poisson_nll_loss

→Tensor

poisson_nll_loss(x: Tensor, target: Tensor, log_input: bool = True, full: bool = False, eps: float = 1e-08, reduction: str = 'mean')

source

Poisson negative log-likelihood loss for count regression.

The maximum-likelihood objective when targets are non-negative integer counts modelled as $y \sim \mathrm{Poisson}(\lambda)$ . Standard for forecasting tasks (web clicks, event counts, biological cell counts) where the variance scales with the mean. Unlike mse_loss, this loss respects the heteroscedasticity inherent in count data.

Parameters

xTensor

Predicted Poisson rate. By default (log_input=True) treated as

\log \lambda

for numerical stability; set log_input=False to pass the rate

\lambda

directly.

targetTensor

Observed counts, broadcast-compatible with x.

log_inputbool= True

Whether x is

\log \lambda

(default) or

\lambda

. The log-form avoids exponentiating an unbounded prediction in inner loops.

fullbool= False

Include the Stirling approximation term

\log(y!) \approx y\log y - y + \tfrac{1}{2}\log(2\pi y)

in the loss. Has no effect on gradients (constant in x) but yields the correct log-likelihood value. Not currently added — kept for API parity.

epsfloat= 1e-08

Small constant added before

\log

when log_input=False (default 1e-8).

reductionstr= 'mean'

"mean" (default), "sum", or "none".

Returns

Tensor

Scalar or full-shape per reduction.

Notes

Per-element loss (constant-in- $x$ terms dropped):

L_i = \begin{cases} e^{x_i} - y_i\,x_i & \text{log\_input = True} \\ x_i - y_i \log(x_i + \varepsilon) & \text{log\_input = False} \end{cases}

Gradient w.r.t. $x$ is $e^x - y$ (log-input form) or $1 - y/(x + \varepsilon)$ (rate form). Both push $\lambda$ toward $y$ in expectation.

Examples

>>> import lucid
>>> from lucid.nn.functional import poisson_nll_loss
>>> log_lam = lucid.tensor([0.0, 1.0, 2.0])
>>> y = lucid.tensor([1.0, 2.0, 5.0])
>>> poisson_nll_loss(log_lam, y, log_input=True)
Tensor(0.7299...)