Joe Suk

Joe Suk, Samory Kpotufe

We study nonparametric contextual bandits where Lipschitz mean reward functions may change over time. We first establish the minimax dynamic regret rate in this less understood setting in terms of number of changes $L$ and total-variation $V$, both capturing all changes in distribution over context space, and argue that state-of-the-art procedures are suboptimal in this setting. Next, we tend to the question of an adaptivity for this setting, i.e. achieving the minimax rate without knowledge of $L$ or $V$. Quite importantly, we posit that the bandit problem, viewed locally at a given context $X_t$, should not be affected by reward changes in other parts of context space $\cal X$. We therefore propose a notion of change, which we term experienced significant shifts, that better accounts for locality, and thus counts considerably less changes than $L$ and $V$. Furthermore, similar to recent work on non-stationary MAB (Suk & Kpotufe, 2022), experienced significant shifts only count the most significant changes in mean rewards, e.g., severe best-arm changes relevant to observed contexts. Our main result is to show that this more tolerant notion of change can in fact be adapted to.

Joe Suk, Arpit Agarwal

The $K$-armed dueling bandits problem, where the feedback is in the form of noisy pairwise preferences, has been widely studied due its applications in information retrieval, recommendation systems, etc. Motivated by concerns that user preferences/tastes can evolve over time, we consider the problem of dueling bandits with distribution shifts. Specifically, we study the recent notion of significant shifts (Suk and Kpotufe, 2022), and ask whether one can design an adaptive algorithm for the dueling problem with $O(\sqrt{K\tilde{L}T})$ dynamic regret, where $\tilde{L}$ is the (unknown) number of significant shifts in preferences. We show that the answer to this question depends on the properties of underlying preference distributions. Firstly, we give an impossibility result that rules out any algorithm with $O(\sqrt{K\tilde{L}T})$ dynamic regret under the well-studied Condorcet and SST classes of preference distributions. Secondly, we show that $\text{SST} \cap \text{STI}$ is the largest amongst popular classes of preference distributions where it is possible to design such an algorithm. Overall, our results provides an almost complete resolution of the above question for the hierarchy of distribution classes.

Joe Suk

We study a $K$-armed non-stationary bandit model where rewards change smoothly, as captured by Hölder class assumptions on rewards as functions of time. Such smooth changes are parametrized by a Hölder exponent $β$ and coefficient $λ$. While various sub-cases of this general model have been studied in isolation, we first establish the minimax dynamic regret rate generally for all $K,β,λ$. Next, we show this optimal dynamic regret can be attained adaptively, without knowledge of $β,λ$. To contrast, even with parameter knowledge, upper bounds were only previously known for limited regimes $β\leq 1$ and $β=2$ (Slivkins, 2014; Krishnamurthy and Gopalan, 2021; Manegueu et al., 2021; Jia et al.,2023). Thus, our work resolves open questions raised by these disparate threads of the literature. We also study the problem of attaining faster gap-dependent regret rates in non-stationary bandits. While such rates are long known to be impossible in general (Garivier and Moulines, 2011), we show that environments admitting a safe arm (Suk and Kpotufe, 2022) allow for much faster rates than the worst-case scaling with $\sqrt{T}$. While previous works in this direction focused on attaining the usual logarithmic regret bounds, as summed over stationary periods, our new gap-dependent rates reveal new optimistic regimes of non-stationarity where even the logarithmic bounds are pessimistic. We show our new gap-dependent rate is tight and that its achievability (i.e., as made possible by a safe arm) has a surprisingly simple and clean characterization within the smooth Hölder class model.

Joe Suk, Jung-hun Kim

We study an infinite-armed bandit problem where actions' mean rewards are initially sampled from a reservoir distribution. Most prior works in this setting focused on stationary rewards (Berry et al., 1997; Wang et al., 2008; Bonald and Proutiere, 2013; Carpentier and Valko, 2015) with the more challenging adversarial/non-stationary variant only recently studied in the context of rotting/decreasing rewards (Kim et al., 2022; 2024). Furthermore, optimal regret upper bounds were only achieved using parameter knowledge of non-stationarity and only known for certain regimes of regularity of the reservoir. This work shows the first parameter-free optimal regret bounds for all regimes while also relaxing distributional assumptions on the reservoir. We first introduce a blackbox scheme to convert a finite-armed MAB algorithm designed for near-stationary environments into a parameter-free algorithm for the infinite-armed non-stationary problem with optimal regret guarantees. We next study a natural notion of significant shift for this problem inspired by recent developments in finite-armed MAB (Suk & Kpotufe, 2022). We show that tighter regret bounds in terms of significant shifts can be adaptively attained by employing a randomized variant of elimination within our blackbox scheme. Our enhanced rates only depend on the rotting non-stationarity and thus exhibit an interesting phenomenon for this problem where rising rewards do not factor into the difficulty of non-stationarity.

