Non-Stationary Bandits

Open systems, where agents arrive and depart over time, are growing in digital platforms. Existing work on online learning in such systems, however, often relies on structural assumptions that may be restrictive in practice. In particular, heterogeneous rewards and general agent patterns create three new challenges: arrivals induce endogenous non-stationarity, agent patterns govern the rate of information accumulation, and uncertainty brought by new agents accumulates over time and scales regret. To this end, we formulate a learning problem for open systems that accommodates both forms of generality. We introduce new concepts to capture the inherent complexities: the pre-training degree of new agents quantifies how much information an agent carries upon entry, stability measures the impact of population changes on the global optimum, and global dynamic regret compares the cumulative expected reward with that of the varying optimal arms. We also study a manifold representation of agents and introduce a dynamic population embedding that directly encodes the population functional for each arm. We develop two UCB-based methods. The first combines entry information with online observations through agent-wise confidence bounds and aggregates the estimates and uncertainties for arm selection. The second pools agent-level observations through kernel regression on the manifold and constructs UCBs by projecting the confidence sets onto the current population embedding. Entry uncertainty enters the regret linearly through the pre-training error, while in stable regimes, regret is governed by the time needed to identify the stable optimal arm. With the manifold as a handle, regret is governed by the information gained from agent representations. These bounds all depend on population dynamics, but in different ways.

In many application domains (e.g., recommender systems, intelligent tutoring systems), the rewards associated to the actions tend to decrease over time. This decay is either caused by the actions executed in the past (e.g., a user may get bored when songs of the same genre are recommended over and over) or by an external factor (e.g., content becomes outdated). These two situations can be modeled as specific instances of the rested and restless bandit settings, where arms are rotting (i.e., their value decrease over time). These problems were thought to be significantly different, since Levine et al. (2017) showed that state-of-the-art algorithms for restless bandit perform poorly in the rested rotting setting. In this paper, we introduce a novel algorithm, Rotting Adaptive Window UCB (RAW-UCB), that achieves near-optimal regret in both rotting rested and restless bandit, without any prior knowledge of the setting (rested or restless) and the type of non-stationarity (e.g., piece-wise constant, bounded variation). This is in striking contrast with previous negative results showing that no algorithm can achieve similar results as soon as rewards are allowed to increase. We confirm our theoretical findings on a number of synthetic and dataset-based experiments.

We study the piecewise-stationary restless multi-armed bandit (PS-RMAB) problem, where each arm evolves as a Markov chain but mean rewards may change across unknown segments. To address the resulting exploration--detection delay trade-off, we propose a modular framework that integrates arbitrary RMAB base algorithms with change detection and a novel diminishing exploration mechanism. This design enables flexible plug-and-play use of existing solvers and detectors, while efficiently adapting to mean changes without prior knowledge of their number. To evaluate performance, we introduce a refined regret notion that measures the excess regret due to exploration and detection, benchmarked against an oracle that restarts the base algorithm at the true change points. Under this metric, we prove a regret bound of $\tilde{O}(\sqrt{LMKT})$, where $L$ denotes the maximum mixing time of the Markov chains across all arms and segments, $M$ the number of segments, $K$ the number of arms, and $T$ the horizon. Simulations confirm that our framework achieves regret close to that of the segment oracle and consistently outperforms base solvers that do not incorporate any mechanism to handle environmental changes.

Multi-model LLM serving operates in a non-stationary, noisy environment: providers revise pricing, model quality can shift or regress without notice, and new models arrive regularly. More than a dozen recent methods have proposed learned routers to navigate the resulting quality--cost tradeoff across portfolios spanning a $\sim$530$\times$ cost range. Despite this activity, two gaps in the current solution space limit routing effectiveness under these conditions: no existing router enforces a dollar-denominated cost ceiling in closed loop over an open-ended request stream, and none provides principled online adaptation to post-deployment shifts in pricing or model quality. We present ParetoBandit, an open-source adaptive router built on cost-aware contextual bandits that addresses both gaps. Its core contributions are: (1) an online primal--dual budget pacer that enforces a per-request cost ceiling without a known horizon, and (2) geometric forgetting on sufficient statistics that gives the bandit bounded memory for tracking quality and cost shifts. A hot-swap model registry further supports runtime model changes with budget-controlled exploration. On 1,824 benchmark prompts with a three-model portfolio, the router maintains budget compliance within 0.4%, adapts to price and quality shifts with up to +0.071 quality lift, and integrates a cold-started model within $\sim$142 steps.

