The source files for all examples can be found in /examples.
Lasso
This example uses the LassoSolver to solve a lasso regression problem. We also show how to use the MLSolver.
The lasso regression problem is
\[\begin{array}{ll} \text{minimize} & (1/2)\|Ax - b\|_2^2 + \lambda \|x\|_1. \end{array}\]
using GeNIOS
using Random, LinearAlgebra, SparseArraysGenerating the problem data
Random.seed!(1)
m, n = 200, 400
A = randn(m, n)
A .-= sum(A, dims=1) ./ m
normalize!.(eachcol(A))
xstar = sprandn(n, 0.1)
b = A*xstar + 1e-3*randn(m)
λ = 0.05*norm(A'*b, Inf)0.10902103801870822LassoSolver interface
The easiest interface for this problem is the LassoSolver, where we just need to specify the regularization parameter (in addition to the problem data).
λ1 = λ
solver = GeNIOS.LassoSolver(λ1, A, b)
res = solve!(solver; options=GeNIOS.SolverOptions(use_dual_gap=true, dual_gap_tol=1e-4, verbose=true))
rmse = sqrt(1/m*norm(A*solver.zk - b, 2)^2)
println("Final RMSE: $(round(rmse, digits=8))")Starting setup...
Setup in 0.123s
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
Iteration Objective RMSE Dual Gap r_primal r_dual ρ Time
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
0 1.682e+01 2.900e-01 9.256e+00 Inf Inf 1.000e+00 0.000
1 1.682e+01 2.900e-01 9.256e+00 0.000e+00 9.692e+00 1.000e+00 0.105
20 2.963e+00 3.613e-02 7.393e-02 1.897e-02 6.309e-02 1.000e+00 0.109
39 2.963e+00 3.602e-02 9.329e-05 1.210e-04 2.713e-04 1.000e+00 0.115
SOLVED in 0.115s, 39 iterations
Total time: 0.238s
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
Final RMSE: 0.05094243MLSolver interface
Under the hood, this is just a wrapper around the MLSolver interface. This interface is more general, and allows us to specify the per-sample loss used in the machine learning problem. Specifically, it solves problems with the form
\[\begin{array}{ll} \text{minimize} & \sum_{i=1}^N f(a_i^Tx - b_i) + \lambda_1 \|x\|_1 + (\lambda_2/2) \|x\|_2^2. \end{array}\]
It's easy to see that the lasso problem is a special case.
f(x) = 0.5*x^2
fconj(x) = 0.5*x^2
λ1 = λ
λ2 = 0.0
solver = GeNIOS.MLSolver(f, λ1, λ2, A, b; fconj=fconj)
res = solve!(solver; options=GeNIOS.SolverOptions(relax=true, use_dual_gap=true, dual_gap_tol=1e-3, verbose=true))
rmse = sqrt(1/m*norm(A*solver.zk - b, 2)^2)
println("Final RMSE: $(round(rmse, digits=8))")Starting setup...
Setup in 0.129s
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
Iteration Objective RMSE Dual Gap r_primal r_dual ρ Time
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
0 1.682e+01 2.900e-01 9.256e+00 Inf Inf 1.000e+00 0.000
1 1.682e+01 2.900e-01 9.256e+00 0.000e+00 9.692e+00 1.000e+00 0.162
20 2.963e+00 3.613e-02 7.393e-02 1.897e-02 6.309e-02 1.000e+00 0.173
35 2.963e+00 3.602e-02 6.247e-04 6.305e-04 1.270e-03 1.000e+00 0.181
SOLVED in 0.181s, 35 iterations
Total time: 0.309s
──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
Final RMSE: 0.05093825Note that we also defined the conjugate function of $f$, defined as
\[f^*(y) = \sup_x \{yx - f(x)\},\]
which allows us to use the dual gap as a stopping criterion (see our paper for a derivation). Specifying the conjugate function is optional, and the solver will fall back to using the primal and dual residuals if it is not specified.
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