Time-Varying Gaussian Process Bandit Optimization

We consider the sequential Bayesian optimization problem with bandit feedback, adopting a formulation that allows for the reward function to vary with time. We model the reward function using a Gaussian process whose evolution obeys a simple Markov model. We introduce two natural extensions of the classical Gaussian process upper confidence bound (GP-UCB) algorithm. The first, R-GP-UCB, resets GP-UCB at regular intervals. The second, TV-GP-UCB, instead forgets about old data in a smooth fashion. Our main contribution comprises of novel regret bounds for these algorithms, providing an explicit characterization of the trade-off between the time horizon and the rate at which the function varies. We illustrate the performance of the algorithms on both synthetic and real data, and we find the gradual forgetting of TV-GP-UCB to perform favorably compared to the sharp resetting of R-GP-UCB. Moreover, both algorithms significantly outperform classical GP-UCB, since it treats stale and fresh data equally.Comment: To appear in AISTATS 201

Robust Submodular Maximization: A Non-Uniform Partitioning Approach

We study the problem of maximizing a monotone submodular function subject to a cardinality constraint $k$, with the added twist that a number of items $\tau$ from the returned set may be removed. We focus on the worst-case setting considered in (Orlin et al., 2016), in which a constant-factor approximation guarantee was given for $\tau = o(\sqrt{k})$. In this paper, we solve a key open problem raised therein, presenting a new Partitioned Robust (PRo) submodular maximization algorithm that achieves the same guarantee for more general $\tau = o(k)$. Our algorithm constructs partitions consisting of buckets with exponentially increasing sizes, and applies standard submodular optimization subroutines on the buckets in order to construct the robust solution. We numerically demonstrate the performance of PRo in data summarization and influence maximization, demonstrating gains over both the greedy algorithm and the algorithm of (Orlin et al., 2016).Comment: Accepted to ICML 201

Adversarially Robust Optimization with Gaussian Processes

In this paper, we consider the problem of Gaussian process (GP) optimization with an added robustness requirement: The returned point may be perturbed by an adversary, and we require the function value to remain as high as possible even after this perturbation. This problem is motivated by settings in which the underlying functions during optimization and implementation stages are different, or when one is interested in finding an entire region of good inputs rather than only a single point. We show that standard GP optimization algorithms do not exhibit the desired robustness properties, and provide a novel confidence-bound based algorithm StableOpt for this purpose. We rigorously establish the required number of samples for StableOpt to find a near-optimal point, and we complement this guarantee with an algorithm-independent lower bound. We experimentally demonstrate several potential applications of interest using real-world data sets, and we show that StableOpt consistently succeeds in finding a stable maximizer where several baseline methods fail.Comment: Corrected typo

Truncated Variance Reduction: A Unified Approach to Bayesian Optimization and Level-Set Estimation

We present a new algorithm, truncated variance reduction (TruVaR), that treats Bayesian optimization (BO) and level-set estimation (LSE) with Gaussian processes in a unified fashion. The algorithm greedily shrinks a sum of truncated variances within a set of potential maximizers (BO) or unclassified points (LSE), which is updated based on confidence bounds. TruVaR is effective in several important settings that are typically non-trivial to incorporate into myopic algorithms, including pointwise costs and heteroscedastic noise. We provide a general theoretical guarantee for TruVaR covering these aspects, and use it to recover and strengthen existing results on BO and LSE. Moreover, we provide a new result for a setting where one can select from a number of noise levels having associated costs. We demonstrate the effectiveness of the algorithm on both synthetic and real-world data sets.Comment: Accepted to NIPS 201

