Optim Julia parameter meaning

I'm trying to use Optim in Julia to solve a two variable minimization problem, similar to the following

x = [1.0, 2.0, 3.0]
y = 1.0 .+ 2.0 .* x .+ [-0.3, 0.3, -0.1]

function sqerror(betas, X, Y)
    err = 0.0
    for i in 1:length(X)
        pred_i = betas[1] + betas[2] * X[i]
        err += (Y[i] - pred_i)^2
    end
    return err
end

res = optimize(b -> sqerror(b, x, y), [0.0,0.0])
res.minimizer

I do not quite understand what [0.0,0.0] means. By looking at the document http://julianlsolvers.github.io/Optim.jl/v0.9.3/user/minimization/. My understanding is that it is the initial condition. However, if I change that to [0.0,0., 0.0], the algorithm still work despite the fact that I only have two unknowns, and the algorithm gives me three instead of two minimizer. I was wondering if anyone knows what[0.0,0.0] really stands for.

Solution

It is initial value. optimize by itself cannot know how many values your sqerror function takes. You specify it by passing this initial value.

For example if you add dimensionality check to sqerror you will get a proper error:

julia> function sqerror(betas::AbstractVector, X::AbstractVector, Y::AbstractVector)
           @assert length(betas) == 2
           err = 0.0
           for i in eachindex(X, Y)
               pred_i = betas[1] + betas[2] * X[i]
               err += (Y[i] - pred_i)^2
           end
           return err
       end
sqerror (generic function with 2 methods)

julia> optimize(b -> sqerror(b, x, y), [0.0,0.0,0.0])
ERROR: AssertionError: length(betas) == 2

Note that I also changed the loop condition to eachindex(X, Y) to ensure that your function checks if X and Y vectors have aligned indices.

Finally if you want performance and reduce compilation cost (so e.g. assuming you do this optimization many times) it would be better to define your optimized function like this:

objective_factory(x, y) = b -> sqerror(b, x, y)
optimize(objective_factory(x, y), [0.0,0.0])