gaussian-splatting-my/envs/gaussian_splatting/Lib/site-packages/torch/nn/modules/loss.py

import warnings

from .distance import PairwiseDistance
from .module import Module
from .. import functional as F
from .. import _reduction as _Reduction

from torch import Tensor
from typing import Callable, Optional


class _Loss(Module):
    reduction: str

    def __init__(self, size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(_Loss, self).__init__()
        if size_average is not None or reduce is not None:
            self.reduction: str = _Reduction.legacy_get_string(size_average, reduce)
        else:
            self.reduction = reduction


class _WeightedLoss(_Loss):
    def __init__(self, weight: Optional[Tensor] = None, size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(_WeightedLoss, self).__init__(size_average, reduce, reduction)
        self.register_buffer('weight', weight)
        self.weight: Optional[Tensor]


class L1Loss(_Loss):
    r"""Creates a criterion that measures the mean absolute error (MAE) between each element in
    the input :math:`x` and target :math:`y`.

    The unreduced (i.e. with :attr:`reduction` set to ``'none'``) loss can be described as:

    .. math::
        \ell(x, y) = L = \{l_1,\dots,l_N\}^\top, \quad
        l_n = \left| x_n - y_n \right|,

    where :math:`N` is the batch size. If :attr:`reduction` is not ``'none'``
    (default ``'mean'``), then:

    .. math::
        \ell(x, y) =
        \begin{cases}
            \operatorname{mean}(L), & \text{if reduction} = \text{`mean';}\\
            \operatorname{sum}(L),  & \text{if reduction} = \text{`sum'.}
        \end{cases}

    :math:`x` and :math:`y` are tensors of arbitrary shapes with a total
    of :math:`n` elements each.

    The sum operation still operates over all the elements, and divides by :math:`n`.

    The division by :math:`n` can be avoided if one sets ``reduction = 'sum'``.

    Supports real-valued and complex-valued inputs.

    Args:
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input: :math:`(*)`, where :math:`*` means any number of dimensions.
        - Target: :math:`(*)`, same shape as the input.
        - Output: scalar. If :attr:`reduction` is ``'none'``, then
          :math:`(*)`, same shape as the input.

    Examples::

        >>> loss = nn.L1Loss()
        >>> input = torch.randn(3, 5, requires_grad=True)
        >>> target = torch.randn(3, 5)
        >>> output = loss(input, target)
        >>> output.backward()
    """
    __constants__ = ['reduction']

    def __init__(self, size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(L1Loss, self).__init__(size_average, reduce, reduction)

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.l1_loss(input, target, reduction=self.reduction)


class NLLLoss(_WeightedLoss):
    r"""The negative log likelihood loss. It is useful to train a classification
    problem with `C` classes.

    If provided, the optional argument :attr:`weight` should be a 1D Tensor assigning
    weight to each of the classes. This is particularly useful when you have an
    unbalanced training set.

    The `input` given through a forward call is expected to contain
    log-probabilities of each class. `input` has to be a Tensor of size either
    :math:`(minibatch, C)` or :math:`(minibatch, C, d_1, d_2, ..., d_K)`
    with :math:`K \geq 1` for the `K`-dimensional case. The latter is useful for
    higher dimension inputs, such as computing NLL loss per-pixel for 2D images.

    Obtaining log-probabilities in a neural network is easily achieved by
    adding a  `LogSoftmax`  layer in the last layer of your network.
    You may use `CrossEntropyLoss` instead, if you prefer not to add an extra
    layer.

    The `target` that this loss expects should be a class index in the range :math:`[0, C-1]`
    where `C = number of classes`; if `ignore_index` is specified, this loss also accepts
    this class index (this index may not necessarily be in the class range).

    The unreduced (i.e. with :attr:`reduction` set to ``'none'``) loss can be described as:

    .. math::
        \ell(x, y) = L = \{l_1,\dots,l_N\}^\top, \quad
        l_n = - w_{y_n} x_{n,y_n}, \quad
        w_{c} = \text{weight}[c] \cdot \mathbb{1}\{c \not= \text{ignore\_index}\},

    where :math:`x` is the input, :math:`y` is the target, :math:`w` is the weight, and
    :math:`N` is the batch size. If :attr:`reduction` is not ``'none'``
    (default ``'mean'``), then

    .. math::
        \ell(x, y) = \begin{cases}
            \sum_{n=1}^N \frac{1}{\sum_{n=1}^N w_{y_n}} l_n, &
            \text{if reduction} = \text{`mean';}\\
            \sum_{n=1}^N l_n,  &
            \text{if reduction} = \text{`sum'.}
        \end{cases}

    Args:
        weight (Tensor, optional): a manual rescaling weight given to each
            class. If given, it has to be a Tensor of size `C`. Otherwise, it is
            treated as if having all ones.
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``None``
        ignore_index (int, optional): Specifies a target value that is ignored
            and does not contribute to the input gradient. When
            :attr:`size_average` is ``True``, the loss is averaged over
            non-ignored targets.
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``None``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will
            be applied, ``'mean'``: the weighted mean of the output is taken,
            ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in
            the meantime, specifying either of those two args will override
            :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input: :math:`(N, C)` or :math:`(C)`, where `C = number of classes`, or
          :math:`(N, C, d_1, d_2, ..., d_K)` with :math:`K \geq 1`
          in the case of `K`-dimensional loss.
        - Target: :math:`(N)` or :math:`()`, where each value is
          :math:`0 \leq \text{targets}[i] \leq C-1`, or
          :math:`(N, d_1, d_2, ..., d_K)` with :math:`K \geq 1` in the case of
          K-dimensional loss.
        - Output: If :attr:`reduction` is ``'none'``, shape :math:`(N)` or
          :math:`(N, d_1, d_2, ..., d_K)` with :math:`K \geq 1` in the case of K-dimensional loss.
          Otherwise, scalar.

    Examples::

        >>> m = nn.LogSoftmax(dim=1)
        >>> loss = nn.NLLLoss()
        >>> # input is of size N x C = 3 x 5
        >>> input = torch.randn(3, 5, requires_grad=True)
        >>> # each element in target has to have 0 <= value < C
        >>> target = torch.tensor([1, 0, 4])
        >>> output = loss(m(input), target)
        >>> output.backward()
        >>>
        >>>
        >>> # 2D loss example (used, for example, with image inputs)
        >>> N, C = 5, 4
        >>> loss = nn.NLLLoss()
        >>> # input is of size N x C x height x width
        >>> data = torch.randn(N, 16, 10, 10)
        >>> conv = nn.Conv2d(16, C, (3, 3))
        >>> m = nn.LogSoftmax(dim=1)
        >>> # each element in target has to have 0 <= value < C
        >>> target = torch.empty(N, 8, 8, dtype=torch.long).random_(0, C)
        >>> output = loss(m(conv(data)), target)
        >>> output.backward()
    """
    __constants__ = ['ignore_index', 'reduction']
    ignore_index: int

    def __init__(self, weight: Optional[Tensor] = None, size_average=None, ignore_index: int = -100,
                 reduce=None, reduction: str = 'mean') -> None:
        super(NLLLoss, self).__init__(weight, size_average, reduce, reduction)
        self.ignore_index = ignore_index

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.nll_loss(input, target, weight=self.weight, ignore_index=self.ignore_index, reduction=self.reduction)


class NLLLoss2d(NLLLoss):
    def __init__(self, weight: Optional[Tensor] = None, size_average=None, ignore_index: int = -100,
                 reduce=None, reduction: str = 'mean') -> None:
        warnings.warn("NLLLoss2d has been deprecated. "
                      "Please use NLLLoss instead as a drop-in replacement and see "
                      "https://pytorch.org/docs/master/nn.html#torch.nn.NLLLoss for more details.")
        super(NLLLoss2d, self).__init__(weight, size_average, ignore_index, reduce, reduction)


class PoissonNLLLoss(_Loss):
    r"""Negative log likelihood loss with Poisson distribution of target.

    The loss can be described as:

    .. math::
        \text{target} \sim \mathrm{Poisson}(\text{input})

        \text{loss}(\text{input}, \text{target}) = \text{input} - \text{target} * \log(\text{input})
                                    + \log(\text{target!})

    The last term can be omitted or approximated with Stirling formula. The
    approximation is used for target values more than 1. For targets less or
    equal to 1 zeros are added to the loss.

