arXiv subjects

Nicholas Bambos

Publications and source records attributed to Nicholas Bambos.

At least 19 recordsLinked to original sources

arXiv2026

Policy Gradient Methods for Non-Markovian Reinforcement Learning

We study policy gradient methods for reinforcement learning in non-Markovian decision processes (NMDPs), where observations and rewards depend on the entire interaction history. To handle this dependence, the agent maintains an internal state that is recursively updated to provide a compact summary of past observations and actions. In contrast to approaches that treat the agent state dynamics as fixed or learn it via predictive objectives, we propose a reward-centric formulation that jointly optimizes the agent state dynamics and the control policy to maximize the expected cumulative reward. To this end, we consider a class of Agent State-Markov (ASM) policies, comprising an agent state dynamics and a control policy that maps the agent state to actions. We establish a novel policy gradient theorem for ASM policies, extending the classical policy gradient results from the Markovian setting to episodic and infinite-horizon discounted NMDPs. Building on this gradient expression, we propose the Agent State-Markov Policy Gradient (ASMPG) algorithm, which leverages the recursive structure of the agent state dynamics for efficient optimization. We establish finite-time and almost sure convergence guarantees, and empirically demonstrate that, on a range of non-Markovian tasks, ASMPG outperforms baselines that learn state representations via predictive objectives.

cs.LG↗

arXiv2026

Last-Iterate Guarantees for Learning in Co-coercive Games

We establish finite-time last-iterate guarantees for vanilla stochastic gradient descent in co-coercive games under noisy feedback. This is a broad class of games that is more general than strongly monotone games, allows for multiple Nash equilibria, and includes examples such as quadratic games with negative semidefinite interaction matrices and potential games with smooth concave potentials. Prior work in this setting has relied on relative noise models, where the noise vanishes as iterates approach equilibrium, an assumption that is often unrealistic in practice. We work instead under a substantially more general noise model in which the second moment of the noise is allowed to scale affinely with the squared norm of the iterates, an assumption natural in learning with unbounded action spaces. Under this model, we prove a last-iterate bound of order $O(\log(t)/t^{1/3})$, the first such bound for co-coercive games under non-vanishing noise. We additionally establish almost sure convergence of the iterates to the set of Nash equilibria and derive time-average convergence guarantees.

cs.GT↗

arXiv2026

Heavy-Tailed and Long-Range Dependent Noise in Stochastic Approximation: A Finite-Time Analysis

Stochastic approximation (SA) is a fundamental iterative framework with broad applications in reinforcement learning and optimization. Classical analyses typically rely on martingale difference or Markov noise with bounded second moments, but many practical settings, including finance and communications, frequently encounter heavy-tailed and long-range dependent (LRD) noise. In this work, we study SA for finding the root of a strongly monotone operator under these non-classical noise models. We establish the first finite-time moment bounds in both settings, providing explicit convergence rates that quantify the impact of heavy tails and temporal dependence. Our analysis employs a noise-averaging argument that regularizes the impact of noise without modifying the iteration. Finally, we apply our general framework to stochastic gradient descent (SGD) and gradient play, and corroborate our finite-time analysis through numerical experiments.

cs.LG↗

arXiv2026

High-Probability Bounds for SGD under the Polyak-Lojasiewicz Condition with Markovian Noise

We present the first uniform-in-time high-probability bound for SGD under the PL condition, where the gradient noise contains both Markovian and martingale difference components. This significantly broadens the scope of finite-time guarantees, as the PL condition arises in many machine learning and deep learning models while Markovian noise naturally arises in decentralized optimization and online system identification problems. We further allow the magnitude of noise to grow with the function value, enabling the analysis of many practical sampling strategies. In addition to the high-probability guarantee, we establish a matching $1/k$ decay rate for the expected suboptimality. Our proof technique relies on the Poisson equation to handle the Markovian noise and a probabilistic induction argument to address the lack of almost-sure bounds on the objective. Finally, we demonstrate the applicability of our framework by analyzing three practical optimization problems: token-based decentralized linear regression, supervised learning with subsampling for privacy amplification, and online system identification.

