Source code for jax._src.nn.functions

# Copyright 2019 The JAX Authors.
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#     https://www.apache.org/licenses/LICENSE-2.0
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"""Shared neural network activations and other functions."""

from __future__ import annotations

from functools import partial
import operator
import numpy as np
from typing import Any

import jax
import jax.numpy as jnp
from jax import custom_jvp
from jax import lax
from jax._src import config
from jax._src import core
from jax._src import dtypes
from jax._src import util
from jax._src.core import AxisName
from jax._src.numpy import util as numpy_util
from jax._src.typing import Array, ArrayLike
from jax._src.ops.special import logsumexp as _logsumexp


# activations

[docs] @custom_jvp @jax.jit def relu(x: ArrayLike) -> Array: r"""Rectified linear unit activation function. Computes the element-wise function: .. math:: \mathrm{relu}(x) = \max(x, 0) except under differentiation, we take: .. math:: \nabla \mathrm{relu}(0) = 0 For more information see `Numerical influence of ReLU’(0) on backpropagation <https://openreview.net/forum?id=urrcVI-_jRm>`_. Args: x : input array Returns: An array. Example: >>> jax.nn.relu(jax.numpy.array([-2., -1., -0.5, 0, 0.5, 1., 2.])) Array([0. , 0. , 0. , 0. , 0.5, 1. , 2. ], dtype=float32) See also: :func:`relu6` """ return jnp.maximum(x, 0)
# For behavior at 0, see https://openreview.net/forum?id=urrcVI-_jRm relu.defjvps(lambda g, ans, x: lax.select(x > 0, g, lax.full_like(g, 0)))
[docs] @jax.jit def squareplus(x: ArrayLike, b: ArrayLike = 4) -> Array: r"""Squareplus activation function. Computes the element-wise function .. math:: \mathrm{squareplus}(x) = \frac{x + \sqrt{x^2 + b}}{2} as described in https://arxiv.org/abs/2112.11687. Args: x : input array b : smoothness parameter """ numpy_util.check_arraylike("squareplus", x) numpy_util.check_arraylike("squareplus", b) x = jnp.asarray(x) b = jnp.asarray(b) y = x + jnp.sqrt(jnp.square(x) + b) return y / 2
[docs] @jax.jit def softplus(x: ArrayLike) -> Array: r"""Softplus activation function. Computes the element-wise function .. math:: \mathrm{softplus}(x) = \log(1 + e^x) Args: x : input array """ return jnp.logaddexp(x, 0)
[docs] @jax.jit def soft_sign(x: ArrayLike) -> Array: r"""Soft-sign activation function. Computes the element-wise function .. math:: \mathrm{soft\_sign}(x) = \frac{x}{|x| + 1} Args: x : input array """ numpy_util.check_arraylike("soft_sign", x) x_arr = jnp.asarray(x) return x_arr / (jnp.abs(x_arr) + 1)
[docs] @partial(jax.jit, inline=True) def sigmoid(x: ArrayLike) -> Array: r"""Sigmoid activation function. Computes the element-wise function: .. math:: \mathrm{sigmoid}(x) = \frac{1}{1 + e^{-x}} Args: x : input array Returns: An array. See also: :func:`log_sigmoid` """ return lax.logistic(x)
[docs] @jax.jit def silu(x: ArrayLike) -> Array: r"""SiLU (a.k.a. swish) activation function. Computes the element-wise function: .. math:: \mathrm{silu}(x) = x \cdot \mathrm{sigmoid}(x) = \frac{x}{1 + e^{-x}} :func:`swish` and :func:`silu` are both aliases for the same function. Args: x : input array Returns: An array. See also: :func:`sigmoid` """ numpy_util.check_arraylike("silu", x) x_arr = jnp.asarray(x) return x_arr * sigmoid(x_arr)
swish = silu
