numpy手撕逻辑回归分类MNIST公式+代码 (2024)

import numpy as npx = np.array([1,5,10,15])y = np.exp(x) # array([2.71828183e+00, 1.48413159e+02, 2.20264658e+04, 3.26901737e+06])S = np.exp(x)/sum(y) # array([8.25925493e-07, 4.50940040e-05, 6.69254359e-03, 9.93261536e-01])# 下面求不带指数形式的Z = x/sum(x) # array([0.03225806, 0.16129032, 0.32258065, 0.48387097])

def L_Gra(W, b, x, y): """ W, b为当前的权重和偏置参数，x,y为样本数据和该样本对应的标签 """ #初始化 W_G = np.zeros(W.shape) b_G = np.zeros(b.shape) S = softmax(np.dot(W,x) + b) W_row=W.shape[0] W_column=W.shape[1] b_column=b.shape[0] #对Wij和bi分别求梯度 for i in range(W_row): for j in range(W_column): W_G[i][j]=(S[i] - 1) * x[j] if y==i else S[i]*x[j] for i in range(b_column): b_G[i]=S[i]-1 if y==i else S[i] #返回W和b的梯度  return W_G, b_G

#!/usr/bin/env python# -*- coding:utf-8 -*-# @FileName :lr_mnist.py# @Time :2021/10/22 14:44# @Author :ninimport timeimport numpy as npimport pandas as pdimport gzip as gzimport matplotlib.pyplot as pltfrom tqdm import tqdmfrom tqdm import trange# ------------读取数据-------------- ## filename为文件名，kind为'data'或'lable'def load_data(filename,kind): with gz.open(filename, 'rb') as fo: # 使用gzip模块打开压缩文件 buf = fo.read() # 把压缩的内容读进缓存中 b' ' 表示这是一个 bytes 对象 index = 0 if kind == 'data': # 因为数据结构中前4行的数据类型都是32位整型，所以采用i格式，需要读取前4行数据，所以需要4个i header = np.frombuffer(buf, '>i', 4, index) index += header.size * header.itemsize data = np.frombuffer(buf, '>B', header[1] * header[2] * header[3], index).reshape(header[1], -1) # 2-d矩阵 elif kind == 'lable': # 因为数据结构中前2行的数据类型都是32位整型，所以采用i格式，需要读取前2行数据，所以需要2个i header = np.frombuffer(buf, '>i', 2, 0) index += header.size * header.itemsize data = np.frombuffer(buf,'>B', header[1], index) return data# -----------加载数据------------- #X_train = load_data('train-images-idx3-ubyte.gz','data')y_train = load_data('train-labels-idx1-ubyte.gz','lable')X_test = load_data('t10k-images-idx3-ubyte.gz','data')y_test = load_data('t10k-labels-idx1-ubyte.gz','lable')# -----------查看数据格式------------- #print('Train data shape:')print(X_train.shape, y_train.shape)print('Test data shape:')print(X_test.shape, y_test.shape)"""# -----------数据可视化------------- #index_1 = 1024plt.imshow(np.reshape(X_train[index_1], (28,28)), cmap='gray')plt.title('Index:{}'.format(index_1))plt.show()print("Label: {}".format(y_train[index_1]))index_2 = 2048plt.imshow(np.reshape(X_train[index_2], (28,28)), cmap='gray')plt.title('Index:{}'.format(index_2))plt.show()print('Label: {}'.format(y_train[index_2]))"""# -----------定义常量------------- #x_dim = 28 * 28 # data shape, 28*28y_dim = 10 # output shape, 10*1W_dim = (y_dim, x_dim) # 矩阵Wb_dim = y_dim # index b# -----------定义logistic regression 中的必要函数------------- ## Softmax函数def softmax(x): """ :param x: :return: 经softmax变换后的向量 """ return np.exp(x) / np.exp(x).sum()# Loss函数# L = -log(Si(yi)), yi为预测值def loss(W,b,x,y): """ W, b为当前的权重和偏置参数，x，y为样本数据和该样本对应的标签 """ return -np.log(softmax(np.dot(W,x) + b)[y]) # 预测值与标签相同的概率# 单样本损失函数梯度def L_Gra(W, b, x, y): """ W,b 为当前的权重和偏置参数，x，y为样本数据和该样本对应的标签 """ # 初始化 W_G = np.zeros(W.shape) b_G = np.zeros(b.shape) S = softmax(np.dot(W,x) + b) W_row = W.shape[0] W_column = W.shape[1] b_column = b.shape[0] # 对Wij和bi分别求梯度 for i in range(W_row): for j in range(W_column): W_G[i][j] = (S[i] - 1) * x[j] if y == i else S[i] * x[j] # S: softmax函数 为什么没用到loss？