The Kullback-Leibler divergence loss measure Kullback-Leibler divergence is a useful distance measure for continuous distributions and is often useful when performing direct regression over the space of (discretely sampled) continuous output distributions.

## Usage

nn_kl_div_loss(reduction = "mean")

## Arguments

reduction

(string, optional): Specifies the reduction to apply to the output: 'none' | 'batchmean' | 'sum' | 'mean'. 'none': no reduction will be applied. 'batchmean': the sum of the output will be divided by batchsize. 'sum': the output will be summed. 'mean': the output will be divided by the number of elements in the output. Default: 'mean'

## Details

As with nn_nll_loss(), the input given is expected to contain log-probabilities and is not restricted to a 2D Tensor.

The targets are interpreted as probabilities by default, but could be considered as log-probabilities with log_target set to TRUE.

This criterion expects a target Tensor of the same size as the input Tensor.

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

$$l(x,y) = L = \{ l_1,\dots,l_N \}, \quad l_n = y_n \cdot \left( \log y_n - x_n \right)$$

where the index $$N$$ spans all dimensions of input and $$L$$ has the same shape as input. If reduction is not 'none' (default 'mean'), then:

$$\ell(x, y) = \begin{array}{ll} \mbox{mean}(L), & \mbox{if reduction} = \mbox{'mean';} \\ \mbox{sum}(L), & \mbox{if reduction} = \mbox{'sum'.} \end{array}$$

In default reduction mode 'mean', the losses are averaged for each minibatch over observations as well as over dimensions. 'batchmean' mode gives the correct KL divergence where losses are averaged over batch dimension only. 'mean' mode's behavior will be changed to the same as 'batchmean' in the next major release.

## Note

reduction = 'mean' doesn't return the true kl divergence value, please use reduction = 'batchmean' which aligns with KL math definition. In the next major release, 'mean' will be changed to be the same as 'batchmean'.

## Shape

• Input: $$(N, *)$$ where $$*$$ means, any number of additional dimensions

• Target: $$(N, *)$$, same shape as the input

• Output: scalar by default. If reduction is 'none', then $$(N, *)$$, the same shape as the input