[docs] @jax.jit def log_sigmoid(x: ArrayLike) -> Array: r"""Log-sigmoid activation function. Computes the element-wise function: .. math:: \mathrm{log\_sigmoid}(x) = \log(\mathrm{sigmoid}(x)) = -\log(1 + e^{-x}) Args: x : input array Returns: An array. See also: :func:`sigmoid` """ numpy_util.check_arraylike("log_sigmoid", x) x_arr = jnp.asarray(x) return -softplus(-x_arr)
[docs] @jax.jit def elu(x: ArrayLike, alpha: ArrayLike = 1.0) -> Array: r"""Exponential linear unit activation function. Computes the element-wise function: .. math:: \mathrm{elu}(x) = \begin{cases} x, & x > 0\\ \alpha \left(\exp(x) - 1\right), & x \le 0 \end{cases} Args: x : input array alpha : scalar or array of alpha values (default: 1.0) Returns: An array. See also: :func:`selu` """ numpy_util.check_arraylike("elu", x) x_arr = jnp.asarray(x) return jnp.where(x_arr > 0, x_arr, alpha * jnp.expm1(jnp.where(x_arr > 0, 0., x_arr)))
[docs] @jax.jit def leaky_relu(x: ArrayLike, negative_slope: ArrayLike = 1e-2) -> Array: r"""Leaky rectified linear unit activation function. Computes the element-wise function: .. math:: \mathrm{leaky\_relu}(x) = \begin{cases} x, & x \ge 0\\ \alpha x, & x < 0 \end{cases} where :math:`\alpha` = :code:`negative_slope`. Args: x : input array negative_slope : array or scalar specifying the negative slope (default: 0.01) Returns: An array. See also: :func:`relu` """ numpy_util.check_arraylike("leaky_relu", x) x_arr = jnp.asarray(x) return jnp.where(x_arr >= 0, x_arr, negative_slope * x_arr)
[docs] @jax.jit def hard_tanh(x: ArrayLike) -> Array: r"""Hard :math:`\mathrm{tanh}` activation function. Computes the element-wise function: .. math:: \mathrm{hard\_tanh}(x) = \begin{cases} -1, & x < -1\\ x, & -1 \le x \le 1\\ 1, & 1 < x \end{cases} Args: x : input array Returns: An array. """ numpy_util.check_arraylike("hard_tanh", x) x_arr = jnp.asarray(x) return jnp.where(x_arr > 1, 1, jnp.where(x_arr < -1, -1, x_arr))
[docs] @jax.jit def celu(x: ArrayLike, alpha: ArrayLike = 1.0) -> Array: r"""Continuously-differentiable exponential linear unit activation. Computes the element-wise function: .. math:: \mathrm{celu}(x) = \begin{cases} x, & x > 0\\ \alpha \left(\exp(\frac{x}{\alpha}) - 1\right), & x \le 0 \end{cases} For more information, see `Continuously Differentiable Exponential Linear Units <https://arxiv.org/pdf/1704.07483.pdf>`_. Args: x : input array alpha : array or scalar (default: 1.0) Returns: An array. """ return jnp.maximum(x, 0.0) + alpha * jnp.expm1(jnp.minimum(x, 0.0) / alpha)
[docs] @jax.jit def selu(x: ArrayLike) -> Array: r"""Scaled exponential linear unit activation. Computes the element-wise function: .. math:: \mathrm{selu}(x) = \lambda \begin{cases} x, & x > 0\\ \alpha e^x - \alpha, & x \le 0 \end{cases} where :math:`\lambda = 1.0507009873554804934193349852946` and :math:`\alpha = 1.6732632423543772848170429916717`. For more information, see `Self-Normalizing Neural Networks <https://papers.nips.cc/paper/6698-self-normalizing-neural-networks.pdf>`_. Args: x : input array Returns: An array. See also: :func:`elu` """ alpha = 1.6732632423543772848170429916717 scale = 1.0507009873554804934193349852946 return scale * elu(x, alpha)
