Best-of-Both-Worlds Bandits

We study online learning with directed weakly observable feedback graphs and propose a Follow-the-Perturbed-Leader (FTPL) policy that achieves best-of-both-worlds guarantee with nearly linear expected time complexity. To preserve the simple structure of FTPL, we develop a novel graph-based loss estimator that combines forced exploration, conditional geometric resampling and adaptive learning rates with surrogate probabilities. Our policy achieves $\tilde{\mathcal{O}}(δ_*^{1/3} T^{2/3})$ regret for adversarial regimes as well as $\tilde{\mathcal{O}}(δ_* \log T / Δ_{\min}^2)$ regret for stochastic regimes, where $δ_*$ and $Δ$, respectively, denote the fractional weak domination number and the suboptimality gap. These results show that refined guarantees for problems with minimax regret of $Θ(T^{2/3})$ can be obtained through appropriately designed adaptive learning rates and loss estimators without sacrificing the simplicity of FTPL.

We study online channel scheduling for a single queue served by $K$ channels under bandit feedback. We propose a simple best-of-both-worlds policy that explores when the queue is empty. During busy periods, it selects the empirical best channel while the prescribed conditions hold and, if any condition is violated, switches to a robust bandit policy to clear the queue. For arbitrary arrivals and services determined by an oblivious adversary, our policy achieves $\tilde{\mathcal{O}}(\sqrt{K L_T})$ worst-case queue length regret with high probability, where $L_T$ is the longest busy-period length. This result improves the previous $T^{3/4}$ dependence and matches the worst-case lower bound up to logarithmic factors. Under bounded i.i.d. stochastic models with a unique best stabilizing channel, we show that the cumulative queue length regret converges to a finite problem-dependent constant, and further establish the logarithmic expected worst-case queue length regret bound.

We study best-of-both-worlds (BOBW) algorithms for linear bandits, which aim to simultaneously guarantee $\gO(\sqrt T)$ regret in the adversarial regime and $\gO(\log T)$ regret in the stochastic regime over $T$ rounds. For finite action sets, we introduce information-acquisition follow-the-regularized-leader (\IAFTRL{}), the first BOBW algorithm based purely on the follow-the-regularized-leader (FTRL) framework that attains an $\gO(\cstar\log T)$-type regret in the stochastic regime, where $\cstar$ is the information-acquisition complexity that characterizes instance-dependent optimality in linear bandits, while retaining the $\gO(\sqrt{T})$ regret guarantee in the adversarial regime. This resolves the open problem posed in~\cite{LeeEtAl2021,KongZ023,ItoT23,ZhaoItoLi2025}.

We study first-order and best-of-both-worlds regret guarantees in multi-armed bandits. Pogodin and Lattimore (2019) proposed an algorithm that combines minimax-optimal worst-case regret with a first-order guarantee whose loss-dependent term is $\mathcal{O}(\sqrt{K L^\star}\log T)$, where $T$ denotes the horizon, $L^\star$ denotes the smallest expected cumulative loss among fixed arms, and $K$ denotes the number of arms. However, a logarithmic gap remains between this loss-dependent term and the optimal first-order scale. In this paper, we remove this logarithmic factor by proposing an efficient and parameter-free algorithm that achieves the minimax optimal first-order regret guarantee $\mathcal{O}(\sqrt{K L^\star})$. Technically, our algorithm is built upon FTRL with a Tsallis entropy regularizer (Zimmert and Seldin,2021), with a learning rate adapted to the learner's cumulative loss and a sampling distribution determined by a scalar fixed point. This construction controls the combined estimation and stability cost in terms of the learner's loss. Moreover, we show that the same algorithm achieves a best-of-both-worlds guarantee, with $\mathcal{O}(\sum_{i\ne i^\star}\frac{\log T}{Δ_i})$ regret for i.i.d. losses with a unique best arm $i^\star$, where $Δ_i$ denotes the suboptimality gap of arm $i$.

Improving on a baseline requires exploring away from it, precisely where its performance may be lost. We propose DUET (Doubling Under Empirical Tests), a horizon-free phased mixing scheme for combinatorial semi-bandits with arbitrary finite action sets. Two observable tests decide when to double the learning weight and when to double the empirical-regret budget and restart. We instantiate the scheme with entropy-FTRL in mixed phases and hybrid-FTRL in pure phases. Assuming the baseline selects every arm with positive probability, this instantiation keeps expected baseline regret logarithmic in $T$, while its expected regret to every fixed action is $\widetilde{O}(\sqrt{T})$ under non-anticipating adversarial losses and polylogarithmic, gap-dependent, under i.i.d. losses with a unique optimal action. All three guarantees hold at every deterministic horizon, and the algorithm knows neither the horizon, nor the gap, nor the environment type. Baseline protection, adversarial robustness, and stochastic adaptivity thus come from one scheme and two base learners, at the price of a logarithm.

