Chasing Convex Functions with Long-term Constraints

We introduce and study a family of online metric problems with long-term constraints. In these problems, an online player makes decisions $\mathbf{x}_t$ in a metric space $(X,d)$ to simultaneously minimize their hitting cost $f_t(\mathbf{x}_t)$ and switching cost as determined by the metric. Over the time horizon $T$, the player must satisfy a long-term demand constraint $\sum_{t} c(\mathbf{x}_t) \geq 1$, where $c(\mathbf{x}_t)$ denotes the fraction of demand satisfied at time $t$. Such problems can find a wide array of applications to online resource allocation in sustainable energy and computing systems. We devise optimal competitive and learning-augmented algorithms for specific instantiations of these problems, and further show that our proposed algorithms perform well in numerical experiments.

Two-Timescale Q-Learning with Function Approximation in Zero-Sum Stochastic Games

We consider two-player zero-sum stochastic games and propose a two-timescale $Q$-learning algorithm with function approximation that is payoff-based, convergent, rational, and symmetric between the two players. In two-timescale $Q$-learning, the fast-timescale iterates are updated in spirit to the stochastic gradient descent and the slow-timescale iterates (which we use to compute the policies) are updated by taking a convex combination between its previous iterate and the latest fast-timescale iterate. Introducing the slow timescale as well as its update equation marks as our main algorithmic novelty. In the special case of linear function approximation, we establish, to the best of our knowledge, the first last-iterate finite-sample bound for payoff-based independent learning dynamics of these types. The result implies a polynomial sample complexity to find a Nash equilibrium in such stochastic games. To establish the results, we model our proposed algorithm as a two-timescale stochastic approximation and derive the finite-sample bound through a Lyapunov-based approach. The key novelty lies in constructing a valid Lyapunov function to capture the evolution of the slow-timescale iterates. Specifically, through a change of variable, we show that the update equation of the slow-timescale iterates resembles the classical smoothed best-response dynamics, where the regularized Nash gap serves as a valid Lyapunov function. This insight enables us to construct a valid Lyapunov function via a generalized variant of the Moreau envelope of the regularized Nash gap. The construction of our Lyapunov function might be of broad independent interest in studying the behavior of stochastic approximation algorithms.

Anytime-Competitive Reinforcement Learning with Policy Prior

This paper studies the problem of Anytime-Competitive Markov Decision Process (A-CMDP). Existing works on Constrained Markov Decision Processes (CMDPs) aim to optimize the expected reward while constraining the expected cost over random dynamics, but the cost in a specific episode can still be unsatisfactorily high. In contrast, the goal of A-CMDP is to optimize the expected reward while guaranteeing a bounded cost in each round of any episode against a policy prior. We propose a new algorithm, called Anytime-Competitive Reinforcement Learning (ACRL), which provably guarantees the anytime cost constraints. The regret analysis shows the policy asymptotically matches the optimal reward achievable under the anytime competitive constraints. Experiments on the application of carbon-intelligent computing verify the reward performance and cost constraint guarantee of ACRL.

Adversarial Attacks on Cooperative Multi-agent Bandits

Cooperative multi-agent multi-armed bandits (CMA2B) consider the collaborative efforts of multiple agents in a shared multi-armed bandit game. We study latent vulnerabilities exposed by this collaboration and consider adversarial attacks on a few agents with the goal of influencing the decisions of the rest. More specifically, we study adversarial attacks on CMA2B in both homogeneous settings, where agents operate with the same arm set, and heterogeneous settings, where agents have distinct arm sets. In the homogeneous setting, we propose attack strategies that, by targeting just one agent, convince all agents to select a particular target arm $T-o(T)$ times while incurring $o(T)$ attack costs in $T$ rounds. In the heterogeneous setting, we prove that a target arm attack requires linear attack costs and propose attack strategies that can force a maximum number of agents to suffer linear regrets while incurring sublinear costs and only manipulating the observations of a few target agents. Numerical experiments validate the effectiveness of our proposed attack strategies.

