arXiv subjects

Julien Grand-Clément

Publications and source records attributed to Julien Grand-Clément.

At least 19 recordsLinked to original sources

arXiv2026

Sharper bounds on the thresholds for sensitive and Blackwell optimality

In perfect-information two-player stochastic games, the notions of Blackwell and sensitive optimality provide generalizations of the classical mean-payoff optimality and discount optimality criteria to account for more farsighted preferences. We provide bounds on the Blackwell threshold $α_{\sf bw}$ and the $d$-sensitive thresholds $α_{\sf d}$, defined as the smallest discount factors above which discount optimal policies coincide with Blackwell optimal policies and $d$-sensitive optimal policies respectively. Our refined bounds improve upon prior work by focusing on ``reduced'' families of polynomials and, crucially, our bounds are tight in terms of controlling the degrees and heights of the minimal polynomials of the Blackwell thresholds. We apply classical root separation based on Mahler's and Cauchy's bounds to our reduced families to derive the strongest upper and lower bounds on $α_{\sf bw}$ in the literature, and we are the first to obtain bounds on $α_{\sf d}$ in the multichain stochastic setting.

cs.GT↗

arXiv2026

Dynamic Programming-Compatible Uncertainty Sets in Robust Markov Decision Processes

In this paper, we investigate the compatibility of robust Markov Decision Processes (RMDPs) with dynamic programming under various assumptions on the uncertainty set, i.e., we investigate when one can solve an RMDP by solving a fixed point equation. We show that in all generality, s-rectangular and sa-rectangular uncertainty sets are the only models of uncertainty that are compatible with dynamic programming. Our analysis shows that existing non-rectangular models, including r-rectangularity, are only weakly compatible with dynamic programming, as they require the assumption that rewards do not depend on the next state. In this case, our results imply that one can always construct a rectangular uncertainty set that is equivalent, for both policy evaluation and optimization, to the dynamic programming-compatible non-rectangular model. This highlights a key limitation: dynamic-programming-compatible non-rectangular uncertainty sets, although practically relevant for uncertainty quantification, do not provide a genuinely distinct assessment of policy performance. Interestingly, our proof techniques rely on identifying a novel simultaneous solvability property, which we show is central to several important properties of RMDPs, including the existence of stationary optimal policies and dynamic programming-based formulations. The simultaneous solvability property enables a unified approach to studying all existing models of uncertainty, rectangular and non-rectangular alike.

math.OC↗

arXiv2026

Dynamic Programming for Epistemic Uncertainty in Markov Decision Processes

In this paper, we propose a general theory of ambiguity-averse MDPs, which treats the uncertain transition probabilities as random variables and evaluates a policy via a risk measure applied to its random return. This ambiguity-averse MDP framework unifies several models of MDPs with epistemic uncertainty for specific choices of risk measures. We extend the concepts of value functions and Bellman operators to our setting. Based on these objects, we establish the consequences of dynamic programming principles in this framework (existence of stationary policies, value and policy iteration algorithms), and we completely characterize law-invariant risk measures compatible with dynamic programming. Our work draws connections among several variants of MDP models and fully delineates what is possible under the dynamic programming paradigm and which risk measures require leaving it.

cs.GT↗

arXiv2026

Ex-Post Equilibria: Structure and Computation

This paper studies ex-post equilibria (EPEs) in simultaneous-move games with parameter uncertainty. We first compare EPEs with existing notions of robust equilibrium and provide a general foundation for EPEs as a solution concept for games with parameter uncertainty, and we show that EPEs are fully characterized by two game-theoretic properties (monotonicity and set-consistency). Since ex-post equilibria may fail to exist, we introduce the notion of an optimal approximate ex-post equilibrium, in which players adopt approximate best responses while minimizing the degree of suboptimality. We study the problem of computing EPEs and optimal approximate EPEs, focusing on two important classes of games: zero-sum games and concave potential games. We provide several hardness results, as well as a general class of computational approaches based on auxiliary minimax formulations.

