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Eytan Bakshy

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20 papers

2 author rows

TMLR Journal 2026 Journal Article

Distilled Thompson Sampling: Practical and Efficient Thompson Sampling via Imitation Learning

Hongseok Namkoong
Samuel Daulton
Eytan Bakshy

Thompson sampling (TS) has emerged as a robust technique for contextual bandit problems. However, TS requires posterior inference and optimization for action generation, prohibiting its use in many online platforms where latency and ease of deployment are of concern. We operationalize TS by proposing a novel imitation-learning-based algorithm that distills a TS policy into an explicit policy representation, allowing fast decision-making and easy deployment in mobile and server-based environments. Using batched data collected under the imitation policy, our algorithm iteratively performs offline updates to the TS policy, and learns a new explicit policy representation to imitate it. Empirically, our imitation policy achieves performance comparable to batch TS while allowing more than an order of magnitude reduction in decision-time latency. Buoyed by low latency and simplicity of implementation, our algorithm has been successfully deployed in multiple video upload systems for Meta. Using a randomized controlled trial, we show our algorithm resulted in significant improvements in video quality and watch time.

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NeurIPS Conference 2025 Conference Paper

Informed Initialization for Bayesian Optimization and Active Learning

Carl Hvarfner
David Eriksson
Eytan Bakshy
Maximilian Balandat

Bayesian Optimization (BO) is a widely used method for optimizing expensive black-box functions, relying on probabilistic surrogate models such as Gaussian Processes (GPs). The quality of the surrogate model is crucial for good optimization performance, especially in the few-shot setting where only a small number of batches of points can be evaluated. In this setting, the initialization plays a critical role in shaping the surrogate's predictive quality and guiding subsequent optimization. Despite this, practitioners typically rely on (quasi-)random designs to cover the input space. However, such approaches neglect two key factors: (a) random designs may not be space-filling, and (b) efficient hyperparameter learning during initialization is essential for high-quality prediction, which may conflict with space-filling designs. To address these limitations, we propose Hyperparameter-Informed Predictive Exploration (HIPE), a novel acquisition strategy that balances space-filling exploration with hyperparameter learning using information-theoretic principles. We derive a closed-form expression for HIPE in the GP setting and demonstrate its effectiveness through extensive experiments in active learning and few-shot BO. Our results show that HIPE outperforms standard initialization strategies in terms of predictive accuracy, hyperparameter identification, and optimization performance, particularly in large-batch, few-shot settings relevant to many real-world BO applications.

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ICML Conference 2025 Conference Paper

Scalable Gaussian Processes with Latent Kronecker Structure

Jihao Andreas Lin
Sebastian Ament
Maximilian Balandat
David Eriksson
José Miguel Hernández-Lobato
Eytan Bakshy

Applying Gaussian processes (GPs) to very large datasets remains a challenge due to limited computational scalability. Matrix structures, such as the Kronecker product, can accelerate operations significantly, but their application commonly entails approximations or unrealistic assumptions. In particular, the most common path to creating a Kronecker-structured kernel matrix is by evaluating a product kernel on gridded inputs that can be expressed as a Cartesian product. However, this structure is lost if any observation is missing, breaking the Cartesian product structure, which frequently occurs in real-world data such as time series. To address this limitation, we propose leveraging latent Kronecker structure, by expressing the kernel matrix of observed values as the projection of a latent Kronecker product. In combination with iterative linear system solvers and pathwise conditioning, our method facilitates inference of exact GPs while requiring substantially fewer computational resources than standard iterative methods. We demonstrate that our method outperforms state-of-the-art sparse and variational GPs on real-world datasets with up to five million examples, including robotics, automated machine learning, and climate applications.

