摘要：前言本文基于官網的寫成。輸入數據是，全稱是，是一組由這個機構搜集的手寫數字掃描文件和每個文件對應標簽的數據集，經過一定的修改使其適合機器學習算法讀取。這個數據集可以從牛的不行的教授的網站獲取。

本文基于TensorFlow官網的Tutorial寫成。輸入數據是MNIST，全稱是Modified National Institute of Standards and Technology，是一組由這個機構搜集的手寫數字掃描文件和每個文件對應標簽的數據集，經過一定的修改使其適合機器學習算法讀取。這個數據集可以從牛的不行的Yann LeCun教授的網站獲取。

本文首先使用sklearn的LogisticRegression()進行訓練，得到的參數繪制效果如下（紅色表示參數估計結果為負，藍色表示參數估計結果為正，綠色代表參數估計結果為零）：

從圖形效果看，我們發現藍色點組成的輪廓與對應的數字輪廓還是比較接近的。

然后本文使用tensorflow對同樣的數據集進行了softmax regression的訓練，得到的參數繪制效果如下：

藍色點組成的輪廓與對應的數字輪廓比較接近。但是對比上下兩幅截圖，感覺tensorflow的效果更平滑一些。不過從測試集的準確率來看，二者都在92%左右，sklearn稍微好一點。注意，92%的準確率看起來不錯，但其實是一個很低的準確率，按照官網教程的說法，應該要感到羞愧。

#!/usr/bin/env python
# -*- coding=utf-8 -*-
# @author: 陳水平
# @date: 2017-01-10
# @description: implement a softmax regression model upon MNIST handwritten digits
# @ref: http://yann.lecun.com/exdb/mnist/

import gzip
import struct
import numpy as np
from sklearn.linear_model import LogisticRegression
from sklearn import preprocessing
from sklearn.metrics import accuracy_score
import tensorflow as tf

# MNIST data is stored in binary format, 
# and we transform them into numpy ndarray objects by the following two utility functions
def read_image(file_name):
    with gzip.open(file_name, "rb") as f:
        buf = f.read()
        index = 0
        magic, images, rows, columns = struct.unpack_from(">IIII" , buf , index)
        index += struct.calcsize(">IIII")

        image_size = ">" + str(images*rows*columns) + "B"
        ims = struct.unpack_from(image_size, buf, index)
        
        im_array = np.array(ims).reshape(images, rows, columns)
        return im_array

def read_label(file_name):
    with gzip.open(file_name, "rb") as f:
        buf = f.read()
        index = 0
        magic, labels = struct.unpack_from(">II", buf, index)
        index += struct.calcsize(">II")
        
        label_size = ">" + str(labels) + "B"
        labels = struct.unpack_from(label_size, buf, index)

        label_array = np.array(labels)
        return label_array

print "Start processing MNIST handwritten digits data..."
train_x_data = read_image("MNIST_data/train-images-idx3-ubyte.gz")
train_x_data = train_x_data.reshape(train_x_data.shape[0], -1).astype(np.float32)
train_y_data = read_label("MNIST_data/train-labels-idx1-ubyte.gz")
test_x_data = read_image("MNIST_data/t10k-images-idx3-ubyte.gz")
test_x_data = test_x_data.reshape(test_x_data.shape[0], -1).astype(np.float32)
test_y_data = read_label("MNIST_data/t10k-labels-idx1-ubyte.gz")

train_x_minmax = train_x_data / 255.0
test_x_minmax = test_x_data / 255.0

# Of course you can also use the utility function to read in MNIST provided by tensorflow
# from tensorflow.examples.tutorials.mnist import input_data
# mnist = input_data.read_data_sets("MNIST_data/", one_hot=False)
# train_x_minmax = mnist.train.images
# train_y_data = mnist.train.labels
# test_x_minmax = mnist.test.images
# test_y_data = mnist.test.labels

