损失函数及其梯度

目录

• Typical Loss
• MSE
  • Derivative
  • MSE Gradient
• Softmax
  • Derivative

Typical Loss

• Mean Squared Error

• Cross Entropy Loss

  • binary
  • multi-class
  • +softmax

MSE

• loss=∑[y−(xw+b)]2$loss=\sum [y-(xw+b){]}^2$

• L2−norm=||y−(xw+b)||2$L_{2-norm}=||y-(xw+b)|{|}_2$

• loss=norm(y−(xw+b))2$loss=norm(y-(xw+b){)}^2$

Derivative

• loss=∑[y−fθ(x)]2$loss=\sum [y-f_{\theta }(x){]}^2$

• ∇loss∇θ=2∑[y−fθ(x)]∗∇fθ(x)∇θ$\frac{\mathrm{\nabla }\text{loss}}{\mathrm{\nabla }\theta }=2\sum [y-f_{\theta }(x)]\ast \frac{\mathrm{\nabla }{f}_{\theta }(x)}{\mathrm{\nabla }\theta }$

MSE Gradient

import tensorflow as tf

x = tf.random.normal([2, 4])
w = tf.random.normal([4, 3])
b = tf.zeros([3])
y = tf.constant([2, 0])
with tf.GradientTape() as tape:

tape.watch([w, b])

prob = tf.nn.softmax(x @ w + b, axis=1)

loss = tf.reduce_mean(tf.losses.MSE(tf.one_hot(y, depth=3), prob))
grads = tape.gradient(loss, [w, b])

grads[0]

<tf.Tensor: id=92, shape=(4, 3), dtype=float32, numpy=
array([[ 0.01156707, -0.00927749, -0.00228957],
       [ 0.03556816, -0.03894382,  0.00337564],
       [-0.02537526,  0.01924876,  0.00612648],
       [-0.0074787 ,  0.00161515,  0.00586352]], dtype=float32)>

grads[1]

<tf.Tensor: id=90, shape=(3,), dtype=float32, numpy=array([-0.01552947,  0.01993286, -0.00440337], dtype=float32)>

Softmax

• soft version of max

• 大的越来越大，小的越来越小、越密集

Derivative

pi=eai∑Nk=1eak$$p_i=\frac{e^{a_i}}{\sum _{k=1}^N{e}^{a_k}}$$

• i=j

∂pi∂aj=∂eai∑Nk=1eak∂aj=pi(1−pj)$$\frac{\mathrm{\partial }{p}_i}{\mathrm{\partial }{a}_j}=\frac{\mathrm{\partial }\frac{e^{a_i}}{\sum _{k=1}^N{e}^{a_k}}}{\mathrm{\partial }{a}_j}=p_i(1-p_j)$$

• i≠j$i\ne j$

∂pi∂aj=∂eai∑Nk=1eak∂aj=−pj∗pi$$\frac{\mathrm{\partial }{p}_i}{\mathrm{\partial }{a}_j}=\frac{\mathrm{\partial }\frac{e^{a_i}}{\sum _{k=1}^N{e}^{a_k}}}{\mathrm{\partial }{a}_j}=-p_j\ast p_i$$

x = tf.random.normal([2, 4])
w = tf.random.normal([4, 3])
b = tf.zeros([3])
y = tf.constant([2, 0])
with tf.GradientTape() as tape:

tape.watch([w, b])

logits =x @ w + b

loss = tf.reduce_mean(

tf.losses.categorical_crossentropy(tf.one_hot(y, depth=3),

logits,

from_logits=True))
grads = tape.gradient(loss, [w, b])

grads[0]

<tf.Tensor: id=226, shape=(4, 3), dtype=float32, numpy=
array([[-0.38076094,  0.33844548,  0.04231545],
       [-1.0262716 , -0.6730384 ,  1.69931   ],
       [ 0.20613424, -0.50421923,  0.298085  ],
       [ 0.5800004 , -0.22329211, -0.35670823]], dtype=float32)>

grads[1]

<tf.Tensor: id=224, shape=(3,), dtype=float32, numpy=array([-0.3719653 ,  0.53269935, -0.16073406], dtype=float32)>