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NeurIPS Conference 2024 Conference Paper

Active Learning for Derivative-Based Global Sensitivity Analysis with Gaussian Processes

Syrine Belakaria
Benjamin Letham
Janardhan R. Doppa
Barbara Engelhardt
Stefano Ermon
Eytan Bakshy

We consider the problem of active learning for global sensitivity analysis of expensive black-box functions. Our aim is to efficiently learn the importance of different input variables, e. g. , in vehicle safety experimentation, we study the impact of the thickness of various components on safety objectives. Since function evaluations are expensive, we use active learning to prioritize experimental resources where they yield the most value. We propose novel active learning acquisition functions that directly target key quantities of derivative-based global sensitivity measures (DGSMs) under Gaussian process surrogate models. We showcase the first application of active learning directly to DGSMs, and develop tractable uncertainty reduction and information gain acquisition functions for these measures. Through comprehensive evaluation on synthetic and real-world problems, our study demonstrates how these active learning acquisition strategies substantially enhance the sample efficiency of DGSM estimation, particularly with limited evaluation budgets. Our work paves the way for more efficient and accurate sensitivity analysis in various scientific and engineering applications.

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ICML Conference 2024 Conference Paper

Joint Composite Latent Space Bayesian Optimization

Natalie Maus
Zhiyuan (Jerry) Lin
Maximilian Balandat
Eytan Bakshy

Bayesian Optimization (BO) is a technique for sample-efficient black-box optimization that employs probabilistic models to identify promising input for evaluation. When dealing with composite-structured functions, such as $f=g \circ h$, evaluating a specific location $x$ yields observations of both the final outcome $f(x) = g(h(x))$ as well as the intermediate output(s) $h(x)$. Previous research has shown that integrating information from these intermediate outputs can enhance BO performance substantially. However, existing methods struggle if the outputs $h(x)$ are high-dimensional. Many relevant problems fall into this setting, including in the context of generative AI, molecular design, or robotics. To effectively tackle these challenges, we introduce Joint Composite Latent Space Bayesian Optimization (JoCo), a novel framework that jointly trains neural network encoders and probabilistic models to adaptively compress high-dimensional input and output spaces into manageable latent representations. This enables effective BO on these compressed representations, allowing JoCo to outperform other state-of-the-art methods in high-dimensional BO on a wide variety of simulated and real-world problems.

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UAI Conference 2024 Conference Paper

Response Time Improves Gaussian Process Models for Perception and Preferences

Michael Shvartsman
Benjamin Letham
Eytan Bakshy
Stephen Keeley

Models for human choice prediction in preference learning and perception science often use binary response data, requiring many samples to accurately learn latent utilities or perceptual intensities. The response time (RT) to make each choice captures additional information about the decision process, but existing models incorporating RTs for choice prediction do so in a fully parametric way or over discrete inputs. At the same time, state-of-the-art Gaussian process (GP) models of perception and preferences operate on choices only, ignoring RTs. We propose two approaches for incorporating RTs into GP preference and perception models. The first is based on stacking GP models, and the second uses a novel differentiable approximation to the likelihood of the diffusion decision model (DDM), the de-facto standard model for choice RTs. Our RT-choice GPs enable better latent value estimation and held-out choice prediction relative to baselines, which we demonstrate on three real-world multivariate datasets covering both human psychophysics and preference learning.

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NeurIPS Conference 2024 Conference Paper

Robust Gaussian Processes via Relevance Pursuit

Sebastian Ament
Elizabeth Santorella
David Eriksson
Ben Letham
Maximilian Balandat
Eytan Bakshy

Gaussian processes (GPs) are non-parametric probabilistic regression models that are popular due to their flexibility, data efficiency, and well-calibrated uncertainty estimates. However, standard GP models assume homoskedastic Gaussian noise, while many real-world applications are subject to non-Gaussian corruptions. Variants of GPs that are more robust to alternative noise models have been proposed, and entail significant trade-offs between accuracy and robustness, and between computational requirements and theoretical guarantees. In this work, we propose and study a GP model that achieves robustness against sparse outliers by inferring data-point-specific noise levels with a sequential selection procedure maximizing the log marginal likelihood that we refer to as relevance pursuit. We show, surprisingly, that the model can be parameterized such that the associated log marginal likelihood is strongly concave in the data-point-specific noise variances, a property rarely found in either robust regression objectives or GP marginal likelihoods. This in turn implies the weak submodularity of the corresponding subset selection problem, and thereby proves approximation guarantees for the proposed algorithm. We compare the model’s performance relative to other approaches on diverse regression and Bayesian optimization tasks, including the challenging but common setting of sparse corruptions of the labels within or close to the function range.

