The goal of active learning is to achieve the same accuracy achievable by passive learning, while using much fewer labels. Exponential savings in label complexity are provably guaranteed in very special cases, but fundamental lower bounds show that such improvements are impossible in general. This suggests a need to explore alternative goals for active learning. Learning with abstention is one such alternative. In this setting, the active learning algorithm may abstain from prediction in certain cases and incur an error that is marginally smaller than 1 2 . We develop the first computationally efficient active learning algorithm with abstention. Furthermore, the algorithm is guaranteed to only abstain on hard examples (where the true label distribution is close to a fair coin), a novel property we term "proper abstention" that also leads to a host of other desirable characteristics. The option to abstain reduces the label complexity by an exponential factor, with no assumptions on the distribution, relative to passive learning algorithms and/or active learning that are not allowed to abstain. A key feature of the algorithm is that it avoids the undesirable "noise-seeking" behavior often seen in active learning. We also explore extensions that achieve constant label complexity and deal with model misspecification.

We study pure exploration in bandits, where the dimension of the feature representation can be much larger than the number of arms. To overcome the curse of dimensionality, we propose to adaptively embed the feature representation of each arm into a lower-dimensional space and carefully deal with the induced model misspecifications. Our approach is conceptually very different from existing works that can either only handle low-dimensional linear bandits or passively deal with model misspecifications. We showcase the application of our approach to two pure exploration settings that were previously under-studied: (1) the reward function belongs to a possibly infinite-dimensional Reproducing Kernel Hilbert Space, and (2) the reward function is nonlinear and can be approximated by neural networks. Our main results provide sample complexity guarantees that only depend on the effective dimension of the feature spaces in the kernel or neural representations. Extensive experiments conducted on both synthetic and real-world datasets demonstrate the efficacy of our methods.

No abstract

No abstract

We study regret minimization problem with the existence of multiple best/nearoptimal arms in the multi-armed bandit setting. We consider the case where the number of arms/actions is comparable or much larger than the time horizon, and make no assumptions about the structure of the bandit instance. Our goal is to design algorithms that can automatically adapt to the unknown hardness of the problem, i.e., the number of best arms. Our setting captures many modern applications of bandit algorithms where the action space is enormous and the information about the underlying instance/structure is unavailable. We first propose an adaptive algorithm that is agnostic to the hardness level and theoretically derive its regret bound. We then prove a lower bound for our problem setting, which indicates: (1) no algorithm can be optimal simultaneously over all hardness levels; and (2) our algorithm achieves an adaptive rate function that is Pareto optimal. With additional knowledge of the expected reward of the best arm, we propose another adaptive algorithm that is minimax optimal, up to polylog factors, over all hardness levels. Experimental results confirm our theoretical guarantees and show advantages of our algorithms over the previous state-of-the-art.Preprint. Under review.