使用Optim.jl在Julia中进行逻辑回归 [英] Logistic regression in Julia using Optim.jl

问题描述

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，但是我无法使它正常工作.

I'm trying to implement a simple regularized logistic regression algorithm in Julia. I'd like to use Optim.jl library to minimize my cost function, but I can't get it to work.

我的成本函数和梯度如下:

My cost function and gradient are as follows:

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，而lambda是正则化参数.)

(Where theta is a vector of parameters for the hypothesis function and lambda is a regularization parameter.)

然后，按照此处给出的说明进行操作: https://github.com/JuliaOpt/Optim.jl 我尝试这样调用优化函数:

Then, according to the instructions given here: https://github.com/JuliaOpt/Optim.jl I try to call the optimization function like this:

# 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，它都表示无法收敛，并且输出看起来有点尴尬:

In both cases it says that it fails to converge and the output looks kind of awkard:

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

(从我的第一个代码清单中)我的方法不正确吗?还是我滥用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?

Question

Is my method (from my first code listing) incorrect? Or am I misusing Optim.jl functions? Either way, what is the proper way to define and minimize the cost function here?

这是我第一次与朱莉娅(Julia)在一起，可能我做错了非常严重的事，但我无法确切说出什么.任何帮助将不胜感激！

It's my first time with Julia and probably I'm doing something terribly wrong, but I can't tell what exactly. Any help will be appreciated!

X和y是训练集，X是90x5矩阵，y是90x1向量(即，我的训练集取自 Iris -我不认为不重要).

X and y are the training set, X is a 90x5 matrix, y a 90x1 vector (namely, my training set is taken from Iris - I don't think it matters).

推荐答案

在下面，我发现我使用闭包和curring进行Logistic回归的成本和梯度计算函数(对于那些习惯于返回成本和梯度的函数的版本) :

Below you find my cost and gradient computation functions for Logistic Regression using closures and currying (version for those who got used to a function that returns the cost and gradient):

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 function implementation:

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

要在Optim.jl中应用cost_gradient，请执行以下操作:

To apply cost_gradient in Optim.jl do the following:

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):

Now, you can easily predict (e.g. training set validation):

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

要么尝试我的方法，要么将其与您的实施方案进行比较.

Either try my approach or compare it with your implementation.

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