cs.GT↗

arXiv2025

Thresholds for sensitive optimality and Blackwell optimality in stochastic games

We investigate refinements of the mean-payoff criterion in two-player zero-sum perfect-information stochastic games. A strategy is Blackwell optimal if it is optimal in the discounted game for all discount factors sufficiently close to $1$. The notion of $d$-sensitive optimality interpolates between mean-payoff optimality (corresponding to the case $d=-1$) and Blackwell optimality ($d=+\infty$). The Blackwell threshold $α_{\sf Bw} \in [0,1[$ is the discount factor above which all optimal strategies in the discounted game are guaranteed to be Blackwell optimal. The $d$-sensitive threshold $α_{\sf d} \in [0,1[$ is defined analogously. Bounding $α_{\sf Bw}$ and $α_{\sf d}$ are fundamental problems in algorithmic game theory, since these thresholds control the complexity for computing Blackwell and $d$-sensitive optimal strategies, by reduction to discounted games which can be solved in $O\left((1-α)^{-1}\right)$ iterations. We provide the first bounds on the $d$-sensitive threshold $α_{\sf d}$ beyond the case $d=-1$, and we establish improved bounds for the Blackwell threshold $α_{\sf Bw}$. This is achieved by leveraging separation bounds on algebraic numbers, relying on Lagrange bounds and more advanced techniques based on Mahler measures and multiplicity theorems.

cs.GT↗

arXiv2025

Playing against a stationary opponent

This paper investigates properties of Blackwell $ε$-optimal strategies in zero-sum stochastic games when the adversary is restricted to stationary strategies, motivated by applications to robust Markov decision processes. For a class of absorbing games, we show that Markovian Blackwell $ε$-optimal strategies may fail to exist, yet we prove the existence of Blackwell $ε$-optimal strategies that can be implemented by a two-state automaton whose internal transitions are independent of actions. For more general absorbing games, however, there need not exist Blackwell $ε$-optimal strategies that are independent of the adversary's decisions. Our findings point to a contrast between absorbing games and generalized Big Match games, and provide new insights into the properties of optimal policies for robust Markov decision processes.

cs.GT↗

arXiv2025

Last-Iterate Convergence Properties of Regret-Matching Algorithms in Games

We study last-iterate convergence properties of algorithms for solving two-player zero-sum games based on Regret Matching$^+$ (RM$^+$). Despite their widespread use for solving real games, virtually nothing is known about their last-iterate convergence. A major obstacle to analyzing RM-type dynamics is that their regret operators lack Lipschitzness and (pseudo)monotonicity. We start by showing numerically that several variants used in practice, such as RM$^+$, predictive RM$^+$ and alternating RM$^+$, all lack last-iterate convergence guarantees even on a simple $3\times 3$ matrix game. We then prove that recent variants of these algorithms based on a smoothing technique, extragradient RM$^{+}$ and smooth Predictive RM$^+$, enjoy asymptotic last-iterate convergence (without a rate), $1/\sqrt{t}$ best-iterate convergence, and when combined with restarting, linear-rate last-iterate convergence. Our analysis builds on a new characterization of the geometric structure of the limit points of our algorithms, marking a significant departure from most of the literature on last-iterate convergence. We believe that our analysis may be of independent interest and offers a fresh perspective for studying last-iterate convergence in algorithms based on non-monotone operators.

cs.GT↗

arXiv2025

On Separation Between Best-Iterate, Random-Iterate, and Last-Iterate Convergence of Learning in Games

