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Graphic explanation mysterious gradient descent algorithm principle (with Python code)

編輯：Python

1 Cited example

Given a function as shown in the figure , How to program through computer algorithm to find $f (x)_{m i n} f(x)_{min}$ f(x)min？

2 Numerical solution

The traditional approach is Numerical solution , As shown in the figure

Follow the following steps to iterate the cycle until optimal ：

1. Any given initial value $x_{0} x_0$ x0;
2. Randomly generate incremental direction , Combined step size generation $Δ x \varDelta x$ Δx;
3. Calculation comparison $f (x_{0}) f\left( x_0 \right)$ f(x0) And $f (x_{0} + Δ x) f\left( x_0+\varDelta x \right)$ f(x0+Δx) Size , if $f (x_{0} + Δ x) < f (x_{0}) f\left( x_0+\varDelta x \right) <f\left( x_0 \right)$ f(x0+Δx)<f(x0) Update location , Otherwise regenerate $Δ x \varDelta x$ Δx;
4. repeat ②③ Until it converges to the optimal $f (x)_{m i n} f(x)_{min}$ f(x)min.

The biggest advantage of numerical solution is the simplicity of programming , But the flaws are obvious ：

1. The setting of the initial value has a great influence on the convergence speed of the result ;
2. Incremental direction random generation , Low efficiency ;
3. It is easy to fall into local optimal solution ;
4. Unable to deal with “ plateau ” Type function .

So-called Fall into local optimal solution It refers to when the iteration enters a minimum or its neighborhood , Because the step size is not selected properly , Whether positive or negative , The learning effect is not as good as the current , As a result, it is impossible to iterate to the global optimum . For this problem, it is shown in the figure , When iteration falls into $x = x_{j} x=x_j$ x=xj when , Because of the learning step $s t e p step$ step The limitation of , Unable to make $f (x_{j} \pm S t e p) < f (x_{j}) f\left( x_j\pm Step \right) <f(x_j)$ f(xj±Step)<f(xj), Therefore, the iteration is locked in the red section of the graph . It can be seen that $x = x_{j} x=x_j$ x=xj It is not the desired global optimum .

If the following figure appears “ plateau ” function , It may also prevent the iteration from being updated .

3 Gradient descent algorithm

Gradient descent algorithm It can be regarded as an improvement of numerical solution , The explanation is as follows ：

Note No $k k$ k After iteration , The argument is updated to $x = x_{k} x=x_k$ x=xk, Let the objective function $f (x) f(x)$ f(x) stay $x = x_{k} x=x_k$ x=xk Taylor expansion ：

f (x) = f (x_{k}) + f^{'} (x_{k}) (x - x_{k}) + o (x) f\left( x \right) =f\left( x_k \right) +f'\left( x_k \right) \left( x-x_k \right) +o(x)

f(x)=f(xk)+f′(xk)(x−xk)+o(x)

Investigate $f (x)_{m i n} f(x)_{min}$ f(x)min, Expect $f (x_{k + 1}) < f (x_{k}) f\left( x_{k+1} \right) <f\left( x_k \right)$ f(xk+1)<f(xk), thus ：

f (x_{k + 1}) - f (x_{k}) = f^{'} (x_{k}) (x_{k + 1} - x_{k}) < 0 f\left( x_{k+1} \right) -f\left( x_k \right) =f'\left( x_k \right) \left( x_{k+1}-x_k \right) <0

f(xk+1)−f(xk)=f′(xk)(xk+1−xk)<0

if $f^{'} (x_{k}) > 0 f'\left( x_k \right) >0$ f′(xk)>0 be $x_{k + 1} < x_{k} x_{k+1}<x_k$ xk+1<xk, That is, the iteration direction is negative ; The opposite is positive . Might as well set $x_{k + 1} - x_{k} = - f^{'} (x_{k}) x_{k+1}-x_k=-f'(x_k)$ xk+1−xk=−f′(xk), To ensure that $f (x_{k + 1}) - f (x_{k}) < 0 f\left( x_{k+1} \right) -f\left( x_k \right) <0$ f(xk+1)−f(xk)<0. It must be pointed out , Taylor's formula holds only if $x \to x_{0} x\rightarrow x_0$ x→x0, so $∣ f^{'} (x_{k}) ∣ |f'\left( x_k \right) |$ ∣f′(xk)∣ Not too big , otherwise $x_{k + 1} x_{k+1}$ xk+1 And $x_{k} x_{k}$ xk The distance is too far, resulting in residual error . So introduce Learning rate $γ \in (0, 1) \gamma \in \left( 0, 1 \right)$ γ∈(0,1) To reduce the offset , namely $x_{k + 1} - x_{k} = - γ f^{'} (x_{k}) x_{k+1}-x_k=-\gamma f'(x_k)$ xk+1−xk=−γf′(xk)