Joe Suk, Samory Kpotufe

We study outlier (a.k.a., anomaly) detection for single-pass non-stationary streaming data. In the well-studied offline or batch outlier detection problem, traditional methods such as kernel One-Class SVM (OCSVM) are both computationally heavy and prone to large false-negative (Type II) errors under non-stationarity. To remedy this, we introduce SONAR, an efficient SGD-based OCSVM solver with strongly convex regularization. We show novel theoretical guarantees on the Type I/II errors of SONAR, superior to those known for OCSVM, and further prove that SONAR ensures favorable lifelong learning guarantees under benign distribution shifts. In the more challenging problem of adversarial non-stationary data, we show that SONAR can be used within an ensemble method and equipped with changepoint detection to achieve adaptive guarantees, ensuring small Type I/II errors on each phase of data. We validate our theoretical findings on synthetic and real-world datasets.

Joe Suk, Yaqi Duan

Reinforcement Learning with Verifiable Rewards (RLVR), which uses simple binary feedback to post-train large language models, has shown significant empirical success. However, a principled understanding of why it works has been lacking. This paper builds a theoretical foundation for RLVR by analyzing its training process at both the full-response (trajectory) and token levels. Central to our analysis is a quantity called the Gradient Gap, which formalizes the direction of improvement from low-reward to high-reward regions of the response space. We prove that convergence critically depends on aligning the update direction with this Gradient Gap. Moreover, we derive a sharp step-size threshold based on the magnitude of the Gradient Gap: below it, learning converges, whereas above it, performance collapses. Our theory further predicts how the critical step size must scale with response length and the success rate, thereby explaining why practical heuristics such as length normalization improve stability and showing that, with a fixed learning rate, the success rate can stagnate strictly below $100\%$. We validate these predictions through controlled bandit simulations.

Joe Suk, Arpit Agarwal

In $K$-armed dueling bandits, the learner receives preference feedback between arms, and the regret of an arm is defined in terms of its suboptimality to a $\textit{winner}$ arm. The $\textit{non-stationary}$ variant of the problem, motivated by concerns of changing user preferences, has received recent interest (Saha and Gupta, 2022; Buening and Saha, 2023; Suk and Agarwal, 2023). The goal here is to design algorithms with low {\em dynamic regret}, ideally without foreknowledge of the amount of change. The notion of regret here is tied to a notion of winner arm, most typically taken to be a so-called Condorcet winner or a Borda winner. However, the aforementioned results mostly focus on the Condorcet winner. In comparison, the Borda version of this problem has received less attention which is the focus of this work. We establish the first optimal and adaptive dynamic regret upper bound $\tilde{O}(\tilde{L}^{1/3} K^{1/3} T^{2/3} )$, where $\tilde{L}$ is the unknown number of significant Borda winner switches. We also introduce a novel $\textit{weighted Borda score}$ framework which generalizes both the Borda and Condorcet problems. This framework surprisingly allows a Borda-style regret analysis of the Condorcet problem and establishes improved bounds over the theoretical state-of-art in regimes with a large number of arms or many spurious changes in Condorcet winner. Such a generalization was not known and could be of independent interest.

Joe Suk, Samory Kpotufe

In bandit with distribution shifts, one aims to automatically adapt to unknown changes in reward distribution, and restart exploration when necessary. While this problem has been studied for many years, a recent breakthrough of Auer et al. (2018, 2019) provides the first adaptive procedure to guarantee an optimal (dynamic) regret $\sqrt{LT}$, for $T$ rounds, and an unknown number $L$ of changes. However, while this rate is tight in the worst case, it remained open whether faster rates are possible, without prior knowledge, if few changes in distribution are actually severe. To resolve this question, we propose a new notion of significant shift, which only counts very severe changes that clearly necessitate a restart: roughly, these are changes involving not only best arm switches, but also involving large aggregate differences in reward overtime. Thus, our resulting procedure adaptively achieves rates always faster (sometimes significantly) than $O(\sqrt{ST})$, where $S\ll L$ only counts best arm switches, while at the same time, always faster than the optimal $O(V^{\frac{1}{3}}T^{\frac{2}{3}})$ when expressed in terms of total variation $V$ (which aggregates differences overtime). Our results are expressed in enough generality to also capture non-stochastic adversarial settings.