# TODO(phawkins): this jit was found to change numerics in a test. Debug this. # @partial(jax.jit, static_argnames=("approximate",))
[docs] def gelu(x: ArrayLike, approximate: bool = True) -> Array: r"""Gaussian error linear unit activation function. If ``approximate=False``, computes the element-wise function: .. math:: \mathrm{gelu}(x) = \frac{x}{2} \left(1 + \mathrm{erf} \left( \frac{x}{\sqrt{2}} \right) \right) If ``approximate=True``, uses the approximate formulation of GELU: .. math:: \mathrm{gelu}(x) = \frac{x}{2} \left(1 + \mathrm{tanh} \left( \sqrt{\frac{2}{\pi}} \left(x + 0.044715 x^3 \right) \right) \right) For more information, see `Gaussian Error Linear Units (GELUs) <https://arxiv.org/abs/1606.08415>`_, section 2. Args: x : input array approximate: whether to use the approximate or exact formulation. """ [x_arr] = numpy_util.promote_args_inexact("gelu", x) if approximate: sqrt_2_over_pi = np.sqrt(2 / np.pi).astype(x_arr.dtype) cdf = 0.5 * (1.0 + jnp.tanh(sqrt_2_over_pi * (x_arr + 0.044715 * (x_arr ** 3)))) return x_arr * cdf else: sqrt_2 = np.sqrt(2).astype(x_arr.dtype) return jnp.array(x_arr * (lax.erf(x_arr / sqrt_2) + 1) / 2, dtype=x_arr.dtype)
[docs] @partial(jax.jit, static_argnames=("axis",)) def glu(x: ArrayLike, axis: int = -1) -> Array: r"""Gated linear unit activation function. Computes the function: .. math:: \mathrm{glu}(x) = x\left[\ldots, 0:\frac{n}{2}, \ldots\right] \cdot \mathrm{sigmoid} \left( x\left[\ldots, \frac{n}{2}:n, \ldots\right] \right) where the array is split into two along ``axis``. The size of the ``axis`` dimension must be divisible by two. Args: x : input array axis: the axis along which the split should be computed (default: -1) Returns: An array. See also: :func:`sigmoid` """ numpy_util.check_arraylike("glu", x) x_arr = jnp.asarray(x) size = x_arr.shape[axis] assert size % 2 == 0, "axis size must be divisible by 2" x1, x2 = jnp.split(x_arr, 2, axis) return x1 * sigmoid(x2)
# other functions logsumexp = _logsumexp
[docs] @partial(jax.jit, static_argnames=("axis",)) def log_softmax(x: ArrayLike, axis: int | tuple[int, ...] | None = -1, where: ArrayLike | None = None, initial: ArrayLike | None = None) -> Array: r"""Log-Softmax function. Computes the logarithm of the :code:`softmax` function, which rescales elements to the range :math:`[-\infty, 0)`. .. math :: \mathrm{log\_softmax}(x)_i = \log \left( \frac{\exp(x_i)}{\sum_j \exp(x_j)} \right) Args: x : input array axis: the axis or axes along which the :code:`log_softmax` should be computed. Either an integer or a tuple of integers. where: Elements to include in the :code:`log_softmax`. initial: The minimum value used to shift the input array. Must be present when :code:`where` is not None. Returns: An array. See also: :func:`softmax` """ numpy_util.check_arraylike("log_softmax", x) x_arr = jnp.asarray(x) x_max = jnp.max(x_arr, axis, where=where, initial=initial, keepdims=True) x_safe = x_arr if where is None else jnp.where(where, x_arr, initial) shifted = x_safe - lax.stop_gradient(x_max) shifted_logsumexp = jnp.log( jnp.sum(jnp.exp(shifted), axis, where=where, keepdims=True)) result = shifted - shifted_logsumexp if where is not None: return jnp.where(where, result, -jnp.inf) return result
