Bayesian optimization (BO) is a powerful framework to optimize black-box expensive-to-evaluate functions via sequential interactions. In several important problems (e.g. drug discovery, circuit design, neural architecture search, etc.), though, such functions are defined over large $\textit{combinatorial and unstructured}$ spaces. This makes existing BO algorithms not feasible due to the intractable maximization of the acquisition function over these domains. To address this issue, we propose $\textbf{GameOpt}$, a novel game-theoretical approach to combinatorial BO. $\textbf{GameOpt}$ establishes a cooperative game between the different optimization variables, and selects points that are game $\textit{equilibria}$ of an upper confidence bound acquisition function. These are stable configurations from which no variable has an incentive to deviate$-$ analog to local optima in continuous domains. Crucially, this allows us to efficiently break down the complexity of the combinatorial domain into individual decision sets, making $\textbf{GameOpt}$ scalable to large combinatorial spaces. We demonstrate the application of $\textbf{GameOpt}$ to the challenging $\textit{protein design}$ problem and validate its performance on four real-world protein datasets. Each protein can take up to $20^{X}$ possible configurations, where $X$ is the length of a protein, making standard BO methods infeasible. Instead, our approach iteratively selects informative protein configurations and very quickly discovers highly active protein variants compared to other baselines.

In reinforcement learning (RL), rewards of states are typically considered additive, and following the Markov assumption, they are independent of states visited previously. In many important applications, such as coverage control, experiment design and informative path planning, rewards naturally have diminishing returns, i.e., their value decreases in light of similar states visited previously. To tackle this, we propose Submodular RL (subRL), a paradigm which seeks to optimize more general, non-additive (and history-dependent) rewards modelled via submodular set functions, which capture diminishing returns. Unfortunately, in general, even in tabular settings, we show that the resulting optimization problem is hard to approximate. On the other hand, motivated by the success of greedy algorithms in classical submodular optimization, we propose subPO, a simple policy gradient-based algorithm for subRL that handles non-additive rewards by greedily maximizing marginal gains. Indeed, under some assumptions on the underlying Markov Decision Process (MDP), subPO recovers optimal constant factor approximations of submodular bandits. Moreover, we derive a natural policy gradient approach for locally optimizing subRL instances even in large state- and action- spaces. We showcase the versatility of our approach by applying subPO to several applications, such as biodiversity monitoring, Bayesian experiment design, informative path planning, and coverage maximization. Our results demonstrate sample efficiency, as well as scalability to high-dimensional state-action spaces.

Bayesian optimization is a methodology to optimize black-box functions. Traditionally, it focuses on the setting where you can arbitrarily query the search space. However, many real-life problems do not offer this flexibility; in particular, the search space of the next query may depend on previous ones. Example challenges arise in the physical sciences in the form of local movement constraints, required monotonicity in certain variables, and transitions influencing the accuracy of measurements. Altogether, such *transition constraints* necessitate a form of planning. This work extends classical Bayesian optimization via the framework of Markov Decision Processes. We iteratively solve a tractable linearization of our utility function using reinforcement learning to obtain a policy that plans ahead for the entire horizon. This is a parallel to the optimization of an *acquisition function in policy space*. The resulting policy is potentially history-dependent and non-Markovian. We showcase applications in chemical reactor optimization, informative path planning, machine calibration, and other synthetic examples.

The increasing availability of massive data sets poses various challenges for machine learning. Prominent among these is learning models under hardware or human resource constraints. In such resource-constrained settings, a simple yet powerful approach is operating on small subsets of the data. Coresets are weighted subsets of the data that provide approximation guarantees for the optimization objective. However, existing coreset constructions are highly model-specific and are limited to simple models such as linear regression, logistic regression, and k-means. In this work, we propose a generic coreset construction framework that formulates the coreset selection as a cardinality-constrained bilevel optimization problem. In contrast to existing approaches, our framework does not require model-specific adaptations and applies to any twice differentiable model, including neural networks. We show the effectiveness of our framework for a wide range of models in various settings, including training non-convex models online and batch active learning.

A key challenge in science and engineering is to design experiments to learn about some unknown quantity of interest. Classical experimental design optimally allocates the experimental budget into measurements to maximize a notion of utility (e.g., reduction in uncertainty about the unknown quantity). We consider a rich setting, where the experiments are associated with states in a Markov chain, and we can only choose them by selecting a policy controlling the state transitions. This problem captures important applications, from exploration in reinforcement learning to spatial monitoring tasks. We propose an algorithm – markov-design – that efficiently selects policies whose measurement allocation provably converges to the optimal one. The algorithm is sequential in nature, adapting its choice of policies (experiments) using past measurements. In addition to our theoretical analysis, we demonstrate our framework on applications in ecological surveillance and pharmacology.

