相对于自适应神经网络、感知器，softmax巧妙低使用简单的方法来实现多分类问题。

• 功能上，完成从N维向量到M维向量的映射
• 输出的结果范围是[0,1]，对于一个sample的结果所有输出总和等于1
• 输出结果，可以隐含地表达该类别的概率

softmax的损失函数是采用了多分类问题中常见的交叉熵，注意经常有2个表达的形式

• 经典的交叉熵形式：L=-sum(y_right * log(y_pred))， 具体
• 简单版本是: L = -Log(y_pred),具体

这两个版本在求导过程有点不同，但是结果都是一样的，同时损失表达的意思也是相同的，因为在第一种表达形式中，当y不是

正确分类时,y_right等于0，当y是正确分类时,y_right等于1。

下面基于mnist数据做了一个多分类的实验，整体能达到85%的精度。

'''
softmax classifier for mnist 

created on 2019.9.28
author: vince
'''
import math
import logging
import numpy
import random
import matplotlib.pyplot as plt
from tensorflow.contrib.learn.python.learn.datasets.mnist import read_data_sets
from sklearn.metrics import accuracy_score

def loss_max_right_class_prob(predictions,y):
	return -predictions[numpy.argmax(y)];

def loss_cross_entropy(predictions,y):
	return -numpy.dot(y,numpy.log(predictions));

'''
Softmax classifier
linear classifier
'''
class Softmax:

	def __init__(self,iter_num = 100000,batch_size = 1):
		self.__iter_num = iter_num;
		self.__batch_size = batch_size;

	def train(self,train_X,train_Y):
		X = numpy.c_[train_X,numpy.ones(train_X.shape[0])];
		Y = numpy.copy(train_Y);

		self.L = [];

		#initialize parameters
		self.__weight = numpy.random.rand(X.shape[1],10) * 2 - 1.0;
		self.__step_len = 1e-3; 

		logging.info("weight:%s" % (self.__weight));

		for iter_index in range(self.__iter_num):
			if iter_index % 1000 == 0:
				logging.info("-----iter:%s-----" % (iter_index));
			if iter_index % 100 == 0:
				l = 0;
				for i in range(0,len(X),100):
					predictions = self.forward_pass(X[i]);
					#l += loss_max_right_class_prob(predictions,Y[i]);
					l += loss_cross_entropy(predictions,Y[i]);
				l /= len(X);
				self.L.append(l);

			sample_index = random.randint(0,len(X) - 1);
			logging.debug("-----select sample %s-----" % (sample_index));

			z = numpy.dot(X[sample_index],self.__weight);
			z = z - numpy.max(z);
			predictions = numpy.exp(z) / numpy.sum(numpy.exp(z));
			dw = self.__step_len * X[sample_index].reshape(-1,1).dot((predictions - Y[sample_index]).reshape(1,-1));
#			dw = self.__step_len * X[sample_index].reshape(-1,1).dot(predictions.reshape(1,-1));
#			dw[range(X.shape[1]),numpy.argmax(Y[sample_index])] -= X[sample_index] * self.__step_len;

			self.__weight -= dw;

			logging.debug("weight:%s" % (self.__weight));
			logging.debug("loss:%s" % (l));
		logging.info("weight:%s" % (self.__weight));
		logging.info("L:%s" % (self.L));

	def forward_pass(self,x):
		net = numpy.dot(x,self.__weight);
		net = net - numpy.max(net);
		net = numpy.exp(net) / numpy.sum(numpy.exp(net));
		return net;

	def predict(self,x):
		x = numpy.append(x,1.0);
		return self.forward_pass(x);

def main():
	logging.basicConfig(level = logging.INFO,format = '%(asctime)s %(filename)s[line:%(lineno)d] %(levelname)s %(message)s',datefmt = '%a,%d %b %Y %H:%M:%S');

	logging.info("trainning begin.");

	mnist = read_data_sets('../data/MNIST',one_hot=True)  # MNIST_data指的是存放数据的文件夹路径，one_hot=True 为采用one_hot的编码方式编码标签

	#load data
	train_X = mnist.train.images        #训练集样本
	validation_X = mnist.validation.images   #验证集样本
	test_X = mnist.test.images         #测试集样本
	#labels
	train_Y = mnist.train.labels        #训练集标签
	validation_Y = mnist.validation.labels   #验证集标签
	test_Y = mnist.test.labels         #测试集标签

	classifier = Softmax();
	classifier.train(train_X,train_Y);

	logging.info("trainning end. predict begin.");

	test_predict = numpy.array([]);
	test_right = numpy.array([]);
	for i in range(len(test_X)):
		predict_label = numpy.argmax(classifier.predict(test_X[i]));
		test_predict = numpy.append(test_predict,predict_label);
		right_label = numpy.argmax(test_Y[i]);
		test_right = numpy.append(test_right,right_label);

	logging.info("right:%s,predict:%s" % (test_right,test_predict));
	score = accuracy_score(test_right,test_predict);
	logging.info("The accruacy score is: %s "% (str(score)));

	plt.plot(classifier.L)
	plt.show();

if __name__ == "__main__":
	main();

损失函数收敛情况

Sun,29 Sep 2019 18:08:08 softmax.py[line:104] INFO trainning end. predict begin.
Sun,29 Sep 2019 18:08:08 softmax.py[line:114] INFO right:[7. 2. 1. ... 4. 5. 6.],predict:[7. 2. 1. ... 4. 8. 6.]
Sun,29 Sep 2019 18:08:08 softmax.py[line:116] INFO The accruacy score is: 0.8486 

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