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# SmoothL1Loss¶

class torch.nn.SmoothL1Loss(size_average=None, reduce=None, reduction='mean', beta=1.0)[source]

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 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 $N$ , the unreduced loss can be described as:

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

with

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

If reduction is not none, then:

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

Note

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

Note

Smooth L1 loss is closely related to HuberLoss, being equivalent to $huber(x, y) / beta$ (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 L1Loss, while HuberLoss converges to a constant 0 loss.

• As beta -> $+\infty$ , Smooth L1 loss converges to a constant 0 loss, while HuberLoss converges to MSELoss.

• For Smooth L1 loss, as beta varies, the L1 segment of the loss has a constant slope of 1. For HuberLoss, the slope of the L1 segment is beta.

Parameters
• size_average (bool, optional) – Deprecated (see 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 size_average is set to False, the losses are instead summed for each minibatch. Ignored when reduce is False. Default: True

• reduce (bool, optional) – Deprecated (see reduction). By default, the losses are averaged or summed over observations for each minibatch depending on size_average. When reduce is False, returns a loss per batch element instead and ignores 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: size_average and reduce are in the process of being deprecated, and in the meantime, specifying either of those two args will override 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: $(N, *)$ where $*$ means any number of additional dimensions

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

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

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