# TODO(phawkins): this jit was found to change numerics in a test. Debug this. #@partial(jax.jit, static_argnames=("axis",))
[docs] def softmax(x: ArrayLike, axis: int | tuple[int, ...] | None = -1, where: ArrayLike | None = None, initial: ArrayLike | None = None) -> Array: r"""Softmax function. Computes the function which rescales elements to the range :math:`[0, 1]` such that the elements along :code:`axis` sum to :math:`1`. .. math :: \mathrm{softmax}(x) = \frac{\exp(x_i)}{\sum_j \exp(x_j)} Args: x : input array axis: the axis or axes along which the softmax should be computed. The softmax output summed across these dimensions should sum to :math:`1`. Either an integer or a tuple of integers. where: Elements to include in the :code:`softmax`. initial: The minimum value used to shift the input array. Must be present when :code:`where` is not None. Returns: An array. See also: :func:`log_softmax` """ if config.softmax_custom_jvp.value: # mypy is confused by the `functools.partial` application in the definition # of `_softmax` and incorrectly concludes that `_softmax` returns # `ReturnValue` -- the unsubstituted type parameter of `custom_jvp`. return _softmax(x, axis, where, initial) # type: ignore[return-value] else: return _softmax_deprecated(x, axis, where, initial)
# TODO(mattjj): replace softmax with _softmax when deprecation flag is removed @partial(jax.custom_jvp, nondiff_argnums=(1,)) def _softmax( x: ArrayLike, axis: int | tuple[int, ...] | None = -1, where: ArrayLike | None = None, initial: ArrayLike | None = None) -> Array: x_max = jnp.max(x, axis, where=where, initial=initial, keepdims=True) x_safe = x if where is None else jnp.where(where, x, initial) unnormalized = jnp.exp(x_safe - x_max) result = unnormalized / jnp.sum(unnormalized, axis, where=where, keepdims=True) if where is not None: result = jnp.where(where, result, 0) return result @_softmax.defjvp def _softmax_jvp(axis, primals, tangents): (x, where, initial), (x_dot, _, _) = primals, tangents y = _softmax(x, axis, where, initial) return y, y * (x_dot - (y * x_dot).sum(axis, where=where, keepdims=True)) def _softmax_deprecated( x: ArrayLike, axis: int | tuple[int, ...] | None = -1, where: ArrayLike | None = None, initial: ArrayLike | None = None) -> Array: x_max = jnp.max(x, axis, where=where, initial=initial, keepdims=True) x_safe = x if where is None else jnp.where(where, x, initial) unnormalized = jnp.exp(x_safe - lax.stop_gradient(x_max)) result = unnormalized / jnp.sum(unnormalized, axis, where=where, keepdims=True) if where is not None: result = jnp.where(where, result, 0) return result
[docs] @partial(jax.jit, static_argnames=("axis",)) def standardize(x: ArrayLike, axis: int | tuple[int, ...] | None = -1, mean: ArrayLike | None = None, variance: ArrayLike | None = None, epsilon: ArrayLike = 1e-5, where: ArrayLike | None = None) -> Array: r"""Normalizes an array by subtracting ``mean`` and dividing by :math:`\sqrt{\mathrm{variance}}`.""" numpy_util.check_arraylike("standardize", x) numpy_util.check_arraylike_or_none("standardize", mean, variance, where) if mean is None: mean = jnp.mean(x, axis, keepdims=True, where=where) if variance is None: # this definition is traditionally seen as less accurate than jnp.var's # mean((x - mean(x))**2) but may be faster and even, given typical # activation distributions and low-precision arithmetic, more accurate # when used in neural network normalization layers variance = jnp.mean( jnp.square(x), axis, keepdims=True, where=where) - jnp.square(mean) return jnp.subtract(x, jnp.asarray(mean)) * lax.rsqrt(jnp.asarray(variance) + epsilon)