cs.LG↗

arXiv2026

Regret and Sample Complexity of Online Q-Learning via Concentration of Stochastic Approximation with Time-Inhomogeneous Markov Chains

We present the first regret bound for classical online Q-learning in infinite-horizon discounted Markov decision processes (MDPs), without relying on optimism or bonus terms. We first analyze Boltzmann Q-learning with decaying temperature and show that its regret depends critically on the suboptimality gap of the MDP: for sufficiently large gaps, the regret is sublinear, while for small gaps it deteriorates and can approach linear growth. To address this limitation, we study a Smoothed $\epsilon_n$-Greedy exploration scheme that combines $\epsilon_n$-greedy and Boltzmann exploration, for which we prove a gap-robust regret bound of near-$\tilde{O}(N^{9/10})$. We also obtain sample complexity guarantees, with both regret and sample complexity bounds holding with high probability. To analyze these algorithms, we develop a high-probability concentration bound for contractive Markovian stochastic approximation with iterate- and time-dependent transition dynamics. This bound may be of independent interest as the contraction factor in our framework is allowed to converge to one asymptotically.

cs.LG↗

arXiv2025

Multi-agent learning under uncertainty: Recurrence vs. concentration

In this paper, we examine the convergence landscape of multi-agent learning under uncertainty. Specifically, we analyze two stochastic models of regularized learning in continuous games -- one in continuous and one in discrete time with the aim of characterizing the long-run behavior of the induced sequence of play. In stark contrast to deterministic, full-information models of learning (or models with a vanishing learning rate), we show that the resulting dynamics do not converge in general. In lieu of this, we ask instead which actions are played more often in the long run, and by how much. We show that, in strongly monotone games, the dynamics of regularized learning may wander away from equilibrium infinitely often, but they always return to its vicinity in finite time (which we estimate), and their long-run distribution is sharply concentrated around a neighborhood thereof. We quantify the degree of this concentration, and we show that these favorable properties may all break down if the underlying game is not strongly monotone -- underscoring in this way the limits of regularized learning in the presence of persistent randomness and uncertainty.

cs.GT↗

arXiv2025

Robust equilibria in continuous games: From strategic to dynamic robustness

In this paper, we examine the robustness of Nash equilibria in continuous games, under both strategic and dynamic uncertainty. Starting with the former, we introduce the notion of a robust equilibrium as those equilibria that remain invariant to small -- but otherwise arbitrary -- perturbations to the game's payoff structure, and we provide a crisp geometric characterization thereof. Subsequently, we turn to the question of dynamic robustness, and we examine which equilibria may arise as stable limit points of the dynamics of "follow the regularized leader" (FTRL) in the presence of randomness and uncertainty. Despite their very distinct origins, we establish a structural correspondence between these two notions of robustness: strategic robustness implies dynamic robustness, and, conversely, the requirement of strategic robustness cannot be relaxed if dynamic robustness is to be maintained. Finally, we examine the rate of convergence to robust equilibria as a function of the underlying regularizer, and we show that entropically regularized learning converges at a geometric rate in games with affinely constrained action spaces.

cs.GT↗

arXiv2025

Choose Your Battles: Distributed Learning Over Multiple Tug of War Games

Consider $N$ players and $K$ games taking place simultaneously. Each of these games is modeled as a Tug-of-War (ToW) game where increasing the action of one player decreases the reward for all other players. Each player participates in only one game at any given time. At each time step, a player decides the game in which they wish to participate in and the action they take in that game. Their reward depends on the actions of all players that are in the same game. This system of $K$ games is termed a 'Meta Tug-of-War' (Meta-ToW) game. These games can model scenarios such as power control, distributed task allocation, and activation in sensor networks. We propose the Meta Tug-of-Peace algorithm, a distributed algorithm where the action updates are done using a simple stochastic approximation algorithm, and the decision to switch games is made using an infrequent 1-bit communication between the players. We prove that in Meta-ToW games, our algorithm converges to an equilibrium that satisfies a target Quality of Service reward vector for the players. We then demonstrate the efficacy of our algorithm through simulations for the scenarios mentioned above.