    Args:
        log_input (bool, optional): if ``True`` the loss is computed as
            :math:`\exp(\text{input}) - \text{target}*\text{input}`, if ``False`` the loss is
            :math:`\text{input} - \text{target}*\log(\text{input}+\text{eps})`.
        full (bool, optional): whether to compute full loss, i. e. to add the
            Stirling approximation term

            .. math::
                \text{target}*\log(\text{target}) - \text{target} + 0.5 * \log(2\pi\text{target}).
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        eps (float, optional): Small value to avoid evaluation of :math:`\log(0)` when
            :attr:`log_input = False`. Default: 1e-8
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Examples::

        >>> loss = nn.PoissonNLLLoss()
        >>> log_input = torch.randn(5, 2, requires_grad=True)
        >>> target = torch.randn(5, 2)
        >>> output = loss(log_input, target)
        >>> output.backward()

    Shape:
        - Input: :math:`(*)`, where :math:`*` means any number of dimensions.
        - Target: :math:`(*)`, same shape as the input.
        - Output: scalar by default. If :attr:`reduction` is ``'none'``, then :math:`(*)`,
          the same shape as the input.
    """
    __constants__ = ['log_input', 'full', 'eps', 'reduction']
    log_input: bool
    full: bool
    eps: float

    def __init__(self, log_input: bool = True, full: bool = False, size_average=None,
                 eps: float = 1e-8, reduce=None, reduction: str = 'mean') -> None:
        super(PoissonNLLLoss, self).__init__(size_average, reduce, reduction)
        self.log_input = log_input
        self.full = full
        self.eps = eps

    def forward(self, log_input: Tensor, target: Tensor) -> Tensor:
        return F.poisson_nll_loss(log_input, target, log_input=self.log_input, full=self.full,
                                  eps=self.eps, reduction=self.reduction)


class GaussianNLLLoss(_Loss):
    r"""Gaussian negative log likelihood loss.

    The targets are treated as samples from Gaussian distributions with
    expectations and variances predicted by the neural network. For a
    ``target`` tensor modelled as having Gaussian distribution with a tensor
    of expectations ``input`` and a tensor of positive variances ``var`` the loss is:

    .. math::
        \text{loss} = \frac{1}{2}\left(\log\left(\text{max}\left(\text{var},
        \ \text{eps}\right)\right) + \frac{\left(\text{input} - \text{target}\right)^2}
        {\text{max}\left(\text{var}, \ \text{eps}\right)}\right) + \text{const.}

    where :attr:`eps` is used for stability. By default, the constant term of
    the loss function is omitted unless :attr:`full` is ``True``. If ``var`` is not the same
    size as ``input`` (due to a homoscedastic assumption), it must either have a final dimension
    of 1 or have one fewer dimension (with all other sizes being the same) for correct broadcasting.

    Args:
        full (bool, optional): include the constant term in the loss
            calculation. Default: ``False``.
        eps (float, optional): value used to clamp ``var`` (see note below), for
            stability. Default: 1e-6.
        reduction (string, optional): specifies the reduction to apply to the
            output:``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction
            will be applied, ``'mean'``: the output is the average of all batch
            member losses, ``'sum'``: the output is the sum of all batch member
            losses. Default: ``'mean'``.

    Shape:
        - Input: :math:`(N, *)` or :math:`(*)` where :math:`*` means any number of additional
          dimensions
        - Target: :math:`(N, *)` or :math:`(*)`, same shape as the input, or same shape as the input
          but with one dimension equal to 1 (to allow for broadcasting)
        - Var: :math:`(N, *)` or :math:`(*)`, same shape as the input, or same shape as the input but
          with one dimension equal to 1, or same shape as the input but with one fewer
          dimension (to allow for broadcasting)
        - Output: scalar if :attr:`reduction` is ``'mean'`` (default) or
          ``'sum'``. If :attr:`reduction` is ``'none'``, then :math:`(N, *)`, same
          shape as the input

    Examples::
        >>> loss = nn.GaussianNLLLoss()
        >>> input = torch.randn(5, 2, requires_grad=True)
        >>> target = torch.randn(5, 2)
        >>> var = torch.ones(5, 2, requires_grad=True) #heteroscedastic
        >>> output = loss(input, target, var)
        >>> output.backward()

        >>> loss = nn.GaussianNLLLoss()
        >>> input = torch.randn(5, 2, requires_grad=True)
        >>> target = torch.randn(5, 2)
        >>> var = torch.ones(5, 1, requires_grad=True) #homoscedastic
        >>> output = loss(input, target, var)
        >>> output.backward()

    Note:
        The clamping of ``var`` is ignored with respect to autograd, and so the
        gradients are unaffected by it.

    Reference:
        Nix, D. A. and Weigend, A. S., "Estimating the mean and variance of the
        target probability distribution", Proceedings of 1994 IEEE International
        Conference on Neural Networks (ICNN'94), Orlando, FL, USA, 1994, pp. 55-60
        vol.1, doi: 10.1109/ICNN.1994.374138.
    """
    __constants__ = ['full', 'eps', 'reduction']
    full: bool
    eps: float

    def __init__(self, *, full: bool = False, eps: float = 1e-6, reduction: str = 'mean') -> None:
        super(GaussianNLLLoss, self).__init__(None, None, reduction)
        self.full = full
        self.eps = eps

    def forward(self, input: Tensor, target: Tensor, var: Tensor) -> Tensor:
        return F.gaussian_nll_loss(input, target, var, full=self.full, eps=self.eps, reduction=self.reduction)


class KLDivLoss(_Loss):
    r"""The Kullback-Leibler divergence loss.

    For tensors of the same shape :math:`y_{\text{pred}},\ y_{\text{true}}`,
    where :math:`y_{\text{pred}}` is the :attr:`input` and :math:`y_{\text{true}}` is the
    :attr:`target`, we define the **pointwise KL-divergence** as

    .. math::

        L(y_{\text{pred}},\ y_{\text{true}})
            = y_{\text{true}} \cdot \log \frac{y_{\text{true}}}{y_{\text{pred}}}
            = y_{\text{true}} \cdot (\log y_{\text{true}} - \log y_{\text{pred}})

    To avoid underflow issues when computing this quantity, this loss expects the argument
    :attr:`input` in the log-space. The argument :attr:`target` may also be provided in the
    log-space if :attr:`log_target`\ `= True`.

    To summarise, this function is roughly equivalent to computing

    .. code-block:: python

        if not log_target: # default
            loss_pointwise = target * (target.log() - input)
        else:
            loss_pointwise = target.exp() * (target - input)

    and then reducing this result depending on the argument :attr:`reduction` as

    .. code-block:: python

        if reduction == "mean":  # default
            loss = loss_pointwise.mean()
        elif reduction == "batchmean":  # mathematically correct
            loss = loss_pointwise.sum() / input.size(0)
        elif reduction == "sum":
            loss = loss_pointwise.sum()
        else:  # reduction == "none"
            loss = loss_pointwise

    .. note::
        As all the other losses in PyTorch, this function expects the first argument,
        :attr:`input`, to be the output of the model (e.g. the neural network)
        and the second, :attr:`target`, to be the observations in the dataset.
        This differs from the standard mathematical notation :math:`KL(P\ ||\ Q)` where
        :math:`P` denotes the distribution of the observations and :math:`Q` denotes the model.

    .. warning::
        :attr:`reduction`\ `= "mean"` doesn't return the true KL divergence value, please use
        :attr:`reduction`\ `= "batchmean"` which aligns with the mathematical definition.
        In a future release, `"mean"` will be changed to be the same as `"batchmean"`.

    Args:
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to `False`, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is `False`. Default: `True`
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is `False`, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: `True`
        reduction (string, optional): Specifies the reduction to apply to the output. Default: `"mean"`
        log_target (bool, optional): Specifies whether `target` is the log space. Default: `False`

    Shape:
        - Input: :math:`(*)`, where :math:`*` means any number of dimensions.
        - Target: :math:`(*)`, same shape as the input.
        - Output: scalar by default. If :attr:`reduction` is `'none'`, then :math:`(*)`,
          same shape as the input.

