24、神经网络优化算法比较
为高效找到使损失函数的值最小的参数,关于最优化(optimization)提了很多方法。
其中包括:
SGD(stochastic gradient descent,随机梯度下降)
Momentum(冲量算法)
Adagrad
Adam
各优化算法比较实验(python)
# -*- coding: utf-8 -*- import numpy as np import matplotlib.pyplot as plt from collections import OrderedDict class SGD: """随机梯度下降法(Stochastic Gradient Descent)""" def __init__(self, lr=0.01): self.lr = lr def update(self, params, grads): for key in params.keys(): params[key] -= self.lr * grads[key] class Momentum: """Momentum SGD""" def __init__(self, lr=0.01, momentum=0.9): self.lr = lr self.momentum = momentum self.v = None def update(self, params, grads): if self.v is None: self.v = {} for key, val in params.items(): self.v[key] = np.zeros_like(val) for key in params.keys(): self.v[key] = self.momentum*self.v[key] - self.lr*grads[key] params[key] += self.v[key] class Nesterov: """Nesterov‘s Accelerated Gradient (http://arxiv.org/abs/1212.0901)""" def __init__(self, lr=0.01, momentum=0.9): self.lr = lr self.momentum = momentum self.v = None def update(self, params, grads): if self.v is None: self.v = {} for key, val in params.items(): self.v[key] = np.zeros_like(val) for key in params.keys(): self.v[key] *= self.momentum self.v[key] -= self.lr * grads[key] params[key] += self.momentum * self.momentum * self.v[key] params[key] -= (1 + self.momentum) * self.lr * grads[key] class AdaGrad: """AdaGrad""" def __init__(self, lr=0.01): self.lr = lr self.h = None def update(self, params, grads): if self.h is None: self.h = {} for key, val in params.items(): self.h[key] = np.zeros_like(val) for key in params.keys(): self.h[key] += grads[key] * grads[key] params[key] -= self.lr * grads[key] / (np.sqrt(self.h[key]) + 1e-7) class RMSprop: """RMSprop""" def __init__(self, lr=0.01, decay_rate = 0.99): self.lr = lr self.decay_rate = decay_rate self.h = None def update(self, params, grads): if self.h is None: self.h = {} for key, val in params.items(): self.h[key] = np.zeros_like(val) for key in params.keys(): self.h[key] *= self.decay_rate self.h[key] += (1 - self.decay_rate) * grads[key] * grads[key] params[key] -= self.lr * grads[key] / (np.sqrt(self.h[key]) + 1e-7) class Adam: """Adam (http://arxiv.org/abs/1412.6980v8)""" def __init__(self, lr=0.001, beta1=0.9, beta2=0.999): self.lr = lr self.beta1 = beta1 self.beta2 = beta2 self.iter = 0 self.m = None self.v = None def update(self, params, grads): if self.m is None: self.m, self.v = {}, {} for key, val in params.items(): self.m[key] = np.zeros_like(val) self.v[key] = np.zeros_like(val) self.iter += 1 lr_t = self.lr * np.sqrt(1.0 - self.beta2**self.iter) / (1.0 - self.beta1**self.iter) for key in params.keys(): self.m[key] += (1 - self.beta1) * (grads[key] - self.m[key]) self.v[key] += (1 - self.beta2) * (grads[key]**2 - self.v[key]) params[key] -= lr_t * self.m[key] / (np.sqrt(self.v[key]) + 1e-7) def f(x, y): return x**2 / 20.0 + y**2 def df(x, y): return x / 10.0, 2.0*y init_pos = (-7.0, 2.0) params = {} params[‘x‘], params[‘y‘] = init_pos[0], init_pos[1] grads = {} grads[‘x‘], grads[‘y‘] = 0, 0 optimizers = OrderedDict() optimizers["SGD"] = SGD(lr=0.95) optimizers["Momentum"] = Momentum(lr=0.1) optimizers["AdaGrad"] = AdaGrad(lr=1.5) optimizers["Adam"] = Adam(lr=0.3) idx = 1 for key in optimizers: optimizer = optimizers[key] x_history = [] y_history = [] params[‘x‘], params[‘y‘] = init_pos[0], init_pos[1] for i in range(30): x_history.append(params[‘x‘]) y_history.append(params[‘y‘]) grads[‘x‘], grads[‘y‘] = df(params[‘x‘], params[‘y‘]) optimizer.update(params, grads) x = np.arange(-10, 10, 0.01) y = np.arange(-5, 5, 0.01) X, Y = np.meshgrid(x, y) Z = f(X, Y) # for simple contour line mask = Z > 7 Z[mask] = 0 # plot plt.subplot(2, 2, idx) idx += 1 plt.plot(x_history, y_history, ‘o-‘, color="red") plt.contour(X, Y, Z)#绘制等高线 plt.ylim(-10, 10) plt.xlim(-10, 10) plt.plot(0, 0, ‘+‘) plt.title(key) plt.xlabel("x") plt.ylabel("y") plt.subplots_adjust(wspace =0, hspace =0)#调整子图间距 plt.show()
实验结果
参考文献
[1] https://blog.csdn.net/fjssharpsword/article/details/85257635
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