Double descent in the regularized signal-only risk

Determine whether the instance-specific risk for the signal-only least squares regression model with ridge regularization exhibits double descent as the proportional ratio c = d/n varies. The signal-only model is defined by X = Z + A with Z = θ v u^T (rank-one spike), A having i.i.d. rotationally bi-invariant entries with mean zero and variance τ_A^2/d, targets y_i = z_i^T β_* + ε_i, and estimator β_so solving min_β ||y − Xβ||_2^2 + μ^2 ||β||_2^2. Ascertain, in the proportional asymptotic regime, whether the risk curve in c displays the characteristic double-descent behavior when μ > 0.

Background

In the proportional asymptotic regime, many least squares generalization risks exhibit double descent as the parameter ratio c = d/n varies, with a peak near the interpolation threshold. For the unregularized settings analyzed in the paper, double descent is apparent and mathematically characterized.

For the signal-only model with ridge regularization (μ > 0), the derived risk expression (Theorem for the signal-only case) is complex, and its behavior with respect to c is not immediately transparent. Although empirical evidence in the paper indicates double descent and suggests a shifted peak location depending on μ and τ_A, a formal theoretical determination of double descent for this regularized case is explicitly identified as unclear.