    Examples::

        >>> kl_loss = nn.KLDivLoss(reduction="batchmean")
        >>> # input should be a distribution in the log space
        >>> input = F.log_softmax(torch.randn(3, 5, requires_grad=True))
        >>> # Sample a batch of distributions. Usually this would come from the dataset
        >>> target = F.softmax(torch.rand(3, 5))
        >>> output = kl_loss(input, target)

        >>> kl_loss = nn.KLDivLoss(reduction="batchmean", log_target=True)
        >>> log_target = F.log_softmax(torch.rand(3, 5))
        >>> output = kl_loss(input, log_target)
    """
    __constants__ = ['reduction']

    def __init__(self, size_average=None, reduce=None, reduction: str = 'mean', log_target: bool = False) -> None:
        super(KLDivLoss, self).__init__(size_average, reduce, reduction)
        self.log_target = log_target

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.kl_div(input, target, reduction=self.reduction, log_target=self.log_target)


class MSELoss(_Loss):
    r"""Creates a criterion that measures the mean squared error (squared L2 norm) between
    each element in the input :math:`x` and target :math:`y`.

    The unreduced (i.e. with :attr:`reduction` set to ``'none'``) loss can be described as:

    .. math::
        \ell(x, y) = L = \{l_1,\dots,l_N\}^\top, \quad
        l_n = \left( x_n - y_n \right)^2,

    where :math:`N` is the batch size. If :attr:`reduction` is not ``'none'``
    (default ``'mean'``), then:

    .. math::
        \ell(x, y) =
        \begin{cases}
            \operatorname{mean}(L), &  \text{if reduction} = \text{`mean';}\\
            \operatorname{sum}(L),  &  \text{if reduction} = \text{`sum'.}
        \end{cases}

    :math:`x` and :math:`y` are tensors of arbitrary shapes with a total
    of :math:`n` elements each.

    The mean operation still operates over all the elements, and divides by :math:`n`.

    The division by :math:`n` can be avoided if one sets ``reduction = 'sum'``.

    Args:
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input: :math:`(*)`, where :math:`*` means any number of dimensions.
        - Target: :math:`(*)`, same shape as the input.

    Examples::

        >>> loss = nn.MSELoss()
        >>> input = torch.randn(3, 5, requires_grad=True)
        >>> target = torch.randn(3, 5)
        >>> output = loss(input, target)
        >>> output.backward()
    """
    __constants__ = ['reduction']

    def __init__(self, size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(MSELoss, self).__init__(size_average, reduce, reduction)

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.mse_loss(input, target, reduction=self.reduction)


class BCELoss(_WeightedLoss):
    r"""Creates a criterion that measures the Binary Cross Entropy between the target and
    the input probabilities:

    The unreduced (i.e. with :attr:`reduction` set to ``'none'``) loss can be described as:

    .. math::
        \ell(x, y) = L = \{l_1,\dots,l_N\}^\top, \quad
        l_n = - w_n \left[ y_n \cdot \log x_n + (1 - y_n) \cdot \log (1 - x_n) \right],

    where :math:`N` is the batch size. If :attr:`reduction` is not ``'none'``
    (default ``'mean'``), then

    .. math::
        \ell(x, y) = \begin{cases}
            \operatorname{mean}(L), & \text{if reduction} = \text{`mean';}\\
            \operatorname{sum}(L),  & \text{if reduction} = \text{`sum'.}
        \end{cases}

    This is used for measuring the error of a reconstruction in for example
    an auto-encoder. Note that the targets :math:`y` should be numbers
    between 0 and 1.

    Notice that if :math:`x_n` is either 0 or 1, one of the log terms would be
    mathematically undefined in the above loss equation. PyTorch chooses to set
    :math:`\log (0) = -\infty`, since :math:`\lim_{x\to 0} \log (x) = -\infty`.
    However, an infinite term in the loss equation is not desirable for several reasons.

    For one, if either :math:`y_n = 0` or :math:`(1 - y_n) = 0`, then we would be
    multiplying 0 with infinity. Secondly, if we have an infinite loss value, then
    we would also have an infinite term in our gradient, since
    :math:`\lim_{x\to 0} \frac{d}{dx} \log (x) = \infty`.
    This would make BCELoss's backward method nonlinear with respect to :math:`x_n`,
    and using it for things like linear regression would not be straight-forward.

    Our solution is that BCELoss clamps its log function outputs to be greater than
    or equal to -100. This way, we can always have a finite loss value and a linear
    backward method.


    Args:
        weight (Tensor, optional): a manual rescaling weight given to the loss
            of each batch element. If given, has to be a Tensor of size `nbatch`.
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input: :math:`(*)`, where :math:`*` means any number of dimensions.
        - Target: :math:`(*)`, same shape as the input.
        - Output: scalar. If :attr:`reduction` is ``'none'``, then :math:`(*)`, same
          shape as input.

    Examples::

        >>> m = nn.Sigmoid()
        >>> loss = nn.BCELoss()
        >>> input = torch.randn(3, requires_grad=True)
        >>> target = torch.empty(3).random_(2)
        >>> output = loss(m(input), target)
        >>> output.backward()
    """
    __constants__ = ['reduction']

    def __init__(self, weight: Optional[Tensor] = None, size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(BCELoss, self).__init__(weight, size_average, reduce, reduction)

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.binary_cross_entropy(input, target, weight=self.weight, reduction=self.reduction)


class BCEWithLogitsLoss(_Loss):
    r"""This loss combines a `Sigmoid` layer and the `BCELoss` in one single
    class. This version is more numerically stable than using a plain `Sigmoid`
    followed by a `BCELoss` as, by combining the operations into one layer,
    we take advantage of the log-sum-exp trick for numerical stability.

    The unreduced (i.e. with :attr:`reduction` set to ``'none'``) loss can be described as:

    .. math::
        \ell(x, y) = L = \{l_1,\dots,l_N\}^\top, \quad
        l_n = - w_n \left[ y_n \cdot \log \sigma(x_n)
        + (1 - y_n) \cdot \log (1 - \sigma(x_n)) \right],

    where :math:`N` is the batch size. If :attr:`reduction` is not ``'none'``
    (default ``'mean'``), then

    .. math::
        \ell(x, y) = \begin{cases}
            \operatorname{mean}(L), & \text{if reduction} = \text{`mean';}\\
            \operatorname{sum}(L),  & \text{if reduction} = \text{`sum'.}
        \end{cases}

    This is used for measuring the error of a reconstruction in for example
    an auto-encoder. Note that the targets `t[i]` should be numbers
    between 0 and 1.

    It's possible to trade off recall and precision by adding weights to positive examples.
    In the case of multi-label classification the loss can be described as:

    .. math::
        \ell_c(x, y) = L_c = \{l_{1,c},\dots,l_{N,c}\}^\top, \quad
        l_{n,c} = - w_{n,c} \left[ p_c y_{n,c} \cdot \log \sigma(x_{n,c})
        + (1 - y_{n,c}) \cdot \log (1 - \sigma(x_{n,c})) \right],

    where :math:`c` is the class number (:math:`c > 1` for multi-label binary classification,
    :math:`c = 1` for single-label binary classification),
    :math:`n` is the number of the sample in the batch and
    :math:`p_c` is the weight of the positive answer for the class :math:`c`.

    :math:`p_c > 1` increases the recall, :math:`p_c < 1` increases the precision.

    For example, if a dataset contains 100 positive and 300 negative examples of a single class,
    then `pos_weight` for the class should be equal to :math:`\frac{300}{100}=3`.
    The loss would act as if the dataset contains :math:`3\times 100=300` positive examples.

    Examples::

        >>> target = torch.ones([10, 64], dtype=torch.float32)  # 64 classes, batch size = 10
        >>> output = torch.full([10, 64], 1.5)  # A prediction (logit)
        >>> pos_weight = torch.ones([64])  # All weights are equal to 1
        >>> criterion = torch.nn.BCEWithLogitsLoss(pos_weight=pos_weight)
        >>> criterion(output, target)  # -log(sigmoid(1.5))
        tensor(0.2014)

    Args:
        weight (Tensor, optional): a manual rescaling weight given to the loss
            of each batch element. If given, has to be a Tensor of size `nbatch`.
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``
        pos_weight (Tensor, optional): a weight of positive examples.
                Must be a vector with length equal to the number of classes.

    Shape:
        - Input: :math:`(*)`, where :math:`*` means any number of dimensions.
        - Target: :math:`(*)`, same shape as the input.
        - Output: scalar. If :attr:`reduction` is ``'none'``, then :math:`(*)`, same
          shape as input.

