Min-Max Optimization

Deploying autonomous driving systems requires robustness against long-tail scenarios that are rare but safety-critical. While adversarial training offers a promising solution, existing methods typically decouple scenario generation from policy optimization and rely on heuristic surrogates. This leads to objective misalignment and fails to capture the shifting failure modes of evolving policies. This paper presents ADV-0, a closed-loop min-max optimization framework that treats the interaction between driving policy (defender) and adversarial agent (attacker) as a zero-sum Markov game. By aligning the attacker's utility directly with the defender's objective, we reveal the optimal adversary distribution. To make this tractable, we cast dynamic adversary evolution as iterative preference learning, efficiently approximating this optimum and offering an algorithm-agnostic solution to the game. Theoretically, ADV-0 converges to a Nash Equilibrium and maximizes a certified lower bound on real-world performance. Experiments indicate that it effectively exposes diverse safety-critical failures and greatly enhances the generalizability of both learned policies and motion planners against unseen long-tail risks.

Sharpness-Aware Minimization (SAM) enhances generalization by minimizing the maximum training loss within a predefined neighborhood around the parameters. However, its practical implementation approximates this as gradient ascent(s) followed by applying the gradient at the ascent point to update the current parameters. This practice can be justified as approximately optimizing the objective by neglecting the (full) derivative of the ascent point with respect to the current parameters. Nevertheless, a direct and intuitive understanding of why using the gradient at the ascent point to update the current parameters works superiorly is still lacking. Our work bridges this gap by proposing a novel and intuitive interpretation. We show that the gradient at the single-step ascent point, \uline{when applied to the current parameters}, provides a better approximation of the direction from the current parameters toward the maximum within the local neighborhood than the local gradient. This improved approximation thereby enables a more direct escape from the maximum within the local neighborhood. Nevertheless, our analysis further reveals two issues. First, the approximation by the gradient at the single-step ascent point is often inaccurate. Second, the approximation quality may degrade as the number of ascent steps increases. To address these limitations, we propose in this paper eXplicit Sharpness-Aware Minimization (XSAM). It tackles the first by explicitly estimating the direction of the maximum during training, while addressing the second by crafting a search space that effectively leverages the gradient information at the multi-step ascent point. XSAM features a unified formulation that applies to both single-step and multi-step settings and only incurs negligible computational overhead. Extensive experiments demonstrate the consistent superiority of XSAM against existing counterparts.

This work establishes a rigorous connection between stability properties of discrete-time algorithms (DTAs) and corresponding continuous-time dynamical systems derived through $O(s^r)$-resolution ordinary differential equations (ODEs). We show that for discrete- and continuous-time dynamical systems satisfying a mild error assumption, exponential stability of a common equilibrium with respect to the continuous time dynamics implies exponential stability of the corresponding equilibrium for the discrete-time dynamics, provided that the step size is chosen sufficiently small. We extend this result to common compact invariant sets. We prove that if an equilibrium is exponentially stable for the $O(s^r)$-resolution ODE, then it is also exponentially stable for the associated DTA. We apply this framework to analyse the limit point properties of several prominent optimisation algorithms, including Two-Timescale Gradient Descent--Ascent (TT-GDA), Generalised Extragradient (GEG), Two-Timescale Proximal Point (TT-PPM), Damped Newton (DN), Regularised Damped Newton (RDN), and the Jacobian method (JM), by studying their $O(1)$- and $O(s)$-resolution ODEs. We show that under a proper choice of hyperparameters, the set of saddle points of the objective function is a subset of the set of exponentially stable equilibria of GEG, TT-PPM, DN, and RDN. We relax the common Hessian invariance assumption through direct analysis of the resolution ODEs, broadening the applicability of our results. Numerical examples illustrate the theoretical findings.

Real-world deployments routinely face distribution shifts, group imbalances, and adversarial perturbations, under which the traditional Empirical Risk Minimization (ERM) framework can degrade severely. Distributionally Robust Optimization (DRO) addresses this issue by optimizing the worst-case expected loss over an uncertainty set of distributions, offering a principled approach to robustness. Meanwhile, as training data in DRO always involves sensitive information, safeguarding it against leakage under Differential Privacy (DP) is essential. In contrast to classical DP-ERM, DP-DRO has received much less attention due to its minimax optimization structure with uncertainty constraint. To bridge the gap, we provide a comprehensive study of DP-(finite-sum)-DRO with $ψ$-divergence and non-convex loss. First, we study DRO with general $ψ$-divergence by reformulating it as a minimization problem, and develop a novel $(\varepsilon, δ)$-DP optimization method, called DP Double-Spider, tailored to this structure. Under mild assumptions, we show that it achieves a utility bound of $\mathcal{O}(\frac{1}{\sqrt{n}}+ (\frac{\sqrt{d \log (1/δ)}}{n \varepsilon})^{2/3})$ in terms of the gradient norm, where $n$ denotes the data size and $d$ denotes the model dimension. We further improve the utility rate for specific divergences. In particular, for DP-DRO with KL-divergence, by transforming the problem into a compositional finite-sum optimization problem, we develop a DP Recursive-Spider method and show that it achieves a utility bound of $\mathcal{O}((\frac{\sqrt{d \log(1/δ)}}{n\varepsilon})^{2/3} )$, matching the best-known result for non-convex DP-ERM. Experimentally, we demonstrate that our proposed methods outperform existing approaches for DP minimax optimization.