Non-ergodic convergence of learning dynamics in games is widely studied recently because of its importance in both theory and practice. Recent work (Cai et al., 2024) showed that a broad class of learning dynamics, including Optimistic Multiplicative Weights Update (OMWU), can exhibit arbitrarily slow last-iterate convergence even in simple $2 \times 2$ matrix games, despite many of these dynamics being known to converge asymptotically in the last iterate. It remains unclear, however, whether these algorithms achieve fast non-ergodic convergence under weaker criteria, such as best-iterate convergence. We show that for $2\times 2$ matrix games, OMWU achieves an $O(T^{-1/6})$ best-iterate convergence rate, in stark contrast to its slow last-iterate convergence in the same class of games. Furthermore, we establish a lower bound showing that OMWU does not achieve any polynomial random-iterate convergence rate, measured by the expected duality gaps across all iterates. This result challenges the conventional wisdom that random-iterate convergence is essentially equivalent to best-iterate convergence, with the former often used as a proxy for establishing the latter. Our analysis uncovers a new connection to dynamic regret and presents a novel two-phase approach to best-iterate convergence, which could be of independent interest.

cs.LG↗

arXiv2025

Fast Last-Iterate Convergence of Learning in Games Requires Forgetful Algorithms

Self-play via online learning is one of the premier ways to solve large-scale two-player zero-sum games, both in theory and practice. Particularly popular algorithms include optimistic multiplicative weights update (OMWU) and optimistic gradient-descent-ascent (OGDA). While both algorithms enjoy $O(1/T)$ ergodic convergence to Nash equilibrium in two-player zero-sum games, OMWU offers several advantages including logarithmic dependence on the size of the payoff matrix and $\widetilde{O}(1/T)$ convergence to coarse correlated equilibria even in general-sum games. However, in terms of last-iterate convergence in two-player zero-sum games, an increasingly popular topic in this area, OGDA guarantees that the duality gap shrinks at a rate of $O(1/\sqrt{T})$, while the best existing last-iterate convergence for OMWU depends on some game-dependent constant that could be arbitrarily large. This begs the question: is this potentially slow last-iterate convergence an inherent disadvantage of OMWU, or is the current analysis too loose? Somewhat surprisingly, we show that the former is true. More generally, we prove that a broad class of algorithms that do not forget the past quickly all suffer the same issue: for any arbitrarily small $δ>0$, there exists a $2\times 2$ matrix game such that the algorithm admits a constant duality gap even after $1/δ$ rounds. This class of algorithms includes OMWU and other standard optimistic follow-the-regularized-leader algorithms.

cs.GT↗

arXiv2025

Beyond discounted returns: Robust Markov decision processes with average and Blackwell optimality

Robust Markov Decision Processes (RMDPs) are a widely used framework for sequential decision-making under parameter uncertainty. RMDPs have been extensively studied when the objective is to maximize the discounted return, but little is known for average optimality (optimizing the long-run average of the rewards obtained over time) and Blackwell optimality (remaining discount optimal for all discount factors sufficiently close to ). In this paper, we prove several foundational results for RMDPs beyond the discounted return. We show that average optimal policies can be chosen stationary and deterministic for sa-rectangular RMDPs but, perhaps surprisingly, we show that for s-rectangular RMDPs average optimal policies may not exist, and if they exist, may need to be history-dependent (Markovian). We also study Blackwell optimality for sa-rectangular RMDPs, where we show that $ε$-Blackwell optimal policies always exist, although Blackwell optimal policies may not exist. We also provide a sufficient condition for their existence, which encompasses virtually any examples from the literature. We then discuss the connection between average and Blackwell optimality, and we describe several algorithms to compute the optimal average return. Interestingly, our approach leverages the connections between RMDPs and stochastic games. Overall, our paper emphasizes the superior practical properties of distance-based sa-rectangular models over s-rectangular models for average and Blackwell optimality.

math.OC↗

arXiv2024

Risk-averse Total-reward MDPs with ERM and EVaR

Optimizing risk-averse objectives in discounted MDPs is challenging because most models do not admit direct dynamic programming equations and require complex history-dependent policies. In this paper, we show that the risk-averse {\em total reward criterion}, under the Entropic Risk Measure (ERM) and Entropic Value at Risk (EVaR) risk measures, can be optimized by a stationary policy, making it simple to analyze, interpret, and deploy. We propose exponential value iteration, policy iteration, and linear programming to compute optimal policies. Compared with prior work, our results only require the relatively mild condition of transient MDPs and allow for {\em both} positive and negative rewards. Our results indicate that the total reward criterion may be preferable to the discounted criterion in a broad range of risk-averse reinforcement learning domains.

