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<!DOCTYPE html><HTML lang="en"><head><meta charset="UTF-8"/><meta content="width=device-width, initial-scale=1.0" name="viewport"/><title>SVM with L^1 regularization · Convex.jl</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../../.."</script><script data-main="../../../assets/documenter.js" src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js"></script><script src="../../../siteinfo.js"></script><script src="../../../../versions.js"></script><link class="docs-theme-link" data-theme-name="documenter-dark" href="../../../assets/themes/documenter-dark.css" rel="stylesheet" type="text/css"/><link class="docs-theme-link" data-theme-name="documenter-light" data-theme-primary="" href="../../../assets/themes/documenter-light.css" rel="stylesheet" type="text/css"/><script src="../../../assets/themeswap.js"></script><script data-outdated-warner="">function maybeAddWarning () {
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using Random
Random.seed!(3);
n = 20;
m = 1000;
TEST = m;
DENSITY = 0.2;
beta_true = randn(n,1);
idxs = randperm(n)[1:round(Int, (1-DENSITY)*n)];
beta_true[idxs] .= 0
offset = 0;
sigma = 45;
X = 5 * randn(m, n);
Y = sign.(X * beta_true .+ offset .+ sigma * randn(m,1));
X_test = 5 * randn(TEST, n);</code></pre><pre><code class="language-julia"># Form SVM with L1 regularization problem.
using Convex, SCS, ECOS
beta = Variable(n);
v = Variable();
loss = sum(pos(1 - Y .* (X*beta - v)));
reg = norm(beta, 1);
# Compute a trade-off curve and record train and test error.
TRIALS = 100
train_error = zeros(TRIALS);
test_error = zeros(TRIALS);
lambda_vals = exp10.(range(-2, stop=0, length=TRIALS);)
beta_vals = zeros(length(beta), TRIALS);
for i = 1:TRIALS
lambda = lambda_vals[i];
problem = minimize(loss/m + lambda*reg);
solve!(problem, () -> ECOS.Optimizer(verbose=0));
# solve!(problem, SCS.Optimizer(verbose=0,linear_solver=SCS.Direct, eps=1e-3))
train_error[i] = sum(float(sign.(X*beta_true .+ offset) .!= sign.(evaluate(X*beta - v))))/m;
test_error[i] = sum(float(sign.(X_test*beta_true .+ offset) .!= sign.(evaluate(X_test*beta - v))))/TEST;
beta_vals[:, i] = evaluate(beta);
end</code></pre><p>Plot the train and test error over the trade-off curve.</p><pre><code class="language-julia">using Plots
plot(lambda_vals, train_error, label="Train error");
plot!(lambda_vals, test_error, label="Test error");
plot!(xscale=:log, yscale=:log, ylabel="errors", xlabel="lambda")</code></pre><svg height="400" viewBox="0 0 2400 1600" width="600" xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><defs><clipPath id="clip3700"><rect height="1600" width="2400" x="0" y="0"></rect></clipPath></defs><path clip-path="url(#clip3700)" d="
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for i = 1:n
plot!(lambda_vals, vec(beta_vals[i,:]), label="beta$i")
end
plot!(xscale=:log, ylabel="betas", xlabel="lambda")</code></pre><svg height="400" viewBox="0 0 2400 1600" width="600" xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><defs><clipPath id="clip4300"><rect height="1600" width="2400" x="0" y="0"></rect></clipPath></defs><path clip-path="url(#clip4300)" d="
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