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gradient_descent.py

Loss surface L(w)L(\mathbf{w})

step 0|loss 24.250

  • ∇L(w)\nabla L(\mathbf{w}) is correct
  • L(wk+1)<L(wk)L(w_{k+1}) < L(w_k) every step
  • w→0\mathbf{w} \to 0 converges

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linear_regression.py

loss L(w)|rolling down the gradient

import numpy as np

def linear_regression(X, y, lr=0.01):
    w, b = 0.0, 0.0
    for _ in range(100):
        err = w * X + b - y
        w -= lr * (err * X).mean()
        b -= lr * err.mean()
    return w, b

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Cosine similarity

Direction reveals how similar two vectors are. Change the angle and compare their alignment.

cos⁡(a,b)=a⋅b∥a∥∥b∥\cos(\mathbf{a},\mathbf{b})=\frac{\mathbf{a}\cdot\mathbf{b}}{\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert}
ab45°
Cosine similarity0.7071
Vector lengths1

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