cs.LG↗

arXiv2024

Interpretable Machine Learning for Resource Allocation with Application to Ventilator Triage

Rationing of healthcare resources is a challenging decision that policy makers and providers may be forced to make during a pandemic, natural disaster, or mass casualty event. Well-defined guidelines to triage scarce life-saving resources must be designed to promote transparency, trust, and consistency. To facilitate buy-in and use during high-stress situations, these guidelines need to be interpretable and operational. We propose a novel data-driven model to compute interpretable triage guidelines based on policies for Markov Decision Process that can be represented as simple sequences of decision trees ("tree policies"). In particular, we characterize the properties of optimal tree policies and present an algorithm based on dynamic programming recursions to compute good tree policies. We utilize this methodology to obtain simple, novel triage guidelines for ventilator allocations for COVID-19 patients, based on real patient data from Montefiore hospitals. We also compare the performance of our guidelines to the official New York State guidelines that were developed in 2015 (well before the COVID-19 pandemic). Our empirical study shows that the number of excess deaths associated with ventilator shortages could be reduced significantly using our policy. Our work highlights the limitations of the existing official triage guidelines, which need to be adapted specifically to COVID-19 before being successfully deployed.

cs.LG↗

arXiv2024

Extensive-Form Game Solving via Blackwell Approachability on Treeplexes

In this paper, we introduce the first algorithmic framework for Blackwell approachability on the sequence-form polytope, the class of convex polytopes capturing the strategies of players in extensive-form games (EFGs). This leads to a new class of regret-minimization algorithms that are stepsize-invariant, in the same sense as the Regret Matching and Regret Matching$^+$ algorithms for the simplex. Our modular framework can be combined with any existing regret minimizer over cones to compute a Nash equilibrium in two-player zero-sum EFGs with perfect recall, through the self-play framework. Leveraging predictive online mirror descent, we introduce Predictive Treeplex Blackwell$^+$ (PTB$^+$), and show a $O(1/\sqrt{T})$ convergence rate to Nash equilibrium in self-play. We then show how to stabilize PTB$^+$ with a stepsize, resulting in an algorithm with a state-of-the-art $O(1/T)$ convergence rate. We provide an extensive set of experiments to compare our framework with several algorithmic benchmarks, including CFR$^+$ and its predictive variant, and we highlight interesting connections between practical performance and the stepsize-dependence or stepsize-invariance properties of classical algorithms.

cs.GT↗

arXiv2023

On the convex formulations of robust Markov decision processes

Robust Markov decision processes (MDPs) are used for applications of dynamic optimization in uncertain environments and have been studied extensively. Many of the main properties and algorithms of MDPs, such as value iteration and policy iteration, extend directly to RMDPs. Surprisingly, there is no known analog of the MDP convex optimization formulation for solving RMDPs. This work describes the first convex optimization formulation of RMDPs under the classical sa-rectangularity and s-rectangularity assumptions. By using entropic regularization and exponential change of variables, we derive a convex formulation with a number of variables and constraints polynomial in the number of states and actions, but with large coefficients in the constraints. We further simplify the formulation for RMDPs with polyhedral, ellipsoidal, or entropy-based uncertainty sets, showing that, in these cases, RMDPs can be reformulated as conic programs based on exponential cones, quadratic cones, and non-negative orthants. Our work opens a new research direction for RMDPs and can serve as a first step toward obtaining a tractable convex formulation of RMDPs.

math.OC↗

arXiv2023

The Best Decisions Are Not the Best Advice: Making Adherence-Aware Recommendations

