# -*- coding: utf-8 -*-
from typing import Union
import jax
import jax.numpy as jnp
import jax.scipy as sci
from jax.core import Primitive
from jax.interpreters import batching, ad, mlir
from brainpy._src.math.interoperability import as_jax
from brainpy._src.math.ndarray import Array
__all__ = [
'Sigmoid',
'sigmoid',
'PiecewiseQuadratic',
'piecewise_quadratic',
'PiecewiseExp',
'piecewise_exp',
'SoftSign',
'soft_sign',
'Arctan',
'arctan',
'NonzeroSignLog',
'nonzero_sign_log',
'ERF',
'erf',
'PiecewiseLeakyRelu',
'piecewise_leaky_relu',
'SquarewaveFourierSeries',
'squarewave_fourier_series',
'S2NN',
's2nn',
'QPseudoSpike',
'q_pseudo_spike',
'LeakyRelu',
'leaky_relu',
'LogTailedRelu',
'log_tailed_relu',
'ReluGrad',
'relu_grad',
'GaussianGrad',
'gaussian_grad',
'InvSquareGrad',
'inv_square_grad',
'MultiGaussianGrad',
'multi_gaussian_grad',
'SlayerGrad',
'slayer_grad',
]
def _heaviside_abstract(x, dx):
return [x]
def _heaviside_imp(x, dx):
z = jnp.asarray(x >= 0, dtype=x.dtype)
return [z]
def _heaviside_batching(args, axes):
return heaviside_p.bind(*args), axes
def _heaviside_jvp(primals, tangents):
x, dx = primals
tx, tdx = tangents
primal_outs = heaviside_p.bind(x, dx)
tangent_outs = [dx * tx, ]
return primal_outs, tangent_outs
heaviside_p = Primitive('heaviside_p')
heaviside_p.multiple_results = True
heaviside_p.def_abstract_eval(_heaviside_abstract)
heaviside_p.def_impl(_heaviside_imp)
batching.primitive_batchers[heaviside_p] = _heaviside_batching
ad.primitive_jvps[heaviside_p] = _heaviside_jvp
mlir.register_lowering(heaviside_p, mlir.lower_fun(_heaviside_imp, multiple_results=True))
def _is_bp_array(x):
return isinstance(x, Array)
def _as_jax(x):
return x.value if _is_bp_array(x) else x
[docs]
class Surrogate(object):
"""The base surrograte gradient function.
To customize a surrogate gradient function, you can inherit this class and
implement the `surrogate_fun` and `surrogate_grad` methods.
Examples
--------
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import jax.numpy as jnp
>>> class MySurrogate(bm.Surrogate):
... def __init__(self, alpha=1.):
... super().__init__()
... self.alpha = alpha
...
... def surrogate_fun(self, x):
... return jnp.sin(x) * self.alpha
...
... def surrogate_grad(self, x):
... return jnp.cos(x) * self.alpha
"""
def __call__(self, x):
x = _as_jax(x)
dx = self.surrogate_grad(x)
return heaviside_p.bind(x, dx)[0]
def __repr__(self):
return f'{self.__class__.__name__}()'
def surrogate_fun(self, x) -> jax.Array:
"""The surrogate function."""
raise NotImplementedError
def surrogate_grad(self, x) -> jax.Array:
"""The gradient function of the surrogate function."""
raise NotImplementedError
[docs]
class Sigmoid(Surrogate):
"""Spike function with the sigmoid-shaped surrogate gradient.
See Also
--------
sigmoid
"""
[docs]
def __init__(self, alpha: float = 4.):
super().__init__()
self.alpha = alpha
def surrogate_fun(self, x):
return sci.special.expit(self.alpha * x)
def surrogate_grad(self, x):
sgax = sci.special.expit(x * self.alpha)
dx = (1. - sgax) * sgax * self.alpha
return dx
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def sigmoid(
x: Union[jax.Array, Array],
alpha: float = 4.,
):
r"""Spike function with the sigmoid-shaped surrogate gradient.
If `origin=False`, return the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
g(x) = \mathrm{sigmoid}(\alpha x) = \frac{1}{1+e^{-\alpha x}}
Backward function:
.. math::
g'(x) = \alpha * (1 - \mathrm{sigmoid} (\alpha x)) \mathrm{sigmoid} (\alpha x)
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-2, 2, 1000)
>>> for alpha in [1., 2., 4.]:
>>> grads = bm.vector_grad(bm.surrogate.sigmoid)(xs, alpha)
>>> plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
Parameter to control smoothness of gradient
Returns
-------
out: jax.Array
The spiking state.
"""
return Sigmoid(alpha=alpha)(x)
[docs]
class PiecewiseQuadratic(Surrogate):
"""Judge spiking state with a piecewise quadratic function.
