When Does $\ell_2$-Boosting Overfit Benignly? High-Dimensional Risk Asymptotics and the $\ell_1$ Implicit Bias

• URL: http://arxiv.org/abs/2605.06314v2
• Date: Tue, 12 May 2026 05:32:02 GMT
• Title: When Does $\ell_2$-Boosting Overfit Benignly? High-Dimensional Risk Asymptotics and the $\ell_1$ Implicit Bias
• Abstract summary: We show that benign overfitting fails at the linear rate.<n>We show that while this localization mechanism should persist in the presence of signals, the exact signal-noise decomposition remains an open problem.
• Score: 15.113649527486276
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Benign overfitting is well-characterized in $\ell_2$ geometries, but its behavior under the $\ell_1$ implicit bias of greedy ensembles remains challenging. The analytical barrier stems from the non-linear coupling of coordinate selection thresholds, which invalidates standard spectral resolvent tools. To isolate this algorithmic bias, we characterize the high-dimensional risk of continuous-time $\ell_2$-Boosting over $p$ features and $n$ samples. By coupling the Convex Gaussian Minimax Theorem with delicate asymptotic expansions of double-sided truncated Gaussian moments, we analytically resolve the non-smooth $\ell_1$ interpolant. Under an isotropic pure-noise model, we prove that benign overfitting fails at the linear rate: greedy selection localizes noise into sparse active sets, and the excess variance decays at a logarithmic rate $Θ(σ^2/\log(p/n))$ for noise variance $σ^2$. We remark that while this localization mechanism should persist in the presence of signals, the exact signal-noise decomposition remains an open problem. For spiked-isotropic designs with $k^*$ head eigenvalues and $r_2 = p - k^*$ tail dimensions, the risk converges to zero when $r_{2} \gg n$, but only at a logarithmic rate $Θ(σ^2/\log(r_2/n))$, which is slower than the linear decay observed in $\ell_2$ geometries. To avoid this slow convergence, we analyze the non-smooth subdifferential dynamics of the boosting flow. This yields a tuning-free early stopping rule that, under a bounded $\ell_1$-path condition, recovers the Lasso basic inequality and attains the minimax-optimal empirical prediction rate for $\ell_1$-bounded signals.

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