Pythonで学ぶニューラルネットワークの最適化手法：SGD、Momentum、Adam

import math

# Minimise f(x) = (x - 3)² with three optimizers. The gradient is 2(x - 3).
grad = lambda x: 2 * (x - 3)

def sgd(steps, lr=0.1):
    x = 0.0
    for _ in range(steps):
        x -= lr * grad(x)
    return x

def momentum(steps, lr=0.1, beta=0.9):
    x, v = 0.0, 0.0
    for _ in range(steps):
        v = beta * v + grad(x)        # remember the direction we were going
        x -= lr * v
    return x

def adam(steps, lr=0.3, b1=0.9, b2=0.999, eps=1e-8):
    x, m, v = 0.0, 0.0, 0.0
    for t in range(1, steps + 1):
        g = grad(x)
        m = b1 * m + (1 - b1) * g     # running mean of gradients
        v = b2 * v + (1 - b2) * g * g # running mean of squared gradients
        m_hat, v_hat = m / (1 - b1 ** t), v / (1 - b2 ** t)
        x -= lr * m_hat / (math.sqrt(v_hat) + eps)
    return x

for steps in (5, 20, 50):
    print(f"{steps:>2} steps  sgd={sgd(steps):.3f}  momentum={momentum(steps):.3f}  adam={adam(steps):.3f}")