See Also
--------
piecewise_quadratic
"""
[docs]
def __init__(self, alpha: float = 1.):
super().__init__()
self.alpha = alpha
def surrogate_fun(self, x):
x = as_jax(x)
z = jnp.where(x < -1 / self.alpha,
0.,
jnp.where(x > 1 / self.alpha,
1.,
(-self.alpha * jnp.abs(x) / 2 + 1) * self.alpha * x + 0.5))
return z
def surrogate_grad(self, x):
x = as_jax(x)
dx = jnp.where(jnp.abs(x) > 1 / self.alpha, 0., (-(self.alpha * x) ** 2 + self.alpha))
return dx
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def piecewise_quadratic(
x: Union[jax.Array, Array],
alpha: float = 1.,
):
r"""Judge spiking state with a piecewise quadratic function [1]_ [2]_ [3]_ [4]_ [5]_.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
g(x) =
\begin{cases}
0, & x < -\frac{1}{\alpha} \\
-\frac{1}{2}\alpha^2|x|x + \alpha x + \frac{1}{2}, & |x| \leq \frac{1}{\alpha} \\
1, & x > \frac{1}{\alpha} \\
\end{cases}
Backward function:
.. math::
g'(x) =
\begin{cases}
0, & |x| > \frac{1}{\alpha} \\
-\alpha^2|x|+\alpha, & |x| \leq \frac{1}{\alpha}
\end{cases}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> for alpha in [0.5, 1., 2., 4.]:
>>> grads = bm.vector_grad(bm.surrogate.piecewise_quadratic)(xs, alpha)
>>> plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
Parameter to control smoothness of gradient
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Esser S K, Merolla P A, Arthur J V, et al. Convolutional networks for fast, energy-efficient neuromorphic computing[J]. Proceedings of the national academy of sciences, 2016, 113(41): 11441-11446.
.. [2] Wu Y, Deng L, Li G, et al. Spatio-temporal backpropagation for training high-performance spiking neural networks[J]. Frontiers in neuroscience, 2018, 12: 331.
.. [3] Bellec G, Salaj D, Subramoney A, et al. Long short-term memory and learning-to-learn in networks of spiking neurons[C]//Proceedings of the 32nd International Conference on Neural Information Processing Systems. 2018: 795-805.
.. [4] Neftci E O, Mostafa H, Zenke F. Surrogate gradient learning in spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks[J]. IEEE Signal Processing Magazine, 2019, 36(6): 51-63.
.. [5] Panda P, Aketi S A, Roy K. Toward scalable, efficient, and accurate deep spiking neural networks with backward residual connections, stochastic softmax, and hybridization[J]. Frontiers in Neuroscience, 2020, 14.
"""
return PiecewiseQuadratic(alpha=alpha)(x)
[docs]
class PiecewiseExp(Surrogate):
"""Judge spiking state with a piecewise exponential function.
See Also
--------
piecewise_exp
"""
[docs]
def __init__(self, alpha: float = 1.):
super().__init__()
self.alpha = alpha
def surrogate_grad(self, x):
x = as_jax(x)
dx = (self.alpha / 2) * jnp.exp(-self.alpha * jnp.abs(x))
return dx
def surrogate_fun(self, x):
x = as_jax(x)
return jnp.where(x < 0, jnp.exp(self.alpha * x) / 2, 1 - jnp.exp(-self.alpha * x) / 2)
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def piecewise_exp(
x: Union[jax.Array, Array],
alpha: float = 1.,
):
r"""Judge spiking state with a piecewise exponential function [1]_.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
g(x) = \begin{cases}
\frac{1}{2}e^{\alpha x}, & x < 0 \\
1 - \frac{1}{2}e^{-\alpha x}, & x \geq 0
\end{cases}
Backward function:
.. math::
g'(x) = \frac{\alpha}{2}e^{-\alpha |x|}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> for alpha in [0.5, 1., 2., 4.]:
>>> grads = bm.vector_grad(bm.surrogate.piecewise_exp)(xs, alpha)
>>> plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
Parameter to control smoothness of gradient
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Neftci E O, Mostafa H, Zenke F. Surrogate gradient learning in spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks[J]. IEEE Signal Processing Magazine, 2019, 36(6): 51-63.
"""
return PiecewiseExp(alpha=alpha)(x)
[docs]
class SoftSign(Surrogate):
"""Judge spiking state with a soft sign function.