# TODO(slebedev): Change the type of `x` to `ArrayLike`. @partial(jax.jit, static_argnames=("num_classes", "dtype", "axis")) def _one_hot(x: Any, num_classes: int, *, dtype: Any, axis: int | AxisName) -> Array: num_classes = core.concrete_dim_or_error( num_classes, "The error arose in jax.nn.one_hot argument `num_classes`.") dtype = dtypes.canonicalize_dtype(dtype) x_arr = jnp.asarray(x) try: output_pos_axis = util.canonicalize_axis(axis, x_arr.ndim + 1) except TypeError: axis_size = lax.psum(1, axis) if num_classes != axis_size: raise ValueError(f"Expected num_classes to match the size of axis {axis}, " f"but {num_classes} != {axis_size}") from None axis_idx = lax.axis_index(axis) return jnp.asarray(x_arr == axis_idx, dtype=dtype) axis = operator.index(axis) # type: ignore[arg-type] lhs = lax.expand_dims(x_arr, (axis,)) rhs_shape = [1] * x_arr.ndim rhs_shape.insert(output_pos_axis, num_classes) rhs = lax.broadcasted_iota(x_arr.dtype, rhs_shape, output_pos_axis) return jnp.asarray(lhs == rhs, dtype=dtype) # TODO(slebedev): Change the type of `x` to `ArrayLike`.
[docs] def one_hot(x: Any, num_classes: int, *, dtype: Any = jnp.float_, axis: int | AxisName = -1) -> Array: """One-hot encodes the given indices. Each index in the input ``x`` is encoded as a vector of zeros of length ``num_classes`` with the element at ``index`` set to one:: >>> jax.nn.one_hot(jnp.array([0, 1, 2]), 3) Array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]], dtype=float32) Indices outside the range [0, num_classes) will be encoded as zeros:: >>> jax.nn.one_hot(jnp.array([-1, 3]), 3) Array([[0., 0., 0.], [0., 0., 0.]], dtype=float32) Args: x: A tensor of indices. num_classes: Number of classes in the one-hot dimension. dtype: optional, a float dtype for the returned values (default :obj:`jnp.float_`). axis: the axis or axes along which the function should be computed. """ num_classes = core.concrete_dim_or_error( num_classes, "The error arose in jax.nn.one_hot argument `num_classes`.") return _one_hot(x, num_classes, dtype=dtype, axis=axis)
[docs] @jax.custom_jvp @jax.jit def relu6(x: ArrayLike) -> Array: r"""Rectified Linear Unit 6 activation function. Computes the element-wise function .. math:: \mathrm{relu6}(x) = \min(\max(x, 0), 6) except under differentiation, we take: .. math:: \nabla \mathrm{relu}(0) = 0 and .. math:: \nabla \mathrm{relu}(6) = 0 Args: x : input array Returns: An array. See also: :func:`relu` """ return jnp.minimum(jnp.maximum(x, 0), 6.)
relu6.defjvps(lambda g, ans, x: lax.select((x > 0) & (x < 6), g, lax.full_like(g, 0)))
[docs] @jax.jit def hard_sigmoid(x: ArrayLike) -> Array: r"""Hard Sigmoid activation function. Computes the element-wise function .. math:: \mathrm{hard\_sigmoid}(x) = \frac{\mathrm{relu6}(x + 3)}{6} Args: x : input array Returns: An array. See also: :func:`relu6` """ return relu6(x + 3.) / 6.
[docs] @jax.jit def hard_silu(x: ArrayLike) -> Array: r"""Hard SiLU (swish) activation function Computes the element-wise function .. math:: \mathrm{hard\_silu}(x) = x \cdot \mathrm{hard\_sigmoid}(x) Both :func:`hard_silu` and :func:`hard_swish` are aliases for the same function. Args: x : input array Returns: An array. See also: :func:`hard_sigmoid` """ numpy_util.check_arraylike("hard_silu", x) x_arr = jnp.asarray(x) return x_arr * hard_sigmoid(x_arr)
hard_swish = hard_silu