Bidding in the European Frequency Containment Reserve (FCR) market is challenging for flexibility providers because competing offers are hidden and bidders observe only partial feedback form the market, such as, clearing price and awarded quantity. For a participant active in a single country, we show that the multi-country FCR clearing problem can be recast as a repeated multi-unit uniform-price auction against an endogenous vector of opposing bids. This reformulation yields an online learning problem and allows us to adapt a Best-of-Both-Worlds combinatorial semi-bandit algorithm implementable from this standard market feedback. The resulting bidder achieves logarithmic pseudo-regret in stochastic environments and $\mathcal{O}(\sqrt{T})$ regret in adversarial ones. Synthetic experiments confirm the expected scaling, and backtests on historical European FCR data show competitive performance in practice: the method performs especially well on stable products, while EXP3-type baselines can be safer under stronger non-stationarity. Overall, the results show that learning-based bidding in FCR markets is theoretically grounded and practically useful when the learning rule matches product-level market stability.

Multi-dueling bandits, where a learner selects $m \geq 2$ arms per round and observes only the winner, arise naturally in many applications including ranking and recommendation systems, yet a fundamental question has remained open: can a single algorithm perform optimally in both stochastic and adversarial environments, without knowing which regime it faces? We answer this affirmatively, providing the first best-of-both-worlds algorithms for multi-dueling bandits under both Condorcet and Borda objectives. For the Condorcet setting, we propose $\texttt{MetaDueling}$, a black-box reduction that converts any dueling bandit algorithm into a multi-dueling bandit algorithm by transforming multi-way winner feedback into an unbiased pairwise signal. Instantiating our reduction with $\texttt{Versatile-DB}$ yields the first best-of-both-worlds algorithm for multi-dueling bandits: it achieves $O(\sqrt{KT})$ pseudo-regret against adversarial preferences and the instance-optimal $O\left(\sum_{i \neq a^\star} \frac{\log T}{Δ_i}\right)$ pseudo-regret under stochastic preferences, both simultaneously and without prior knowledge of the regime. For the Borda setting, we propose $\texttt{SA-MiDEX}$, a stochastic-and-adversarial algorithm that achieves $O\left(K^2 \log KT + K \log^2 T + \sum_{i: Δ_i^{\mathrm{B}} > 0} \frac{K\log KT}{(Δ_i^{\mathrm{B}})^2}\right)$ regret in stochastic environments and $O\left(K \sqrt{T \log KT} + K^{1/3} T^{2/3} (\log K)^{1/3}\right)$ regret against adversaries, again without prior knowledge of the regime. We complement our upper bounds with matching lower bounds for the Condorcet setting. For the Borda setting, our upper bounds are near-optimal with respect to the lower bounds (within a factor of $K$) and match the best-known results in the literature.

This paper studies the optimality and complexity of Follow-the-Perturbed-Leader (FTPL) policy in $m$-set semi-bandit problems. FTPL has been studied extensively as a promising candidate of an efficient algorithm with favorable regret for adversarial combinatorial semi-bandits. Nevertheless, the optimality of FTPL has still been unknown unlike Follow-the-Regularized-Leader (FTRL) whose optimality has been proved for various tasks of online learning. In this paper, we extend the analysis of FTPL with geometric resampling (GR) to $m$-set semi-bandits, which is a special case of combinatorial semi-bandits, showing that FTPL with Fréchet and Pareto distributions with certain parameters achieves the best possible regret of $O(\sqrt{mdT})$ in adversarial setting. We also show that FTPL with Fréchet and Pareto distributions with a certain parameter achieves a logarithmic regret for stochastic setting, meaning the Best-of-Both-Worlds optimality of FTPL for $m$-set semi-bandit problems. Furthermore, we extend the conditional geometric resampling to $m$-set semi-bandits for efficient loss estimation in FTPL, reducing the computational complexity from $O(d^2)$ of the original geometric resampling to $O(md(\log(d/m)+1))$ without sacrificing the regret performance.

We study linear contextual bandits with paid observations, where at each round the learner observes a context, selects an action, and may pay a fixed cost to observe feedback from a subset of arms. We propose two Follow-the-Regularized-Leader algorithms with Best-of-Both-Worlds guarantees. The first, Agg-SPB, extends the SPB-matching framework of Tsuchiya and Ito (2024) by aggregating context-dependent stability terms, achieving the characteristic $T^{2/3}$ adversarial regret rate and logarithmic dependence on $T$ in stochastic environments. The second, CE-SPB, combines arm-dependent observation probabilities with an entropy-adaptive learning rate inspired by Kuroki et al. (2024). It achieves an entropy-adaptive $\widetilde{O}(T^{2/3})$ adversarial guarantee and polylogarithmic stochastic regret, while avoiding the minimum-context-mass dependence arising in the stochastic analysis of Agg-SPB. Both algorithms further extend to corrupted stochastic environments with explicit corruption-dependent guarantees. These results establish that logarithmic stochastic regret is compatible with the $T^{2/3}$ adversarial regime for linear contextual bandits with paid observations, while highlighting a tradeoff between sharper horizon dependence in stochastic settings and path-dependent matching without explicit minimum-context-mass dependence.