See Also
--------
soft_sign
"""
[docs]
def __init__(self, alpha=1.):
super().__init__()
self.alpha = alpha
def surrogate_grad(self, x):
x = as_jax(x)
dx = self.alpha * 0.5 / (1 + jnp.abs(self.alpha * x)) ** 2
return dx
def surrogate_fun(self, x):
x = as_jax(x)
return x / (2 / self.alpha + 2 * jnp.abs(x)) + 0.5
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def soft_sign(
x: Union[jax.Array, Array],
alpha: float = 1.,
):
r"""Judge spiking state with a soft sign function.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
g(x) = \frac{1}{2} (\frac{\alpha x}{1 + |\alpha x|} + 1)
= \frac{1}{2} (\frac{x}{\frac{1}{\alpha} + |x|} + 1)
Backward function:
.. math::
g'(x) = \frac{\alpha}{2(1 + |\alpha x|)^{2}} = \frac{1}{2\alpha(\frac{1}{\alpha} + |x|)^{2}}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> for alpha in [0.5, 1., 2., 4.]:
>>> grads = bm.vector_grad(bm.surrogate.soft_sign)(xs, alpha)
>>> plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
Parameter to control smoothness of gradient
Returns
-------
out: jax.Array
The spiking state.
"""
return SoftSign(alpha=alpha)(x)
[docs]
class Arctan(Surrogate):
"""Judge spiking state with an arctan function.
See Also
--------
arctan
"""
[docs]
def __init__(self, alpha=1.):
super().__init__()
self.alpha = alpha
def surrogate_grad(self, x):
x = as_jax(x)
dx = self.alpha * 0.5 / (1 + (jnp.pi / 2 * self.alpha * x) ** 2)
return dx
def surrogate_fun(self, x):
x = as_jax(x)
return jnp.arctan2(jnp.pi / 2 * self.alpha * x) / jnp.pi + 0.5
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def arctan(
x: Union[jax.Array, Array],
alpha: float = 1.,
):
r"""Judge spiking state with an arctan function.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
g(x) = \frac{1}{\pi} \arctan(\frac{\pi}{2}\alpha x) + \frac{1}{2}
Backward function:
.. math::
g'(x) = \frac{\alpha}{2(1 + (\frac{\pi}{2}\alpha x)^2)}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> for alpha in [0.5, 1., 2., 4.]:
>>> grads = bm.vector_grad(bm.surrogate.arctan)(xs, alpha)
>>> plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
Parameter to control smoothness of gradient
Returns
-------
out: jax.Array
The spiking state.
"""
return Arctan(alpha=alpha)(x)
[docs]
class NonzeroSignLog(Surrogate):
"""Judge spiking state with a nonzero sign log function.
See Also
--------
nonzero_sign_log
"""
[docs]
def __init__(self, alpha=1.):
super().__init__()
self.alpha = alpha
def surrogate_grad(self, x):
x = as_jax(x)
dx = 1. / (1 / self.alpha + jnp.abs(x))
return dx
def surrogate_fun(self, x):
x = as_jax(x)
return jnp.where(x < 0, -1., 1.) * jnp.log(jnp.abs(self.alpha * x) + 1)
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def nonzero_sign_log(
x: Union[jax.Array, Array],
alpha: float = 1.,
):
r"""Judge spiking state with a nonzero sign log function.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
g(x) = \mathrm{NonzeroSign}(x) \log (|\alpha x| + 1)
where
.. math::
\begin{split}\mathrm{NonzeroSign}(x) =
\begin{cases}
1, & x \geq 0 \\
-1, & x < 0 \\
\end{cases}\end{split}
Backward function:
.. math::
g'(x) = \frac{\alpha}{1 + |\alpha x|} = \frac{1}{\frac{1}{\alpha} + |x|}
This surrogate function has the advantage of low computation cost during the backward.
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> for alpha in [0.5, 1., 2., 4.]:
>>> grads = bm.vector_grad(bm.surrogate.nonzero_sign_log)(xs, alpha)
>>> plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
Parameter to control smoothness of gradient
Returns
-------
out: jax.Array
The spiking state.
"""
return NonzeroSignLog(alpha=alpha)(x)
[docs]
class ERF(Surrogate):
"""Judge spiking state with an erf function.
See Also
--------
erf
"""
[docs]
def __init__(self, alpha=1.):
super().__init__()
self.alpha = alpha
def surrogate_grad(self, x):
x = as_jax(x)
dx = (self.alpha / jnp.sqrt(jnp.pi)) * jnp.exp(-jnp.power(self.alpha, 2) * x * x)
return dx
def surrogate_fun(self, x):
x = as_jax(x)
return sci.special.erf(-self.alpha * x) * 0.5
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def erf(
x: Union[jax.Array, Array],
alpha: float = 1.,
):
r"""Judge spiking state with an erf function [1]_ [2]_ [3]_.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
\begin{split}
g(x) &= \frac{1}{2}(1-\text{erf}(-\alpha x)) \\
&= \frac{1}{2} \text{erfc}(-\alpha x) \\
&= \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\alpha x}e^{-t^2}dt
\end{split}
Backward function:
.. math::
g'(x) = \frac{\alpha}{\sqrt{\pi}}e^{-\alpha^2x^2}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> for alpha in [0.5, 1., 2., 4.]:
>>> grads = bm.vector_grad(bm.surrogate.nonzero_sign_log)(xs, alpha)
>>> plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
Parameter to control smoothness of gradient
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Esser S K, Appuswamy R, Merolla P, et al. Backpropagation for energy-efficient neuromorphic computing[J]. Advances in neural information processing systems, 2015, 28: 1117-1125.