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ICML Conference 2023 Conference Paper

Hypervolume Knowledge Gradient: A Lookahead Approach for Multi-Objective Bayesian Optimization with Partial Information

Samuel Daulton
Maximilian Balandat
Eytan Bakshy

Bayesian optimization is a popular method for sample efficient multi-objective optimization. However, existing Bayesian optimization techniques fail to effectively exploit common and often-neglected problem structure such as decoupled evaluations, where objectives can be queried independently from one another and each may consume different resources, or multi-fidelity evaluations, where lower fidelity-proxies of the objectives can be evaluated at lower cost. In this work, we propose a general one-step lookahead acquisition function based on the Knowledge Gradient that addresses the complex question of what to evaluate when and at which design points in a principled Bayesian decision-theoretic fashion. Hence, our approach naturally addresses decoupled, multi-fidelity, and standard multi-objective optimization settings in a unified Bayesian decision making framework. By construction, our method is the one-step Bayes-optimal policy for hypervolume maximization. Empirically, we demonstrate that our method improves sample efficiency in a wide variety of synthetic and real-world problems. Furthermore, we show that our method is general-purpose and yields competitive performance in standard (potentially noisy) multi-objective optimization.

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NeurIPS Conference 2023 Conference Paper

Unexpected Improvements to Expected Improvement for Bayesian Optimization

Sebastian Ament
Samuel Daulton
David Eriksson
Maximilian Balandat
Eytan Bakshy

Expected Improvement (EI) is arguably the most popular acquisition function in Bayesian optimization and has found countless successful applications, but its performance is often exceeded by that of more recent methods. Notably, EI and its variants, including for the parallel and multi-objective settings, are challenging to optimize because their acquisition values vanish numerically in many regions. This difficulty generally increases as the number of observations, dimensionality of the search space, or the number of constraints grow, resulting in performance that is inconsistent across the literature and most often sub-optimal. Herein, we propose LogEI, a new family of acquisition functions whose members either have identical or approximately equal optima as their canonical counterparts, but are substantially easier to optimize numerically. We demonstrate that numerical pathologies manifest themselves in “classic” analytic EI, Expected Hypervolume Improvement (EHVI), as well as their constrained, noisy, and parallel variants, and propose corresponding reformulations that remedy these pathologies. Our empirical results show that members of the LogEI family of acquisition functions substantially improve on the optimization performance of their canonical counterparts and surprisingly, are on par with or exceed the performance of recent state-of-the-art acquisition functions, highlighting the understated role of numerical optimization in the literature.

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NeurIPS Conference 2022 Conference Paper

Bayesian Optimization over Discrete and Mixed Spaces via Probabilistic Reparameterization

Samuel Daulton
Xingchen Wan
David Eriksson
Maximilian Balandat
Michael A Osborne
Eytan Bakshy

Optimizing expensive-to-evaluate black-box functions of discrete (and potentially continuous) design parameters is a ubiquitous problem in scientific and engineering applications. Bayesian optimization (BO) is a popular, sample-efficient method that leverages a probabilistic surrogate model and an acquisition function (AF) to select promising designs to evaluate. However, maximizing the AF over mixed or high-cardinality discrete search spaces is challenging standard gradient-based methods cannot be used directly or evaluating the AF at every point in the search space would be computationally prohibitive. To address this issue, we propose using probabilistic reparameterization (PR). Instead of directly optimizing the AF over the search space containing discrete parameters, we instead maximize the expectation of the AF over a probability distribution defined by continuous parameters. We prove that under suitable reparameterizations, the BO policy that maximizes the probabilistic objective is the same as that which maximizes the AF, and therefore, PR enjoys the same regret bounds as the original BO policy using the underlying AF. Moreover, our approach provably converges to a stationary point of the probabilistic objective under gradient ascent using scalable, unbiased estimators of both the probabilistic objective and its gradient. Therefore, as the number of starting points and gradient steps increase, our approach will recover of a maximizer of the AF (an often-neglected requisite for commonly used BO regret bounds). We validate our approach empirically and demonstrate state-of-the-art optimization performance on a wide range of real-world applications. PR is complementary to (and benefits) recent work and naturally generalizes to settings with multiple objectives and black-box constraints.

