# ADAM : A METHOD FOR STOCHASTIC OPTIMIZATION

### Article catalog

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

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

## General

a great reputation.

## primary coverage

F (theta) f (theta) f (theta) is used to represent the objective function. The stochastic optimization usually needs to minimize e (f (theta)) \ matchb {e} (f(\ theta)) e (f (theta)), but because we take a small batch every time, we actually deal with F1 (theta) ,ft(θ)f_1(\theta),\ldots, f_T(\theta)f1​(θ),… , ft (θ). Use gt = ∇ θ ft (θ) g_ t=\nabla_ {\theta}f_ T (theta) gt = ∧ ft (θ) represents the gradient corresponding to step ttt

Adam method estimates gradient e (GT) and mathbb {e} (g) respectively_ t) The origin of the first and second moments of E (GT)

### algorithm

Note: the following algorithms are all element wise operations

### Select appropriate parameters

First, analyze why there are
m^t←mt/(1−β2t),v^t←vt/(1−β2t).(A.1) \tag{A.1} \hat{m}_t \leftarrow m_t / (1-\beta_2^t), \\ \hat{v}_t \leftarrow v_t / (1-\beta_2^t). m^t​←mt​/(1−β2t​),v^t​←vt​/(1−β2t​).(A.1)

Can be proved by induction
mt=(1−β1)∑i=1tβ1t−i⋅givt=(1−β2)∑i=1tβ2t−i⋅gi2.(A.2) \tag{A.2} m_t = (1-\beta_1) \sum_{i=1}^t \beta_1^{t-i} \cdot g_i \\ v_t = (1-\beta_2) \sum_{i=1}^t \beta_2^{t-i} \cdot g_i^2. mt​=(1−β1​)i=1∑t​β1t−i​⋅gi​vt​=(1−β2​)i=1∑t​β2t−i​⋅gi2​.(A.2)
If the distribution is stable: E[gt]=E[g],E[gt2]=E[g2]\mathbb{E}[g_t]=\mathbb{E}[g],\mathbb{E}[g_t^2]=\mathbb{E}[g^2]E[gt] = E [g], E [G T 2] = E[g2], then
E[mt]=E[g]⋅(1−β1t)E[vt]=E[g2]⋅(1−β2t).(A.3) \tag{A.3} \mathbb{E}[m_t]=\mathbb{E}[g] \cdot(1-\beta_1^t) \\ \mathbb{E}[v_t]= \mathbb{E}[g^2] \cdot (1- \beta_2^t). E[mt​]=E[g]⋅(1−β1t​)E[vt​]=E[g2]⋅(1−β2t​).(A.3)
That's why there is (A.1) this step

A large application scenario when Adam proposed is dropout, which often requires us to take a larger β 2\beta_2 β 2 (it can be understood as offsetting random factors)

Since E[g]/E[g2] < 1 \ mathb {e} [g] / \ sqrt {\ mathb {e} [G ^ 2]} \ Le 1E[g]/E[g2] < 1, we can understand step α as a trust region (since ∣ Δ t ∣ < a | De lt a_ t| \frac{<}{\approx} a∣Δt​∣≈<​a).

Another important property is that, for example, the function expands (or shrinks) ccc times cfcf, and the gradient is cgcgcg, which corresponds to
c⋅m^tc2⋅v^t=m^tv^t, \frac{c \cdot \hat{m}_t}{\sqrt{c^2 \cdot \hat{v}_t}}= \frac{\hat{m}_t}{\sqrt{\hat{v}_t}}, c2⋅v^t​​c⋅m^t​​=v^t​​m^t​​,
There is no change

### Some other optimization algorithms

θt+1=θt−α⋅1∑i=1tgt2+ϵgt. \theta_{t+1} = \theta_t -\alpha \cdot \frac{1}{\sqrt{\sum_{i=1}^tg_t^2}+\epsilon} g_t. θt+1​=θt​−α⋅∑i=1t​gt2​​+ϵ1​gt​.

RMSprop:

vt=β2vt−1+(1−β2)gt2θt+1=θt−α⋅1vt+ϵgt. v_t = \beta_2 v_{t-1} + (1-\beta_2) g_t^2 \\ \theta_{t+1} = \theta_t -\alpha \cdot \frac{1}{\sqrt{v_t+\epsilon}}g_t. vt​=β2​vt−1​+(1−β2​)gt2​θt+1​=θt​−α⋅vt​+ϵ​1​gt​.

vt=β2vt−1+(1−β2)gt2θt+1=θt−α⋅mt−1+ϵvt+ϵgtmt=β1mt−1+(1−β1)[θt+1−θt]2. v_t = \beta_2 v_{t-1} + (1-\beta_2) g_t^2 \\ \theta_{t+1} = \theta_t -\alpha \cdot \frac{\sqrt{m_{t-1}+\epsilon}}{\sqrt{v_t+\epsilon}}g_t \\ m_t = \beta_1 m_{t-1}+(1-\beta_1)[\theta_{t+1}-\theta_t]^2. vt​=β2​vt−1​+(1−β2​)gt2​θt+1​=θt​−α⋅vt​+ϵ​mt−1​+ϵ​​gt​mt​=β1​mt−1​+(1−β1​)[θt+1​−θt​]2.

Note: item by item

Another algorithm is proposed in this paper

### theory

I don't want to talk about it. I feel that there are many mistakes

## code



import numpy as np

def __init__(self, instance, alpha=0.001, beta1=0.9, beta2=0.999,
epsilon=1e-8, beta_decay=1., alpha_decay=False):
:param instance: the theta in paper, should have the grad method to call 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):
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 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:

self.parameters = parameters

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

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

if __name__ == "__main__":

x = np.array([10., 10.])
xs = []
ys = []
for i in range(100):
xs.append(x.parameters.copy())
y = f(x.parameters)
ys.append(y)
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()



Posted by danleighton on Thu, 04 Jun 2020 08:08:43 -0700