.. [2] Wu Y, Deng L, Li G, et al. Spatio-temporal backpropagation for training high-performance spiking neural networks[J]. Frontiers in neuroscience, 2018, 12: 331.
.. [3] Yin B, Corradi F, Bohté S M. Effective and efficient computation with multiple-timescale spiking recurrent neural networks[C]//International Conference on Neuromorphic Systems 2020. 2020: 1-8.
"""
return ERF(alpha=alpha)(x)
[docs]
class PiecewiseLeakyRelu(Surrogate):
"""Judge spiking state with a piecewise leaky relu function.
See Also
--------
piecewise_leaky_relu
"""
[docs]
def __init__(self, c=0.01, w=1.):
super().__init__()
self.c = c
self.w = w
def surrogate_fun(self, x):
x = as_jax(x)
z = jnp.where(x < -self.w,
self.c * x + self.c * self.w,
jnp.where(x > self.w,
self.c * x - self.c * self.w + 1,
0.5 * x / self.w + 0.5))
return z
def surrogate_grad(self, x):
x = as_jax(x)
dx = jnp.where(jnp.abs(x) > self.w, self.c, 1 / self.w)
return dx
def __repr__(self):
return f'{self.__class__.__name__}(c={self.c}, w={self.w})'
[docs]
def piecewise_leaky_relu(
x: Union[jax.Array, Array],
c: float = 0.01,
w: float = 1.,
):
r"""Judge spiking state with a piecewise leaky relu function [1]_ [2]_ [3]_ [4]_ [5]_ [6]_ [7]_ [8]_.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
\begin{split}g(x) =
\begin{cases}
cx + cw, & x < -w \\
\frac{1}{2w}x + \frac{1}{2}, & -w \leq x \leq w \\
cx - cw + 1, & x > w \\
\end{cases}\end{split}
Backward function:
.. math::
\begin{split}g'(x) =
\begin{cases}
\frac{1}{w}, & |x| \leq w \\
c, & |x| > w
\end{cases}\end{split}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> for c in [0.01, 0.05, 0.1]:
>>> for w in [1., 2.]:
>>> grads1 = bm.vector_grad(bm.surrogate.piecewise_leaky_relu)(xs, c=c, w=w)
>>> plt.plot(bm.as_numpy(xs), bm.as_numpy(grads1), label=f'x={c}, w={w}')
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
c: float
When :math:`|x| > w` the gradient is `c`.
w: float
When :math:`|x| <= w` the gradient is `1 / w`.
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Yin S, Venkataramanaiah S K, Chen G K, et al. Algorithm and hardware design of discrete-time spiking neural networks based on back propagation with binary activations[C]//2017 IEEE Biomedical Circuits and Systems Conference (BioCAS). IEEE, 2017: 1-5.
.. [2] Wu Y, Deng L, Li G, et al. Spatio-temporal backpropagation for training high-performance spiking neural networks[J]. Frontiers in neuroscience, 2018, 12: 331.
.. [3] Huh D, Sejnowski T J. Gradient descent for spiking neural networks[C]//Proceedings of the 32nd International Conference on Neural Information Processing Systems. 2018: 1440-1450.
.. [4] Wu Y, Deng L, Li G, et al. Direct training for spiking neural networks: Faster, larger, better[C]//Proceedings of the AAAI Conference on Artificial Intelligence. 2019, 33(01): 1311-1318.
.. [5] Gu P, Xiao R, Pan G, et al. STCA: Spatio-Temporal Credit Assignment with Delayed Feedback in Deep Spiking Neural Networks[C]//IJCAI. 2019: 1366-1372.
.. [6] Roy D, Chakraborty I, Roy K. Scaling deep spiking neural networks with binary stochastic activations[C]//2019 IEEE International Conference on Cognitive Computing (ICCC). IEEE, 2019: 50-58.
.. [7] Cheng X, Hao Y, Xu J, et al. LISNN: Improving Spiking Neural Networks with Lateral Interactions for Robust Object Recognition[C]//IJCAI. 1519-1525.
.. [8] Kaiser J, Mostafa H, Neftci E. Synaptic plasticity dynamics for deep continuous local learning (DECOLLE)[J]. Frontiers in Neuroscience, 2020, 14: 424.
