import symjax.tensor as T
[docs]def relu(x):
r"""Rectified linear unit activation function.
Computes the element-wise function:
.. math::
\mathrm{relu}(x) = \max(x, 0)
"""
return T.maximum(x, 0)
[docs]def softplus(x):
r"""Softplus activation function.
Computes the element-wise function
.. math::
\mathrm{softplus}(x) = \log(1 + e^x)
"""
return T.logaddexp(x, 0)
[docs]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 / (T.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 T.expit(x)
[docs]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 * T.sigmoid(x)
[docs]def swish(x, beta):
r"""Swish activation function.
Computes the element-wise function:
.. math::
\mathrm{silu}(x) = x \cdot \mathrm{sigmoid}(x) = \frac{x}{1 + e^{-\beta * x}}
"""
return x * T.sigmoid(beta * x)
def leaky_swish(x, beta, negative_slope=1e-2):
r"""Swish activation function associated to leaky relu.
Computes the element-wise function:
.. math::
\mathrm{silu}(x) = x \cdot \mathrm{sigmoid}(x) = \frac{x}{1 + e^{-\beta * x}}
"""
feature = T.stack([negative_slope * x, x], -1)
return (feature * softmax(feature * beta)).sum(-1)
[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 = T.where(x > 0, 0.0, x)
return T.where(x > 0, x, alpha * T.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 T.where(x >= 0, x, negative_slope * x)
[docs]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 T.where(x > 1, 1, T.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 T.where(x > 0, x, alpha * T.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)
[docs]def gelu(x, approximate: bool = True):
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:
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 + T.tanh(sqrt_2_over_pi * (x + 0.044715 * (x ** 3))))
return x * cdf
else:
raise NotImplemented
# return x * (lax.erf(x / np.sqrt(2)) + 1) / 2, dtype=x.dtype)
[docs]def glu(linear_x, gated_x, axis=-1):
"""Gated linear unit activation function."""
return linear_x * sigmoid(gated_x)
[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 - T.stop_gradient(x.max(axis, keepdims=True))
return shifted - T.log(T.sum(T.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 = T.exp(x - T.stop_gradient(x.max(axis, keepdims=True)))
return unnormalized / unnormalized.sum(axis, keepdims=True)
[docs]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 = T.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 = T.mean(T.square(x), axis, keepdims=True) - T.square(mean)
return (x - mean) * T.rsqrt(variance + epsilon)
[docs]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 T.minimum(T.maximum(x, 0), 6.0)
[docs]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 + 3.0) / 6.0
[docs]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)
def log_1_minus_sigmoid(x):
return -module.__dict__["softplus"](x)