     Examples::

        >>> loss = nn.BCEWithLogitsLoss()
        >>> input = torch.randn(3, requires_grad=True)
        >>> target = torch.empty(3).random_(2)
        >>> output = loss(input, target)
        >>> output.backward()
    """
    def __init__(self, weight: Optional[Tensor] = None, size_average=None, reduce=None, reduction: str = 'mean',
                 pos_weight: Optional[Tensor] = None) -> None:
        super(BCEWithLogitsLoss, self).__init__(size_average, reduce, reduction)
        self.register_buffer('weight', weight)
        self.register_buffer('pos_weight', pos_weight)
        self.weight: Optional[Tensor]
        self.pos_weight: Optional[Tensor]

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.binary_cross_entropy_with_logits(input, target,
                                                  self.weight,
                                                  pos_weight=self.pos_weight,
                                                  reduction=self.reduction)


class HingeEmbeddingLoss(_Loss):
    r"""Measures the loss given an input tensor :math:`x` and a labels tensor :math:`y`
    (containing 1 or -1).
    This is usually used for measuring whether two inputs are similar or
    dissimilar, e.g. using the L1 pairwise distance as :math:`x`, and is typically
    used for learning nonlinear embeddings or semi-supervised learning.

    The loss function for :math:`n`-th sample in the mini-batch is

    .. math::
        l_n = \begin{cases}
            x_n, & \text{if}\; y_n = 1,\\
            \max \{0, \Delta - x_n\}, & \text{if}\; y_n = -1,
        \end{cases}

    and the total loss functions is

    .. math::
        \ell(x, y) = \begin{cases}
            \operatorname{mean}(L), & \text{if reduction} = \text{`mean';}\\
            \operatorname{sum}(L),  & \text{if reduction} = \text{`sum'.}
        \end{cases}

    where :math:`L = \{l_1,\dots,l_N\}^\top`.

    Args:
        margin (float, optional): Has a default value of `1`.
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input: :math:`(*)` where :math:`*` means, any number of dimensions. The sum operation
          operates over all the elements.
        - Target: :math:`(*)`, same shape as the input
        - Output: scalar. If :attr:`reduction` is ``'none'``, then same shape as the input
    """
    __constants__ = ['margin', 'reduction']
    margin: float

    def __init__(self, margin: float = 1.0, size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(HingeEmbeddingLoss, self).__init__(size_average, reduce, reduction)
        self.margin = margin

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.hinge_embedding_loss(input, target, margin=self.margin, reduction=self.reduction)


class MultiLabelMarginLoss(_Loss):
    r"""Creates a criterion that optimizes a multi-class multi-classification
    hinge loss (margin-based loss) between input :math:`x` (a 2D mini-batch `Tensor`)
    and output :math:`y` (which is a 2D `Tensor` of target class indices).
    For each sample in the mini-batch:

    .. math::
        \text{loss}(x, y) = \sum_{ij}\frac{\max(0, 1 - (x[y[j]] - x[i]))}{\text{x.size}(0)}

    where :math:`x \in \left\{0, \; \cdots , \; \text{x.size}(0) - 1\right\}`, \
    :math:`y \in \left\{0, \; \cdots , \; \text{y.size}(0) - 1\right\}`, \
    :math:`0 \leq y[j] \leq \text{x.size}(0)-1`, \
    and :math:`i \neq y[j]` for all :math:`i` and :math:`j`.

    :math:`y` and :math:`x` must have the same size.

    The criterion only considers a contiguous block of non-negative targets that
    starts at the front.

    This allows for different samples to have variable amounts of target classes.

    Args:
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input: :math:`(C)` or :math:`(N, C)` where `N` is the batch size and `C`
          is the number of classes.
        - Target: :math:`(C)` or :math:`(N, C)`, label targets padded by -1 ensuring same shape as the input.
        - Output: scalar. If :attr:`reduction` is ``'none'``, then :math:`(N)`.

    Examples::

        >>> loss = nn.MultiLabelMarginLoss()
        >>> x = torch.FloatTensor([[0.1, 0.2, 0.4, 0.8]])
        >>> # for target y, only consider labels 3 and 0, not after label -1
        >>> y = torch.LongTensor([[3, 0, -1, 1]])
        >>> loss(x, y)
        >>> # 0.25 * ((1-(0.1-0.2)) + (1-(0.1-0.4)) + (1-(0.8-0.2)) + (1-(0.8-0.4)))
        tensor(0.8500)

    """
    __constants__ = ['reduction']

    def __init__(self, size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(MultiLabelMarginLoss, self).__init__(size_average, reduce, reduction)

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.multilabel_margin_loss(input, target, reduction=self.reduction)


class SmoothL1Loss(_Loss):
    r"""Creates a criterion that uses a squared term if the absolute
    element-wise error falls below beta and an L1 term otherwise.
    It is less sensitive to outliers than :class:`torch.nn.MSELoss` and in some cases
    prevents exploding gradients (e.g. see the paper `Fast R-CNN`_ by Ross Girshick).

    For a batch of size :math:`N`, the unreduced loss can be described as:

    .. math::
        \ell(x, y) = L = \{l_1, ..., l_N\}^T

    with

    .. math::
        l_n = \begin{cases}
        0.5 (x_n - y_n)^2 / beta, & \text{if } |x_n - y_n| < beta \\
        |x_n - y_n| - 0.5 * beta, & \text{otherwise }
        \end{cases}

    If `reduction` is not `none`, then:

    .. math::
        \ell(x, y) =
        \begin{cases}
            \operatorname{mean}(L), &  \text{if reduction} = \text{`mean';}\\
            \operatorname{sum}(L),  &  \text{if reduction} = \text{`sum'.}
        \end{cases}

    .. note::
        Smooth L1 loss can be seen as exactly :class:`L1Loss`, but with the :math:`|x - y| < beta`
        portion replaced with a quadratic function such that its slope is 1 at :math:`|x - y| = beta`.
        The quadratic segment smooths the L1 loss near :math:`|x - y| = 0`.

    .. note::
        Smooth L1 loss is closely related to :class:`HuberLoss`, being
        equivalent to :math:`huber(x, y) / beta` (note that Smooth L1's beta hyper-parameter is
        also known as delta for Huber). This leads to the following differences:

        * As beta -> 0, Smooth L1 loss converges to :class:`L1Loss`, while :class:`HuberLoss`
          converges to a constant 0 loss. When beta is 0, Smooth L1 loss is equivalent to L1 loss.
        * As beta -> :math:`+\infty`, Smooth L1 loss converges to a constant 0 loss, while
          :class:`HuberLoss` converges to :class:`MSELoss`.
        * For Smooth L1 loss, as beta varies, the L1 segment of the loss has a constant slope of 1.
          For :class:`HuberLoss`, the slope of the L1 segment is beta.

    .. _`Fast R-CNN`: https://arxiv.org/abs/1504.08083

    Args:
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``
        beta (float, optional): Specifies the threshold at which to change between L1 and L2 loss.
            The value must be non-negative. Default: 1.0

    Shape:
        - Input: :math:`(*)`, where :math:`*` means any number of dimensions.
        - Target: :math:`(*)`, same shape as the input.
        - Output: scalar. If :attr:`reduction` is ``'none'``, then :math:`(*)`, same shape as the input.
    """
    __constants__ = ['reduction']

    def __init__(self, size_average=None, reduce=None, reduction: str = 'mean', beta: float = 1.0) -> None:
        super(SmoothL1Loss, self).__init__(size_average, reduce, reduction)
        self.beta = beta

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.smooth_l1_loss(input, target, reduction=self.reduction, beta=self.beta)


class HuberLoss(_Loss):
    r"""Creates a criterion that uses a squared term if the absolute
    element-wise error falls below delta and a delta-scaled L1 term otherwise.
    This loss combines advantages of both :class:`L1Loss` and :class:`MSELoss`; the
    delta-scaled L1 region makes the loss less sensitive to outliers than :class:`MSELoss`,
    while the L2 region provides smoothness over :class:`L1Loss` near 0. See
    `Huber loss <https://en.wikipedia.org/wiki/Huber_loss>`_ for more information.

