In many scientific and engineering applications, we are tasked with the optimisation of an expensive to evaluate black box function $f$. Traditional settings for this problem assume just the availability of this single function. However, in many cases, cheap approximations to $f$ may be obtainable. For example, the expensive real world behaviour of a robot can be approximated by a cheap computer simulation. We can use these approximations to eliminate low function value regions cheaply and use the expensive evaluations of $f$ in a small but promising region and speedily identify the optimum. We formalise this task as a \emph{multi-fidelity} bandit problem where the target function and its approximations are sampled from a Gaussian process. We develop MF-GP-UCB, a novel method based on upper confidence bound techniques. In our theoretical analysis we demonstrate that it exhibits precisely the above behaviour, and achieves better regret than strategies which ignore multi-fidelity information. Empirically, MF-GP-UCB outperforms such naive strategies and other multi-fidelity methods on several synthetic and real experiments.

Bandit methods for black-box optimisation, such as Bayesian optimisation, are used in a variety of applications including hyper-parameter tuning and experiment design. Recently, \emph{multi-fidelity} methods have garnered considerable attention since function evaluations have become increasingly expensive in such applications. Multi-fidelity methods use cheap approximations to the function of interest to speed up the overall optimisation process. However, most multi-fidelity methods assume only a finite number of approximations. In many practical applications however, a continuous spectrum of approximations might be available. For instance, when tuning an expensive neural network, one might choose to approximate the cross validation performance using less data $N$ and/or few training iterations $T$. Here, the approximations are best viewed as arising out of a continuous two dimensional space $(N,T)$. In this work, we develop a Bayesian optimisation method, BOCA, for this setting. We characterise its theoretical properties and show that it achieves better regret than than strategies which ignore the approximations. BOCA outperforms several other baselines in synthetic and real experiments.

Sophisticated gated recurrent neural network architectures like LSTMs and GRUs have been shown to be highly effective in a myriad of applications. We develop an un-gated unit, the statistical recurrent unit (SRU), that is able to learn long term dependencies in data by only keeping moving averages of statistics. The SRU's architecture is simple, un-gated, and contains a comparable number of parameters to LSTMs; yet, SRUs perform favorably to more sophisticated LSTM and GRU alternatives, often outperforming one or both in various tasks. We show the efficacy of SRUs as compared to LSTMs and GRUs in an unbiased manner by optimizing respective architectures' hyperparameters in a Bayesian optimization scheme for both synthetic and real-world tasks.

We introduce a simple permutation equivariant layer for deep learning with set structure.This type of layer, obtained by parameter-sharing, has a simple implementation and linear-time complexity in the size of each set. We use deep permutation-invariant networks to perform point-could classification and MNIST-digit summation, where in both cases the output is invariant to permutations of the input. In a semi-supervised setting, where the goal is make predictions for each instance within a set, we demonstrate the usefulness of this type of layer in set-outlier detection as well as semi-supervised learning with clustering side-information.

Autonomous systems can be used to search for sparse signals in a large space; e.g., aerial robots can be deployed to localize threats, detect gas leaks, or respond to distress calls. Intuitively, search algorithms may increase efficiency by collecting aggregate measurements summarizing large contiguous regions. However, most existing search methods either ignore the possibility of such region observations (e.g., Bayesian optimization and multi-armed bandits) or make strong assumptions about the sensing mechanism that allow each measurement to arbitrarily encode all signals in the entire environment (e.g., compressive sensing). We propose an algorithm that actively collects data to search for sparse signals using only noisy measurements of the average values on rectangular regions (including single points), based on the greedy maximization of information gain. We analyze our algorithm in 1d and show that it requires $\tilde{O}(\frac{n}{\mu^2}+k^2)$ measurements to recover all of $k$ signal locations with small Bayes error, where $\mu$ and $n$ are the signal strength and the size of the search space, respectively. We also show that active designs can be fundamentally more efficient than passive designs with region sensing, contrasting with the results of Arias-Castro, Candes, and Davenport (2013). We demonstrate the empirical performance of our algorithm on a search problem using satellite image data and in high dimensions.

