arXiv subjects

Pierre-Louis Cauvin

Publications and source records attributed to Pierre-Louis Cauvin.

3 recordsLinked to original sources

arXiv2026

Bregman meets Lévy: Stochastic mirror descent with heavy-tailed noise in continuous and discrete time

We study the robustness of stochastic mirror descent (SMD) under heavy-tailed noise, focusing on whether the method retains its convergence guarantees when run with infinite-variance stochastic gradient input. To address this question in a principled manner, we begin by introducing a continuous-time model of SMD as a stochastic differential equation (SDE) driven by a centered Lévy noise process with finite $p$-th order moments, $1 < p \leq 2$. This scheme -- which we call the Lévy mirror flow (LMF) -- arises naturally as the scaling limit of SMD in the presence of heavy-tailed noise. In particular, when $p < 2$ -- the heavy noise regime -- the trajectories of LMF generically exhibit jump discontinuities of arbitrary magnitude which, if frequent enough, lead to infinite variance. Nonetheless, despite this highly singular behavior, we show that LMF attains $ε$-optimality within $\mathcal{O}(ε^{-p/(p-1)})$ time in the convex case, and within $\mathcal{\tilde O}(ε^{-1/(p-1)})$ time for (relatively) strongly convex objectives. These guarantees provide a transparent characterization of the impact of frequent long jumps on the convergence of the process, and percolate to a series of matching discrete-time guarantees for several variants of SMD under heavy-tailed noise.

math.OC↗

arXiv2025

The impact of uncertainty on regularized learning in games

In this paper, we investigate how randomness and uncertainty influence learning in games. Specifically, we examine a perturbed variant of the dynamics of "follow-the-regularized-leader" (FTRL), where the players' payoff observations and strategy updates are continually impacted by random shocks. Our findings reveal that, in a fairly precise sense, "uncertainty favors extremes": in any game, regardless of the noise level, every player's trajectory of play reaches an arbitrarily small neighborhood of a pure strategy in finite time (which we estimate). Moreover, even if the player does not ultimately settle at this strategy, they return arbitrarily close to some (possibly different) pure strategy infinitely often. This prompts the question of which sets of pure strategies emerge as robust predictions of learning under uncertainty. We show that (a) the only possible limits of the FTRL dynamics under uncertainty are pure Nash equilibria; and (b) a span of pure strategies is stable and attracting if and only if it is closed under better replies. Finally, we turn to games where the deterministic dynamics are recurrent - such as zero-sum games with interior equilibria - and we show that randomness disrupts this behavior, causing the stochastic dynamics to drift toward the boundary on average.

cs.GT↗

arXiv2022

Edgeworth expansion for Bernoulli weighted mean

In this work, we derive an Edgeworth expansion for the Bernoulli weighted mean $\hatμ = \frac{\sum_{i=1}^n Y_i T_i}{\sum_{i=1}^n T_i}$ in the case where $Y_1, \dots, Y_n$ are i.i.d. non semi-lattice random variables and $T_1, \dots, T_n$ are Bernoulli distributed random variables with parameter $p$. We also define the notion of a semi-lattice distribution, which gives a more geometrical equivalence to the classical Cramér's condition in dimensions bigger than 1. Our result provides a first step into the generalization of classical Edgeworth expansion theorems for random vectors that contain both semi-lattice and non semi-lattice variables, in order to prove consistency of bootstrap methods in more realistic setups, for instance in the use case of online AB testing.

math.PR↗