梯度下降Python代码调试求助：输出结果与预期不符

我正在为课程编写梯度下降的Python代码，使用Google Colab完成编程项目，但代码运行结果与预期不符，怀疑问题出在求和部分，请求帮助排查错误。

代码 #

import numpy as np
#Parameters
lamb= 0.1
beta = 1
alpha = 1
#Label at z is +1 and -z is -1
#Optimal w = z / ||z||^2 and b = 0
z = np.array([1,2])
#Data Matrix and Labels
x = np.vstack((z,-z))
y = np.array([1,-1])
#Random Initialization
w = np.array([-0.34637426,  1.16720011])
b = np.array([-0.24424188])
m=len(z)
sumw=0
sumb=0
#Insert your code here to compute the gradients and loss. You can
#Use as many additional lines of code as needed (i.e., don't try too
#hard to put the whole computation in one line)
for i in range(1000):
  for j in range(m):
    sumb+=(y[j])/(1+np.exp((-beta)*(1-y[j]*(x[j]@w-b))))
    sumw+=(y[j]*x[j])/(1+np.exp((-beta)*(1-y[j]*(x[j]@w-b))))
  grad_w = 2 * lamb * w - (1/2)*sumw
  grad_b=(1/m)*sumb
  b-=alpha*grad_b
  w-=alpha*grad_w
  if i % 100 == 0:
        print(w,' ', b)

当前输出 #

[-0.00311842  1.48172206]   [-0.19604564]
[2.52108931 5.04217862]   [5.53009896]
[2.7530958  5.50619161]   [3.85357184]
[2.76527666 5.53055331]   [0.82437199]
[2.76756568 5.53513137]   [-2.25691088]
[2.79916861 5.59833721]   [-5.09974894]
[2.93920254 5.87840507]   [-6.08686155]
[3.00843697 6.01687395]   [-4.78457688]
[3.01993034 6.03986068]   [-2.86787546]
[3.02152889 6.04305778]   [-0.86030793]

预期输出 #

[-0.34637426  1.16720011] [-0.24424188] 
[0.5989022  1.19780439]  [-3.17437461e-06] 
[0.5989022  1.19780439]  [-4.60193897e-11] 
[0.5989022  1.19780439]  [-5.82867088e-16] 
[0.5989022  1.19780439]  [-1.94289029e-16] 
[0.5989022  1.19780439]  [-1.94289029e-16] 
[0.5989022  1.19780439]  [-1.94289029e-16] 
[0.5989022  1.19780439]  [-1.94289029e-16] 
[0.5989022  1.19780439]  [-1.94289029e-16] 
[0.5989022  1.19780439]  [-1.94289029e-16] 

问题背景（对应原Problem 4 iii） #

该问题是带L2正则化的逻辑回归变种，目标是通过梯度下降优化参数w和b，最优解为w = z / ||z||²，b = 0。

错误排查与修复 #

1. 求和变量未重置：sumw和sumb在循环外初始化后，每次迭代没有重置为0，导致梯度计算持续累加历史求和结果，引发参数发散。需将sumw=0和sumb=0移到外层循环内部，每次迭代前重置。
2. 梯度符号错误：grad_b的计算缺少负号，根据损失函数梯度推导，正确梯度应为-(1/m)*sumb，否则参数更新方向错误，无法收敛到最优解。
3. 预测项符号错误：逻辑回归预测项应为x[j]@w + b，原代码写成x[j]@w - b，导致模型预测逻辑错误，影响梯度计算正确性。

修复后的代码：

import numpy as np
#Parameters
lamb= 0.1
beta = 1
alpha = 1
#Label at z is +1 and -z is -1
#Optimal w = z / ||z||^2 and b = 0
z = np.array([1,2])
#Data Matrix and Labels
x = np.vstack((z,-z))
y = np.array([1,-1])
#Random Initialization
w = np.array([-0.34637426,  1.16720011])
b = np.array([-0.24424188])
m = len(y)  # 明确样本数
#Insert your code here to compute the gradients and loss. You can
#Use as many additional lines of code as needed (i.e., don't try too
#hard to put the whole computation in one line)
for i in range(1000):
  sumw = 0  # 每次迭代前重置求和变量
  sumb = 0
  for j in range(m):
    # 修正预测项符号
    exp_term = np.exp((-beta)*(1 - y[j]*(x[j]@w + b)))
    sumb += y[j] / (1 + exp_term)
    sumw += y[j] * x[j] / (1 + exp_term)
  grad_w = 2 * lamb * w - (1/m)*sumw
  grad_b = -(1/m)*sumb  # 修正梯度符号
  b -= alpha*grad_b
  w -= alpha*grad_w
  if i % 100 == 0:
        print(w,' ', b)

运行修复后的代码，参数会快速收敛到预期的最优值。

内容的提问来源于stack exchange，提问作者Cole Hendrickson