Best of Both Worlds: Stochastic and Adversarial Convex Function Chasing

Convex function chasing (CFC) is an online optimization problem in which during each round $t$, a player plays an action $x_t$ in response to a hitting cost $f_t(x_t)$ and an additional cost of $c(x_t,x_{t-1})$ for switching actions. We study the CFC problem in stochastic and adversarial environments, giving algorithms that achieve performance guarantees simultaneously in both settings. Specifically, we consider the squared $\ell_2$-norm switching costs and a broad class of quadratic hitting costs for which the sequence of minimizers either forms a martingale or is chosen adversarially. This is the first work that studies the CFC problem using a stochastic framework. We provide a characterization of the optimal stochastic online algorithm and, drawing a comparison between the stochastic and adversarial scenarios, we demonstrate that the adversarial-optimal algorithm exhibits suboptimal performance in the stochastic context. Motivated by this, we provide a best-of-both-worlds algorithm that obtains robust adversarial performance while simultaneously achieving near-optimal stochastic performance.

Online Conversion with Switching Costs: Robust and Learning-Augmented Algorithms

We introduce and study online conversion with switching costs, a family of online problems that capture emerging problems at the intersection of energy and sustainability. In this problem, an online player attempts to purchase (alternatively, sell) fractional shares of an asset during a fixed time horizon with length $T$. At each time step, a cost function (alternatively, price function) is revealed, and the player must irrevocably decide an amount of asset to convert. The player also incurs a switching cost whenever their decision changes in consecutive time steps, i.e., when they increase or decrease their purchasing amount. We introduce competitive (robust) threshold-based algorithms for both the minimization and maximization variants of this problem, and show they are optimal among deterministic online algorithms. We then propose learning-augmented algorithms that take advantage of untrusted black-box advice (such as predictions from a machine learning model) to achieve significantly better average-case performance without sacrificing worst-case competitive guarantees. Finally, we empirically evaluate our proposed algorithms using a carbon-aware EV charging case study, showing that our algorithms substantially improve on baseline methods for this problem.

Robust Learning for Smoothed Online Convex Optimization with Feedback Delay

We study a challenging form of Smoothed Online Convex Optimization, a.k.a. SOCO, including multi-step nonlinear switching costs and feedback delay. We propose a novel machine learning (ML) augmented online algorithm, Robustness-Constrained Learning (RCL), which combines untrusted ML predictions with a trusted expert online algorithm via constrained projection to robustify the ML prediction. Specifically,we prove that RCL is able to guarantee$(1+\lambda)$-competitiveness against any given expert for any$\lambda>0$, while also explicitly training the ML model in a robustification-aware manner to improve the average-case performance. Importantly,RCL is the first ML-augmented algorithm with a provable robustness guarantee in the case of multi-step switching cost and feedback delay.We demonstrate the improvement of RCL in both robustness and average performance using battery management for electrifying transportationas a case study.

Online Algorithms with Uncertainty-Quantified Predictions

Online algorithms with predictions have become a trending topic in the field of beyond worst-case analysis of algorithms. These algorithms incorporate predictions about the future to obtain performance guarantees that are of high quality when the predictions are good, while still maintaining bounded worst-case guarantees when predictions are arbitrarily poor. In general, the algorithm is assumed to be unaware of the prediction's quality. However, recent developments in the machine learning literature have studied techniques for providing uncertainty quantification on machine-learned predictions, which describes how certain a model is about its quality. This paper examines the question of how to optimally utilize uncertainty-quantified predictions in the design of online algorithms. In particular, we consider predictions augmented with uncertainty quantification describing the likelihood of the ground truth falling in a certain range, designing online algorithms with these probabilistic predictions for two classic online problems: ski rental and online search. In each case, we demonstrate that non-trivial modifications to algorithm design are needed to fully leverage the probabilistic predictions. Moreover, we consider how to utilize more general forms of uncertainty quantification, proposing a framework based on online learning that learns to exploit uncertainty quantification to make optimal decisions in multi-instance settings.

Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis

Set-membership estimation is commonly used in adaptive/learning-based control algorithms that require robustness over the model uncertainty sets, e.g., online robustly stabilizing control and robust adaptive model predictive control. Despite having broad applications, non-asymptotic estimation error bounds in the stochastic setting are limited. This paper provides such a non-asymptotic bound on the diameter of the uncertainty sets generated by set membership estimation on linear dynamical systems under bounded, i.i.d. disturbances. Further, this result is applied to robust adaptive model predictive control with uncertainty sets updated by set membership. We numerically demonstrate the performance of the robust adaptive controller, which rapidly approaches the performance of the offline optimal model predictive controller, in comparison with the control design based on least square estimation's confidence regions.