Accelerating Linear Bandits with Persistent Candidate Sets

We study online learning with an additional offline dataset in the stochastic linear bandit setting. Although this problem arises frequently in practice, the offline-to-online tradeoff remains poorly understood in structured environments. We propose a linear bandit algorithm that balances this tradeoff: it relies on offline data during early rounds, and increasingly favors exploration as the horizon grows. We establish regret bounds showing that our method is simultaneously competitive with both purely online and purely offline solutions. In particular, it achieves sublinear regret relative to the optimal action in the number of online interactions, while its regret relative to an offline reference decreases as the number of offline samples grows. Empirical results further demonstrate its effectiveness across various problem parameters.

We study linear bandits with memory, where past actions induce endogenous nonstationarity through an arbitrary known, bounded matrix-valued memory map. To trade off exploration and exploitation while accounting for the memory dynamics, we develop RSM-LinUCB, a Bellman-centric algorithm that learns as in linear bandits and plans as in reinforcement learning. This design admits a novel regret decomposition which separates the memory-induced error from the cumulative reward estimation error along the learner's trajectory. We prove a high-probability regret bound of $\widetilde O\big(dRS(M+1)+σd\sqrt T\big)$, where $T$ is the learning horizon, $d$ is the parameter dimension, $M$ is the memory length, $R$ and $S$ bound the memory-map operator norm and reward-parameter norm, respectively, and $σ$ is the sub-Gaussian noise scale. Our results reveal that the multiplicative memory-horizon coupling in prior bounds is not intrinsic: memory only contributes an additive cost, up to logarithmic factors. We also prove a matching minimax lower bound, establishing near-optimality. We further extend the algorithm to generalized linear rewards, preserving this separation with near-optimal memory and leading statistical dependence. Our algorithms outperform the baselines in numerical experiments on synthetic instances and semi-synthetic KV- and semantic-cache tasks.

We give the first adversarial linear bandit algorithm based on Follow-the-Perturbed-Leader (FTPL) that achieves $\sqrt{T}$ regret over general convex action sets. Our algorithm attains $\widetilde{\mathcal{O}}(d^{5/4}\sqrt{T})$ regret using only linear optimization oracles, and the near-optimal $\widetilde{\mathcal{O}}(d\sqrt{T})$ regret when an optimal-design exploration distribution is further available. This improves on the $T^{2/3}$ regret of the FTPL algorithm of Dani and Hayes (2006) for general convex sets, and generalizes the $\sqrt{T}$ regret of Lévy et al. (2026) over two special action sets (hypercube and Euclidean ball). Compared to previous oracle-efficient linear bandit algorithms based on continuous exponential weights, the FTPL approach generally has better computational efficiency, and is much simpler to implement. The key algorithmic element is a perturbation distribution that adapts to the estimated rewards. Its simplest form, an adaptive Gaussian perturbation, is straightforward to implement and already attains the regret bound above. We further extend it to two general constructions, random-weighting and elliptical perturbations, each covering distributions from light tails (Gaussian) to heavy tails (Student-$t_2$). On the technical side, we develop a systematic way to analyze the stability term in the FTPL regret through the Fisher information of the perturbation distribution. This builds a general framework for FTPL beyond existing ones such as the hazard rate and the FTRL-FTPL duality.