"""
return PiecewiseLeakyRelu(c=c, w=w)(x)
[docs]
class SquarewaveFourierSeries(Surrogate):
"""Judge spiking state with a squarewave fourier series.
See Also
--------
squarewave_fourier_series
"""
[docs]
def __init__(self, n=2, t_period=8.):
super().__init__()
self.n = n
self.t_period = t_period
def surrogate_grad(self, x):
x = as_jax(x)
w = jnp.pi * 2. / self.t_period
dx = jnp.cos(w * x)
for i in range(2, self.n):
dx += jnp.cos((2 * i - 1.) * w * x)
dx *= 4. / self.t_period
return dx
def surrogate_fun(self, x):
x = as_jax(x)
w = jnp.pi * 2. / self.t_period
ret = jnp.sin(w * x)
for i in range(2, self.n):
c = (2 * i - 1.)
ret += jnp.sin(c * w * x) / c
z = 0.5 + 2. / jnp.pi * ret
return z
def __repr__(self):
return f'{self.__class__.__name__}(n={self.n}, t_period={self.t_period})'
[docs]
def squarewave_fourier_series(
x: Union[jax.Array, Array],
n: int = 2,
t_period: float = 8.,
):
r"""Judge spiking state with a squarewave fourier series.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
g(x) = 0.5 + \frac{1}{\pi}*\sum_{i=1}^n {\sin\left({(2i-1)*2\pi}*x/T\right) \over 2i-1 }
Backward function:
.. math::
g'(x) = \sum_{i=1}^n\frac{4\cos\left((2 * i - 1.) * 2\pi * x / T\right)}{T}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> for n in [2, 4, 8]:
>>> f = bm.surrogate.SquarewaveFourierSeries(n=n)
>>> grads1 = bm.vector_grad(f)(xs)
>>> plt.plot(bm.as_numpy(xs), bm.as_numpy(grads1), label=f'n={n}')
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
n: int
t_period: float
Returns
-------
out: jax.Array
The spiking state.
"""
return SquarewaveFourierSeries(n=n, t_period=t_period)(x)
[docs]
class S2NN(Surrogate):
"""Judge spiking state with the S2NN surrogate spiking function.
See Also
--------
s2nn
"""
[docs]
def __init__(self, alpha=4., beta=1., epsilon=1e-8):
super().__init__()
self.alpha = alpha
self.beta = beta
self.epsilon = epsilon
def surrogate_fun(self, x):
x = as_jax(x)
z = jnp.where(x < 0.,
sci.special.expit(x * self.alpha),
self.beta * jnp.log(jnp.abs((x + 1.)) + self.epsilon) + 0.5)
return z
def surrogate_grad(self, x):
x = as_jax(x)
sg = sci.special.expit(self.alpha * x)
dx = jnp.where(x < 0., self.alpha * sg * (1. - sg), self.beta / (x + 1.))
return dx
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha}, beta={self.beta}, epsilon={self.epsilon})'
[docs]
def s2nn(
x: Union[jax.Array, Array],
alpha: float = 4.,
beta: float = 1.,
epsilon: float = 1e-8,
):
r"""Judge spiking state with the S2NN surrogate spiking function [1]_.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
\begin{split}g(x) = \begin{cases}
\mathrm{sigmoid} (\alpha x), x < 0 \\
\beta \ln(|x + 1|) + 0.5, x \ge 0
\end{cases}\end{split}
Backward function:
.. math::
\begin{split}g'(x) = \begin{cases}
\alpha * (1 - \mathrm{sigmoid} (\alpha x)) \mathrm{sigmoid} (\alpha x), x < 0 \\
\frac{\beta}{(x + 1)}, x \ge 0
\end{cases}\end{split}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> grads = bm.vector_grad(bm.surrogate.s2nn)(xs, 4., 1.)
>>> plt.plot(bm.as_numpy(xs), bm.as_numpy(grads), label=r'$\alpha=4, \beta=1$')
>>> grads = bm.vector_grad(bm.surrogate.s2nn)(xs, 8., 2.)
>>> plt.plot(bm.as_numpy(xs), bm.as_numpy(grads), label=r'$\alpha=8, \beta=2$')
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
The param that controls the gradient when ``x < 0``.
beta: float
The param that controls the gradient when ``x >= 0``
epsilon: float
Avoid nan
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Suetake, Kazuma et al. “S2NN: Time Step Reduction of Spiking Surrogate Gradients for Training Energy Efficient Single-Step Neural Networks.” ArXiv abs/2201.10879 (2022): n. pag.
"""
return S2NN(alpha=alpha, beta=beta, epsilon=epsilon)(x)
[docs]
class QPseudoSpike(Surrogate):
"""Judge spiking state with the q-PseudoSpike surrogate function.