    For a batch of size :math:`N`, the unreduced loss can be described as:

    .. math::
        \ell(x, y) = L = \{l_1, ..., l_N\}^T

    with

    .. math::
        l_n = \begin{cases}
        0.5 (x_n - y_n)^2, & \text{if } |x_n - y_n| < delta \\
        delta * (|x_n - y_n| - 0.5 * delta), & \text{otherwise }
        \end{cases}

    If `reduction` is not `none`, then:

    .. math::
        \ell(x, y) =
        \begin{cases}
            \operatorname{mean}(L), &  \text{if reduction} = \text{`mean';}\\
            \operatorname{sum}(L),  &  \text{if reduction} = \text{`sum'.}
        \end{cases}

    .. note::
        When delta is set to 1, this loss is equivalent to :class:`SmoothL1Loss`.
        In general, this loss differs from :class:`SmoothL1Loss` by a factor of delta (AKA beta
        in Smooth L1).
        See :class:`SmoothL1Loss` for additional discussion on the differences in behavior
        between the two losses.

    Args:
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Default: ``'mean'``
        delta (float, optional): Specifies the threshold at which to change between delta-scaled L1 and L2 loss.
            The value must be positive.  Default: 1.0

    Shape:
        - Input: :math:`(*)` where :math:`*` means any number of dimensions.
        - Target: :math:`(*)`, same shape as the input.
        - Output: scalar. If :attr:`reduction` is ``'none'``, then :math:`(*)`, same shape as the input.
    """
    __constants__ = ['reduction', 'delta']

    def __init__(self, reduction: str = 'mean', delta: float = 1.0) -> None:
        super().__init__(reduction=reduction)
        self.delta = delta

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.huber_loss(input, target, reduction=self.reduction, delta=self.delta)


class SoftMarginLoss(_Loss):
    r"""Creates a criterion that optimizes a two-class classification
    logistic loss between input tensor :math:`x` and target tensor :math:`y`
    (containing 1 or -1).

    .. math::
        \text{loss}(x, y) = \sum_i \frac{\log(1 + \exp(-y[i]*x[i]))}{\text{x.nelement}()}

    Args:
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input: :math:`(*)`, where :math:`*` means any number of dimensions.
        - Target: :math:`(*)`, same shape as the input.
        - Output: scalar. If :attr:`reduction` is ``'none'``, then :math:`(*)`, same
          shape as input.

    """
    __constants__ = ['reduction']

    def __init__(self, size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(SoftMarginLoss, self).__init__(size_average, reduce, reduction)

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.soft_margin_loss(input, target, reduction=self.reduction)


class CrossEntropyLoss(_WeightedLoss):
    r"""This criterion computes the cross entropy loss between input and target.

    It is useful when training a classification problem with `C` classes.
    If provided, the optional argument :attr:`weight` should be a 1D `Tensor`
    assigning weight to each of the classes.
    This is particularly useful when you have an unbalanced training set.

    The `input` is expected to contain raw, unnormalized scores for each class.
    `input` has to be a Tensor of size :math:`(C)` for unbatched input,
    :math:`(minibatch, C)` or :math:`(minibatch, C, d_1, d_2, ..., d_K)` with :math:`K \geq 1` for the
    `K`-dimensional case. The last being useful for higher dimension inputs, such
    as computing cross entropy loss per-pixel for 2D images.

    The `target` that this criterion expects should contain either:

    - Class indices in the range :math:`[0, C)` where :math:`C` is the number of classes; if
      `ignore_index` is specified, this loss also accepts this class index (this index
      may not necessarily be in the class range). The unreduced (i.e. with :attr:`reduction`
      set to ``'none'``) loss for this case can be described as:

      .. math::
          \ell(x, y) = L = \{l_1,\dots,l_N\}^\top, \quad
          l_n = - w_{y_n} \log \frac{\exp(x_{n,y_n})}{\sum_{c=1}^C \exp(x_{n,c})}
          \cdot \mathbb{1}\{y_n \not= \text{ignore\_index}\}

      where :math:`x` is the input, :math:`y` is the target, :math:`w` is the weight,
      :math:`C` is the number of classes, and :math:`N` spans the minibatch dimension as well as
      :math:`d_1, ..., d_k` for the `K`-dimensional case. If
      :attr:`reduction` is not ``'none'`` (default ``'mean'``), then

      .. math::
          \ell(x, y) = \begin{cases}
              \sum_{n=1}^N \frac{1}{\sum_{n=1}^N w_{y_n} \cdot \mathbb{1}\{y_n \not= \text{ignore\_index}\}} l_n, &
               \text{if reduction} = \text{`mean';}\\
                \sum_{n=1}^N l_n,  &
                \text{if reduction} = \text{`sum'.}
            \end{cases}

      Note that this case is equivalent to the combination of :class:`~torch.nn.LogSoftmax` and
      :class:`~torch.nn.NLLLoss`.

    - Probabilities for each class; useful when labels beyond a single class per minibatch item
      are required, such as for blended labels, label smoothing, etc. The unreduced (i.e. with
      :attr:`reduction` set to ``'none'``) loss for this case can be described as:

      .. math::
          \ell(x, y) = L = \{l_1,\dots,l_N\}^\top, \quad
          l_n = - \sum_{c=1}^C w_c \log \frac{\exp(x_{n,c})}{\sum_{i=1}^C \exp(x_{n,i})} y_{n,c}

      where :math:`x` is the input, :math:`y` is the target, :math:`w` is the weight,
      :math:`C` is the number of classes, and :math:`N` spans the minibatch dimension as well as
      :math:`d_1, ..., d_k` for the `K`-dimensional case. If
      :attr:`reduction` is not ``'none'`` (default ``'mean'``), then

      .. math::
          \ell(x, y) = \begin{cases}
              \frac{\sum_{n=1}^N l_n}{N}, &
               \text{if reduction} = \text{`mean';}\\
                \sum_{n=1}^N l_n,  &
                \text{if reduction} = \text{`sum'.}
            \end{cases}

    .. note::
        The performance of this criterion is generally better when `target` contains class
        indices, as this allows for optimized computation. Consider providing `target` as
        class probabilities only when a single class label per minibatch item is too restrictive.

    Args:
        weight (Tensor, optional): a manual rescaling weight given to each class.
            If given, has to be a Tensor of size `C`
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        ignore_index (int, optional): Specifies a target value that is ignored
            and does not contribute to the input gradient. When :attr:`size_average` is
            ``True``, the loss is averaged over non-ignored targets. Note that
            :attr:`ignore_index` is only applicable when the target contains class indices.
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will
            be applied, ``'mean'``: the weighted mean of the output is taken,
            ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in
            the meantime, specifying either of those two args will override
            :attr:`reduction`. Default: ``'mean'``
        label_smoothing (float, optional): A float in [0.0, 1.0]. Specifies the amount
            of smoothing when computing the loss, where 0.0 means no smoothing. The targets
            become a mixture of the original ground truth and a uniform distribution as described in
            `Rethinking the Inception Architecture for Computer Vision <https://arxiv.org/abs/1512.00567>`__. Default: :math:`0.0`.

    Shape:
        - Input: Shape :math:`(C)`, :math:`(N, C)` or :math:`(N, C, d_1, d_2, ..., d_K)` with :math:`K \geq 1`
          in the case of `K`-dimensional loss.
        - Target: If containing class indices, shape :math:`()`, :math:`(N)` or :math:`(N, d_1, d_2, ..., d_K)` with
          :math:`K \geq 1` in the case of K-dimensional loss where each value should be between :math:`[0, C)`.
          If containing class probabilities, same shape as the input and each value should be between :math:`[0, 1]`.
        - Output: If reduction is 'none', same shape as the target. Otherwise, scalar.

        where:

        .. math::
            \begin{aligned}
                C ={} & \text{number of classes} \\
                N ={} & \text{batch size} \\
            \end{aligned}

    Examples::

        >>> # Example of target with class indices
        >>> loss = nn.CrossEntropyLoss()
        >>> input = torch.randn(3, 5, requires_grad=True)
        >>> target = torch.empty(3, dtype=torch.long).random_(5)
        >>> output = loss(input, target)
        >>> output.backward()
        >>>
        >>> # Example of target with class probabilities
        >>> input = torch.randn(3, 5, requires_grad=True)
        >>> target = torch.randn(3, 5).softmax(dim=1)
        >>> output = loss(input, target)
        >>> output.backward()
    """
    __constants__ = ['ignore_index', 'reduction', 'label_smoothing']
    ignore_index: int
    label_smoothing: float

    def __init__(self, weight: Optional[Tensor] = None, size_average=None, ignore_index: int = -100,
                 reduce=None, reduction: str = 'mean', label_smoothing: float = 0.0) -> None:
        super(CrossEntropyLoss, self).__init__(weight, size_average, reduce, reduction)
        self.ignore_index = ignore_index
        self.label_smoothing = label_smoothing