Understanding the nature of dark energy, the mysterious force driving the accelerated expansion of the Universe, is a major challenge of modern cosmology. The next generation of cosmological surveys, specifically designed to address this issue, rely on accurate measurements of the apparent shapes of distant galaxies. However, shape measurement methods suffer from various unavoidable biases and therefore will rely on a precise calibration to meet the accuracy requirements of the science analysis. This calibration process remains an open challenge as it requires large sets of high quality galaxy images. To this end, we study the application of deep conditional generative models in generating realistic galaxy images. In particular we consider variations on conditional variational autoencoder and introduce a new adversarial objective for training of conditional generative networks. Our results suggest a reliable alternative to the acquisition of expensive high quality observations for generating the calibration data needed by the next generation of cosmological surveys.

A common problem in disciplines of applied Statistics research such as Astrostatistics is of estimating the posterior distribution of relevant parameters. Typically, the likelihoods for such models are computed via expensive experiments such as cosmological simulations of the universe. An urgent challenge in these research domains is to develop methods that can estimate the posterior with few likelihood evaluations. In this paper, we study active posterior estimation in a Bayesian setting when the likelihood is expensive to evaluate. Existing techniques for posterior estimation are based on generating samples representative of the posterior. Such methods do not consider efficiency in terms of likelihood evaluations. In order to be query efficient we treat posterior estimation in an active regression framework. We propose two myopic query strategies to choose where to evaluate the likelihood and implement them using Gaussian processes. Via experiments on a series of synthetic and real examples we demonstrate that our approach is significantly more query efficient than existing techniques and other heuristics for posterior estimation.

In many scientific and engineering applications, we are tasked with the optimisation of an expensive to evaluate black box function f. Traditional methods for this problem assume just the availability of this single function. However, in many cases, cheap approximations to f may be obtainable. For example, the expensive real world behaviour of a robot can be approximated by a cheap computer simulation. We can use these approximations to eliminate low function value regions cheaply and use the expensive evaluations of f in a small but promising region and speedily identify the optimum. We formalise this task as a multi-fidelity bandit problem where the target function and its approximations are sampled from a Gaussian process. We develop MF-GP-UCB, a novel method based on upper confidence bound techniques. In our theoretical analysis we demonstrate that it exhibits precisely the above behaviour, and achieves better regret than strategies which ignore multi-fidelity information. MF-GP-UCB outperforms such naive strategies and other multi-fidelity methods on several synthetic and real experiments.

We study a variant of the classical stochastic $K$-armed bandit where observing the outcome of each arm is expensive, but cheap approximations to this outcome are available. For example, in online advertising the performance of an ad can be approximated by displaying it for shorter time periods or to narrower audiences. We formalise this task as a \emph{multi-fidelity} bandit, where, at each time step, the forecaster may choose to play an arm at any one of $M$ fidelities. The highest fidelity (desired outcome) expends cost $\costM$. The $m$\ssth fidelity (an approximation) expends $\costm < \costM$ and returns a biased estimate of the highest fidelity. We develop \mfucb, a novel upper confidence bound procedure for this setting and prove that it naturally adapts to the sequence of available approximations and costs thus attaining better regret than naive strategies which ignore the approximations. For instance, in the above online advertising example, \mfucbs would use the lower fidelities to quickly eliminate suboptimal ads and reserve the larger expensive experiments on a small set of promising candidates. We complement this result with a lower bound and show that \mfucbs is nearly optimal under certain conditions.

Autonomous systems can be used to search for sparse signals in a large space; e.g., aerial robots can be deployed to localize threats, detect gas leaks, or respond to distress calls. Intuitively, search algorithms may increase efficiency by collecting aggregate measurements summarizing large contiguous regions. However, most existing search methods either ignore the possibility of such region observations (e.g., Bayesian optimization and multi-armed bandits) or make strong assumptions about the sensing mechanism that allow each measurement to arbitrarily encode all signals in the entire environment (e.g., compressive sensing). We propose an algorithm that actively collects data to search for sparse signals using only noisy measurements of the average values on rectangular regions (including single points), based on the greedy maximization of information gain. We analyze our algorithm in 1d and show that it requires $\tilde{O}(\frac{n}{\mu^2}+k^2)$ measurements to recover all of $k$ signal locations with small Bayes error, where $\mu$ and $n$ are the signal strength and the size of the search space, respectively. We also show that active designs can be fundamentally more efficient than passive designs with region sensing, contrasting with the results of Arias-Castro, Candes, and Davenport (2013). We demonstrate the empirical performance of our algorithm on a search problem using satellite image data and in high dimensions.