import random

import math

​

def sigmoid(x):  # activation for last layer (probability)

    return 1.0 / (1.0 + math.exp(-x))

​

def dsigmoid(y):  # derivative given output y = sigmoid(x)

    return y * (1.0 - y)

​

def relu(x):

    return x if x > 0 else 0.0

​

def relu_grad(x):

    return 1.0 if x > 0 else 0.0

​

def dot(a, b):

    return sum(x*y for x, y in zip(a, b))

​

def matvec(W, v):

    # W: list of rows, v: vector

    return [dot(row, v) for row in W]

​

def add(v, b):

    return [x + y for x, y in zip(v, b)]

​

def outer(u, v):

    # returns matrix: u * v^T

    return [[ui * vj for vj in v] for ui in u]

​