Adam — Update to RMSprop, storing gradient and square of gradients
Step by step implementation with animation for better understanding
2.7 How does Adam works?
Adam stands for Adaptive Moment Estimation. It is an update to RMSprop in which we store gradients and square of gradients in a restricted manner and we call them first moment and second moment respectively.
Note — In the original paper, the bias correction is included in first and second moment but we will include it in the learning rate and we will call it ‘learning rate hat’
We calculate ‘learning rate hat’ as follow:
Where beta 1 and beta 2 are decay rates for the first and second moment respectively, and ‘i’ is the iteration loop index starting from 0.
And, we calculate update as follow:
You can download the Jupyter Notebook from here.
Note — It is recommended that you have a look at the first post in this chapter.
This post is divided into 3 sections.
- Adam in 1 variable
- Adam animation for 1 variable
- Adam in multi-variable function
Adam in 1 variable
In this method, we store gradients and squares of gradients in a restricted manner and add a bias correction with decay rates in the learning rate calling it learning rate hat.
Adam algorithm in simple language is as follows:
Step 1 - Set starting point and learning rate
Step 2 - initiate beta1 = 0.9
beta2 = 0.999
moment1 = 0
moment2 = 0
epsilon = 10**-8
Step 3 - Initiate loop
Step 3.1 - calculate learning rate hat as stated above
Step 3.2 - calculate moment1 = beta1 * moment1 +
(1 - beta1) * gradient
Step 3.3 - calculate moment2 = beta2 * moment2 +
(1 - beta2) * gradient**2
Step 3.4 - calculate update as stated above
Step 3.5 - add update to pointFirst, let us define the function and its derivative and we start from x = -1
import numpy as np
np.random.seed(42)def f(x): # function definition
return x - x**3def fdash(x): # function derivative definition
return 1 - 3*(x**2)
And now Adam
point = -1 # step 1
learning_rate = 0.01beta1 = 0.9 # step 2
beta2 = 0.999
epsilon = 10**-8
moment1 = 0
moment2 = 0for i in range(1000): # step 3
learning_rate_hat = learning_rate * np.sqrt(1 - beta2**(i + 1))
/ (1 - beta1**(i + 1))
# step 3.1
moment1 = beta1 * moment1 + (1 - beta1) * fdash(point)
# step 3.2
moment2 = beta2 * moment2 + (1 - beta2) * fdash(point)**2
# step 3.3
update = - learning_rate_hat * moment1 / (moment2**0.5 +
epsilon)
# step 3.4
point += update # step 3.5
point # Minima
And, we have successfully implemented Adam in Python.
Adam animation for better understanding
Everything thing is the same as what we did earlier for the animation of the previous 6 optimizers. We will create a list to store starting point and updated points in it and will use the iᵗʰ index value for iᵗʰ frame of the animation.
import matplotlib.pyplot as plt
import matplotlib.animation as animation
from matplotlib.animation import PillowWriterpoint_adam = [-1] # initiating list with
# starting point in itpoint = -1 # step 1
learning_rate = 0.01beta1 = 0.9 # step 2
beta2 = 0.999
epsilon = 10**-8
moment1 = 0
moment2 = 0for i in range(1000): # step 3
learning_rate_hat = learning_rate * np.sqrt(1 - beta2**(i + 1))
/ (1 - beta1**(i + 1))
# step 3.1
moment1 = beta1 * moment1 + (1 - beta1) * fdash(point)
# step 3.2
moment2 = beta2 * moment2 + (1 - beta2) * fdash(point)**2
# step 3.3
update = - learning_rate_hat * moment1 / (moment2**0.5 +
epsilon)
# step 3.4
point += update # step 3.5
point_adam.append(point) # adding updated point to
# the list
point # Minima
We will do some settings for our graph for the animation. You can change them if you want something different.
plt.rcParams.update({'font.size': 22})fig = plt.figure(dpi = 100)fig.set_figheight(10.80)
fig.set_figwidth(19.20)x_ = np.linspace(-5, 5, 10000)
y_ = f(x_)ax = plt.axes()
ax.plot(x_, y_)
ax.grid(alpha = 0.5)
ax.set_xlim(-5, 5)
ax.set_ylim(-5, 5)
ax.set_xlabel('x')
ax.set_ylabel('y', rotation = 0)
ax.scatter(-1, f(-1), color = 'red')
ax.hlines(f(-0.5773502691896256), -5, 5, linestyles = 'dashed', alpha = 0.5)ax.set_title('Adam, learning_rate = 0.01')
Now we will animate the Adam optimizer.
def animate(i):
ax.clear()
ax.plot(x_, y_)
ax.grid(alpha = 0.5)
ax.set_xlim(-5, 5)
ax.set_ylim(-5, 5)
ax.set_xlabel('x')
ax.set_ylabel('y', rotation = 0)
ax.hlines(f(-0.5773502691896256), -5, 5, linestyles = 'dashed', alpha = 0.5)
ax.set_title('Adam, learning_rate = 0.01')
ax.scatter(point_adam[i], f(point_adam[i]), color = 'red')The last line in the code snippet above is using the iᵗʰ index value from the list for iᵗʰ frame in the animation.
anim = animation.FuncAnimation(fig, animate, frames = 200, interval = 20)anim.save('2.7 Adam.gif')
We are creating an animation that only has 200 frames and the gif is at 50 fps or frame interval is 20 ms.
It is to be noted that in less than 200 iterations we have reached the minima.
Adam in multi-variable function (2 variables right now)
Everything is the same, we only have to initialize point (1, 0) and moment1 = 0 and moment2 = 0 but with shape (2, 1) and replace fdash(point) with gradient(point).
But first, let us define the function, its partial derivatives and, gradient array
We know that Minima for this function is at (2, -1)
and we will start from (1, 0)
The partial derivatives are
def f(x, y): # function
return 2*(x**2) + 2*x*y + 2*(y**2) - 6*x # definitiondef fdash_x(x, y): # partial derivative
return 4*x + 2*y - 6 # w.r.t xdef fdash_y(x, y): # partial derivative
return 2*x + 4*y # w.r.t ydef gradient(point):
return np.array([[ fdash_x(point[0][0], point[1][0]) ],
[ fdash_y(point[0][0], point[1][0]) ]], dtype = np.float64) # gradients
Now the steps for Adam in 2 variables are
point = np.array([[ 1 ], # step 1
[ 0 ]], dtype = np.float64)learning_rate = 0.01beta1 = 0.9 # step 2
beta2 = 0.999
epsilon = 10**-8
moment1 = np.array([[ 0 ],
[ 0 ]], dtype = np.float64)
moment2 = np.array([[ 0 ],
[ 0 ]], dtype = np.float64)for i in range(1000): # step 3
learning_rate_hat = learning_rate * np.sqrt(1 - beta2**(i + 1))
/ (1 - beta1**(i + 1))
# step 3.1
moment1 = beta1 * moment1 + (1 - beta1) * gradient(point)
# step 3.2
moment2 = beta2 * moment2 + (1 - beta2) * gradient(point)**2
# step 3.3
update = - learning_rate_hat * moment1 / (moment2**0.5 +
epsilon)
# step 3.4
point += update # step 3.5
point # Minima
I hope now you understand how Adam works.
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