See Also
--------
q_pseudo_spike
"""
[docs]
def __init__(self, alpha=2.):
super().__init__()
self.alpha = alpha
def surrogate_grad(self, x):
x = as_jax(x)
dx = jnp.power(1 + 2 / (self.alpha + 1) * jnp.abs(x), -self.alpha)
return dx
def surrogate_fun(self, x):
x = as_jax(x)
z = jnp.where(x < 0.,
0.5 * jnp.power(1 - 2 / (self.alpha - 1) * jnp.abs(x), 1 - self.alpha),
1. - 0.5 * jnp.power(1 + 2 / (self.alpha - 1) * jnp.abs(x), 1 - self.alpha))
return z
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def q_pseudo_spike(
x: Union[jax.Array, Array],
alpha: float = 2.,
):
r"""Judge spiking state with the q-PseudoSpike surrogate function [1]_.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
\begin{split}g(x) =
\begin{cases}
\frac{1}{2}(1-\frac{2x}{\alpha-1})^{1-\alpha}, & x < 0 \\
1 - \frac{1}{2}(1+\frac{2x}{\alpha-1})^{1-\alpha}, & x \geq 0.
\end{cases}\end{split}
Backward function:
.. math::
g'(x) = (1+\frac{2|x|}{\alpha-1})^{-\alpha}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> xs = bm.linspace(-3, 3, 1000)
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> for alpha in [0.5, 1., 2., 4.]:
>>> grads = bm.vector_grad(bm.surrogate.q_pseudo_spike)(xs, alpha)
>>> plt.plot(bm.as_numpy(xs), bm.as_numpy(grads), label=r'$\alpha=$' + str(alpha))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
The parameter to control tail fatness of gradient.
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Herranz-Celotti, Luca and Jean Rouat. “Surrogate Gradients Design.” ArXiv abs/2202.00282 (2022): n. pag.
"""
return QPseudoSpike(alpha=alpha)(x)
[docs]
class LeakyRelu(Surrogate):
"""Judge spiking state with the Leaky ReLU function.
See Also
--------
leaky_relu
"""
[docs]
def __init__(self, alpha=0.1, beta=1.):
super().__init__()
self.alpha = alpha
self.beta = beta
def surrogate_fun(self, x):
x = as_jax(x)
return jnp.where(x < 0., self.alpha * x, self.beta * x)
def surrogate_grad(self, x):
x = as_jax(x)
dx = jnp.where(x < 0., self.alpha, self.beta)
return dx
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha}, beta={self.beta})'
[docs]
def leaky_relu(
x: Union[jax.Array, Array],
alpha: float = 0.1,
beta: float = 1.,
):
r"""Judge spiking state with the Leaky ReLU function.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
\begin{split}g(x) =
\begin{cases}
\beta \cdot x, & x \geq 0 \\
\alpha \cdot x, & x < 0 \\
\end{cases}\end{split}
Backward function:
.. math::
\begin{split}g'(x) =
\begin{cases}
\beta, & x \geq 0 \\
\alpha, & x < 0 \\
\end{cases}\end{split}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> xs = bm.linspace(-3, 3, 1000)
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> grads = bm.vector_grad(bm.surrogate.leaky_relu)(xs, 0., 1.)
>>> plt.plot(bm.as_numpy(xs), bm.as_numpy(grads), label=r'$\alpha=0., \beta=1.$')
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
The parameter to control the gradient when :math:`x < 0`.
beta: float
The parameter to control the gradient when :math:`x >= 0`.
Returns
-------
out: jax.Array
The spiking state.
"""
return LeakyRelu(alpha=alpha, beta=beta)(x)
[docs]
class LogTailedRelu(Surrogate):
"""Judge spiking state with the Log-tailed ReLU function.
See Also
--------
log_tailed_relu
"""
[docs]
def __init__(self, alpha=0.):
super().__init__()
self.alpha = alpha
def surrogate_fun(self, x):
x = as_jax(x)
z = jnp.where(x > 1,
jnp.log(x),
jnp.where(x > 0,
x,
self.alpha * x))
return z
def surrogate_grad(self, x):
x = as_jax(x)
dx = jnp.where(x > 1,
1 / x,
jnp.where(x > 0,
1.,
self.alpha))
return dx
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def log_tailed_relu(
x: Union[jax.Array, Array],
alpha: float = 0.,
):
r"""Judge spiking state with the Log-tailed ReLU function [1]_.