Many high-stake decisions follow an expert-in-loop structure in that a human operator receives recommendations from an algorithm but is the ultimate decision maker. Hence, the algorithm's recommendation may differ from the actual decision implemented in practice. However, most algorithmic recommendations are obtained by solving an optimization problem that assumes recommendations will be perfectly implemented. We propose an adherence-aware optimization framework to capture the dichotomy between the recommended and the implemented policy and analyze the impact of partial adherence on the optimal recommendation. We show that overlooking the partial adherence phenomenon, as is currently being done by most recommendation engines, can lead to arbitrarily severe performance deterioration, compared with both the current human baseline performance and what is expected by the recommendation algorithm. Our framework also provides useful tools to analyze the structure and to compute optimal recommendation policies that are naturally immune against such human deviations, and are guaranteed to improve upon the baseline policy.

cs.HC↗

arXiv2023

Regret Matching+: (In)Stability and Fast Convergence in Games

Regret Matching+ (RM+) and its variants are important algorithms for solving large-scale games. However, a theoretical understanding of their success in practice is still a mystery. Moreover, recent advances on fast convergence in games are limited to no-regret algorithms such as online mirror descent, which satisfy stability. In this paper, we first give counterexamples showing that RM+ and its predictive version can be unstable, which might cause other players to suffer large regret. We then provide two fixes: restarting and chopping off the positive orthant that RM+ works in. We show that these fixes are sufficient to get $O(T^{1/4})$ individual regret and $O(1)$ social regret in normal-form games via RM+ with predictions. We also apply our stabilizing techniques to clairvoyant updates in the uncoupled learning setting for RM+ and prove desirable results akin to recent works for Clairvoyant online mirror descent. Our experiments show the advantages of our algorithms over vanilla RM+-based algorithms in matrix and extensive-form games.

cs.GT↗

arXiv2023

Reducing Blackwell and Average Optimality to Discounted MDPs via the Blackwell Discount Factor

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell optimality. Many existing approaches to computing average-optimal policies solve for discounted optimal policies with a discount factor close to $1$, but they only work under strong or hard-to-verify assumptions such as ergodicity or weakly communicating MDPs. In this paper, we show that when the discount factor is larger than the Blackwell discount factor $γ_{\mathrm{bw}}$, all discounted optimal policies become Blackwell- and average-optimal, and we derive a general upper bound on $γ_{\mathrm{bw}}$. The upper bound on $γ_{\mathrm{bw}}$ provides the first reduction from average and Blackwell optimality to discounted optimality, without any assumptions, and new polynomial-time algorithms for average- and Blackwell-optimal policies. Our work brings new ideas from the study of polynomials and algebraic numbers to the analysis of MDPs. Our results also apply to robust MDPs, enabling the first algorithms to compute robust Blackwell-optimal policies.

cs.LG↗

arXiv2022

Solving optimization problems with Blackwell approachability

We introduce the Conic Blackwell Algorithm$^+$ (CBA$^+$) regret minimizer, a new parameter- and scale-free regret minimizer for general convex sets. CBA$^+$ is based on Blackwell approachability and attains $O(\sqrt{T})$ regret. We show how to efficiently instantiate CBA$^+$ for many decision sets of interest, including the simplex, $\ell_{p}$ norm balls, and ellipsoidal confidence regions in the simplex. Based on CBA$^+$, we introduce SP-CBA$^+$, a new parameter-free algorithm for solving convex-concave saddle-point problems, which achieves a $O(1/\sqrt{T})$ ergodic rate of convergence. In our simulations, we demonstrate the wide applicability of SP-CBA$^+$ on several standard saddle-point problems, including matrix games, extensive-form games, distributionally robust logistic regression, and Markov decision processes. In each setting, SP-CBA$^+$ achieves state-of-the-art numerical performance, and outperforms classical methods, without the need for any choice of step sizes or other algorithmic parameters.

math.OC↗