cs.GT↗

arXiv2025

Finite-Time Bounds for Two-Time-Scale Stochastic Approximation with Arbitrary Norm Contractions and Markovian Noise

Two-time-scale Stochastic Approximation (SA) is an iterative algorithm with applications in reinforcement learning and optimization. Prior finite time analysis of such algorithms has focused on fixed point iterations with mappings contractive under Euclidean norm. Motivated by applications in reinforcement learning, we give the first mean square bound on non linear two-time-scale SA where the iterations have arbitrary norm contractive mappings and Markovian noise. We show that the mean square error decays at a rate of $O(1/n^{2/3})$ in the general case, and at a rate of $O(1/n)$ in a special case where the slower timescale is noiseless. Our analysis uses the generalized Moreau envelope to handle the arbitrary norm contractions and solutions of Poisson equation to deal with the Markovian noise. By analyzing the SSP Q-Learning algorithm, we give the first $O(1/n)$ bound for an algorithm for asynchronous control of MDPs under the average reward criterion. We also obtain a rate of $O(1/n)$ for Q-Learning with Polyak-averaging and provide an algorithm for learning Generalized Nash Equilibrium (GNE) for strongly monotone games which converges at a rate of $O(1/n^{2/3})$.

cs.LG↗

arXiv2025

Optimal Control for Remote Patient Monitoring with Multidimensional Health States

Selecting the right monitoring level in Remote Patient Monitoring (RPM) systems for e-healthcare is crucial for balancing patient outcomes, various resources, and patient's quality of life. A prior work has used one-dimensional health representations, but patient health is inherently multidimensional and typically consists of many measurable physiological factors. In this paper, we introduce a multidimensional health state model within the RPM framework and use dynamic programming to study optimal monitoring strategies. Our analysis reveals that the optimal control is characterized by switching curves (for two-dimensional health states) or switching hyper-surfaces (in general): patients switch to intensive monitoring when health measurements cross a specific multidimensional surface. We further study how the optimal switching curve varies for different medical conditions and model parameters. This finding of the optimal control structure provides actionable insights for clinicians and aids in resource planning. The tunable modeling framework enhances the applicability and effectiveness of RPM services across various medical conditions.

eess.SY↗

arXiv2024

Accelerated regularized learning in finite N-person games

Motivated by the success of Nesterov's accelerated gradient algorithm for convex minimization problems, we examine whether it is possible to achieve similar performance gains in the context of online learning in games. To that end, we introduce a family of accelerated learning methods, which we call "follow the accelerated leader" (FTXL), and which incorporates the use of momentum within the general framework of regularized learning - and, in particular, the exponential/multiplicative weights algorithm and its variants. Drawing inspiration and techniques from the continuous-time analysis of Nesterov's algorithm, we show that FTXL converges locally to strict Nash equilibria at a superlinear rate, achieving in this way an exponential speed-up over vanilla regularized learning methods (which, by comparison, converge to strict equilibria at a geometric, linear rate). Importantly, FTXL maintains its superlinear convergence rate in a broad range of feedback structures, from deterministic, full information models to stochastic, realization-based ones, and even when run with bandit, payoff-based information, where players are only able to observe their individual realized payoffs.

cs.GT↗

arXiv2024

Learning to Control Unknown Strongly Monotone Games

Consider a strongly monotone game where the players' utility functions include a reward function and a linear term for each dimension, with coefficients that are controlled by the manager. Gradient play converges to a unique Nash equilibrium (NE) that does not optimize the global objective. The global performance at NE can be improved by imposing linear constraints on the NE, also known as a generalized Nash equilibrium (GNE). We therefore want the manager to control the coefficients such that they impose the desired constraint on the NE. However, this requires knowing the players' rewards and action sets. Obtaining this game information is infeasible in a large-scale network and violates user privacy. To overcome this, we propose a simple algorithm that learns to shift the NE to meet the linear constraints by adjusting the controlled coefficients online. Our algorithm only requires the linear constraints violation as feedback and does not need to know the reward functions or the action sets. We prove that our algorithm converges with probability 1 to the set of GNE given by coupled linear constraints. We then prove an L2 convergence rate of near-$O(t^{-1/4})$.