Motivated by the principle of satisficing in decision-making, we study satisficing regret guarantees for nonstationary $K$-armed bandits. We show that in the general realizable, piecewise-stationary setting with $L$ stationary segments, the optimal regret is $Θ(L\log T)$ as long as $L\geq 2$. This stands in sharp contrast to the case of $L=1$ (i.e., the stationary setting), where a $T$-independent $Θ(1)$ satisficing regret is achievable under realizability. In other words, the optimal regret has to scale with $T$ even if just a little nonstationarity presents. A key ingredient in our analysis is a novel Fano-based framework tailored to nonstationary bandits via a post-interaction reference construction. This framework strictly extends the classical Fano method for passive estimation as well as recent interactive Fano techniques for stationary bandits. As a complement, we also discuss a special regime in which constant satisficing regret is again possible.

Contextual bandits are incredibly useful in many practical problems. We go one step further by devising a more realistic problem that combines: (1) contextual bandits with dense arm features, (2) non-linear reward functions, and (3) a generalization of correlated bandits where reward distributions change over time but the degree of correlation maintains. This formulation lends itself to a wider set of applications such as recommendation tasks. To solve this problem, we introduce conditionally coupled contextual C3 Thompson sampling for Bernoulli bandits. It combines an improved Nadaraya-Watson estimator on an embedding space with Thompson sampling that allows online learning without retraining. Empirical results show that C3 outperforms the next best algorithm by 5.7% lower average cumulative regret on four OpenML tabular datasets as well as demonstrating a 12.4% click lift on Microsoft News Dataset (MIND) compared to other algorithms.

We study the fixed-budget best-arm identification (BAI) problem in non-stationary linear bandits. Concretely, given a fixed time budget $T\in \mathbb{N}$, finite arm set $\mathcal{X} \subset \mathbb{R}^d$, and a potentially adversarial sequence of unknown parameters $\lbrace θ_t\rbrace_{t=1}^{T}$ (hence non-stationary), a learner aims to identify the arm with the largest cumulative reward $x_* = \arg\max_{x \in \mathcal{X}} x^\top\sum_{t=1}^T θ_t$ with high probability. In this setting, it is well-known that i.i.d. sampling arms from the G-optimal design yields a minimax-optimal error probability of $\exp\left(-Θ\left(T / H_{G}\right)\right)$, where $H_{G}$ scales proportionally with the dimension $d$. However, this notion of complexity is overly pessimistic, as it is derived from a lower bound in which the arm set consists only of the standard basis vectors, thus masking any potential advantages arising from arm sets with richer geometric structure. To address this, we establish an arm-set-dependent lower bound that, in contrast, holds for any arm set. Motivated by the ideas underlying our lower bound, we propose the Adjacent-optimal design, a specialization of the well-known $\mathcal{X}\mathcal{Y}$-optimal design, and develop the $\textsf{Adjacent-BAI}$ algorithm. We prove that the error probability of $\textsf{Adjacent-BAI}$ matches our lower bound up to constants, verifying the tightness of our lower bound, and establishing the arm-set-dependent complexity of this setting.

We consider stochastic non-stationary linear bandits where the linear parameter connecting contexts to the reward changes over time. Existing algorithms in this setting localize the policy by gradually discarding or down-weighting past data, effectively shrinking the time horizon over which learning can occur. However, in many settings historical data may still carry partial information about the reward model. We propose to leverage such data while adapting to changes, by assuming the reward model decomposes into stationary and non-stationary components. Based on this assumption, we introduce ISD-linUCB, an algorithm that uses past data to learn invariances in the reward model and subsequently exploits them to improve online performance. We show both theoretically and empirically that leveraging invariance reduces the problem dimensionality, yielding significant regret improvements in fast-changing environments when sufficient historical data is available.

We study the constrained variant of the multi-armed bandit (MAB) problem, in which the learner aims not only at minimizing the total loss incurred during the learning dynamic, but also at controlling the violation of multiple unknown constraints, under both full and bandit feedback. We consider a non-stationary environment that subsumes both stochastic and adversarial models and where, at each round, both losses and constraints are drawn from distributions that may change arbitrarily over time. In such a setting, it is provably not possible to guarantee both sublinear regret and sublinear violation. Accordingly, prior work has mainly focused either on settings with stochastic constraints or on relaxing the benchmark with fully adversarial constraints (e.g., via competitive ratios with respect to the optimum). We provide the first algorithms that achieve optimal rates of regret and positive constraint violation when the constraints are stochastic while the losses may vary arbitrarily, and that simultaneously yield guarantees that degrade smoothly with the degree of adversariality of the constraints. Specifically, under full feedback we propose an algorithm attaining $\widetilde{\mathcal{O}}(\sqrt{T}+C)$ regret and $\widetilde{\mathcal{O}}(\sqrt{T}+C)$ {positive} violation, where $C$ quantifies the amount of non-stationarity in the constraints. We then show how to extend these guarantees when only bandit feedback is available for the losses. Finally, when bandit feedback is available for the constraints, we design an algorithm achieving $\widetilde{\mathcal{O}}(\sqrt{T}+C)$ {positive} violation and $\widetilde{\mathcal{O}}(\sqrt{T}+C\sqrt{T})$ regret.