Alternate loss functions and regression models that achieve robustness to outliers by modulating the learning rate

We study loss-based filtering for finite-sum optimization with a subset of corrupted component functions whose gradients may be highly unreliable. Motivated by minimum-loss-based SGD (min-$k$-loss) and quantile-based methods for corrupted linear systems, we propose and analyze a general loss-filtering framework -- Quantile-$k$-Loss SGD (Q$k$L-SGD) -- that samples $k$ component losses at each iteration and updates using an index chosen uniformly from the lower empirical $q$-quantile. We prove linear convergence of this family of methods under standard convexity assumptions, requiring the sample size to scale with the number of corruptions and a subset strong-convexity threshold. For the cases when large enough sampling is impossible or undesirable, we give a complementary small-sample probabilistic analysis that covers any sample size $k$ and the convergence behavior depends on the probability of selecting an outlier and on the curvature of the selected good step. Experiments on polynomial regression, regularized logistic regression, and regularized hinge loss show that intermediate quantiles often outperform both standard SGD and min-$k$-loss SGD. In particular, min-$k$ often stalls by repeatedly selecting nearly solved components, while intermediate quantiles retain robustness and produce more informative updates.

The problem of linear predictions has been extensively studied for the past century under pretty generalized frameworks. Recent advances in the robust statistics literature allow us to analyze robust versions of classical linear models through the prism of Median of Means (MoM). Combining these approaches in a piecemeal way might lead to ad-hoc procedures, and the restricted theoretical conclusions that underpin each individual contribution may no longer be valid. To meet these challenges coherently, in this study, we offer a unified robust framework that includes a broad variety of linear prediction problems on a Hilbert space, coupled with a generic class of loss functions. Notably, we do not require any assumptions on the distribution of the outlying data points ($\mathcal{O}$) nor the compactness of the support of the inlying ones ($\mathcal{I}$). Under mild conditions on the dual norm, we show that for misspecification level $ε$, these estimators achieve an error rate of $O(\max\left\{|\mathcal{O}|^{1/2}n^{-1/2}, |\mathcal{I}|^{1/2}n^{-1} \right\}+ε)$, matching the best-known rates in literature. This rate is slightly slower than the classical rates of $O(n^{-1/2})$, indicating that we need to pay a price in terms of error rates to obtain robust estimates. Additionally, we show that this rate can be improved to achieve so-called "fast rates" under additional assumptions.

The literature has proposed various robust alternatives to empirical risk minimisation to address failure modes such as distribution shift, label noise and finite-sample degeneracies. Examples include distributionally robust optimization, label smoothing, vicinal risk minimization, and Mixup. However, such approaches are typically developed in isolation, forcing practitioners to commit a priori to a single failure mode even when the dominant mode for the task is unclear. To address this, we organize a broad class of existing methods along three common design axes and derive a tractable training procedure that decomposes robust learning into sequential stages (reference distribution enrichment, input-space perturbation, label-space perturbation, and sample-level aggregation), each with a choice of stance (pessimistic, neutral, or optimistic). This results in a unified design space in which joint hyperparameter optimization can compose and configure robustness strategies suited to the task at hand. Across tabular, image, and reward modeling benchmarks, joint hyperparameter optimization is competitive with the best single-method baseline in each setting, offering a reliable default for practitioners who do not know a priori which failure mode dominates their task.