# Copyright 2019 Google LLC
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# Licensed under the Apache License, Version 2.0 (the "License");
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# https://www.apache.org/licenses/LICENSE-2.0
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"""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
@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)
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)
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)
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)
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)
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,
where: Optional[Array] = None,
initial: Optional[Array] = 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) = \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.
"""
x_max = jnp.max(x, axis, where=where, initial=initial, keepdims=True)
shifted = x - lax.stop_gradient(x_max)
shifted_logsumexp = jnp.log(
jnp.sum(jnp.exp(shifted), axis, where=where, keepdims=True))
return shifted - shifted_logsumexp
[docs]def softmax(x: Array,
axis: Optional[Union[int, Tuple[int, ...]]] = -1,
where: Optional[Array] = None,
initial: Optional[Array] = 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.
"""
x_max = jnp.max(x, axis, where=where, initial=initial, keepdims=True)
unnormalized = jnp.exp(x - lax.stop_gradient(x_max))
return unnormalized / jnp.sum(unnormalized, axis, where=where, keepdims=True)
def normalize(x: Array,
axis: Optional[Union[int, Tuple[int, ...]]] = -1,
mean: Optional[Array] = None,
variance: Optional[Array] = None,
epsilon: Array = 1e-5,
where: Optional[Array] = None) -> Array:
"""Normalizes an array by subtracting mean and dividing by sqrt(var)."""
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 (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)
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.)
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.
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