Multiple Descent for Random-Design Ridge Regression under Optimal Regularization and Independent Noise

We study the possible irregularity of sample-wise risk for optimally regularized ridge regression in fixed dimension. For every $d\ge2$, we construct a finite-support random-design linear model in $\mathbb{R}^d$ with bounded homoscedastic noise independent of the design whose optimally regularized excess risk has at least $d-1$ local maxima. Thus, optimizing the ridge parameter at every sample size does not ensure monotonicity, even under standard independence and homoscedasticity assumptions. This removes the design-dependent noise required by the authors' previous construction. It also contrasts with standard proportional asymptotics, where optimal ridge tuning restores monotonicity under mild regularity conditions.