梯度下降簡介

目錄

• Outline
• What's Gradient
• What does it mean？
• How to search
• For instance
• AutoGrad
  • $2^{nd}$-order

Outline

• What's Gradient

• What does it mean

• How to Search

• AutoGrad

What's Gradient

• 導數，derivative，抽象表達

• 偏微分，partial derivative，沿着某個具體的軸運動

• 梯度，gradient，向量

\[\nabla{f} = (\frac{\partial{f}}{\partial{x_1}};\frac{\partial{f}}{{\partial{x_2}}};\cdots;\frac{\partial{f}}{{\partial{x_n}}}) \]

What does it mean？

• 箭頭的方向表示梯度的方向
• 箭頭模的大小表示梯度增大的速率

How to search

• 沿着梯度下降的反方向搜索

For instance

\[\theta_{t+1}=\theta_t-\alpha_t\nabla{f(\theta_t)} \]

AutoGrad

• With Tf.GradientTape() as tape:

  • Build computation graph
  • \(loss = f_\theta{(x)}\)

• [w_grad] = tape.gradient(loss,[w])

import tensorflow as tf

w = tf.constant(1.)
x = tf.constant(2.)
y = x * w

with tf.GradientTape() as tape:
    tape.watch([w])
    y2 = x * w

grad1 = tape.gradient(y, [w])
grad1

[None]

with tf.GradientTape() as tape:
    tape.watch([w])
    y2 = x * w

grad2 = tape.gradient(y2, [w])
grad2

[<tf.Tensor: id=30, shape=(), dtype=float32, numpy=2.0>]

try:
    grad2 = tape.gradient(y2, [w])
except Exception as e:
    print(e)

GradientTape.gradient can only be called once on non-persistent tapes.

• 永久保存grad

with tf.GradientTape(persistent=True) as tape:
    tape.watch([w])
    y2 = x * w

grad2 = tape.gradient(y2, [w])
grad2

[<tf.Tensor: id=35, shape=(), dtype=float32, numpy=2.0>]

grad2 = tape.gradient(y2, [w])
grad2

[<tf.Tensor: id=39, shape=(), dtype=float32, numpy=2.0>]

\(2^{nd}\)-order

• y = xw + b

• \(\frac{\partial{y}}{\partial{w}} = x\)

• \(\frac{\partial^2{y}}{\partial{w^2}} = \frac{\partial{y'}}{\partial{w}} = \frac{\partial{X}}{\partial{w}} = None\)

with tf.GradientTape() as t1:
    with tf.GradientTape() as t2:
        y = x * w + b
    dy_dw, dy_db = t2.gradient(y, [w, b])

d2y_dw2 = t1.gradient(dy_dw, w)