Certifiable, adaptive uncertainty estimates for unknown quantities are an essential ingredient of sequential decision-making algorithms. Standard approaches rely on problem-dependent concentration results and are limited to a specific combination of parameterization, noise family, and estimator. In this paper, we revisit the likelihood-based inference principle and propose to use \emph{likelihood ratios} to construct \emph{any-time valid} confidence sequences without requiring specialized treatment in each application scenario. Our method is especially suitable for problems with well-specified likelihoods, and the resulting sets always maintain the prescribed coverage in a model-agnostic manner. The size of the sets depends on a choice of estimator sequence in the likelihood ratio. We discuss how to provably choose the best sequence of estimators and shed light on connections to online convex optimization with algorithms such as Follow-the-Regularized-Leader. To counteract the initially large bias of the estimators, we propose a reweighting scheme that also opens up deployment in non-parametric settings such as RKHS function classes. We provide a \emph{non-asymptotic} analysis of the likelihood ratio confidence sets size for generalized linear models, using insights from convex duality and online learning. We showcase the practical strength of our method on generalized linear bandit problems, survival analysis, and bandits with various additive noise distributions.

We study adaptive sensing of Cox point processes, a widely used model from spatial statistics. We introduce three tasks: maximization of captured events, search for the maximum of the intensity function and learning level sets of the intensity function. We model the intensity function as a sample from a truncated Gaussian process, represented in a specially constructed positive basis. In this basis, the positivity constraint on the intensity function has a simple form. We show how the minimal description positive basis can be adapted to the covariance kernel, to non-stationarity and make connections to common positive bases from prior works. Our adaptive sensing algorithms use Langevin dynamics and are based on posterior sampling (Cox-Thompson) and top-two posterior sampling (Top2) principles. With latter, the difference between samples serves as a surrogate to the uncertainty. We demonstrate the approach using examples from environmental monitoring and crime rate modeling, and compare it to the classical Bayesian experimental design approach.

In Bayesian Optimization (BO) we study black-box function optimization with noisy point evaluations and Bayesian priors. Convergence of BO can be greatly sped up by batching, where multiple evaluations of the black-box function are performed in a single round. The main difficulty in this setting is to propose at the same time diverse and informative batches of evaluation points. In this work, we introduce DPP-Batch Bayesian Optimization (DPP-BBO), a universal framework for inducing batch diversity in sampling based BO by leveraging the repulsive properties of Determinantal Point Processes (DPP) to naturally diversify the batch sampling procedure. We illustrate this framework by formulating DPP-Thompson Sampling (DPP-TS) as a variant of the popular Thompson Sampling (TS) algorithm and introducing a Markov Chain Monte Carlo procedure to sample from it. We then prove novel Bayesian simple regret bounds for both classical batched TS as well as our counterpart DPP-TS; with the latter bound being tighter. Our real-world, as well as synthetic, experiments demonstrate improved performance of DPP-BBO over classical batching methods with Gaussian process and Cox process models.

Optimal experimental design seeks to determine the most informative allocation of experiments to infer an unknown statistical quantity. In this work, we investigate optimal design of experiments for {\em estimation of linear functionals in reproducing kernel Hilbert spaces (RKHSs)}. This problem has been extensively studied in the linear regression setting under an estimability condition, which allows estimating parameters without bias. We generalize this framework to RKHSs, and allow for the linear functional to be only approximately inferred, i.e., with a fixed bias. This scenario captures many important modern applications such as estimation of gradient maps, integrals and solutions to differential equations. We provide algorithms for constructing bias-aware designs for linear functionals. We derive non-asymptotic confidence sets for fixed and adaptive designs under sub-Gaussian noise, enabling us to certify estimation with bounded error with high probability.

Combinatorial bandits with semi-bandit feedback generalize multi-armed bandits, where the agent chooses sets of arms and observes a noisy reward for each arm contained in the chosen set. The action set satisfies a given structure such as forming a base of a matroid or a path in a graph. We focus on the pure-exploration problem of identifying the best arm with fixed confidence, as well as a more general setting, where the structure of the answer set differs from the one of the action set. Using the recently popularized game framework, we interpret this problem as a sequential zero-sum game and develop a CombGame meta-algorithm whose instances are asymptotically optimal algorithms with finite time guarantees. In addition to comparing two families of learners to instantiate our meta-algorithm, the main contribution of our work is a specific oracle efficient instance for best-arm identification with combinatorial actions. Based on a projection-free online learning algorithm for convex polytopes, it is the first computationally efficient algorithm which is asymptotically optimal and has competitive empirical performance.