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UAI Conference 2022 Conference Paper

Multi-objective Bayesian optimization over high-dimensional search spaces

Samuel Daulton
David Eriksson
Maximilian Balandat
Eytan Bakshy

Many real world scientific and industrial applications require optimizing multiple competing black-box objectives. When the objectives are expensive-to-evaluate, multi-objective Bayesian optimization (BO) is a popular approach because of its high sample efficiency. However, even with recent methodological advances, most existing multi-objective BO methods perform poorly on search spaces with more than a few dozen parameters and rely on global surrogate models that scale cubically with the number of observations. In this work we propose MORBO, a scalable method for multi-objective BO over high-dimensional search spaces. MORBO identifies diverse globally optimal solutions by performing BO in multiple local regions of the design space in parallel using a coordinated strategy. We show that MORBO significantly advances the state-of-the-art in sample efficiency for several high-dimensional synthetic problems and real world applications, including an optical display design problem and a vehicle design problem with 146 and 222 parameters, respectively. On these problems, where existing BO algorithms fail to scale and perform well, MORBO provides practitioners with order-of-magnitude improvements in sample efficiency over the current approach.

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ICML Conference 2022 Conference Paper

Robust Multi-Objective Bayesian Optimization Under Input Noise

Samuel Daulton
Sait Cakmak
Maximilian Balandat
Michael A. Osborne
Enlu Zhou
Eytan Bakshy

Bayesian optimization (BO) is a sample-efficient approach for tuning design parameters to optimize expensive-to-evaluate, black-box performance metrics. In many manufacturing processes, the design parameters are subject to random input noise, resulting in a product that is often less performant than expected. Although BO methods have been proposed for optimizing a single objective under input noise, no existing method addresses the practical scenario where there are multiple objectives that are sensitive to input perturbations. In this work, we propose the first multi-objective BO method that is robust to input noise. We formalize our goal as optimizing the multivariate value-at-risk (MVaR), a risk measure of the uncertain objectives. Since directly optimizing MVaR is computationally infeasible in many settings, we propose a scalable, theoretically-grounded approach for optimizing MVaR using random scalarizations. Empirically, we find that our approach significantly outperforms alternative methods and efficiently identifies optimal robust designs that will satisfy specifications across multiple metrics with high probability.

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NeurIPS Conference 2021 Conference Paper

Bayesian Optimization with High-Dimensional Outputs

Wesley J. Maddox
Maximilian Balandat
Andrew G. Wilson
Eytan Bakshy

Bayesian optimization is a sample-efficient black-box optimization procedure that is typically applied to a small number of independent objectives. However, in practice we often wish to optimize objectives defined over many correlated outcomes (or “tasks”). For example, scientists may want to optimize the coverage of a cell tower network across a dense grid of locations. Similarly, engineers may seek to balance the performance of a robot across dozens of different environments via constrained or robust optimization. However, the Gaussian Process (GP) models typically used as probabilistic surrogates for multi-task Bayesian optimization scale poorly with the number of outcomes, greatly limiting applicability. We devise an efficient technique for exact multi-task GP sampling that combines exploiting Kronecker structure in the covariance matrices with Matheron’s identity, allowing us to perform Bayesian optimization using exact multi-task GP models with tens of thousands of correlated outputs. In doing so, we achieve substantial improvements in sample efficiency compared to existing approaches that model solely the outcome metrics. We demonstrate how this unlocks a new class of applications for Bayesian optimization across a range of tasks in science and engineering, including optimizing interference patterns of an optical interferometer with 65, 000 outputs.