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.cross_entropy(input, target, weight=self.weight,
                               ignore_index=self.ignore_index, reduction=self.reduction,
                               label_smoothing=self.label_smoothing)


class MultiLabelSoftMarginLoss(_WeightedLoss):
    r"""Creates a criterion that optimizes a multi-label one-versus-all
    loss based on max-entropy, between input :math:`x` and target :math:`y` of size
    :math:`(N, C)`.
    For each sample in the minibatch:

    .. math::
        loss(x, y) = - \frac{1}{C} * \sum_i y[i] * \log((1 + \exp(-x[i]))^{-1})
                         + (1-y[i]) * \log\left(\frac{\exp(-x[i])}{(1 + \exp(-x[i]))}\right)

    where :math:`i \in \left\{0, \; \cdots , \; \text{x.nElement}() - 1\right\}`,
    :math:`y[i] \in \left\{0, \; 1\right\}`.

    Args:
        weight (Tensor, optional): a manual rescaling weight given to each
            class. If given, it has to be a Tensor of size `C`. Otherwise, it is
            treated as if having all ones.
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input: :math:`(N, C)` where `N` is the batch size and `C` is the number of classes.
        - Target: :math:`(N, C)`, label targets padded by -1 ensuring same shape as the input.
        - Output: scalar. If :attr:`reduction` is ``'none'``, then :math:`(N)`.
    """
    __constants__ = ['reduction']

    def __init__(self, weight: Optional[Tensor] = None, size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(MultiLabelSoftMarginLoss, self).__init__(weight, size_average, reduce, reduction)

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.multilabel_soft_margin_loss(input, target, weight=self.weight, reduction=self.reduction)


class CosineEmbeddingLoss(_Loss):
    r"""Creates a criterion that measures the loss given input tensors
    :math:`x_1`, :math:`x_2` and a `Tensor` label :math:`y` with values 1 or -1.
    This is used for measuring whether two inputs are similar or dissimilar,
    using the cosine similarity, and is typically used for learning nonlinear
    embeddings or semi-supervised learning.

    The loss function for each sample is:

    .. math::
        \text{loss}(x, y) =
        \begin{cases}
        1 - \cos(x_1, x_2), & \text{if } y = 1 \\
        \max(0, \cos(x_1, x_2) - \text{margin}), & \text{if } y = -1
        \end{cases}

    Args:
        margin (float, optional): Should be a number from :math:`-1` to :math:`1`,
            :math:`0` to :math:`0.5` is suggested. If :attr:`margin` is missing, the
            default value is :math:`0`.
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input1: :math:`(N, D)` or :math:`(D)`, where `N` is the batch size and `D` is the embedding dimension.
        - Input2: :math:`(N, D)` or :math:`(D)`, same shape as Input1.
        - Target: :math:`(N)` or :math:`()`.
        - Output: If :attr:`reduction` is ``'none'``, then :math:`(N)`, otherwise scalar.
    """
    __constants__ = ['margin', 'reduction']
    margin: float

    def __init__(self, margin: float = 0., size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(CosineEmbeddingLoss, self).__init__(size_average, reduce, reduction)
        self.margin = margin

    def forward(self, input1: Tensor, input2: Tensor, target: Tensor) -> Tensor:
        return F.cosine_embedding_loss(input1, input2, target, margin=self.margin, reduction=self.reduction)


class MarginRankingLoss(_Loss):
    r"""Creates a criterion that measures the loss given
    inputs :math:`x1`, :math:`x2`, two 1D mini-batch or 0D `Tensors`,
    and a label 1D mini-batch or 0D `Tensor` :math:`y` (containing 1 or -1).

    If :math:`y = 1` then it assumed the first input should be ranked higher
    (have a larger value) than the second input, and vice-versa for :math:`y = -1`.

    The loss function for each pair of samples in the mini-batch is:

    .. math::
        \text{loss}(x1, x2, y) = \max(0, -y * (x1 - x2) + \text{margin})

    Args:
        margin (float, optional): Has a default value of :math:`0`.
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input1: :math:`(N)` or :math:`()` where `N` is the batch size.
        - Input2: :math:`(N)` or :math:`()`, same shape as the Input1.
        - Target: :math:`(N)` or :math:`()`, same shape as the inputs.
        - Output: scalar. If :attr:`reduction` is ``'none'`` and Input size is not :math:`()`, then :math:`(N)`.

    Examples::

        >>> loss = nn.MarginRankingLoss()
        >>> input1 = torch.randn(3, requires_grad=True)
        >>> input2 = torch.randn(3, requires_grad=True)
        >>> target = torch.randn(3).sign()
        >>> output = loss(input1, input2, target)
        >>> output.backward()
    """
    __constants__ = ['margin', 'reduction']
    margin: float

    def __init__(self, margin: float = 0., size_average=None, reduce=None, reduction: str = 'mean') -> None:
        super(MarginRankingLoss, self).__init__(size_average, reduce, reduction)
        self.margin = margin

    def forward(self, input1: Tensor, input2: Tensor, target: Tensor) -> Tensor:
        return F.margin_ranking_loss(input1, input2, target, margin=self.margin, reduction=self.reduction)


class MultiMarginLoss(_WeightedLoss):
    r"""Creates a criterion that optimizes a multi-class classification hinge
    loss (margin-based loss) between input :math:`x` (a 2D mini-batch `Tensor`) and
    output :math:`y` (which is a 1D tensor of target class indices,
    :math:`0 \leq y \leq \text{x.size}(1)-1`):

    For each mini-batch sample, the loss in terms of the 1D input :math:`x` and scalar
    output :math:`y` is:

    .. math::
        \text{loss}(x, y) = \frac{\sum_i \max(0, \text{margin} - x[y] + x[i])^p}{\text{x.size}(0)}

    where :math:`x \in \left\{0, \; \cdots , \; \text{x.size}(0) - 1\right\}`
    and :math:`i \neq y`.

    Optionally, you can give non-equal weighting on the classes by passing
    a 1D :attr:`weight` tensor into the constructor.

    The loss function then becomes:

    .. math::
        \text{loss}(x, y) = \frac{\sum_i \max(0, w[y] * (\text{margin} - x[y] + x[i]))^p}{\text{x.size}(0)}

    Args:
        p (int, optional): Has a default value of :math:`1`. :math:`1` and :math:`2`
            are the only supported values.
        margin (float, optional): Has a default value of :math:`1`.
        weight (Tensor, optional): a manual rescaling weight given to each
            class. If given, it has to be a Tensor of size `C`. Otherwise, it is
            treated as if having all ones.
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input: :math:`(N, C)` or :math:`(C)`, where :math:`N` is the batch size and :math:`C` is the number of classes.
        - Target: :math:`(N)` or :math:`()`, where each value is :math:`0 \leq \text{targets}[i] \leq C-1`.
        - Output: scalar. If :attr:`reduction` is ``'none'``, then same shape as the target.

    Examples::

        >>> loss = nn.MultiMarginLoss()
        >>> x = torch.tensor([[0.1, 0.2, 0.4, 0.8]])
        >>> y = torch.tensor([3])
        >>> loss(x, y)
        >>> # 0.25 * ((1-(0.8-0.1)) + (1-(0.8-0.2)) + (1-(0.8-0.4)))
        tensor(0.3250)
    """
    __constants__ = ['p', 'margin', 'reduction']
    margin: float
    p: int

    def __init__(self, p: int = 1, margin: float = 1., weight: Optional[Tensor] = None, size_average=None,
                 reduce=None, reduction: str = 'mean') -> None:
        super(MultiMarginLoss, self).__init__(weight, size_average, reduce, reduction)
        if p != 1 and p != 2:
            raise ValueError("only p == 1 and p == 2 supported")
        assert weight is None or weight.dim() == 1
        self.p = p
        self.margin = margin

    def forward(self, input: Tensor, target: Tensor) -> Tensor:
        return F.multi_margin_loss(input, target, p=self.p, margin=self.margin,
                                   weight=self.weight, reduction=self.reduction)


class TripletMarginLoss(_Loss):
    r"""Creates a criterion that measures the triplet loss given an input
    tensors :math:`x1`, :math:`x2`, :math:`x3` and a margin with a value greater than :math:`0`.
    This is used for measuring a relative similarity between samples. A triplet
    is composed by `a`, `p` and `n` (i.e., `anchor`, `positive examples` and `negative
    examples` respectively). The shapes of all input tensors should be
    :math:`(N, D)`.

