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

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

Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization

In this paper, we consider the problem of sequentially optimizing a black-box function $f$ based on noisy samples and bandit feedback. We assume that $f$ is smooth in the sense of having a bounded norm in some reproducing kernel Hilbert space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian process bandit optimization. We provide algorithm-independent lower bounds on the simple regret, measuring the suboptimality of a single point reported after $T$ rounds, and on the cumulative regret, measuring the sum of regrets over the $T$ chosen points. For the isotropic squared-exponential kernel in $d$ dimensions, we find that an average simple regret of $\epsilon$ requires $T = \Omega\big(\frac{1}{\epsilon^2} (\log\frac{1}{\epsilon})^{d/2}\big)$, and the average cumulative regret is at least $\Omega\big( \sqrt{T(\log T)^{d/2}} \big)$, thus matching existing upper bounds up to the replacement of $d/2$ by $2d+O(1)$ in both cases. For the Mat\'ern-$\nu$ kernel, we give analogous bounds of the form $\Omega\big( (\frac{1}{\epsilon})^{2+d/\nu}\big)$ and $\Omega\big( T^{\frac{\nu + d}{2\nu + d}} \big)$, and discuss the resulting gaps to the existing upper bounds.Comment: Appearing in COLT 2017. This version corrects a few minor mistakes in Table I, which summarizes the new and existing regret bound

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

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

Particle sorting by a structured microfluidic ratchet device with tunable selectivity: Theory and Experiment

We theoretically predict and experimentally demonstrate that several different particle species can be separated from each other by means of a ratchet device, consisting of periodically arranged triangular (ratchet) shaped obstacles. We propose an explicit algorithm for suitably tailoring the externally applied, time-dependent voltage protocol so that one or several, arbitrarily selected particle species are forced to migrate oppositely to all the remaining species. As an example we present numerical simulations for a mixture of five species, labelled according to their increasing size, so that species 2 and 4 simultaneously move in one direction and species 1, 3, and 5 in the other. The selection of species to be separated from the others can be changed at any time by simply adapting the voltage protocol. This general theoretical concept to utilize one device for many different sorting tasks is experimentally confirmed for a mixture of three colloidal particle species

Chiral particle separation by a non-chiral micro-lattice

We conceived a model experiment for a continuous separation strategy of chiral molecules (enantiomers) without the need of any chiral selector structure or derivatization agents: Micro-particles that only differ by their chirality are shown to migrate along different directions when driven by a steady fluid flow through a square lattice of cylindrical posts. In accordance with our numerical predictions, the transport directions of the enantiomers depend very sensitively on the orientation of the lattice relatively to the fluid flow

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