Generalized Linear Bandits with Memory

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 study nonstationary generalized linear bandits (GLBs), where the expected reward is modeled through a nonlinear link function with an unknown time-varying parameter. This framework encompasses a broad class of reward models, including linear, Bernoulli, and binomial rewards. Existing approaches are predominantly based on maximum-likelihood estimation (MLE), using sliding-window, restart, or discounting mechanisms to handle nonstationarity. Although these methods achieve statistically efficient regret guarantees, they generally require revisiting past observations at every round, which leads to computation and memory costs that grow with time; moreover, several of them rely on a non-convex projection step. In this paper, we propose DOMD-GLB, a new algorithm for nonstationary GLBs that utilizes discounted online mirror descent (DOMD) for parameter estimation, thereby incurring only $O(1)$ computation and memory costs per round. We prove dynamic regret bounds of order $\tilde{O} \big(c_μ^{-1/2} d^{3/4} P_T^{1/4} T^{3/4}\big)$ in drifting environments and $\tilde{O}\big(c_μ^{-1/3} d^{2/3} Γ_T^{1/3} T^{2/3}\big)$in piecewise-stationary environments, where $d$ denotes the feature dimension, $T$ the time horizon, $P_T$ the path length, $Γ_T$ the number of change points, and $c_μ$ a curvature parameter associated with the link function, while substantially improving computational efficiency over prior work. To the best of our knowledge, this is the first algorithm for nonstationary GLBs with per-round computation and memory costs independent of time.

In this paper we first study the problem of generalized linear bandit (GLB) under heavy-tailed noise. The characteristics of heavy-tailed distributions are widely observed in real-world applications such as personalized recommendation, financial markets, and medical treatments. Based on the online mirror descent (OMD) method, we propose an algorithm EHM that extends the adaptive Huber loss method (Wang et al., 2025) with one-pass update ($\mathcal{O}(1)$ computational complexity with respect to current round $t$ and the time horizon $T$), which simultaneously achieves an almost optimal regret of $\widetilde{\mathcal{O}}(T^{\frac{1}{1+ε}})$ where $T$ is the time horizon. In addition, by utilizing a special property of some link function (Sawarni et al., 2025), our algorithm eliminates the need to know a commonly used parameter. Next, we study the GLB problem under the case when contextual characteristic becomes piecewise constant, and we slightly revised former algorithm to obtain the PGLB-EHM algorithm. After theoretical analysis, we prove that the regret upper bound order stays the same. Furthermore, we look deeper into a special case of nonlinear bandit (NB) and present the NB-EHM algorithm with bisection method and special restriction. Eventually we utilize the affine lifting approach and show that the general NB problem can be applied with NB-EHM to achieve a sublinear regret bound.