Shira Vansover-Hager

Shira Vansover-Hager, Matan Schliserman, Ofir Schlisselberg et al.

Mirror Descent (MD) extends Gradient Descent (GD) beyond Euclidean geometry and has recently reappeared as a lens for KL-regularized policy optimization in reinforcement learning and LLM post-training. This raises a basic robustness question, crucial to reproducibility and reliability: how sensitive are MD dynamics to their inputs? We focus on initialization, often itself a pretrained or previously aligned model. Quadratic-regularized MD, including GD and Mahalanobis geometries, is well-known to be stable for convex smooth objectives. We show a sharp contrast: once the regularizer is non-quadratic, MD can be exponentially more sensitive to initialization than GD, even with a well-conditioned regularizer in Euclidean norm. We give a three-dimensional construction with a convex, smooth objective and a strongly convex, smooth, well-conditioned regularizer where an initial $\varepsilon$ perturbation is quickly amplified to $\min\{\text{polylog}^{-1}(1/\varepsilon), \varepsilon e^{Ω(ηT)}\}$ after $T$ iterations of MD with step size $η$. For canonical KL-regularized MD on the simplex, we show that even linear objectives can amplify an initial $\varepsilon$ perturbation exponentially fast in high-dimensional or near-boundary regimes. Finally, we show that adding a Bregman regularization term toward an anchor point can stabilize the dynamics while largely preserving the optimization guarantees, and that the choice of anchor is crucial: anchoring at the initialization only partially mitigates the instability, whereas anchoring at a fixed point yields a more stable mechanism.

Matan Schliserman, Shira Vansover-Hager, Tomer Koren

Understanding the generalization behavior of learning algorithms is a central goal of learning theory. A recently emerging explanation is that learning algorithms are successful in practice because they converge to flat minima, which have been consistently associated with improved generalization performance. In this work, we study the link between flat minima and generalization in the canonical setting of stochastic convex optimization with a non-negative, $β$-smooth objective. Our first finding is that, even in this fundamental and well-studied setting, flat empirical minima may incur trivial $Ω(1)$ population risk while sharp minima generalizes optimally. Then, we show that this poor generalization behavior extends to two natural ''sharpness-aware'' algorithms originally proposed by Foret et al. (2021), designed to bias optimization toward flat solutions: Sharpness-Aware Gradient Descent (SA-GD) and Sharpness-Aware Minimization (SAM). For SA-GD, which performs gradient steps on the maximal loss in a predefined neighborhood, we prove that while it successfully converges to a flat minimum at a fast rate, the population risk of the solution can still be as large as $Ω(1)$, indicating that even flat minima found algorithmically using a sharpness-aware gradient method might generalize poorly. For SAM, a computationally efficient approximation of SA-GD based on normalized ascent steps, we show that although it minimizes the empirical loss, it may converge to a sharp minimum and also incur population risk $Ω(1)$. Finally, we establish population risk upper bounds for both SA-GD and SAM using algorithmic stability techniques.