# 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.
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# https://www.apache.org/licenses/LICENSE-2.0
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# distributed under the License is distributed on an "AS IS" BASIS,
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"""Shared neural network activations and other functions."""
import numpy as np
from jax import custom_jvp
from jax import dtypes
from jax import lax
from jax import core
from jax.scipy.special import expit
import jax.numpy as jnp
# activations
@custom_jvp
def relu(x):
r"""Rectified linear unit activation function.
Computes the element-wise function:
.. math::
\mathrm{relu}(x) = \max(x, 0)
"""
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):
r"""Softplus activation function.
Computes the element-wise function
.. math::
\mathrm{softplus}(x) = \log(1 + e^x)
"""
return jnp.logaddexp(x, 0)
def soft_sign(x):
r"""Soft-sign activation function.
Computes the element-wise function
.. math::
\mathrm{soft\_sign}(x) = \frac{x}{|x| + 1}
"""
return x / (jnp.abs(x) + 1)
[docs]def sigmoid(x):
r"""Sigmoid activation function.
Computes the element-wise function:
.. math::
\mathrm{sigmoid}(x) = \frac{1}{1 + e^{-x}}
"""
return expit(x)
def silu(x):
r"""SiLU activation function.
Computes the element-wise function:
.. math::
\mathrm{silu}(x) = x \cdot \mathrm{sigmoid}(x) = \frac{x}{1 + e^{-x}}
"""
return x * sigmoid(x)
swish = silu
[docs]def log_sigmoid(x):
r"""Log-sigmoid activation function.
Computes the element-wise function:
.. math::
\mathrm{log\_sigmoid}(x) = \log(\mathrm{sigmoid}(x)) = -\log(1 + e^{-x})
"""
return -softplus(-x)
[docs]def elu(x, alpha=1.0):
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}
"""
safe_x = jnp.where(x > 0, 0., x)
return jnp.where(x > 0, x, alpha * jnp.expm1(safe_x))
[docs]def leaky_relu(x, negative_slope=1e-2):
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`.
"""
return jnp.where(x >= 0, x, negative_slope * x)
def hard_tanh(x):
r"""Hard :math:`\mathrm{tanh}` activation function.
Computes the element-wise function:
.. math::
\mathrm{hard\_tanh}(x) = \begin{cases}
-1, & x < -1\\
x, & 0 \le x \le 1\\
1, & 1 < x
\end{cases}
"""
return jnp.where(x > 1, 1, jnp.where(x < -1, -1, x))
[docs]def celu(x, alpha=1.0):
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>`_."""
return jnp.where(x > 0, x, alpha * jnp.expm1(x / alpha))
[docs]def selu(x):
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>`_.
"""
alpha = 1.6732632423543772848170429916717
scale = 1.0507009873554804934193349852946
return scale * elu(x, alpha)
def gelu(x):
r"""Gaussian error linear unit activation function.
Computes the element-wise function:
.. 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)
We explicitly use the approximation rather than the exact formulation for
speed. For more information, see `Gaussian Error Linear Units (GELUs)
<https://arxiv.org/abs/1606.08415>`_, section 2.
"""
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
def glu(x, axis=-1):
"""Gated linear unit activation function."""
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
[docs]def log_softmax(x, axis=-1):
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:
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, axis=-1):
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:
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)
def normalize(x, axis=-1, mean=None, variance=None, epsilon=1e-5):
"""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, num_classes, *, dtype=jnp.float64):
"""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 float64 if
jax_enable_x64 is true, otherwise float32).
"""
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)
lhs = x[..., jnp.newaxis]
rhs = lax.broadcast_to_rank(jnp.arange(num_classes, dtype=x.dtype), lhs.ndim)
return jnp.array(lhs == rhs, dtype=dtype)
def relu6(x):
r"""Rectified Linear Unit 6 activation function.
Computes the element-wise function
.. math::
\mathrm{relu6}(x) = \min(\max(x, 0), 6)
"""
return jnp.minimum(jnp.maximum(x, 0), 6.)
def hard_sigmoid(x):
r"""Hard Sigmoid activation function.
Computes the element-wise function
.. math::
\mathrm{hard\_sigmoid}(x) = \frac{\mathrm{relu6}(x + 3)}{6}
"""
return relu6(x + 3.) / 6.
def hard_silu(x):
r"""Hard SiLU activation function
Computes the element-wise function
.. math::
\mathrm{hard\_silu}(x) = x \cdot \mathrm{hard\_sigmoid}(x)
"""
return x * hard_sigmoid(x)
hard_swish = hard_silu