Source code for jax._src.nn.functions

# Copyright 2019 Google LLC
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

"""Shared neural network activations and other functions."""


import operator
import numpy as np
from typing import Any, Optional, Tuple, Union

from jax import custom_jvp
from jax._src import dtypes
from jax import lax
from jax import core
from jax.core import AxisName
from jax._src import util
from jax.scipy.special import expit
from jax.scipy.special import logsumexp as _logsumexp
import jax.numpy as jnp

Array = Any

# activations

[docs]@custom_jvp def relu(x: Array) -> Array: r"""Rectified linear unit activation function. Computes the element-wise function: .. math:: \mathrm{relu}(x) = \max(x, 0) Args: x : input array """ return jnp.maximum(x, 0)
relu.defjvps(lambda g, ans, x: lax.select(x > 0, g, lax.full_like(g, 0)))
[docs]def softplus(x: Array) -> 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]def soft_sign(x: Array) -> Array: r"""Soft-sign activation function. Computes the element-wise function .. math:: \mathrm{soft\_sign}(x) = \frac{x}{|x| + 1} Args: x : input array """ return x / (jnp.abs(x) + 1)
[docs]def sigmoid(x: Array) -> Array: r"""Sigmoid activation function. Computes the element-wise function: .. math:: \mathrm{sigmoid}(x) = \frac{1}{1 + e^{-x}} Args: x : input array """ return expit(x)
[docs]def silu(x: Array) -> Array: r"""SiLU activation function. Computes the element-wise function: .. math:: \mathrm{silu}(x) = x \cdot \mathrm{sigmoid}(x) = \frac{x}{1 + e^{-x}} Args: x : input array """ return x * sigmoid(x)
swish = silu
[docs]def log_sigmoid(x: Array) -> 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 """ return -softplus(-x)
[docs]def elu(x: Array, alpha: Array = 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) """ safe_x = jnp.where(x > 0, 0., x) return jnp.where(x > 0, x, alpha * jnp.expm1(safe_x))
[docs]def leaky_relu(x: Array, negative_slope: Array = 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) """ return jnp.where(x >= 0, x, negative_slope * x)
[docs]def hard_tanh(x: Array) -> 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 """ return jnp.where(x > 1, 1, jnp.where(x < -1, -1, x))
[docs]def celu(x: Array, alpha: Array = 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) """ return jnp.where(x > 0, x, alpha * jnp.expm1(x / alpha))
[docs]def selu(x: Array) -> 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 """ alpha = 1.6732632423543772848170429916717 scale = 1.0507009873554804934193349852946 return scale * elu(x, alpha)
[docs]def gelu(x: Array, 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. """ if approximate: sqrt_2_over_pi = np.sqrt(2 / np.pi).astype(x.dtype) cdf = 0.5 * (1.0 + jnp.tanh(sqrt_2_over_pi * (x + 0.044715 * (x ** 3)))) return x * cdf else: return jnp.array(x * (lax.erf(x / np.sqrt(2)) + 1) / 2, dtype=x.dtype)
[docs]def glu(x: Array, axis: int = -1) -> Array: """Gated linear unit activation function. Args: x : input array axis: the axis along which the split should be computed (default: -1) """ size = x.shape[axis] assert size % 2 == 0, "axis size must be divisible by 2" x1, x2 = jnp.split(x, 2, axis) return x1 * sigmoid(x2)
# other functions logsumexp = _logsumexp
[docs]def log_softmax(x: Array, axis: Optional[Union[int, Tuple[int, ...]]] = -1) -> 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) = \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. """ shifted = x - lax.stop_gradient(x.max(axis, keepdims=True)) return shifted - jnp.log(jnp.sum(jnp.exp(shifted), axis, keepdims=True))
[docs]def softmax(x: Array, axis: Optional[Union[int, Tuple[int, ...]]] = -1) -> 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. """ unnormalized = jnp.exp(x - lax.stop_gradient(x.max(axis, keepdims=True))) return unnormalized / unnormalized.sum(axis, keepdims=True)
[docs]def normalize(x: Array, axis: Optional[Union[int, Tuple[int, ...]]] = -1, mean: Optional[Array] = None, variance: Optional[Array] = None, epsilon: Array = 1e-5) -> Array: """Normalizes an array by subtracting mean and dividing by sqrt(var).""" if mean is None: mean = jnp.mean(x, axis, keepdims=True) 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) - jnp.square(mean) return (x - mean) * lax.rsqrt(variance + epsilon)
[docs]def one_hot(x: Array, num_classes: int, *, dtype: Any = jnp.float_, axis: Union[int, AxisName] = -1) -> Array: """One-hot encodes the given indicies. 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) DeviceArray([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]], dtype=float32) Indicies outside the range [0, num_classes) will be encoded as zeros:: >>> jax.nn.one_hot(jnp.array([-1, 3]), 3) DeviceArray([[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_or_error( int, num_classes, "The error arose in jax.nn.one_hot argument `num_classes`.") dtype = dtypes.canonicalize_dtype(dtype) x = jnp.asarray(x) try: output_pos_axis = util.canonicalize_axis(axis, x.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 == axis_idx, dtype=dtype) axis = operator.index(axis) lhs = lax.expand_dims(x, (axis,)) rhs_shape = [1] * x.ndim rhs_shape.insert(output_pos_axis, num_classes) rhs = lax.broadcast_in_dim(jnp.arange(num_classes, dtype=x.dtype), rhs_shape, (output_pos_axis,)) return jnp.asarray(lhs == rhs, dtype=dtype)
[docs]def relu6(x: Array) -> Array: r"""Rectified Linear Unit 6 activation function. Computes the element-wise function .. math:: \mathrm{relu6}(x) = \min(\max(x, 0), 6) Args: x : input array """ return jnp.minimum(jnp.maximum(x, 0), 6.)
[docs]def hard_sigmoid(x: Array) -> 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 """ return relu6(x + 3.) / 6.
[docs]def hard_silu(x: Array) -> Array: r"""Hard SiLU activation function Computes the element-wise function .. math:: \mathrm{hard\_silu}(x) = x \cdot \mathrm{hard\_sigmoid}(x) Args: x : input array """ return x * hard_sigmoid(x)
hard_swish = hard_silu