Local Anti-Concentration Class: Logarithmic Regret for Greedy Linear Contextual Bandit

We study the performance guarantees of exploration-free greedy algorithms for the linear contextual bandit problem. We introduce a novel condition, named the \textit{Local Anti-Concentration} (LAC) condition, which enables a greedy bandit algorithm to achieve provable efficiency. We show that the LAC condition is satisfied by a broad class of distributions, including Gaussian, exponential, uniform, Cauchy, and Student's~$t$ distributions, along with other exponential family distributions and their truncated variants. This significantly expands the class of distributions under which greedy algorithms can perform efficiently. Under our proposed LAC condition, we prove that the cumulative expected regret of the greedy algorithm for the linear contextual bandit is bounded by $O(\operatorname{poly} \log T)$. Our results establish the widest range of distributions known to date that allow a sublinear regret bound for greedy algorithms, further achieving a sharp poly-logarithmic regret.

Improved Regret of Linear Ensemble Sampling

In this work, we close the fundamental gap of theory and practice by providing an improved regret bound for linear ensemble sampling. We prove that with an ensemble size logarithmic in $T$, linear ensemble sampling can achieve a frequentist regret bound of $\tilde{\mathcal{O}}(d^{3/2}\sqrt{T})$, matching state-of-the-art results for randomized linear bandit algorithms, where $d$ and $T$ are the dimension of the parameter and the time horizon respectively. Our approach introduces a general regret analysis framework for linear bandit algorithms. Additionally, we reveal a significant relationship between linear ensemble sampling and Linear Perturbed-History Exploration (LinPHE), showing that LinPHE is a special case of linear ensemble sampling when the ensemble size equals $T$. This insight allows us to derive a new regret bound of $\tilde{\mathcal{O}}(d^{3/2}\sqrt{T})$ for LinPHE, independent of the number of arms. Our contributions advance the theoretical foundation of ensemble sampling, bringing its regret bounds in line with the best known bounds for other randomized exploration algorithms.

Demystifying Linear MDPs and Novel Dynamics Aggregation Framework

In this work, we prove that, in linear MDPs, the feature dimension $d$ is lower bounded by $S/U$ in order to aptly represent transition probabilities, where $S$ is the size of the state space and $U$ is the maximum size of directly reachable states. Hence, $d$ can still scale with $S$ depending on the direct reachability of the environment. To address this limitation of linear MDPs, we propose a novel structural aggregation framework based on dynamics, named as the "dynamics aggregation". For this newly proposed framework, we design a provably efficient hierarchical reinforcement learning algorithm in linear function approximation that leverages aggregated sub-structures. Our proposed algorithm exhibits statistical efficiency, achieving a regret of $ \tilde{O} ( d_{\psi}^{3/2} H^{3/2}\sqrt{ N T} )$, where $d_{\psi}$ represents the feature dimension of aggregated subMDPs and $N$ signifies the number of aggregated subMDPs. We establish that the condition $d_{\psi}^3 N \ll d^{3}$ is readily met in most real-world environments with hierarchical structures, enabling a substantial improvement in the regret bound compared to LSVI-UCB, which enjoys a regret of $ \tilde{O} (d^{3/2} H^{3/2} \sqrt{ T})$. To the best of our knowledge, this work presents the first HRL algorithm with linear function approximation that offers provable guarantees.

Queueing Matching Bandits with Preference Feedback

In this study, we consider multi-class multi-server asymmetric queueing systems consisting of $N$ queues on one side and $K$ servers on the other side, where jobs randomly arrive in queues at each time. The service rate of each job-server assignment is unknown and modeled by a feature-based Multi-nomial Logit (MNL) function. At each time, a scheduler assigns jobs to servers, and each server stochastically serves at most one job based on its preferences over the assigned jobs. The primary goal of the algorithm is to stabilize the queues in the system while learning the service rates of servers. To achieve this goal, we propose algorithms based on UCB and Thompson Sampling, which achieve system stability with an average queue length bound of $O(\min\{N,K\}/\epsilon)$ for a large time horizon $T$, where $\epsilon$ is a traffic slackness of the system. Furthermore, the algorithms achieve sublinear regret bounds of $\tilde{O}(\min\{\sqrt{T} Q_{\max},T^{3/4}\})$, where $Q_{\max}$ represents the maximum queue length over agents and times. Lastly, we provide experimental results to demonstrate the performance of our algorithms.

Magnituder Layers for Implicit Neural Representations in 3D

Improving the efficiency and performance of implicit neural representations in 3D, particularly Neural Radiance Fields (NeRF) and Signed Distance Fields (SDF) is crucial for enabling their use in real-time applications. These models, while capable of generating photo-realistic novel views and detailed 3D reconstructions, often suffer from high computational costs and slow inference times. To address this, we introduce a novel neural network layer called the "magnituder", designed to reduce the number of training parameters in these models without sacrificing their expressive power. By integrating magnituders into standard feed-forward layer stacks, we achieve improved inference speed and adaptability. Furthermore, our approach enables a zero-shot performance boost in trained implicit neural representation models through layer-wise knowledge transfer without backpropagation, leading to more efficient scene reconstruction in dynamic environments.

