ADAM : A METHOD FOR STOCHASTIC OPTIMIZATION

目录

• 概
• 主要内容
  • 算法
  • 选择合适的参数
  • 一些别的优化算法
  • AdaMax
  • 理论
• 代码

Kingma D P, Ba J. Adam: A Method for Stochastic Optimization[J]. arXiv: Learning, 2014.

@article{kingma2014adam:,
title={Adam: A Method for Stochastic Optimization},
author={Kingma, Diederik P and Ba, Jimmy},
journal={arXiv: Learning},
year={2014}}

概

鼎鼎大名.

主要内容

用(f( heta))表示目标函数, 随机最优通常需要最小化(mathbb{E}(f( heta))), 但是因为每一次我们都取的是一个小批次, 故实际上我们处理的是(f_1( heta),ldots, f_T( heta)). 用(g_t= abla_{ heta}f_t( heta))表示第(t)步对应的梯度.

Adam 方法分别估计梯度(mathbb{E}(g_t))的一阶矩和二阶矩(Adam: adaptive moment estimation 名字的由来).

算法

注意: 下面的算法中关于向量的运算都是逐项(element-wise)的运算.

选择合适的参数

首先, 分析为什么会有

[ ag{A.1} hat{m}_t leftarrow m_t / (1-eta_2^t), \ hat{v}_t leftarrow v_t / (1-eta_2^t). ]

可以用归纳法证明

[ ag{A.2} m_t = (1-eta_1) sum_{i=1}^t eta_1^{t-i} cdot g_i \ v_t = (1-eta_2) sum_{i=1}^t eta_2^{t-i} cdot g_i^2. ]

倘若分布稳定: (mathbb{E}[g_t]=mathbb{E}[g],mathbb{E}[g_t^2]=mathbb{E}[g^2]), 则

[ ag{A.3} mathbb{E}[m_t]=mathbb{E}[g] cdot(1-eta_1^t) \ mathbb{E}[v_t]= mathbb{E}[g^2] cdot (1- eta_2^t). ]

这就是为什么会有(A.1)这一步.

Adam提出时的一个很大的应用场景就是dropout(正对梯度是稀疏的情况), 这是往往需要我们取较大的(eta_2)(可理解为抵消随机因素).

既然(mathbb{E}[g]/sqrt{mathbb{E}[g^2]}le 1), 我们可以把步长(alpha)理解为一个信赖域(既然(|Delta_t| frac{<}{approx} a)).

另外一个很重要的性质是, 比如函数扩大(或缩小)(c)倍(cf), 此时梯度相应为(cg), 我们所对应的

[frac{c cdot hat{m}_t}{sqrt{c^2 cdot hat{v}_t}}= frac{hat{m}_t}{sqrt{hat{v}_t}}, ]

并无变化.

一些别的优化算法

AdaGrad:

[ heta_{t+1} = heta_t -alpha cdot frac{1}{sqrt{sum_{i=1}^tg_t^2}+epsilon} g_t. ]

RMSprop:

[v_t = eta_2 v_{t-1} + (1-eta_2) g_t^2 \ heta_{t+1} = heta_t -alpha cdot frac{1}{sqrt{v_t+epsilon}}g_t. ]

AdaDelta:

[v_t = eta_2 v_{t-1} + (1-eta_2) g_t^2 \ heta_{t+1} = heta_t -alpha cdot frac{sqrt{m_{t-1}+epsilon}}{sqrt{v_t+epsilon}}g_t \ m_t = eta_1 m_{t-1}+(1-eta_1)[ heta_{t+1}- heta_t]^2. ]

注: 均为逐项

AdaMax

本文还提出了另外一种算法

理论

不想谈了, 感觉证明有好多错误.

代码

import numpy as np

class Adam:

    def __init__(self, instance, alpha=0.001, beta1=0.9, beta2=0.999,
                 epsilon=1e-8, beta_decay=1., alpha_decay=False):
        """ the Adam using numpy
        :param instance: the theta in paper, should have the grad method to call the grads
                            and the zero_grad method for clearing the grads
        :param alpha: the same as the paper default:0.001
        :param beta1: the same as the paper default:0.9
        :param beta2: the same as the paper default:0.999
        :param epsilon: the same as the paper default:1e-8
        :param beta_decay:
        :param alpha_decay: default False, if True, we will set alpha = alpha / sqrt(t)
        """
        self.instance = instance
        self.alpha = alpha
        self.beta1 = beta1
        self.beta2 = beta2
        self.epsilon = epsilon
        self.beta_decay = beta_decay
        self.alpha_decay = alpha_decay
        self.initialize_paras()

    def initialize_paras(self):
        self.m = 0.
        self.v = 0.
        self.timestep = 0

    def update_paras(self):
        grads = self.instance.grad
        self.beta1 *= self.beta_decay
        self.beta2 *= self.beta_decay
        self.m = self.beta1 * self.m + (1 - self.beta1) * grads
        self.v = self.beta2 * self.v + (1 - self.beta2) * grads ** 2
        self.timestep += 1
        if self.alpha_decay:
            return self.alpha / np.sqrt(self.timestep)
        return self.alpha

    def zero_grad(self):
        self.instance.zero_grad()

    def step(self):
        alpha = self.update_paras()
        betat1 = 1 - self.beta1 ** self.timestep
        betat2 = 1 - self.beta2 ** self.timestep
        temp = alpha * np.sqrt(betat2) / betat1
        self.instance.parameters -= temp * self.m / (np.sqrt(self.v) + self.epsilon)






class PPP:

    def __init__(self, parameters, grad_func):
        self.parameters = parameters
        self.zero_grad()
        self.grad_func = grad_func

    def zero_grad(self):
        self.grad = np.zeros_like(self.parameters)

    def calc_grad(self):
        self.grad += self.grad_func(self.parameters)



def f(x):
    return x[0] ** 2 + 5 * x[1] ** 2

def grad(x):
    return np.array([2 * x[0], 100 * x[1]])


if __name__ == "__main__":

    x = np.array([10., 10.])
    x = PPP(x, grad)
    xs = []
    ys = []
    optim = Adam(x, alpha=0.4)
    for i in range(100):
        xs.append(x.parameters.copy())
        y = f(x.parameters)
        ys.append(y)
        optim.zero_grad()
        x.calc_grad()
        optim.step()
    xs = np.array(xs)
    ys = np.array(ys)
    import matplotlib.pyplot as plt
    fig, (ax0, ax1)= plt.subplots(1, 2)
    ax0.plot(xs[:, 0], xs[:, 1])
    ax0.scatter(xs[:, 0], xs[:, 1])
    ax0.set(title="trajectory", xlabel="x", ylabel="y")
    ax1.plot(np.arange(len(ys)), ys)
    ax1.set(title="loss-iterations", xlabel="iterations", ylabel="loss")
    plt.show()