梯度下降简介

目录

• 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

• 导数，derivative，抽象表达

• 偏微分，partial derivative，沿着某个具体的轴运动

• 梯度，gradient，向量

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

What does it mean？

• 箭头的方向表示梯度的方向
• 箭头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

[ heta_{t+1}= heta_t-alpha_t abla{f( heta_t)} ]

AutoGrad

• With Tf.GradientTape() as tape:

  • Build computation graph
  • (loss = f_ heta{(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)