Inhomogeneous Poisson point processes are widely used models of event occurrences. We address \emph{adaptive sensing of Poisson Point processes}, namely, maximizing the number of captured events subject to sensing costs. We encode prior assumptions on the rate function by modeling it as a member of a known \emph{reproducing kernel Hilbert space} (RKHS). By partitioning the domain into separate small regions, and using heteroscedastic linear regression, we propose a tractable estimator of Poisson process rates for two feedback models: \emph{count-record}, where exact locations of events are observed, and \emph{histogram} feedback, where only counts of events are observed. We derive provably accurate anytime confidence estimates for our estimators for sequentially acquired Poisson count data. Using these, we formulate algorithms based on optimism that provably incur sublinear count-regret. We demonstrate the practicality of the method on problems from crime modeling, revenue maximization as well as environmental monitoring.

Bayesian optimization and kernelized bandit algorithms are widely used techniques for sequential black box function optimization with applications in parameter tuning, control, robotics among many others. To be effective in high dimensional settings, previous approaches make additional assumptions, for example on low-dimensional subspaces or an additive structure. In this work, we go beyond the additivity assumption and use an orthogonal projection pursuit regression model, which strictly generalizes additive models. We present a two-stage algorithm motivated by experimental design to first decorrelate the additive components. Subsequently, the bandit optimization benefits from the statistically efficient additive model. Our method provably decorrelates the fully additive model and achieves optimal sublinear simple regret in terms of the number of function evaluations. To prove the rotation recovery, we derive novel concentration inequalities for linear regression on subspaces. In addition, we specifically address the issue of acquisition function optimization and present two domain dependent efficient algorithms. We validate the algorithm numerically on synthetic as well as real-world optimization problems.

We analyze the convergence rate of the randomized Newton-like method introduced by Qu et. al. (2016) for smooth and convex objectives, which uses random coordinate blocks of a Hessian-over-approximation matrix M instead of the true Hessian. The convergence analysis of the algorithm is challenging because of its complex dependence on the structure of M. However, we show that when the coordinate blocks are sampled with probability proportional to their determinant, the convergence rate depends solely on the eigenvalue distribution of matrix M, and has an analytically tractable form. To do so, we derive a fundamental new expectation formula for determinantal point processes. We show that determinantal sampling allows us to reason about the optimal subset size of blocks in terms of the spectrum of M. Additionally, we provide a numerical evaluation of our analysis, demonstrating cases where determinantal sampling is superior or on par with uniform sampling.

Coresets are small data summaries that are sufficient for model training. They can be maintained online, enabling efficient handling of large data streams under resource constraints. However, existing constructions are limited to simple models such as k-means and logistic regression. In this work, we propose a novel coreset construction via cardinality-constrained bilevel optimization. We show how our framework can efficiently generate coresets for deep neural networks, and demonstrate its empirical benefits in continual learning and in streaming settings.

Bayesian optimization is known to be difficult to scale to high dimensions, because the acquisition step requires solving a non-convex optimization problem in the same search space. In order to scale the method and keep its benefits, we propose an algorithm (LineBO) that restricts the problem to a sequence of iteratively chosen one-dimensional sub-problems that can be solved efficiently. We show that our algorithm converges globally and obtains a fast local rate when the function is strongly convex. Further, if the objective has an invariant subspace, our method automatically adapts to the effective dimension without changing the algorithm. When combined with the SafeOpt algorithm to solve the sub-problems, we obtain the first safe Bayesian optimization algorithm with theoretical guarantees applicable in high-dimensional settings. We evaluate our method on multiple synthetic benchmarks, where we obtain competitive performance. Further, we deploy our algorithm to optimize the beam intensity of the Swiss Free Electron Laser with up to 40 parameters while satisfying safe operation constraints.

We develop an efficient and provably no-regret Bayesian optimization (BO) algorithm for optimization of black-box functions in high dimensions. We assume a generalized additive model with possibly overlapping variable groups. When the groups do not overlap, we are able to provide the first provably no-regret \emph{polynomial time} (in the number of evaluations of the acquisition function) algorithm for solving high dimensional BO. To make the optimization efficient and feasible, we introduce a novel deterministic Fourier Features approximation based on numerical integration with detailed analysis for the squared exponential kernel. The error of this approximation decreases \emph{exponentially} with the number of features, and allows for a precise approximation of both posterior mean and variance. In addition, the kernel matrix inversion improves in its complexity from cubic to essentially linear in the number of data points measured in basic arithmetic operations.