Xingyu Xu

Aarshvi Gajjar, Wai Ming Tai, Xingyu Xu et al.

We study active learning methods for single index models of the form $F({\mathbf x}) = f(\langle {\mathbf w}, {\mathbf x}\rangle)$, where $f:\mathbb{R} \to \mathbb{R}$ and ${\mathbf x,\mathbf w} \in \mathbb{R}^d$. In addition to their theoretical interest as simple examples of non-linear neural networks, single index models have received significant recent attention due to applications in scientific machine learning like surrogate modeling for partial differential equations (PDEs). Such applications require sample-efficient active learning methods that are robust to adversarial noise. I.e., that work even in the challenging agnostic learning setting. We provide two main results on agnostic active learning of single index models. First, when $f$ is known and Lipschitz, we show that $\tilde{O}(d)$ samples collected via {statistical leverage score sampling} are sufficient to learn a near-optimal single index model. Leverage score sampling is simple to implement, efficient, and already widely used for actively learning linear models. Our result requires no assumptions on the data distribution, is optimal up to log factors, and improves quadratically on a recent ${O}(d^{2})$ bound of \cite{gajjar2023active}. Second, we show that $\tilde{O}(d)$ samples suffice even in the more difficult setting when $f$ is \emph{unknown}. Our results leverage tools from high dimensional probability, including Dudley's inequality and dual Sudakov minoration, as well as a novel, distribution-aware discretization of the class of Lipschitz functions.

He Wang, Xingyu Xu, Yuejie Chi

While Bayesian-based exploration often demonstrates superior empirical performance compared to bonus-based methods in model-based reinforcement learning (RL), its theoretical understanding remains limited for model-free settings. Existing provable algorithms either suffer from computational intractability or rely on stage-wise policy updates which reduce responsiveness and slow down the learning process. In this paper, we propose a novel variant of Q-learning algorithm, refereed to as RandomizedQ, which integrates sampling-based exploration with agile, step-wise, policy updates, for episodic tabular RL. We establish an $\widetilde{O}(\sqrt{H^5SAT})$ regret bound, where $S$ is the number of states, $A$ is the number of actions, $H$ is the episode length, and $T$ is the total number of episodes. In addition, we present a logarithmic regret bound under a mild positive sub-optimality condition on the optimal Q-function. Empirically, RandomizedQ exhibits outstanding performance compared to existing Q-learning variants with both bonus-based and Bayesian-based exploration on standard benchmarks.