We consider max-min and min-max problems with objective functions that are possibly non-smooth, submodular with respect to the minimiser and concave with respect to the maximiser. We investigate the performance of a zeroth-order method applied to this problem. The method is based on the subgradient of the Lovász extension of the objective function with respect to the minimiser and based on Gaussian smoothing to estimate the smoothed function gradient with respect to the maximiser. In expectation sense, we prove the convergence of the algorithm to an $ε$-saddle point in the offline case. Moreover, we show that, in the expectation sense, in the online setting, the algorithm achieves $O(\sqrt{N(1+\bar{P}_N)})$ online duality gap, where $N$ is the number of iterations and $\bar{P}_N$ is the path length of the sequence of optimal decisions. The complexity analysis and hyperparameter selection are presented for all the cases. The theoretical results are illustrated via numerical examples.

Efficient computation of min-max problems is a central question in optimization, learning, games, and control. Arguably the most natural algorithm is gradient-descent-ascent (GDA). However, since the 1970s, conventional wisdom has argued that GDA fails to converge even on simple problems. This failure spurred an extensive literature on modifying GDA with additional building blocks such as extragradients, optimism, momentum, anchoring, etc. In contrast, we show that GDA converges in its original form by simply using a judicious choice of stepsizes. The key innovation is the proposal of unconventional stepsize schedules (dubbed slingshot stepsize schedules) that are time-varying, asymmetric, and periodically negative. We show that all three properties are necessary for convergence, and that altogether this enables GDA to converge on the classical counterexamples (e.g., unconstrained convex-concave problems). The core algorithmic intuition is that although negative stepsizes make backward progress, they de-synchronize the min and max variables (overcoming the cycling issue of GDA), and lead to a slingshot phenomenon in which the forward progress in the other iterations is overwhelmingly larger. This results in fast overall convergence. Geometrically, the slingshot dynamics leverage the non-reversibility of gradient flow: positive/negative steps cancel to first order, yielding a second-order net movement in a new direction that leads to convergence and is otherwise impossible for GDA to move in. We interpret this as a second-order finite-differencing algorithm and show that, intriguingly, it approximately implements consensus optimization, an empirically popular algorithm for min-max problems involving deep neural networks (e.g., training GANs).

We study a class of stochastic nonsmooth optimization problems in which an outer variable minimizes the expectation of a pointwise maximum. This minimization--expectation--maximization (minEmax) problem arises in Wasserstein distributionally robust optimization and adversarially robust training, and it cannot in general be reformulated as a finite-dimensional minimax problem when the underlying distribution is not empirical. We propose a stochastic smoothing proximal gradient method based on log-mean-exp smoothing of the value function. Under compactness and Lipschitz-type assumptions, we present nonasymptotic analysis in terms of Goldstein stationarity and show that every almost-sure cluster point generated by our method is a Clarke stationary point; by Clarke regularity, such a point is also directional stationary for the original problem. Numerical experiments on newsvendor, robust regression, and adversarially robust learning problems show that the proposed method is competitive with existing baselines.

Lower-bound analyses for nonconvex strongly-concave minimax optimization problems have shown that stochastic first-order algorithms require at least $\mathcal{O}(\varepsilon^{-4})$ sample complexity to find an $\varepsilon$-stationary point. Some works indicate that this complexity can be improved to $\mathcal{O}(\varepsilon^{-3})$ when the stochastic loss gradient is Lipschitz continuous. The question of achieving enhanced convergence rates under distinct conditions, remains open. In this work, we address this question for optimization problems that are nonconvex in the minimization variable and strongly concave or Polyak-Lojasiewicz (PL) in the maximization variable. We introduce novel bias-corrected momentum algorithms utilizing efficient Hessian-vector products. We establish convergence conditions and demonstrate a lower iteration complexity of $\mathcal{O}(\varepsilon^{-3})$ for the proposed algorithms. The effectiveness of the proposed method is validated through applications to robust logistic regression and robust adaptive cruise control.