If `origin=False`, computes the forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
If `origin=True`, computes the original function:
.. math::
\begin{split}g(x) =
\begin{cases}
\alpha x, & x \leq 0 \\
x, & 0 < x \leq 0 \\
log(x), x > 1 \\
\end{cases}\end{split}
Backward function:
.. math::
\begin{split}g'(x) =
\begin{cases}
\alpha, & x \leq 0 \\
1, & 0 < x \leq 0 \\
\frac{1}{x}, x > 1 \\
\end{cases}\end{split}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> xs = bm.linspace(-3, 3, 1000)
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> grads = bm.vector_grad(bm.surrogate.leaky_relu)(xs, 0., 1.)
>>> plt.plot(bm.as_numpy(xs), bm.as_numpy(grads), label=r'$\alpha=0., \beta=1.$')
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
The parameter to control the gradient.
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Cai, Zhaowei et al. “Deep Learning with Low Precision by Half-Wave Gaussian Quantization.” 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2017): 5406-5414.
"""
return LogTailedRelu(alpha=alpha)(x)
[docs]
class ReluGrad(Surrogate):
"""Judge spiking state with the ReLU gradient function.
See Also
--------
relu_grad
"""
[docs]
def __init__(self, alpha=0.3, width=1.):
super().__init__()
self.alpha = alpha
self.width = width
def surrogate_grad(self, x):
x = as_jax(x)
dx = jnp.maximum(self.alpha * self.width - jnp.abs(x) * self.alpha, 0)
return dx
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha}, width={self.width})'
[docs]
def relu_grad(
x: Union[jax.Array, Array],
alpha: float = 0.3,
width: float = 1.,
):
r"""Spike function with the ReLU gradient function [1]_.
The forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
Backward function:
.. math::
g'(x) = \text{ReLU}(\alpha * (\mathrm{width}-|x|))
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> xs = bm.linspace(-3, 3, 1000)
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> for s in [0.5, 1.]:
>>> for w in [1, 2.]:
>>> grads = bm.vector_grad(bm.surrogate.relu_grad)(xs, s, w)
>>> plt.plot(bm.as_numpy(xs), bm.as_numpy(grads), label=r'$\alpha=$' + f'{s}, width={w}')
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
The parameter to control the gradient.
width: float
The parameter to control the width of the gradient.
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Neftci, E. O., Mostafa, H. & Zenke, F. Surrogate gradient learning in spiking neural networks. IEEE Signal Process. Mag. 36, 61–63 (2019).
"""
return ReluGrad(alpha=alpha, width=width)(x)
[docs]
class GaussianGrad(Surrogate):
"""Judge spiking state with the Gaussian gradient function.
See Also
--------
gaussian_grad
"""
[docs]
def __init__(self, sigma=0.5, alpha=0.5):
super().__init__()
self.sigma = sigma
self.alpha = alpha
def surrogate_grad(self, x):
x = as_jax(x)
dx = jnp.exp(-(x ** 2) / 2 * jnp.power(self.sigma, 2)) / (jnp.sqrt(2 * jnp.pi) * self.sigma)
return self.alpha * dx
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha}, sigma={self.sigma})'
[docs]
def gaussian_grad(
x: Union[jax.Array, Array],
sigma: float = 0.5,
alpha: float = 0.5,
):
r"""Spike function with the Gaussian gradient function [1]_.
The forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
Backward function:
.. math::
g'(x) = \alpha * \text{gaussian}(x, 0., \sigma)
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> xs = bm.linspace(-3, 3, 1000)
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> for s in [0.5, 1., 2.]:
>>> grads = bm.vector_grad(bm.surrogate.gaussian_grad)(xs, s, 0.5)
>>> plt.plot(bm.as_numpy(xs), bm.as_numpy(grads), label=r'$\alpha=0.5, \sigma=$' + str(s))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
sigma: float
The parameter to control the variance of gaussian distribution.
alpha: float
The parameter to control the scale of the gradient.
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Yin, B., Corradi, F. & Bohté, S.M. Accurate and efficient time-domain classification with adaptive spiking recurrent neural networks. Nat Mach Intell 3, 905–913 (2021).
"""
return GaussianGrad(sigma=sigma, alpha=alpha)(x)
[docs]
class MultiGaussianGrad(Surrogate):
"""Judge spiking state with the multi-Gaussian gradient function.