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NeurIPS Conference 2021 Conference Paper

Multi-Step Budgeted Bayesian Optimization with Unknown Evaluation Costs

Raul Astudillo
Daniel Jiang
Maximilian Balandat
Eytan Bakshy
Peter Frazier

Bayesian optimization (BO) is a sample-efficient approach to optimizing costly-to-evaluate black-box functions. Most BO methods ignore how evaluation costs may vary over the optimization domain. However, these costs can be highly heterogeneous and are often unknown in advance in many practical settings, such as hyperparameter tuning of machine learning algorithms or physics-based simulation optimization. Moreover, those few existing methods that acknowledge cost heterogeneity do not naturally accommodate a budget constraint on the total evaluation cost. This combination of unknown costs and a budget constraint introduces a new dimension to the exploration-exploitation trade-off, where learning about the cost incurs a cost itself. Existing methods do not reason about the various trade-offs of this problem in a principled way, leading often to poor performance. We formalize this claim by proving that the expected improvement and the expected improvement per unit of cost, arguably the two most widely used acquisition functions in practice, can be arbitrarily inferior with respect to the optimal non-myopic policy. To overcome the shortcomings of existing approaches, we propose the budgeted multi-step expected improvement, a non-myopic acquisition function that generalizes classical expected improvement to the setting of heterogeneous and unknown evaluation costs. We show that our acquisition function outperforms existing methods in a variety of synthetic and real problems.

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NeurIPS Conference 2021 Conference Paper

Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement

Samuel Daulton
Maximilian Balandat
Eytan Bakshy

Optimizing multiple competing black-box objectives is a challenging problem in many fields, including science, engineering, and machine learning. Multi-objective Bayesian optimization (MOBO) is a sample-efficient approach for identifying the optimal trade-offs between the objectives. However, many existing methods perform poorly when the observations are corrupted by noise. We propose a novel acquisition function, NEHVI, that overcomes this important practical limitation by applying a Bayesian treatment to the popular expected hypervolume improvement (EHVI) criterion and integrating over this uncertainty in the Pareto frontier. We argue that, even in the noiseless setting, generating multiple candidates in parallel is an incarnation of EHVI with uncertainty in the Pareto frontier and therefore can be addressed using the same underlying technique. Through this lens, we derive a natural parallel variant, qNEHVI, that reduces computational complexity of parallel EHVI from exponential to polynomial with respect to the batch size. qNEHVI is one-step Bayes-optimal for hypervolume maximization in both noisy and noiseless environments, and we show that it can be optimized effectively with gradient-based methods via sample average approximation. Empirically, we demonstrate not only that qNEHVI is substantially more robust to observation noise than existing MOBO approaches, but also that it achieves state-of-the-art optimization performance and competitive wall-times in large-batch environments.

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NeurIPS Conference 2020 Conference Paper

BoTorch: A Framework for Efficient Monte-Carlo Bayesian Optimization

Maximilian Balandat
Brian Karrer
Daniel Jiang
Samuel Daulton
Ben Letham
Andrew G. Wilson
Eytan Bakshy

Bayesian optimization provides sample-efficient global optimization for a broad range of applications, including automatic machine learning, engineering, physics, and experimental design. We introduce BoTorch, a modern programming framework for Bayesian optimization that combines Monte-Carlo (MC) acquisition functions, a novel sample average approximation optimization approach, auto-differentiation, and variance reduction techniques. BoTorch's modular design facilitates flexible specification and optimization of probabilistic models written in PyTorch, simplifying implementation of new acquisition functions. Our approach is backed by novel theoretical convergence results and made practical by a distinctive algorithmic foundation that leverages fast predictive distributions, hardware acceleration, and deterministic optimization. We also propose a novel "one-shot" formulation of the Knowledge Gradient, enabled by a combination of our theoretical and software contributions. In experiments, we demonstrate the improved sample efficiency of BoTorch relative to other popular libraries.

