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Gradient Descent

Given a the multi-variable function $\large {F(x)}$ differentiable in a neighborhood of a point $\large a$
$\large F(x)$ decreases fastest if one goes from $\large a$ in the direction of the negative gradient of $\large F$ at $\large a$, $\large -\nabla{F(a)}$

Gradient Descent Algorithm:

Choose a starting point, $\large x_0$
Choose the sequence $\large x_0, x_1, x_2, ...$ such that $ \large x_{n+1} = x_n - \eta \nabla(F(x_n) $

So convergence of the gradient descent depends on the starting point $\large x_0$ and the learning rate $\large \eta$



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from time import sleep

import numpy as np

from ipywidgets import *
import bqplot.pyplot as plt
from bqplot import Toolbar



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f = lambda x: np.exp(-x) * np.sin(5 * x)
df = lambda x: -np.exp(-x) * np.sin(5 * x) + 5 * np.cos(5 *x) * np.exp(-x)



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x = np.linspace(0.5, 2.5, 500)
y = f(x)



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def update_sol_path(x, y):
    with sol_path.hold_sync():
        sol_path.x = x
        sol_path.y = y
    
    with sol_points.hold_sync():
        sol_points.x = x
        sol_points.y = y



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def gradient_descent(x0, f, df, eta=.1, tol=1e-6, num_iters=10):
    x = [x0]
    i = 0
    
    while i < num_iters:
        x_prev = x[-1]
        grad = df(x_prev)
        x_curr = x_prev - eta * grad
        x.append(x_curr)
        sol_lbl.value = sol_lbl_tmpl.format(x_curr)
        sleep(.5)
        
        update_sol_path(x, [f(i) for i in x])
        
        if np.abs(x_curr - x_prev) < tol:
            break
        i += 1



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txt_layout = Layout(width='150px')
x0_box = FloatText(description='x0', layout=txt_layout, value=2.4)
eta_box = FloatText(description='Learning Rate', 
                    style={'description_width':'initial'}, 
                    layout=txt_layout, value=.1)

go_btn = Button(description='GO', button_style='success', layout=Layout(width='50px'))
reset_btn = Button(description='Reset', button_style='success', layout=Layout(width='100px'))

sol_lbl_tmpl = 'x = {:.4f}'
sol_lbl = Label()
# sol_lbl.layout.width = '300px'

# plot of curve and solution
fig_layout = Layout(width='720px', height='500px')
fig = plt.figure(layout=fig_layout, title='Gradient Descent', display_toolbar=True)
fig.pyplot = Toolbar(figure=fig)

curve = plt.plot(x, y, colors=['dodgerblue'], stroke_width=2)
sol_path = plt.plot([], [], colors=['#ccc'], opacities=[.7])
sol_points = plt.plot([], [], 'mo', default_size=20)

def optimize():
    f.marks = [curve]
    gradient_descent(x0_box.value, f, df, eta=eta_box.value)

def reset():
    curve.scales['x'].min = .4
    curve.scales['x'].max = 2.5
    
    curve.scales['y'].min = -.5
    curve.scales['y'].max = .4
    sol_path.x = sol_path.y = []
    sol_points.x = sol_points.y = []
    sol_lbl.value = ''
    
go_btn.on_click(lambda btn: optimize())
reset_btn.on_click(lambda btn: reset())

final_fig = VBox([fig, fig.pyplot], 
                 layout=Layout(overflow_x='hidden'))
HBox([final_fig, VBox([x0_box, eta_box, go_btn, reset_btn, sol_lbl])])



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