# We evaluate the softmax regression model by sklearn first
eval_sklearn = False
if eval_sklearn:
    print "Start evaluating softmax regression model by sklearn..."
    reg = LogisticRegression(solver="lbfgs", multi_class="multinomial")
    reg.fit(train_x_minmax, train_y_data)
    np.savetxt("coef_softmax_sklearn.txt", reg.coef_, fmt="%.6f")  # Save coefficients to a text file
    test_y_predict = reg.predict(test_x_minmax)
    print "Accuracy of test set: %f" % accuracy_score(test_y_data, test_y_predict)

eval_tensorflow = True
batch_gradient = False
if eval_tensorflow:
    print "Start evaluating softmax regression model by tensorflow..."
    # reformat y into one-hot encoding style
    lb = preprocessing.LabelBinarizer()
    lb.fit(train_y_data)
    train_y_data_trans = lb.transform(train_y_data)
    test_y_data_trans = lb.transform(test_y_data)

    x = tf.placeholder(tf.float32, [None, 784])
    W = tf.Variable(tf.zeros([784, 10]))
    b = tf.Variable(tf.zeros([10]))
    V = tf.matmul(x, W) + b
    y = tf.nn.softmax(V)

    y_ = tf.placeholder(tf.float32, [None, 10])

    loss = tf.reduce_mean(-tf.reduce_sum(y_ * tf.log(y), reduction_indices=[1]))
    optimizer = tf.train.GradientDescentOptimizer(0.5)
    train = optimizer.minimize(loss)

    init = tf.initialize_all_variables()

    sess = tf.Session()
    sess.run(init)

    if batch_gradient:
        for step in range(300):
            sess.run(train, feed_dict={x: train_x_minmax, y_: train_y_data_trans})
            if step % 10 == 0:
                print "Batch Gradient Descent processing step %d" % step
        print "Finally we got the estimated results, take such a long time..."
    else:
        for step in range(1000):
            sample_index = np.random.choice(train_x_minmax.shape[0], 100)
            batch_xs = train_x_minmax[sample_index, :]
            batch_ys = train_y_data_trans[sample_index, :]
            sess.run(train, feed_dict={x: batch_xs, y_: batch_ys})
            if step % 100 == 0:
                print "Stochastic Gradient Descent processing step %d" % step
    np.savetxt("coef_softmax_tf.txt", np.transpose(sess.run(W)), fmt="%.6f")  # Save coefficients to a text file
    correct_prediction = tf.equal(tf.argmax(y, 1), tf.argmax(y_, 1))
    accuracy = tf.reduce_mean(tf.cast(correct_prediction, tf.float32))
    print "Accuracy of test set: %f" % sess.run(accuracy, feed_dict={x: test_x_minmax, y_: test_y_data_trans})

輸出如下：

Start processing MNIST handwritten digits data...
Start evaluating softmax regression model by sklearn...
Accuracy of test set: 0.926300
Start evaluating softmax regression model by tensorflow...
Stochastic Gradient Descent processing step 0
Stochastic Gradient Descent processing step 100
Stochastic Gradient Descent processing step 200
Stochastic Gradient Descent processing step 300
Stochastic Gradient Descent processing step 400
Stochastic Gradient Descent processing step 500
Stochastic Gradient Descent processing step 600
Stochastic Gradient Descent processing step 700
Stochastic Gradient Descent processing step 800
Stochastic Gradient Descent processing step 900
Accuracy of test set: 0.917400

sklearn的估計時間有點長，因為每一輪參數更新都是基于全量的訓練集數據算出損失，再算出梯度，然后再改進結果的。

tensorflow采用batch gradient descent估計算法時，時間也比較長，原因同上。

tensorflow采用stochastic gradient descent估計算法時間短，最后的估計結果也挺好，相當于每輪迭代只用到了部分數據集算出損失和梯度，速度變快，但可能bias增加；所以把迭代次數增多，這樣可以降低variance，總體上的誤差相比batch gradient descent并沒有差多少。

參數效果的繪圖采用R實現，示例代碼如下：

library(dplyr)
library(tidyr)
library(ggplot2)

t <- read.table("coef_softmax_tf.txt")

n <- t %>% 
  tibble::rownames_to_column("digit") %>%
  gather(var_name, var_value, -digit) %>%
  mutate(var_name=stringr::str_sub(var_name, 2))
n$var_name <- as.numeric(n$var_name)
n$digit <- as.numeric(n$digit)
n <- n %>% 
  mutate(digit=digit-1, var_name=var_name-1, y=28 - floor(var_name/28), x=var_name %% 28, v=ifelse(var_value>0, 1, ifelse(var_value<0, -1, 0)))

ggplot(n) + geom_point(aes(x=x,y=y,color=as.factor(v))) + facet_wrap(~digit)