See Also
--------
multi_gaussian_grad
"""
[docs]
def __init__(self, h=0.15, s=6.0, sigma=0.5, scale=0.5):
super().__init__()
self.h = h
self.s = s
self.sigma = sigma
self.scale = scale
def surrogate_grad(self, x):
x = as_jax(x)
g1 = jnp.exp(-x ** 2 / (2 * jnp.power(self.sigma, 2))) / (jnp.sqrt(2 * jnp.pi) * self.sigma)
g2 = jnp.exp(-(x - self.sigma) ** 2 / (2 * jnp.power(self.s * self.sigma, 2))
) / (jnp.sqrt(2 * jnp.pi) * self.s * self.sigma)
g3 = jnp.exp(-(x + self.sigma) ** 2 / (2 * jnp.power(self.s * self.sigma, 2))
) / (jnp.sqrt(2 * jnp.pi) * self.s * self.sigma)
dx = g1 * (1. + self.h) - g2 * self.h - g3 * self.h
return self.scale * dx
def __repr__(self):
return f'{self.__class__.__name__}(h={self.h}, s={self.s}, sigma={self.sigma}, scale={self.scale})'
[docs]
def multi_gaussian_grad(
x: Union[jax.Array, Array],
h: float = 0.15,
s: float = 6.0,
sigma: float = 0.5,
scale: float = 0.5,
):
r"""Spike function with the multi-Gaussian gradient function [1]_.
The forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
Backward function:
.. math::
\begin{array}{l}
g'(x)=(1+h){{{\mathcal{N}}}}(x, 0, {\sigma }^{2})
-h{{{\mathcal{N}}}}(x, \sigma,{(s\sigma )}^{2})-
h{{{\mathcal{N}}}}(x, -\sigma ,{(s\sigma )}^{2})
\end{array}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> xs = bm.linspace(-3, 3, 1000)
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> grads = bm.vector_grad(bm.surrogate.multi_gaussian_grad)(xs)
>>> plt.plot(bm.as_numpy(xs), bm.as_numpy(grads))
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
h: float
The hyper-parameters of approximate function
s: float
The hyper-parameters of approximate function
sigma: float
The gaussian sigma.
scale: float
The gradient scale.
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Yin, B., Corradi, F. & Bohté, S.M. Accurate and efficient time-domain classification with adaptive spiking recurrent neural networks. Nat Mach Intell 3, 905–913 (2021).
"""
return MultiGaussianGrad(h=h, s=s, sigma=sigma, scale=scale)(x)
[docs]
class InvSquareGrad(Surrogate):
"""Judge spiking state with the inverse-square surrogate gradient function.
See Also
--------
inv_square_grad
"""
[docs]
def __init__(self, alpha=100.):
super().__init__()
self.alpha = alpha
def surrogate_grad(self, x):
dx = 1. / (self.alpha * jnp.abs(x) + 1.0) ** 2
return dx
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def inv_square_grad(
x: Union[jax.Array, Array],
alpha: float = 100.
):
r"""Spike function with the inverse-square surrogate gradient.
Forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
Backward function:
.. math::
g'(x) = \frac{1}{(\alpha * |x| + 1.) ^ 2}
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> xs = bm.linspace(-1, 1, 1000)
>>> for alpha in [1., 10., 100.]:
>>> grads = bm.vector_grad(bm.surrogate.inv_square_grad)(xs, alpha)
>>> plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
Parameter to control smoothness of gradient
Returns
-------
out: jax.Array
The spiking state.
"""
return InvSquareGrad(alpha=alpha)(x)
[docs]
class SlayerGrad(Surrogate):
"""Judge spiking state with the slayer surrogate gradient function.
See Also
--------
slayer_grad
"""
[docs]
def __init__(self, alpha=1.):
super().__init__()
self.alpha = alpha
def surrogate_grad(self, x):
dx = jnp.exp(-self.alpha * jnp.abs(x))
return dx
def __repr__(self):
return f'{self.__class__.__name__}(alpha={self.alpha})'
[docs]
def slayer_grad(
x: Union[jax.Array, Array],
alpha: float = 1.
):
r"""Spike function with the slayer surrogate gradient function.
Forward function:
.. math::
g(x) = \begin{cases}
1, & x \geq 0 \\
0, & x < 0 \\
\end{cases}
Backward function:
.. math::
g'(x) = \exp(-\alpha |x|)
.. plot::
:include-source: True
>>> import brainpy as bp
>>> import brainpy.math as bm
>>> import matplotlib.pyplot as plt
>>> bp.visualize.get_figure(1, 1, 4, 6)
>>> xs = bm.linspace(-3, 3, 1000)
>>> for alpha in [0.5, 1., 2., 4.]:
>>> grads = bm.vector_grad(bm.surrogate.slayer_grad)(xs, alpha)
>>> plt.plot(xs, grads, label=r'$\alpha$=' + str(alpha))
>>> plt.legend()
>>> plt.show()
Parameters
----------
x: jax.Array, Array
The input data.
alpha: float
Parameter to control smoothness of gradient
Returns
-------
out: jax.Array
The spiking state.
References
----------
.. [1] Shrestha, S. B. & Orchard, G. Slayer: spike layer error reassignment in time. In Advances in Neural Information Processing Systems Vol. 31, 1412–1421 (NeurIPS, 2018).
"""
return SlayerGrad(alpha=alpha)(x)
