the input data is not a formatted `dlarray`

object, then you must
specify the `DataFormat`

option.

For more information, see Deep Learning Data Formats.

**Data Types: **`char`

| `string`

`WeightsFormat`

— Description of dimensions of weights

character vector | string scalar

Description of the dimensions of the weights, specified as a character vector or string scalar.

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.

If `weights`

is a numeric vector and
`Y`

has two or more nonsingleton
dimensions, then you must specify the
`WeightsFormat`

option.

If `weights`

is not a vector, or
`weights`

and
`Y`

are both vectors, then the
default value of `WeightsFormat`

is the same
as the format of `Y`

.

For more information, see Deep Learning Data Formats.

**Data Types: **`char`

| `string`

## Output Arguments

`loss`

— L_{2} loss

`dlarray`

L_{2} loss, returned as an unformatted
`dlarray`

. The output `loss`

is an unformatted
`dlarray`

with the same underlying data type as the input
`Y`

.

The size of `loss`

depends on the `Reduction`

option.

## Algorithms

### L2 Loss

The L_{2} loss operation computes the
L_{2} loss (based on the squared L_{2} norm) given
network predictions and target values. When the `Reduction`

option is
`"sum"`

and the `NormalizationFactor`

option is
`"batch-size"`

, the computed value is known as the mean squared error
(MSE).

For each element *Y _{j}* of the input, the

`l2loss`

function computes the corresponding element-wise loss values using$${\text{loss}}_{j}={\left({Y}_{j}-{T}_{j}\right)}^{2},$$

where *Y _{j}* is a predicted value
and

*T*is the corresponding target value.

_{j}To reduce the loss values to a scalar, the function then reduces the element-wise loss using the formula

$$\text{loss}=\frac{1}{N}{\displaystyle \sum _{j}{m}_{j}{w}_{j}{\text{loss}}_{j},}$$

where *N* is the normalization factor,
*m _{j}* is the mask value for element

*j*, and

*w*is the weight value for element

_{j}*j*.

If you do not opt to reduce the loss, then the function applies the mask and the weights to the loss values directly:

$${\text{loss}}_{j}^{*}={m}_{j}{w}_{j}{\text{loss}}_{j}$$

### Deep Learning Array Formats

Most deep learning networks and functions operate on different dimensions of the input data in different ways.

For example, an LSTM operation iterates over the time dimension of the input data, and a batch normalization operation normalizes over the batch dimension of the input data.

To provide input data with labeled dimensions or input data with additional layout information, you can use *data formats*.

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).

To create formatted input data, create a `dlarray`

object and specify the format using the second argument.

To provide additional layout information with unformatted data, specify the formats using the `DataFormat`

and `WeightsFormat`

arguments.

For more information, see Deep Learning Data Formats.

## Extended Capabilities

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The `l2loss`

function
supports GPU array input with these usage notes and limitations:

When at least one of the following input arguments is a

`gpuArray`

or a`dlarray`

with underlying data of type`gpuArray`

, this function runs on the GPU:`Y`

`targets`

`weights`

`Mask`

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

## Version History

**Introduced in R2021b**

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