cs.MA↗

arXiv2024

Tiered Service Architecture for Remote Patient Monitoring

We develop a remote patient monitoring (RPM) service architecture, which has two tiers of monitoring: ordinary and intensive. The patient's health state improves or worsens in each time period according to certain probabilities, which depend on the monitoring tier. The patient incurs a "loss of quality of life" cost or an "invasiveness" cost, which is higher under intensive monitoring than under ordinary. On the other hand, their health improves faster under intensive monitoring than under ordinary. In each period, the service decides which monitoring tier to use, based on the health of the patient. We investigate the optimal policy for making that choice by formulating the problem using dynamic programming. We first provide analytic conditions for selecting ordinary vs intensive monitoring in the asymptotic regime where the number of health states is large. In the general case, we investigate the optimal policy numerically. We observe a threshold behavior, that is, when the patient's health drops below a certain threshold the service switches them to intensive monitoring, while ordinary monitoring is used during adequately good health states of the patient. The modeling and analysis provides a general framework for managing RPM services for various health conditions with medically/clinically defined system parameters.

eess.SY↗

arXiv2023

Payoff-based learning with matrix multiplicative weights in quantum games

In this paper, we study the problem of learning in quantum games - and other classes of semidefinite games - with scalar, payoff-based feedback. For concreteness, we focus on the widely used matrix multiplicative weights (MMW) algorithm and, instead of requiring players to have full knowledge of the game (and/or each other's chosen states), we introduce a suite of minimal-information matrix multiplicative weights (3MW) methods tailored to different information frameworks. The main difficulty to attaining convergence in this setting is that, in contrast to classical finite games, quantum games have an infinite continuum of pure states (the quantum equivalent of pure strategies), so standard importance-weighting techniques for estimating payoff vectors cannot be employed. Instead, we borrow ideas from bandit convex optimization and we design a zeroth-order gradient sampler adapted to the semidefinite geometry of the problem at hand. As a first result, we show that the 3MW method with deterministic payoff feedback retains the $\mathcal{O}(1/\sqrt{T})$ convergence rate of the vanilla, full information MMW algorithm in quantum min-max games, even though the players only observe a single scalar. Subsequently, we relax the algorithm's information requirements even further and we provide a 3MW method that only requires players to observe a random realization of their payoff observable, and converges to equilibrium at an $\mathcal{O}(T^{-1/4})$ rate. Finally, going beyond zero-sum games, we show that a regularized variant of the proposed 3MW method guarantees local convergence with high probability to all equilibria that satisfy a certain first-order stability condition.

cs.GT↗

arXiv2023

Equilibrium Bandits: Learning Optimal Equilibria of Unknown Dynamics

Consider a decision-maker that can pick one out of $K$ actions to control an unknown system, for $T$ turns. The actions are interpreted as different configurations or policies. Holding the same action fixed, the system asymptotically converges to a unique equilibrium, as a function of this action. The dynamics of the system are unknown to the decision-maker, which can only observe a noisy reward at the end of every turn. The decision-maker wants to maximize its accumulated reward over the $T$ turns. Learning what equilibria are better results in higher rewards, but waiting for the system to converge to equilibrium costs valuable time. Existing bandit algorithms, either stochastic or adversarial, achieve linear (trivial) regret for this problem. We present a novel algorithm, termed Upper Equilibrium Concentration Bound (UECB), that knows to switch an action quickly if it is not worth it to wait until the equilibrium is reached. This is enabled by employing convergence bounds to determine how far the system is from equilibrium. We prove that UECB achieves a regret of $\mathcal{O}(\log(T)+\tau_c\log(\tau_c)+\tau_c\log\log(T))$ for this equilibrium bandit problem where $\tau_c$ is the worst case approximate convergence time to equilibrium. We then show that both epidemic control and game control are special cases of equilibrium bandits, where $\tau_c\log \tau_c$ typically dominates the regret. We then test UECB numerically for both of these applications.