求解过程中已经加入了 for i in range(b_column): b_G[i] = S[i] - 1 if y == i else S[i] return W_G, b_G# 检验模型在测试集上的正确率def test_accurate(W, b, X_test, y_test): num = len(X_test) # results中保存测试结果，正确则添加一个1，错误则添加一个0 results = [] for i in range(num): # 检验softmax(W*x+b)中最大的元素的 下标是否为y， 若是则预测正确 y_i = np.dot(W,X_test[i]) + b # X_test相当于784个特征点，W是权重矩阵，y_i是预测值 res = 1 if softmax(y_i).argmax() == y_test[i] else 0 # 和真实的label值比较判断是否预测正确 results.append(res) accurate_rate = np.mean(results) return accurate_rate# ------------------- ## 梯度下降法# ------------------- #def mini_batch(batch_size, lr, epoches): """ batch_size 为batch的尺寸，lr为步长，epoches为epoch的数量，训练次数 """ accurate_rates = [] # 记录每次迭代的正确率 accurate_rate iters_W = [] # 记录每次迭代的 W iters_b = [] # 记录每次迭代的 b W = np.zeros(W_dim) b = np.zeros(b_dim) # 根据batch_size的尺寸将原本样本和标签分批后 # 初始化 x_batches = np.zeros(((int(X_train.shape[0]/batch_size), batch_size, 784))) y_batches = np.zeros(((int(X_train.shape[0]/batch_size), batch_size))) batch_num = int(X_train.shape[0]/batch_size) # 分批 for i in range(0, X_train.shape[0], batch_size): x_batches[int(i/batch_size)] = X_train[i:i + batch_size] y_batches[int(i/batch_size)] = y_train[i:i + batch_size] print('Start training...') for epoch in range(epoches): print("epoch:{}".format(epoch)) start = time.time() with trange(batch_num) as t: for i in t: # i是batch批次 # 设置进度条左边显示的信息 t.set_description("epoch %i" % epoch) # 设置进度条右边显示的信息 # t.set_postfix(loss=random(), gen=randint(1, 999), str="h", lst=[1, 2]) # init grads W_gradients = np.zeros(W_dim) # (10,784) b_gradients = np.zeros(b_dim) # (10,1) x_batch, y_batch = x_batches[i], y_batches[i] # j为单个样本 for j in range(batch_size): # 每个batch中的每个样本都计算下梯度值 W_g, b_g = L_Gra(W, b, x_batch[j], y_batch[j]) # 计算单个样本的梯度值 W_gradients += W_g # 梯度累积 b_gradients += b_g W_gradients /= batch_size # 累积完 /m 即batch大小 b_gradients /= batch_size # 进行梯度下降 W -= lr * W_gradients b -= lr * b_gradients # 记录每个iteration之后的总W和b的值 iters_W.append(W.copy()) iters_b.append(b.copy()) # epoch每结束一次跑一次测试 end = time.time() time_cost = (end - start) print("time cost:{}s".format(time_cost)) accurate_rates.append(test_accurate(W, b, X_test, y_test)) return W, b, time_cost, accurate_rates, iters_W, iters_bdef run(lr, batch_size, epochs_num): """ W ,b : w和b的最优解 W_s, b_s : 每次迭代后得到的值 """ W, b, time_cost, accuracys, W_s, b_s = mini_batch(batch_size, lr, epochs_num) # 求W和b与最优解的距离，用二阶范数表示距离 iterations = len(W_s) dis_W = [] dis_b = [] for i in range(iterations): # 计算每一次迭代与最终W的欧氏距离 dis_W.append(np.linalg.norm(W_s[i] - W)) dis_b.append(np.linalg.norm(b_s[i] - b)) print("the parameters is: step length alpah:{}; batch size:{}; Epoches:{}".format(lr, batch_size, epochs_num)) print("Result: accuracy:{:.2%},time cost:{:.2f}".format(accuracys[-1], time_cost)) # ----------------作图-------------- # # 精确度随迭代次数的变化 plt.title('The Model accuracy variation chart ') plt.xlabel('Iterations') plt.ylabel('Accuracy') plt.plot(accuracys,'m') plt.grid() plt.show() # W和b距最优解的距离随迭代次数的变化 plt.title('The distance from the optimal solution') plt.xlabel('Iterations') plt.ylabel('Distance') plt.plot(dis_W, 'r', label='distance between W and W*') plt.plot(dis_b, 'g', label='distance between b and b*') plt.legend() plt.grid() plt.show()# ----------参数输入---------------lr = 1e-5batch_size = 10000epochs_num = 20# 运行函数run(lr, batch_size, epochs_num)