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NeurIPS Conference 2020 Conference Paper

Differentiable Expected Hypervolume Improvement for Parallel Multi-Objective Bayesian Optimization

Samuel Daulton
Maximilian Balandat
Eytan Bakshy

In many real-world scenarios, decision makers seek to efficiently optimize multiple competing objectives in a sample-efficient fashion. Multi-objective Bayesian optimization (BO) is a common approach, but many of the best-performing acquisition functions do not have known analytic gradients and suffer from high computational overhead. We leverage recent advances in programming models and hardware acceleration for multi-objective BO using Expected Hypervolume Improvement (EHVI)---an algorithm notorious for its high computational complexity. We derive a novel formulation of q-Expected Hypervolume Improvement (qEHVI), an acquisition function that extends EHVI to the parallel, constrained evaluation setting. qEHVI is an exact computation of the joint EHVI of q new candidate points (up to Monte-Carlo (MC) integration error). Whereas previous EHVI formulations rely on gradient-free acquisition optimization or approximated gradients, we compute exact gradients of the MC estimator via auto-differentiation, thereby enabling efficient and effective optimization using first-order and quasi-second-order methods. Our empirical evaluation demonstrates that qEHVI is computationally tractable in many practical scenarios and outperforms state-of-the-art multi-objective BO algorithms at a fraction of their wall time.

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NeurIPS Conference 2020 Conference Paper

High-Dimensional Contextual Policy Search with Unknown Context Rewards using Bayesian Optimization

Qing Feng
Ben Letham
Hongzi Mao
Eytan Bakshy

Contextual policies are used in many settings to customize system parameters and actions to the specifics of a particular setting. In some real-world settings, such as randomized controlled trials or A/B tests, it may not be possible to measure policy outcomes at the level of context—we observe only aggregate rewards across a distribution of contexts. This makes policy optimization much more difficult because we must solve a high-dimensional optimization problem over the entire space of contextual policies, for which existing optimization methods are not suitable. We develop effective models that leverage the structure of the search space to enable contextual policy optimization directly from the aggregate rewards using Bayesian optimization. We use a collection of simulation studies to characterize the performance and robustness of the models, and show that our approach of inferring a low-dimensional context embedding performs best. Finally, we show successful contextual policy optimization in a real-world video bitrate policy problem.

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NeurIPS Conference 2020 Conference Paper

Re-Examining Linear Embeddings for High-Dimensional Bayesian Optimization

Ben Letham
Roberto Calandra
Akshara Rai
Eytan Bakshy

Bayesian optimization (BO) is a popular approach to optimize expensive-to-evaluate black-box functions. A significant challenge in BO is to scale to high-dimensional parameter spaces while retaining sample efficiency. A solution considered in existing literature is to embed the high-dimensional space in a lower-dimensional manifold, often via a random linear embedding. In this paper, we identify several crucial issues and misconceptions about the use of linear embeddings for BO. We study the properties of linear embeddings from the literature and show that some of the design choices in current approaches adversely impact their performance. We show empirically that properly addressing these issues significantly improves the efficacy of linear embeddings for BO on a range of problems, including learning a gait policy for robot locomotion.

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JMLR Journal 2019 Journal Article

Bayesian Optimization for Policy Search via Online-Offline Experimentation

Benjamin Letham
Eytan Bakshy

Online field experiments are the gold-standard way of evaluating changes to real-world interactive machine learning systems. Yet our ability to explore complex, multi-dimensional policy spaces-such as those found in recommendation and ranking problems-is often constrained by the limited number of experiments that can be run simultaneously. To alleviate these constraints, we augment online experiments with an offline simulator and apply multi-task Bayesian optimization to tune live machine learning systems. We describe practical issues that arise in these types of applications, including biases that arise from using a simulator and assumptions for the multi-task kernel. We measure empirical learning curves which show substantial gains from including data from biased offline experiments, and show how these learning curves are consistent with theoretical results for multi-task Gaussian process generalization. We find that improved kernel inference is a significant driver of multi-task generalization. Finally, we show several examples of Bayesian optimization efficiently tuning a live machine learning system by combining offline and online experiments. [abs] [ pdf ][ bib ] [ appendix ] &copy JMLR 2019. ( edit, beta )

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