regression - 使用 Optim.jl 在 Julia 中进行逻辑回归

Question

我正在尝试在 Julia 中实现一个简单的正则化逻辑回归算法。我想使用 Optim.jl 库来最小化我的成本函数，但我无法让它工作。

我的成本函数和梯度如下:

function cost(X, y, theta, lambda)
    m = length(y)
    h = sigmoid(X * theta)
    reg = (lambda / (2*m)) * sum(theta[2:end].^2)
    J = (1/m) * sum( (-y).*log(h) - (1-y).*log(1-h) ) + reg
    return J
end

function grad(X, y, theta, lambda, gradient)
    m = length(y)
    h = sigmoid(X * theta)
    # gradient = zeros(size(theta))
    gradient = (1/m) * X' * (h - y)
    gradient[2:end] = gradient[2:end] + (lambda/m) * theta[2:end]
    return gradient
end

(其中 theta 是假设函数的参数向量， lambda 是正则化参数。)

然后，根据此处给出的说明: https://github.com/JuliaOpt/Optim.jl我尝试这样调用优化函数:

# those are handle functions I define to pass them as arguments:
c(theta::Vector) = cost(X, y, theta, lambda)
g!(theta::Vector, gradient::Vector) = grad(X, y, theta, lambda, gradient)

# then I do
optimize(c,some_initial_theta) 
# or maybe
optimize(c,g!,initial_theta,method = :l_bfgs) # try a different algorithm

在这两种情况下，它都表示它无法收敛，并且输出看起来有点尴尬:

julia> optimize(c,initial_theta)
Results of Optimization Algorithm
 * Algorithm: Nelder-Mead
 * Starting Point: [0.0,0.0,0.0,0.0,0.0]
 * Minimum: [1.7787162051775145,3.4584135105727145,-6.659680628594007,4.776952006060713,1.5034743945407143]
 * Value of Function at Minimum: -Inf
 * Iterations: 1000
 * Convergence: false
   * |x - x'| < NaN: false
   * |f(x) - f(x')| / |f(x)| < 1.0e-08: false
   * |g(x)| < NaN: false
   * Exceeded Maximum Number of Iterations: true
 * Objective Function Calls: 1013
 * Gradient Call: 0

julia> optimize(c,g!,initial_theta,method = :l_bfgs)
Results of Optimization Algorithm
 * Algorithm: L-BFGS
 * Starting Point: [0.0,0.0,0.0,0.0,0.0]
 * Minimum: [-6.7055e-320,-2.235e-320,-6.7055e-320,-2.244e-320,-6.339759952602652e-7]
 * Value of Function at Minimum: 0.693148
 * Iterations: 1
 * Convergence: false
   * |x - x'| < 1.0e-32: false
   * |f(x) - f(x')| / |f(x)| < 1.0e-08: false
   * |g(x)| < 1.0e-08: false
   * Exceeded Maximum Number of Iterations: false
 * Objective Function Calls: 75
 * Gradient Call: 75

问题

我的方法(来自我的第一个代码 list )不正确吗？还是我滥用了 Optim.jl 函数？无论哪种方式，在这里定义和最小化成本函数的正确方法是什么？

这是我第一次和 Julia 在一起，可能我做错了什么，但我不知 Prop 体是什么。任何帮助将不胜感激!

编辑
X和 y是训练集， X是一个 90x5 矩阵， y一个 90x1 的向量(也就是说，我的训练集取自 Iris - 我认为这并不重要)。

Answer 1

下面是我使用闭包和柯里化(Currying)的逻辑回归的成本和梯度计算函数(适用于那些习惯于返回成本和梯度的函数的人的版本):

function cost_gradient(θ, X, y, λ)
    m = length(y)
    return (θ::Array) -> begin 
        h = sigmoid(X * θ)   
        J = (1 / m) * sum(-y .* log(h) .- (1 - y) .* log(1 - h)) + λ / (2 * m) * sum(θ[2:end] .^ 2)         
    end, (θ::Array, storage::Array) -> begin  
        h = sigmoid(X * θ) 
        storage[:] = (1 / m) * (X' * (h .- y)) + (λ / m) * [0; θ[2:end]]        
    end
end

Sigmoid函数实现:

sigmoid(z) = 1.0 ./ (1.0 + exp(-z))

申请 cost_gradient在 Optim.jl 中执行以下操作:

using Optim
#...
# Prerequisites:
# X size is (m,d), where d is the number of training set features
# y size is (m,1)
# λ as the regularization parameter, e.g 1.5
# ITERATIONS number of iterations, e.g. 1000
X=[ones(size(X,1)) X] #add x_0=1.0 column; now X size is (m,d+1)
initialθ = zeros(size(X,2),1) #initialTheta size is (d+1, 1)
cost, gradient! = cost_gradient(initialθ, X, y, λ)
res = optimize(cost, gradient!, initialθ, method = ConjugateGradient(), iterations = ITERATIONS);
θ = Optim.minimizer(res);

现在，您可以轻松预测(例如训练集验证):

predictions = sigmoid(X * θ) #X size is (m,d+1)

尝试我的方法或将其与您的实现进行比较。