# Source code for fvcore.nn.smooth_l1_loss

```
# Copyright (c) Facebook, Inc. and its affiliates. All Rights Reserved.
import torch
[docs]def smooth_l1_loss(
input: torch.Tensor, target: torch.Tensor, beta: float, reduction: str = "none"
) -> torch.Tensor:
"""
Smooth L1 loss defined in the Fast R-CNN paper as:
::
| 0.5 * x ** 2 / beta if abs(x) < beta
smoothl1(x) = |
| abs(x) - 0.5 * beta otherwise,
where x = input - target.
Smooth L1 loss is related to Huber loss, which is defined as:
::
| 0.5 * x ** 2 if abs(x) < beta
huber(x) = |
| beta * (abs(x) - 0.5 * beta) otherwise
Smooth L1 loss is equal to huber(x) / beta. This leads to the following
differences:
- As beta -> 0, Smooth L1 loss converges to L1 loss, while Huber loss
converges to a constant 0 loss.
- As beta -> +inf, Smooth L1 converges to a constant 0 loss, while Huber loss
converges to L2 loss.
- For Smooth L1 loss, as beta varies, the L1 segment of the loss has a constant
slope of 1. For Huber loss, the slope of the L1 segment is beta.
Smooth L1 loss can be seen as exactly L1 loss, but with the abs(x) < beta
portion replaced with a quadratic function such that at abs(x) = beta, its
slope is 1. The quadratic segment smooths the L1 loss near x = 0.
Args:
input (Tensor): input tensor of any shape
target (Tensor): target value tensor with the same shape as input
beta (float): L1 to L2 change point.
For beta values < 1e-5, L1 loss is computed.
reduction: 'none' | 'mean' | 'sum'
'none': No reduction will be applied to the output.
'mean': The output will be averaged.
'sum': The output will be summed.
Returns:
The loss with the reduction option applied.
Note:
PyTorch's builtin "Smooth L1 loss" implementation does not actually
implement Smooth L1 loss, nor does it implement Huber loss. It implements
the special case of both in which they are equal (beta=1).
See: https://pytorch.org/docs/stable/nn.html#torch.nn.SmoothL1Loss.
"""
if beta < 1e-5:
# if beta == 0, then torch.where will result in nan gradients when
# the chain rule is applied due to pytorch implementation details
# (the False branch "0.5 * n ** 2 / 0" has an incoming gradient of
# zeros, rather than "no gradient"). To avoid this issue, we define
# small values of beta to be exactly l1 loss.
loss = torch.abs(input - target)
else:
n = torch.abs(input - target)
cond = n < beta
loss = torch.where(cond, 0.5 * n ** 2 / beta, n - 0.5 * beta)
if reduction == "mean":
loss = loss.mean() if loss.numel() > 0 else 0.0 * loss.sum()
elif reduction == "sum":
loss = loss.sum()
return loss
```