Lasso Bandit with Compatibility Condition on Optimal Arm

We consider a stochastic sparse linear bandit problem where only a sparse subset of context features affects the expected reward function, i.e., the unknown reward parameter has sparse structure. In the existing Lasso bandit literature, the compatibility conditions together with additional diversity conditions on the context features are imposed to achieve regret bounds that only depend logarithmically on the ambient dimension $d$. In this paper, we demonstrate that even without the additional diversity assumptions, the compatibility condition only on the optimal arm is sufficient to derive a regret bound that depends logarithmically on $d$, and our assumption is strictly weaker than those used in the lasso bandit literature under the single parameter setting. We propose an algorithm that adapts the forced-sampling technique and prove that the proposed algorithm achieves $O(\text{poly}\log dT)$ regret under the margin condition. To our knowledge, the proposed algorithm requires the weakest assumptions among Lasso bandit algorithms under a single parameter setting that achieve $O(\text{poly}\log dT)$ regret. Through the numerical experiments, we confirm the superior performance of our proposed algorithm.

Nearly Minimax Optimal Regret for Multinomial Logistic Bandit

In this paper, we investigate the contextual multinomial logit (MNL) bandit problem in which a learning agent sequentially selects an assortment based on contextual information, and user feedback follows an MNL choice model. There has been a significant discrepancy between lower and upper regret bounds, particularly regarding the feature dimension $d$ and the maximum assortment size $K$. Additionally, the variation in reward structures between these bounds complicates the quest for optimality. Under uniform rewards, where all items have the same expected reward, we establish a regret lower bound of $\Omega(d\sqrt{\smash[b]{T/K}})$ and propose a constant-time algorithm, OFU-MNL+, that achieves a matching upper bound of $\tilde{\mathcal{O}}(d\sqrt{\smash[b]{T/K}})$. Under non-uniform rewards, we prove a lower bound of $\Omega(d\sqrt{T})$ and an upper bound of $\tilde{\mathcal{O}}(d\sqrt{T})$, also achievable by OFU-MNL+. Our empirical studies support these theoretical findings. To the best of our knowledge, this is the first work in the MNL contextual bandit literature to prove minimax optimality -- for either uniform or non-uniform reward setting -- and to propose a computationally efficient algorithm that achieves this optimality up to logarithmic factors.

Follow-the-Perturbed-Leader with Fréchet-type Tail Distributions: Optimality in Adversarial Bandits and Best-of-Both-Worlds

This paper studies the optimality of the Follow-the-Perturbed-Leader (FTPL) policy in both adversarial and stochastic $K$-armed bandits. Despite the widespread use of the Follow-the-Regularized-Leader (FTRL) framework with various choices of regularization, the FTPL framework, which relies on random perturbations, has not received much attention, despite its inherent simplicity. In adversarial bandits, there has been conjecture that FTPL could potentially achieve $\mathcal{O}(\sqrt{KT})$ regrets if perturbations follow a distribution with a Fr\'{e}chet-type tail. Recent work by Honda et al. (2023) showed that FTPL with Fr\'{e}chet distribution with shape $\alpha=2$ indeed attains this bound and, notably logarithmic regret in stochastic bandits, meaning the Best-of-Both-Worlds (BOBW) capability of FTPL. However, this result only partly resolves the above conjecture because their analysis heavily relies on the specific form of the Fr\'{e}chet distribution with this shape. In this paper, we establish a sufficient condition for perturbations to achieve $\mathcal{O}(\sqrt{KT})$ regrets in the adversarial setting, which covers, e.g., Fr\'{e}chet, Pareto, and Student-$t$ distributions. We also demonstrate the BOBW achievability of FTPL with certain Fr\'{e}chet-type tail distributions. Our results contribute not only to resolving existing conjectures through the lens of extreme value theory but also potentially offer insights into the effect of the regularization functions in FTRL through the mapping from FTPL to FTRL.

Learning Uncertainty-Aware Temporally-Extended Actions

In reinforcement learning, temporal abstraction in the action space, exemplified by action repetition, is a technique to facilitate policy learning through extended actions. However, a primary limitation in previous studies of action repetition is its potential to degrade performance, particularly when sub-optimal actions are repeated. This issue often negates the advantages of action repetition. To address this, we propose a novel algorithm named Uncertainty-aware Temporal Extension (UTE). UTE employs ensemble methods to accurately measure uncertainty during action extension. This feature allows policies to strategically choose between emphasizing exploration or adopting an uncertainty-averse approach, tailored to their specific needs. We demonstrate the effectiveness of UTE through experiments in Gridworld and Atari 2600 environments. Our findings show that UTE outperforms existing action repetition algorithms, effectively mitigating their inherent limitations and significantly enhancing policy learning efficiency.

Doubly Perturbed Task-Free Continual Learning

Task-free online continual learning (TF-CL) is a challenging problem where the model incrementally learns tasks without explicit task information. Although training with entire data from the past, present as well as future is considered as the gold standard, naive approaches in TF-CL with the current samples may be conflicted with learning with samples in the future, leading to catastrophic forgetting and poor plasticity. Thus, a proactive consideration of an unseen future sample in TF-CL becomes imperative. Motivated by this intuition, we propose a novel TF-CL framework considering future samples and show that injecting adversarial perturbations on both input data and decision-making is effective. Then, we propose a novel method named Doubly Perturbed Continual Learning (DPCL) to efficiently implement these input and decision-making perturbations. Specifically, for input perturbation, we propose an approximate perturbation method that injects noise into the input data as well as the feature vector and then interpolates the two perturbed samples. For decision-making process perturbation, we devise multiple stochastic classifiers. We also investigate a memory management scheme and learning rate scheduling reflecting our proposed double perturbations. We demonstrate that our proposed method outperforms the state-of-the-art baseline methods by large margins on various TF-CL benchmarks.