Best arm identification (BAI) aims to identify the highest- performance arm among a set of K arms by collecting stochastic samples from each arm. In real-world problems, the best arm needs to satisfy additional feasibility constraints. While there is limited prior work on BAI with feasibility constraints, they typically assume the performance and con- straints are observed simultaneously on each pull of an arm. However, this assumption does not reflect most practical use cases, e.g., in drug discovery, we wish to find the most potent drug whose toxicity and solubility are below certain safety thresholds. These safety experiments can be conducted separately from the potency measurement. Thus, this requires de- signing BAI algorithms that not only decide which arm to pull but also decide whether to test for the arm’s performance or feasibility. In this work, we study feasible BAI which allows a decision-maker to choose a tuple (i, ℓ), where i ∈ [K] de- notes an arm and ℓ denotes whether she wishes to test for its performance (ℓ = 0) or any of its N feasibility constraints (ℓ ∈ [N ]). We focus on the fixed confidence setting, which is to identify the feasible arm with the highest performance, with a probability of at least 1 − δ. We propose an efficient algorithm and upper-bound its sample complexity, showing our algorithm can naturally adapt to the problem’s difficulty and eliminate arms by worse performance or infeasibility, whichever is easier. We complement this upper bound with a lower bound showing that our algorithm is asymptotically (δ → 0) optimal. Finally, we empirically show that our algorithm outperforms other state-of-the-art BAI algorithms in both synthetic and real-world datasets.

We study actively labeling streaming data, where an active learner is faced with a stream of data points and must carefully choose which of these points to label via an expensive experiment. Such problems frequently arise in applications such as healthcare and astronomy. We first study a setting when the data’s inputs belong to one of K discrete distributions and formalize this problem via a loss that captures the labeling cost and the prediction error. When the labeling cost is B, our algorithm, which chooses to label a point if the uncertainty is larger than a time and cost dependent threshold, achieves a worst-case upper bound of $\tilde O(B^{\frac{1}{3}} K^{\frac{1}{3}} T^{\frac{2}{3}})$ on the loss after T rounds. We also provide a more nuanced upper bound which demonstrates that the algorithm can adapt to the arrival pattern, and achieves better performance when the arrival pattern is more favorable. We complement both upper bounds with matching lower bounds. We next study this problem when the inputs belong to a continuous domain and the output of the experiment is a smooth function with bounded RKHS norm. After T rounds in d dimensions, we show that the loss is bounded by $\tilde O(B^{\frac{1}{d+3}} T^{\frac{d+2}{d+3}})$ in an RKHS with a squared exponential kernel and by $\tilde O(B^{\frac{1}{2d+3}} T^{\frac{2d+2}{2d+3}})$ in an RKHS with a Matérn kernel. Our empirical evaluation demonstrates that our method outperforms other baselines in several synthetic experiments and two real experiments in medicine and astronomy.

We consider an adversarially-trained version of the nonnegative matrix factorization, a popular latent dimensionality reduction technique. In our formulation, an attacker adds an arbitrary matrix of bounded norm to the given data matrix. We design efficient algorithms inspired by adversarial training to optimize for dictionary and coefficient matrices with enhanced generalization abilities. Extensive simulations on synthetic and benchmark datasets demonstrate the superior predictive performance on matrix completion tasks of our proposed method compared to state-of-the-art competitors, including other variants of adversarial nonnegative matrix factorization.