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

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

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

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

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)