Sparse Data Augmentation for Optimization with Provable Guarantees

In this paper, we study nonconvex constrained stochastic zeroth-order optimization problems with exact constraints and stochastic objective evaluations. To solve this class of problems, we propose a framework of mirror descent linearized augmented Lagrangian methods that employs two-point stochastic zeroth-order gradient estimators and exploits non-Euclidean mirror descent geometry. Under mild assumptions, we establish oracle complexity guarantees for finding an $ε$-KKT point parameterized by $p \geq 2$. Under Rademacher smoothing, our analysis reveals a trade-off between the variance of the zeroth-order gradient estimators and the smoothness of the mirror map. In the high-accuracy regime, the resulting effective oracle complexity is $\mathcal{O}(p d^{2/p}ε^{-3})$ for $p \in [2,2\ln d]$ and $\mathcal{O}(\ln d\,ε^{-3})$ for $p > 2\ln d$. These bounds reduce the dimension dependence in the leading term. When $p=2$, our method recovers the Euclidean setting with an oracle complexity of $\mathcal{O}(dε^{-3})$, improving the $ε$-dependence over existing methods. Furthermore, to eliminate initial near-feasibility requirements, we introduce a multi-stage scheme that finds an $ε$-KKT point within $\mathcal{O}(1+\log\log(e/ε))$ stages while maintaining the leading-order complexity. Numerical tests on QCQPs, black-box adversarial attacks, and fairness-constrained classification demonstrate the effectiveness of our proposed method.

Learning instability is a long-standing problem across machine learning, but it is especially acute in the overparameterized regime that defines modern deep learning: large models fine-tuned or trained on limited data traverse flat loss landscapes with many nearly-equivalent minima, and stochastic factors (initialization, data order, dropout, hardware non-determinism) can route optimization to very different solutions. The rise of large pretrained models (LPMs) makes the problem more urgent: training cost is high, downstream data is often small, and repeated runs for variance reduction are prohibitive. We introduce \textbf{GRAIN} (\textbf{G}roup \textbf{A}ggregation via m\textbf{IN}-norm objective), a lightweight training algorithm that replaces the mean aggregation used in mini-batch optimization (both across mini-batches and within a mini-batch) with a min-norm convex combination of group-wise gradients. \mName guarantees a non-negative inner product between the aggregated update and every group gradient, resolving intra- and inner-batch gradient conflict, and retains an $\mathcal{O}(1/T)$ convergence rate comparable to SGD. Under mild smoothness and absolute-continuity assumptions, the min-norm solution differs almost surely from the arithmetic mean, which yields a uniform-stability bound for \mName strictly tighter than the standard bound for SGD. Empirically across generation, classification, and regression at LPM scale, \mName delivers consistent improvements in mean performance and reductions in run-to-run variance over a broad suite of tasks, with no extra training-time or storage cost beyond a single backward pass.

Many problems in trustworthy ML can be expressed as constraints on prediction rates across subpopulations, including group fairness constraints (demographic parity, equalized odds, etc.). In this work, we study such constrained minimization problems under differential privacy (DP). Standard DP optimization techniques like DP-SGD rely on objectives that decompose over individual examples, enabling per-example gradient clipping and noise addition. Rate constraints, however, depend on aggregate statistics across groups, creating inter-sample dependencies that violate this decomposability. To address this, we develop RaCO-DP, a DP variant of Stochastic Gradient Descent-Ascent (SGDA) that solves the Lagrangian formulation of rate constraint problems. Through careful design, the extra privacy cost incurred by incorporating these constraints in our approach is limited to that of privately estimating a histogram over each mini-batch at every step. We prove the convergence of our algorithm through a novel analysis of SGDA that leverages the linear structure of the dual parameter. Empirical results show that our method Pareto-dominates existing private learning approaches under group fairness constraints and also achieves strong privacy-utility-fairness performance on neural networks.