cs.LG↗

arXiv2023

Learning in quantum games

In this paper, we introduce a class of learning dynamics for general quantum games, that we call "follow the quantum regularized leader" (FTQL), in reference to the classical "follow the regularized leader" (FTRL) template for learning in finite games. We show that the induced quantum state dynamics decompose into (i) a classical, commutative component which governs the dynamics of the system's eigenvalues in a way analogous to the evolution of mixed strategies under FTRL; and (ii) a non-commutative component for the system's eigenvectors which has no classical counterpart. Despite the complications that this non-classical component entails, we find that the FTQL dynamics incur no more than constant regret in all quantum games. Moreover, adjusting classical notions of stability to account for the nonlinear geometry of the state space of quantum games, we show that only pure quantum equilibria can be stable and attracting under FTQL while, as a partial converse, pure equilibria that satisfy a certain "variational stability" condition are always attracting. Finally, we show that the FTQL dynamics are Poincar\'e recurrent in quantum min-max games, extending in this way a very recent result for the quantum replicator dynamics.

cs.GT↗

arXiv2022

Learning in Games with Quantized Payoff Observations

This paper investigates the impact of feedback quantization on multi-agent learning. In particular, we analyze the equilibrium convergence properties of the well-known "follow the regularized leader" (FTRL) class of algorithms when players can only observe a quantized (and possibly noisy) version of their payoffs. In this information-constrained setting, we show that coarser quantization triggers a qualitative shift in the convergence behavior of FTRL schemes. Specifically, if the quantization error lies below a threshold value (which depends only on the underlying game and not on the level of uncertainty entering the process or the specific FTRL variant under study), then (i) FTRL is attracted to the game's strict Nash equilibria with arbitrarily high probability; and (ii) the algorithm's asymptotic rate of convergence remains the same as in the non-quantized case. Otherwise, for larger quantization levels, these convergence properties are lost altogether: players may fail to learn anything beyond their initial state, even with full information on their payoff vectors. This is in contrast to the impact of quantization in continuous optimization problems, where the quality of the obtained solution degrades smoothly with the quantization level.

cs.GT↗

arXiv2022

No Weighted-Regret Learning in Adversarial Bandits with Delays

Consider a scenario where a player chooses an action in each round $t$ out of $T$ rounds and observes the incurred cost after a delay of $d_{t}$ rounds. The cost functions and the delay sequence are chosen by an adversary. We show that in a non-cooperative game, the expected weighted ergodic distribution of play converges to the set of coarse correlated equilibria if players use algorithms that have "no weighted-regret" in the above scenario, even if they have linear regret due to too large delays. For a two-player zero-sum game, we show that no weighted-regret is sufficient for the weighted ergodic average of play to converge to the set of Nash equilibria. We prove that the FKM algorithm with $n$ dimensions achieves an expected regret of $O\left(nT^{\frac{3}{4}}+\sqrt{n}T^{\frac{1}{3}}D^{\frac{1}{3}}\right)$ and the EXP3 algorithm with $K$ arms achieves an expected regret of $O\left(\sqrt{\log K\left(KT+D\right)}\right)$ even when $D=\sum_{t=1}^{T}d_{t}$ and $T$ are unknown. These bounds use a novel doubling trick that, under mild assumptions, provably retains the regret bound for when $D$ and $T$ are known. Using these bounds, we show that FKM and EXP3 have no weighted-regret even for $d_{t}=O\left(t\log t\right)$. Therefore, algorithms with no weighted-regret can be used to approximate a CCE of a finite or convex unknown game that can only be simulated with bandit feedback, even if the simulation involves significant delays.

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