Gradient Descent Algorithm gradient descent algorithm

2. Gradient Descent Algorithm gradient descent algorithm

Bilibili Video Tutorial Portal: PyTorch Deep Learning Practice - Gradient Descent Algorithm

2.1 Optimization problem

2.2 Formula derivation

2.3 Gradient Descent gradient descent

import matplotlib.pyplot as plt

x_data = [1.0, 2.0, 3.0]
y_data = [2.0, 4.0, 6.0]

w = 1.0


def forward(x):
    return x * w


def cost(xs, ys):
    cost = 0
    for x, y in zip(xs, ys):
        y_pred = forward(x)
        cost += (y_pred - y) ** 2
    return cost / len(xs)


def gradient(xs, ys):
    grad = 0
    for x, y in zip(xs, ys):
        grad += 2 * x * (x * w - y)
    return grad / len(xs)


epoch_list = []
cost_list = []

print('Predict (before training)', 4, forward(4))
for epoch in range(100):
    cost_val = cost(x_data, y_data)
    grad_val = gradient(x_data, y_data)
    w -= 0.01 * grad_val
    print('Epoch:', epoch, 'W=', round(w, 2), 'Loss=', round(cost_val, 2))

    epoch_list.append(epoch)
    cost_list.append(cost_val)

print('Predict (after training)', 4, forward(4))

plt.plot(epoch_list, cost_list)
plt.grid(True, linestyle="--", color="gray", linewidth="0.5", axis="both")
plt.xlabel('Epoch')
plt.ylabel('Cost')
plt.show()

Predict (before training) 4 4.0
Epoch: 0 W= 1.09 Loss= 4.67
...
Epoch: 99 W= 2.0 Loss= 0.0
Predict (after training) 4 7.999777758621207

2.4 Stochastic Gradient Descent stochastic gradient descent

import matplotlib.pyplot as plt

x_data = [1.0, 2.0, 3.0]
y_data = [2.0, 4.0, 6.0]

w = 1.0


def forward(x):
    return x * w


def loss(x, y):
    y_pred = forward(x)
    return (y_pred - y) ** 2


def gradient(x, y):
    return 2 * x * (x * w - y)


epoch_list = []
loss_list = []

print('Predict (before training)', 4, forward(4))

for epoch in range(100):
    for x, y in zip(x_data, y_data):
        grad = gradient(x, y)
        w -= 0.01 * grad
        print("grad:", x, y, grad)
        l = loss(x, y)

    print("progress:", epoch, "w=", round(w, 2), "loss=", round(l, 2))
    epoch_list.append(epoch)
    loss_list.append(l)

print('Predict (after training)', 4, forward(4))

plt.plot(epoch_list, loss_list)
plt.grid(True, linestyle="--", color="gray", linewidth="0.5", axis="both")
plt.xlabel('Epoch')
plt.ylabel('Cost')
plt.show()

Predict (before training) 4 4.0
grad: 1.0 2.0 -2.0
grad: 2.0 4.0 -7.84
grad: 3.0 6.0 -16.2288
progress: 0 w= 1.26 loss= 4.92
...
grad: 1.0 2.0 -2.0650148258027912e-13
grad: 2.0 4.0 -8.100187187665142e-13
grad: 3.0 6.0 -1.6786572132332367e-12
progress: 99 w= 2.0 loss= 0.0
Predict (after training) 4 7.9999999999996945