R中梯度下降和线性模型之间Theta值的差异

Question

我正在使用波士顿数据集作为我的输入，我正在尝试建立一个模型来使用 RM（每个住宅的平均房间数）预测 MEDV（1000 美元的自住住房的中值）

我从 Digitheads blog 中篡改了以下代码，并没有像您看到的那样多。

我的代码如下：

#library(datasets)
#data("Boston")

x <- Boston$rm
y <- Boston$medv

# fit a linear model
res <- lm( y ~ x )
print(res)

Call:
lm(formula = y ~ x)

Coefficients:
(Intercept)            x  
    -34.671        9.102

# plot the data and the model
plot(x,y, col=rgb(0.2,0.4,0.6,0.4), main='Linear regression')
abline(res, col='blue')

# squared error cost function
cost <- function(X, y, theta) {
  sum( (X %*% theta - y)^2 ) / (2*length(y))
}

# learning rate and iteration limit
alpha <- 0.01
num_iters <- 1000

# keep history
cost_history <- double(num_iters)
theta_history <- list(num_iters)

# initialize coefficients
theta <- matrix(c(0,0), nrow=2)

# add a column of 1's for the intercept coefficient
X <- cbind(1, matrix(x))

# gradient descent
for (i in 1:num_iters) {
  error <- (X %*% theta - y)
  delta <- t(X) %*% error / length(y)
  theta <- theta - alpha * delta
  cost_history[i] <- cost(X, y, theta)
  theta_history[[i]] <- theta
}

print(theta)

          [,1]
[1,] -3.431269
[2,]  4.191125

根据 Digitheads 博客，他使用 lm（线性模型）的 theta 值与他的梯度下降值匹配，而我的不匹配。这些数字不应该匹配吗？

正如您从有关 theta 的各种值的绘图中看到的那样，我最终的 y 截距与几行以上的 print(theta) 值不一致？

任何人都可以就我哪里出错提出建议吗？

Answer 1

梯度下降需要一段时间才能收敛。增加迭代次数将使模型收敛到 OLS 值。例如：

# learning rate and iteration limit
alpha <- 0.01
num_iters <- 100000 # Here I increase the number of iterations in your code to 100k. 
# The gd algorithm now takes a minute or so to run on my admittedly 
# middle-of-the-line laptop.

# keep history
cost_history <- double(num_iters)
theta_history <- list(num_iters)

# initialize coefficients
theta <- matrix(c(0,0), nrow=2)

# add a column of 1's for the intercept coefficient
X <- cbind(1, matrix(x))

# gradient descent (now takes a little longer!)
for (i in 1:num_iters) {
  error <- (X %*% theta - y)
  delta <- (t(X) %*% error) / length(y)
  theta <- theta - alpha * delta
  cost_history[i] <- cost(X, y, theta)
  theta_history[[i]] <- theta
}

print(theta)
     [,1]
[1,] -34.670410
[2,]   9.102076

R中梯度下降和线性模型之间Theta值的差异

Difference in Theta values between gradient descent and linear model in R

r

machine-learning

linear-regression

gradient-descent