On the engineering , Learning rate $γ \gamma$ γ We should make a reasonable choice in combination with practical application ,== $γ \gamma$ γ Too large an amount makes the iteration oscillate on both sides of the minimum , The algorithm can't converge ; $γ \gamma$ γ Too small will reduce learning efficiency , The algorithm converges slowly ==.

For vector , The above iterative formula is extended to

x_{k + 1} = x_{k} - γ \nabla_{x_{k}} {\boldsymbol{x}_{\boldsymbol{k}+1}=\boldsymbol{x}_{\boldsymbol{k}}-\gamma \nabla _{\boldsymbol{x}_{\boldsymbol{k}}}}

xk+1=xk−γ∇xk

among $\nabla_{x} = {(\frac{\partial f (x)}{\partial x_{1}}, \frac{\partial f (x)}{\partial x_{2}}, \dots \dots, \frac{\partial f (x)}{\partial x_{n}})}^{T} \nabla _{\boldsymbol{x}}=\left( \frac{\partial f(\boldsymbol{x})}{\partial x_1},\frac{\partial f(\boldsymbol{x})}{\partial x_2},\cdots \cdots ,\frac{\partial f(\boldsymbol{x})}{\partial x_n} \right) ^T$ ∇x=(∂x1∂f(x),∂x2∂f(x),⋯⋯,∂xn∂f(x))T Is the gradient of multivariate function , Therefore, this iterative algorithm is also called Gradient descent algorithm

The gradient descent algorithm determines the direction and step size of each iteration through the function gradient , Improved algorithm efficiency . But in principle, we can know , This algorithm cannot solve the initial value setting in numerical solution 、 The problem of local optimal collapse and partial function locking .

4 Code combat ：Logistic Return to

import pandas as pd
import numpy as np
import os
import matplotlib.pyplot as plt
import matplotlib as mpl
from Logit import Logit
''' * @breif: from CSV Load the specified data in * @param[in]: file -> file name * @param[in]: colName -> Column name to load * @param[in]: mode -> Load mode , set: A dictionary consisting of a column name and its data , df: df type * @retval: mode Return value in mode '''
def loadCsvData(file, colName, mode='df'):
assert mode in ('set', 'df')
df = pd.read_csv(file, encoding='utf-8-sig', usecols=colName)
if mode == 'df':
return df
if mode == 'set':
res = {}
for col in colName:
res[col] = df[col].values
return res
if __name__ == '__main__':
# ============================
# Read CSV data 
# ============================
csvPath = os.path.abspath(os.path.join(__file__, "../../data/dataset3.0alpha.csv"))
dataX = loadCsvData(csvPath, [" Sugar content ", " density "], 'df')
dataY = loadCsvData(csvPath, [" Good melon "], 'df')
label = np.array([
1 if i == " yes " else 0
for i in list(map(lambda s: s.strip(), list(dataY[' Good melon '])))
])
# ============================
# Draw sample points 
# ============================
line_x = np.array([np.min(dataX[' density ']), np.max(dataX[' density '])])
mpl.rcParams['font.sans-serif'] = [u'SimHei']
plt.title(' Log probability regression simulation \nLogistic Regression Simulation')
plt.xlabel('density')
plt.ylabel('sugarRate')
plt.scatter(dataX[' density '][label==0],
dataX[' Sugar content '][label==0],
marker='^',
color='k',
s=100,
label=' Bad melon ')
plt.scatter(dataX[' density '][label==1],
dataX[' Sugar content '][label==1],
marker='^',
color='r',
s=100,
label=' Good melon ')
# ============================
# Instantiate log probability regression model 
# ============================
logit = Logit(dataX, label)
# Gradient descent method 
logit.logitRegression(logit.gradientDescent)
line_y = -logit.w[0, 0] / logit.w[1, 0] * line_x - logit.w[2, 0] / logit.w[1, 0]
plt.plot(line_x, line_y, 'b-', label=" Gradient descent method ")
# mapping 
plt.legend(loc='upper left')
plt.show()
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