Robust Adaptive Decision Making: Bayesian Optimization and Beyond

The central task in many interactive machine learning systems can be formalized as the sequential optimization of a black-box function. Bayesian optimization (BO) is a powerful model-based framework for \emph{adaptive} experimentation, where the primary goal is the optimization of the black-box function via sequentially chosen decisions. In many real-world tasks, it is essential for the decisions to be \emph{robust} against, e.g., adversarial failures and perturbations, dynamic and time-varying phenomena, a mismatch between simulations and reality, etc. Under such requirements, the standard methods and BO algorithms become inadequate. In this dissertation, we consider four research directions with the goal of enhancing robust and adaptive decision making in BO and associated problems. First, we study the related problem of level-set estimation (LSE) with Gaussian Processes (GPs). While in BO the goal is to find a maximizer of the unknown function, in LSE one seeks to find all "sufficiently good" solutions. We propose an efficient confidence-bound based algorithm that treats BO and LSE in a unified fashion. It is effective in settings that are non-trivial to incorporate into existing algorithms, including cases with pointwise costs, heteroscedastic noise, and multi-fidelity setting. Our main result is a general regret guarantee that covers these aspects. Next, we consider GP optimization with robustness requirement: An adversary may perturb the returned design, and so we seek to find a robust maximizer in the case this occurs. This requirement is motivated by, e.g., settings where the functions during optimization and implementation stages are different. We propose a novel robust confidence-bound based algorithm. The rigorous regret guarantees for this algorithm are established and complemented with an algorithm-independent lower bound. We experimentally demonstrate that our robust approach consistently succeeds in finding a robust maximizer while standard BO methods fail. We then investigate the problem of GP optimization in which the reward function varies with time. The setting is motivated by many practical applications in which the function to be optimized is not static. We model the unknown reward function via a GP whose evolution obeys a simple Markov model. Two confidence-bound based algorithms with the ability to "forget" about old data are proposed. We obtain regret bounds for these algorithms that jointly depend on the time horizon and the rate at which the function varies. Finally, we consider the maximization of a set function subject to a cardinality constraint $k$ in the case a number of items $\tau$ from the returned set may be removed. One notable application is in batch BO where we need to select experiments to run, but some of them can fail. Our focus is on the worst-case adversarial setting, and we consider both \emph{submodular} (i.e., satisfies a natural notion of diminishing returns) and \emph{non-submodular} objectives. We propose robust algorithms that achieve constant-factor approximation guarantees. In the submodular case, the result on the maximum number of allowed removals is improved to $\tau = o(k)$ in comparison to the previously known $\tau=o(\sqrt{k})$. In the non-submodular case, we obtain new guarantees in the support selection and batch BO tasks. We empirically demonstrate the robust performance of our algorithms in these, as well as, in data summarization and influence maximization tasks

Near-Optimally Teaching the Crowd to Classify

How should we present training examples to learners to teach them classification rules? This is a natural problem when training workers for crowdsourcing labeling tasks, and is also motivated by challenges in data-driven online education. We propose a natural stochastic model of the learners, modeling them as randomly switching among hypotheses based on observed feedback. We then develop STRICT, an efficient algorithm for selecting examples to teach to workers. Our solution greedily maximizes a submodular surrogate objective function in order to select examples to show to the learners. We prove that our strategy is competitive with the optimal teaching policy. Moreover, for the special case of linear separators, we prove that an exponential reduction in error probability can be achieved. Our experiments on simulated workers as well as three real image annotation tasks on Amazon Mechanical Turk show the effectiveness of our teaching algorithm

Streaming Robust Submodular Maximization: A Partitioned Thresholding Approach

We study the classical problem of maximizing a monotone submodular function subject to a cardinality constraint k, with two additional twists: (i) elements arrive in a streaming fashion, and (ii) m items from the algorithm's memory are removed after the stream is finished. We develop a robust submodular algorithm STAR-T. It is based on a novel partitioning structure and an exponentially decreasing thresholding rule. STAR-T makes one pass over the data and retains a short but robust summary. We show that after the removal of any m elements from the obtained summary, a simple greedy algorithm STAR-T-GREEDY that runs on the remaining elements achieves a constant-factor approximation guarantee. In two different data summarization tasks, we demonstrate that it matches or outperforms existing greedy and streaming methods, even if they are allowed the benefit of knowing the removed subset in advance.Comment: To appear in NIPS 201

Combining Pessimism with Optimism for Robust and Efficient Model-Based Deep Reinforcement Learning

In real-world tasks, reinforcement learning (RL) agents frequently encounter situations that are not present during training time. To ensure reliable performance, the RL agents need to exhibit robustness against worst-case situations. The robust RL framework addresses this challenge via a worst-case optimization between an agent and an adversary. Previous robust RL algorithms are either sample inefficient, lack robustness guarantees, or do not scale to large problems. We propose the Robust Hallucinated Upper-Confidence RL (RH-UCRL) algorithm to provably solve this problem while attaining near-optimal sample complexity guarantees. RH-UCRL is a model-based reinforcement learning (MBRL) algorithm that effectively distinguishes between epistemic and aleatoric uncertainty and efficiently explores both the agent and adversary decision spaces during policy learning. We scale RH-UCRL to complex tasks via neural networks ensemble models as well as neural network policies. Experimentally, we demonstrate that RH-UCRL outperforms other robust deep RL algorithms in a variety of adversarial environments