    The distance swap is described in detail in the paper `Learning shallow
    convolutional feature descriptors with triplet losses`_ by
    V. Balntas, E. Riba et al.

    The loss function for each sample in the mini-batch is:

    .. math::
        L(a, p, n) = \max \{d(a_i, p_i) - d(a_i, n_i) + {\rm margin}, 0\}


    where

    .. math::
        d(x_i, y_i) = \left\lVert {\bf x}_i - {\bf y}_i \right\rVert_p

    See also :class:`~torch.nn.TripletMarginWithDistanceLoss`, which computes the
    triplet margin loss for input tensors using a custom distance function.

    Args:
        margin (float, optional): Default: :math:`1`.
        p (int, optional): The norm degree for pairwise distance. Default: :math:`2`.
        swap (bool, optional): The distance swap is described in detail in the paper
            `Learning shallow convolutional feature descriptors with triplet losses` by
            V. Balntas, E. Riba et al. Default: ``False``.
        size_average (bool, optional): Deprecated (see :attr:`reduction`). By default,
            the losses are averaged over each loss element in the batch. Note that for
            some losses, there are multiple elements per sample. If the field :attr:`size_average`
            is set to ``False``, the losses are instead summed for each minibatch. Ignored
            when :attr:`reduce` is ``False``. Default: ``True``
        reduce (bool, optional): Deprecated (see :attr:`reduction`). By default, the
            losses are averaged or summed over observations for each minibatch depending
            on :attr:`size_average`. When :attr:`reduce` is ``False``, returns a loss per
            batch element instead and ignores :attr:`size_average`. Default: ``True``
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Note: :attr:`size_average`
            and :attr:`reduce` are in the process of being deprecated, and in the meantime,
            specifying either of those two args will override :attr:`reduction`. Default: ``'mean'``

    Shape:
        - Input: :math:`(N, D)` or :math:`(D)` where :math:`D` is the vector dimension.
        - Output: A Tensor of shape :math:`(N)` if :attr:`reduction` is ``'none'`` and
          input shape is :math:`(N, D)`; a scalar otherwise.

    Examples::

    >>> triplet_loss = nn.TripletMarginLoss(margin=1.0, p=2)
    >>> anchor = torch.randn(100, 128, requires_grad=True)
    >>> positive = torch.randn(100, 128, requires_grad=True)
    >>> negative = torch.randn(100, 128, requires_grad=True)
    >>> output = triplet_loss(anchor, positive, negative)
    >>> output.backward()

    .. _Learning shallow convolutional feature descriptors with triplet losses:
        http://www.bmva.org/bmvc/2016/papers/paper119/index.html
    """
    __constants__ = ['margin', 'p', 'eps', 'swap', 'reduction']
    margin: float
    p: float
    eps: float
    swap: bool

    def __init__(self, margin: float = 1.0, p: float = 2., eps: float = 1e-6, swap: bool = False, size_average=None,
                 reduce=None, reduction: str = 'mean'):
        super(TripletMarginLoss, self).__init__(size_average, reduce, reduction)
        self.margin = margin
        self.p = p
        self.eps = eps
        self.swap = swap

    def forward(self, anchor: Tensor, positive: Tensor, negative: Tensor) -> Tensor:
        return F.triplet_margin_loss(anchor, positive, negative, margin=self.margin, p=self.p,
                                     eps=self.eps, swap=self.swap, reduction=self.reduction)


class TripletMarginWithDistanceLoss(_Loss):
    r"""Creates a criterion that measures the triplet loss given input
    tensors :math:`a`, :math:`p`, and :math:`n` (representing anchor,
    positive, and negative examples, respectively), and a nonnegative,
    real-valued function ("distance function") used to compute the relationship
    between the anchor and positive example ("positive distance") and the
    anchor and negative example ("negative distance").

    The unreduced loss (i.e., with :attr:`reduction` set to ``'none'``)
    can be described as:

    .. math::
        \ell(a, p, n) = L = \{l_1,\dots,l_N\}^\top, \quad
        l_i = \max \{d(a_i, p_i) - d(a_i, n_i) + {\rm margin}, 0\}

    where :math:`N` is the batch size; :math:`d` is a nonnegative, real-valued function
    quantifying the closeness of two tensors, referred to as the :attr:`distance_function`;
    and :math:`margin` is a nonnegative margin representing the minimum difference
    between the positive and negative distances that is required for the loss to
    be 0.  The input tensors have :math:`N` elements each and can be of any shape
    that the distance function can handle.

    If :attr:`reduction` is not ``'none'``
    (default ``'mean'``), then:

    .. math::
        \ell(x, y) =
        \begin{cases}
            \operatorname{mean}(L), &  \text{if reduction} = \text{`mean';}\\
            \operatorname{sum}(L),  &  \text{if reduction} = \text{`sum'.}
        \end{cases}

    See also :class:`~torch.nn.TripletMarginLoss`, which computes the triplet
    loss for input tensors using the :math:`l_p` distance as the distance function.

    Args:
        distance_function (callable, optional): A nonnegative, real-valued function that
            quantifies the closeness of two tensors. If not specified,
            `nn.PairwiseDistance` will be used.  Default: ``None``
        margin (float, optional): A nonnegative margin representing the minimum difference
            between the positive and negative distances required for the loss to be 0. Larger
            margins penalize cases where the negative examples are not distant enough from the
            anchors, relative to the positives. Default: :math:`1`.
        swap (bool, optional): Whether to use the distance swap described in the paper
            `Learning shallow convolutional feature descriptors with triplet losses` by
            V. Balntas, E. Riba et al. If True, and if the positive example is closer to the
            negative example than the anchor is, swaps the positive example and the anchor in
            the loss computation. Default: ``False``.
        reduction (string, optional): Specifies the (optional) reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the sum of the output will be divided by the number of
            elements in the output, ``'sum'``: the output will be summed. Default: ``'mean'``


    Shape:
        - Input: :math:`(N, *)` where :math:`*` represents any number of additional dimensions
          as supported by the distance function.
        - Output: A Tensor of shape :math:`(N)` if :attr:`reduction` is ``'none'``, or a scalar
          otherwise.

    Examples::

    >>> # Initialize embeddings
    >>> embedding = nn.Embedding(1000, 128)
    >>> anchor_ids = torch.randint(0, 1000, (1,))
    >>> positive_ids = torch.randint(0, 1000, (1,))
    >>> negative_ids = torch.randint(0, 1000, (1,))
    >>> anchor = embedding(anchor_ids)
    >>> positive = embedding(positive_ids)
    >>> negative = embedding(negative_ids)
    >>>
    >>> # Built-in Distance Function
    >>> triplet_loss = \
    >>>     nn.TripletMarginWithDistanceLoss(distance_function=nn.PairwiseDistance())
    >>> output = triplet_loss(anchor, positive, negative)
    >>> output.backward()
    >>>
    >>> # Custom Distance Function
    >>> def l_infinity(x1, x2):
    >>>     return torch.max(torch.abs(x1 - x2), dim=1).values
    >>>
    >>> triplet_loss = \
    >>>     nn.TripletMarginWithDistanceLoss(distance_function=l_infinity, margin=1.5)
    >>> output = triplet_loss(anchor, positive, negative)
    >>> output.backward()
    >>>
    >>> # Custom Distance Function (Lambda)
    >>> triplet_loss = \
    >>>     nn.TripletMarginWithDistanceLoss(
    >>>         distance_function=lambda x, y: 1.0 - F.cosine_similarity(x, y))
    >>> output = triplet_loss(anchor, positive, negative)
    >>> output.backward()

    Reference:
        V. Balntas, et al.: Learning shallow convolutional feature descriptors with triplet losses:
        http://www.bmva.org/bmvc/2016/papers/paper119/index.html
    """
    __constants__ = ['margin', 'swap', 'reduction']
    margin: float
    swap: bool

    def __init__(self, *, distance_function: Optional[Callable[[Tensor, Tensor], Tensor]] = None,
                 margin: float = 1.0, swap: bool = False, reduction: str = 'mean'):
        super(TripletMarginWithDistanceLoss, self).__init__(size_average=None, reduce=None, reduction=reduction)
        self.distance_function: Optional[Callable[[Tensor, Tensor], Tensor]] = \
            distance_function if distance_function is not None else PairwiseDistance()
        self.margin = margin
        self.swap = swap

    def forward(self, anchor: Tensor, positive: Tensor, negative: Tensor) -> Tensor:
        return F.triplet_margin_with_distance_loss(anchor, positive, negative,
                                                   distance_function=self.distance_function,
                                                   margin=self.margin, swap=self.swap, reduction=self.reduction)


class CTCLoss(_Loss):
    r"""The Connectionist Temporal Classification loss.

