We study the multinomial logit (MNL) bandit problem, where at each time step, the seller offers an assortment of size at most $K$ from a pool of $N$ items, and the buyer purchases an item from the assortment according to a MNL choice model. The objective is to learn the model parameters and maximize the expected revenue. We present (i) an algorithm that identifies the optimal assortment $S^*$ within $\widetilde{O}(\sum_{i = 1}^N \Delta_i^{-2})$ time steps with high probability, and (ii) an algorithm that incurs $O(\sum_{i \notin S^*} K\Delta_i^{-1} \log T)$ regret in $T$ time steps. To our knowledge, our algorithms are the \emph{first} to achieve gap-dependent bounds that \emph{fully} depends on the suboptimality gaps of \emph{all} items. Our technical contributions include an algorithmic framework that relates the MNL-bandit problem to a variant of the top-$K$ arm identification problem in multi-armed bandits, a generalized epoch-based offering procedure, and a layer-based adaptive estimation procedure.

This paper studies model-based bandit and reinforcement learning (RL) with nonlinear function approximations. We propose to study convergence to approximate local maxima because we show that global convergence is statistically intractable even for one-layer neural net bandit with a deterministic reward. For both nonlinear bandit and RL, the paper presents a model-based algorithm, Virtual Ascent with Online Model Learner (ViOlin), which provably converges to a local maximum with sample complexity that only depends on the sequential Rademacher complexity of the model class. Our results imply novel global or local regret bounds on several concrete settings such as linear bandit with finite or sparse model class, and two-layer neural net bandit. A key algorithmic insight is that optimism may lead to over-exploration even for two-layer neural net model class. On the other hand, for convergence to local maxima, it suffices to maximize the virtual return if the model can also reasonably predict the gradient and Hessian of the real return.

Deep Reinforcement Learning (RL) powered by neural net approximation of the Q function has had enormous empirical success. While the theory of RL has traditionally focused on linear function approximation (or eluder dimension) approaches, little is known about nonlinear RL with neural net approximations of the Q functions. This is the focus of this work, where we study function approximation with two-layer neural networks (considering both ReLU and polynomial activation functions). Our first result is a computationally and statistically efficient algorithm in the generative model setting under completeness for two-layer neural networks. Our second result considers this setting but under only realizability of the neural net function class. Here, assuming deterministic dynamics, the sample complexity scales linearly in the algebraic dimension. In all cases, our results significantly improve upon what can be attained with linear (or eluder dimension) methods.

Bandit problems with linear or concave reward have been extensively studied, but relatively few works have studied bandits with non-concave reward. This work considers a large family of bandit problems where the unknown underlying reward function is non-concave, including the low-rank generalized linear bandit problems and two-layer neural network with polynomial activation bandit problem.For the low-rank generalized linear bandit problem, we provide a minimax-optimal algorithm in the dimension, refuting both conjectures in \cite{lu2021low,jun2019bilinear}. Our algorithms are based on a unified zeroth-order optimization paradigm that applies in great generality and attains optimal rates in several structured polynomial settings (in the dimension). We further demonstrate the applicability of our algorithms in RL in the generative model setting, resulting in improved sample complexity over prior approaches.Finally, we show that the standard optimistic algorithms (e.g., UCB) are sub-optimal by dimension factors. In the neural net setting (with polynomial activation functions) with noiseless reward, we provide a bandit algorithm with sample complexity equal to the intrinsic algebraic dimension. Again, we show that optimistic approaches have worse sample complexity, polynomial in the extrinsic dimension (which could be exponentially worse in the polynomial degree).

This paper presents new \emph{variance-aware} confidence sets for linear bandits and linear mixture Markov Decision Processes (MDPs).With the new confidence sets, we obtain the follow regret bounds:For linear bandits, we obtain an $\widetilde{O}(\mathrm{poly}(d)\sqrt{1 + \sum_{k=1}^{K}\sigma_k^2})$ data-dependent regret bound, where $d$ is the feature dimension, $K$ is the number of rounds, and $\sigma_k^2$ is the \emph{unknown} variance of the reward at the $k$-th round. This is the first regret bound that only scales with the variance and the dimension but \emph{no explicit polynomial dependency on $K$}.When variances are small, this bound can be significantly smaller than the $\widetilde{\Theta}\left(d\sqrt{K}\right)$ worst-case regret bound.For linear mixture MDPs, we obtain an $\widetilde{O}(\mathrm{poly}(d, \log H)\sqrt{K})$ regret bound, where $d$ is the number of base models, $K$ is the number of episodes, and $H$ is the planning horizon. This is the first regret bound that only scales \emph{logarithmically} with $H$ in the reinforcement learning with linear function approximation setting, thus \emph{exponentially improving} existing results, and resolving an open problem in \citep{zhou2020nearly}.We develop three technical ideas that may be of independent interest:1) applications of the peeling technique to both the input norm and the variance magnitude, 2) a recursion-based estimator for the variance, and 3) a new convex potential lemma that generalizes the seminal elliptical potential lemma.

Motivated by practical needs such as large-scale learning, we study the impact of adaptivity constraints to linear contextual bandits, a central problem in online learning and decision making. We consider two popular limited adaptivity models in literature: batch learning and rare policy switches. We show that, when the context vectors are adversarially chosen in d-dimensional linear contextual bandits, the learner needs O(d logd logT) policy switches to achieve the minimax-optimal regret, and this is optimal up to poly(logd, loglogT) factors; for stochastic context vectors, even in the more restricted batch learning model, only O(loglogT) batches are needed to achieve the optimal regret. Together with the known results in literature, our results present a complete picture about the adaptivity constraints in linear contextual bandits. Along the way, we propose the distributional optimal design, a natural extension of the optimal experiment design, and provide a both statistically and computationally efficient learning algorithm for the problem, which may be of independent interest.