    Calculates loss between a continuous (unsegmented) time series and a target sequence. CTCLoss sums over the
    probability of possible alignments of input to target, producing a loss value which is differentiable
    with respect to each input node. The alignment of input to target is assumed to be "many-to-one", which
    limits the length of the target sequence such that it must be :math:`\leq` the input length.

    Args:
        blank (int, optional): blank label. Default :math:`0`.
        reduction (string, optional): Specifies the reduction to apply to the output:
            ``'none'`` | ``'mean'`` | ``'sum'``. ``'none'``: no reduction will be applied,
            ``'mean'``: the output losses will be divided by the target lengths and
            then the mean over the batch is taken. Default: ``'mean'``
        zero_infinity (bool, optional):
            Whether to zero infinite losses and the associated gradients.
            Default: ``False``
            Infinite losses mainly occur when the inputs are too short
            to be aligned to the targets.

    Shape:
        - Log_probs: Tensor of size :math:`(T, N, C)` or :math:`(T, C)`,
          where :math:`T = \text{input length}`,
          :math:`N = \text{batch size}`, and
          :math:`C = \text{number of classes (including blank)}`.
          The logarithmized probabilities of the outputs (e.g. obtained with
          :func:`torch.nn.functional.log_softmax`).
        - Targets: Tensor of size :math:`(N, S)` or
          :math:`(\operatorname{sum}(\text{target\_lengths}))`,
          where :math:`N = \text{batch size}` and
          :math:`S = \text{max target length, if shape is } (N, S)`.
          It represent the target sequences. Each element in the target
          sequence is a class index. And the target index cannot be blank (default=0).
          In the :math:`(N, S)` form, targets are padded to the
          length of the longest sequence, and stacked.
          In the :math:`(\operatorname{sum}(\text{target\_lengths}))` form,
          the targets are assumed to be un-padded and
          concatenated within 1 dimension.
        - Input_lengths: Tuple or tensor of size :math:`(N)` or :math:`()`,
          where :math:`N = \text{batch size}`. It represent the lengths of the
          inputs (must each be :math:`\leq T`). And the lengths are specified
          for each sequence to achieve masking under the assumption that sequences
          are padded to equal lengths.
        - Target_lengths: Tuple or tensor of size :math:`(N)` or :math:`()`,
          where :math:`N = \text{batch size}`. It represent lengths of the targets.
          Lengths are specified for each sequence to achieve masking under the
          assumption that sequences are padded to equal lengths. If target shape is
          :math:`(N,S)`, target_lengths are effectively the stop index
          :math:`s_n` for each target sequence, such that ``target_n = targets[n,0:s_n]`` for
          each target in a batch. Lengths must each be :math:`\leq S`
          If the targets are given as a 1d tensor that is the concatenation of individual
          targets, the target_lengths must add up to the total length of the tensor.
        - Output: scalar. If :attr:`reduction` is ``'none'``, then
          :math:`(N)` if input is batched or :math:`()` if input is unbatched, where :math:`N = \text{batch size}`.

    Examples::

        >>> # Target are to be padded
        >>> T = 50      # Input sequence length
        >>> C = 20      # Number of classes (including blank)
        >>> N = 16      # Batch size
        >>> S = 30      # Target sequence length of longest target in batch (padding length)
        >>> S_min = 10  # Minimum target length, for demonstration purposes
        >>>
        >>> # Initialize random batch of input vectors, for *size = (T,N,C)
        >>> input = torch.randn(T, N, C).log_softmax(2).detach().requires_grad_()
        >>>
        >>> # Initialize random batch of targets (0 = blank, 1:C = classes)
        >>> target = torch.randint(low=1, high=C, size=(N, S), dtype=torch.long)
        >>>
        >>> input_lengths = torch.full(size=(N,), fill_value=T, dtype=torch.long)
        >>> target_lengths = torch.randint(low=S_min, high=S, size=(N,), dtype=torch.long)
        >>> ctc_loss = nn.CTCLoss()
        >>> loss = ctc_loss(input, target, input_lengths, target_lengths)
        >>> loss.backward()
        >>>
        >>>
        >>> # Target are to be un-padded
        >>> T = 50      # Input sequence length
        >>> C = 20      # Number of classes (including blank)
        >>> N = 16      # Batch size
        >>>
        >>> # Initialize random batch of input vectors, for *size = (T,N,C)
        >>> input = torch.randn(T, N, C).log_softmax(2).detach().requires_grad_()
        >>> input_lengths = torch.full(size=(N,), fill_value=T, dtype=torch.long)
        >>>
        >>> # Initialize random batch of targets (0 = blank, 1:C = classes)
        >>> target_lengths = torch.randint(low=1, high=T, size=(N,), dtype=torch.long)
        >>> target = torch.randint(low=1, high=C, size=(sum(target_lengths),), dtype=torch.long)
        >>> ctc_loss = nn.CTCLoss()
        >>> loss = ctc_loss(input, target, input_lengths, target_lengths)
        >>> loss.backward()
        >>>
        >>>
        >>> # Target are to be un-padded and unbatched (effectively N=1)
        >>> T = 50      # Input sequence length
        >>> C = 20      # Number of classes (including blank)
        >>>
        >>> # Initialize random batch of input vectors, for *size = (T,C)
        >>> input = torch.randn(T, C).log_softmax(2).detach().requires_grad_()
        >>> input_lengths = torch.tensor(T, dtype=torch.long)
        >>>
        >>> # Initialize random batch of targets (0 = blank, 1:C = classes)
        >>> target_lengths = torch.randint(low=1, high=T, size=(), dtype=torch.long)
        >>> target = torch.randint(low=1, high=C, size=(target_lengths,), dtype=torch.long)
        >>> ctc_loss = nn.CTCLoss()
        >>> loss = ctc_loss(input, target, input_lengths, target_lengths)
        >>> loss.backward()

    Reference:
        A. Graves et al.: Connectionist Temporal Classification:
        Labelling Unsegmented Sequence Data with Recurrent Neural Networks:
        https://www.cs.toronto.edu/~graves/icml_2006.pdf

    Note:
        In order to use CuDNN, the following must be satisfied: :attr:`targets` must be
        in concatenated format, all :attr:`input_lengths` must be `T`.  :math:`blank=0`,
        :attr:`target_lengths` :math:`\leq 256`, the integer arguments must be of
        dtype :attr:`torch.int32`.

        The regular implementation uses the (more common in PyTorch) `torch.long` dtype.


    Note:
        In some circumstances when using the CUDA backend with CuDNN, this operator
        may select a nondeterministic algorithm to increase performance. If this is
        undesirable, you can try to make the operation deterministic (potentially at
        a performance cost) by setting ``torch.backends.cudnn.deterministic =
        True``.
        Please see the notes on :doc:`/notes/randomness` for background.
    """
    __constants__ = ['blank', 'reduction']
    blank: int
    zero_infinity: bool

    def __init__(self, blank: int = 0, reduction: str = 'mean', zero_infinity: bool = False):
        super(CTCLoss, self).__init__(reduction=reduction)
        self.blank = blank
        self.zero_infinity = zero_infinity

    def forward(self, log_probs: Tensor, targets: Tensor, input_lengths: Tensor, target_lengths: Tensor) -> Tensor:
        return F.ctc_loss(log_probs, targets, input_lengths, target_lengths, self.blank, self.reduction,
                          self.zero_infinity)

# TODO: L1HingeEmbeddingCriterion
# TODO: MSECriterion weight
# TODO: ClassSimplexCriterion