Accelerating Look-ahead in Bayesian Optimization: Multilevel Monte Carlo is All you Need

We leverage multilevel Monte Carlo (MLMC) to improve the performance of multi-step look-ahead Bayesian optimization (BO) methods that involve nested expectations and maximizations. The complexity rate of naive Monte Carlo degrades for nested operations, whereas MLMC is capable of achieving the canonical Monte Carlo convergence rate for this type of problem, independently of dimension and without any smoothness assumptions. Our theoretical study focuses on the approximation improvements for one- and two-step look-ahead acquisition functions, but, as we discuss, the approach is generalizable in various ways, including beyond the context of BO. Findings are verified numerically and the benefits of MLMC for BO are illustrated on several benchmark examples. Code is available here https://github.com/Shangda-Yang/MLMCBO.

Joint Composite Latent Space Bayesian Optimization

Bayesian Optimization (BO) is a technique for sample-efficient black-box optimization that employs probabilistic models to identify promising input locations for evaluation. When dealing with composite-structured functions, such as f=g o 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 viable 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.

Bayesian Optimization of Function Networks with Partial Evaluations

Bayesian optimization is a framework for optimizing functions that are costly or time-consuming to evaluate. Recent work has considered Bayesian optimization of function networks (BOFN), where the objective function is computed via a network of functions, each taking as input the output of previous nodes in the network and additional parameters. Exploiting this network structure has been shown to yield significant performance improvements. Existing BOFN algorithms for general-purpose networks are required to evaluate the full network at each iteration. However, many real-world applications allow evaluating nodes individually. To take advantage of this opportunity, we propose a novel knowledge gradient acquisition function for BOFN that chooses which node to evaluate as well as the inputs for that node in a cost-aware fashion. This approach can dramatically reduce query costs by allowing the evaluation of part of the network at a lower cost relative to evaluating the entire network. We provide an efficient approach to optimizing our acquisition function and show it outperforms existing BOFN methods and other benchmarks across several synthetic and real-world problems. Our acquisition function is the first to enable cost-aware optimization of a broad class of function networks.

Unexpected Improvements to Expected Improvement for Bayesian Optimization

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.

Bayesian Optimization over High-Dimensional Combinatorial Spaces via Dictionary-based Embeddings

We consider the problem of optimizing expensive black-box functions over high-dimensional combinatorial spaces which arises in many science, engineering, and ML applications. We use Bayesian Optimization (BO) and propose a novel surrogate modeling approach for efficiently handling a large number of binary and categorical parameters. The key idea is to select a number of discrete structures from the input space (the dictionary) and use them to define an ordinal embedding for high-dimensional combinatorial structures. This allows us to use existing Gaussian process models for continuous spaces. We develop a principled approach based on binary wavelets to construct dictionaries for binary spaces, and propose a randomized construction method that generalizes to categorical spaces. We provide theoretical justification to support the effectiveness of the dictionary-based embeddings. Our experiments on diverse real-world benchmarks demonstrate the effectiveness of our proposed surrogate modeling approach over state-of-the-art BO methods.

Bayesian Optimization over Discrete and Mixed Spaces via Probabilistic Reparameterization

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.

Robust Multi-Objective Bayesian Optimization Under Input Noise

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.

Multi-Step Budgeted Bayesian Optimization with Unknown Evaluation Costs

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. This occurs 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 the 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. Finally, we show that our acquisition function outperforms existing methods in a variety of synthetic and real problems.

Sustainable AI: Environmental Implications, Challenges and Opportunities

This paper explores the environmental impact of the super-linear growth trends for AI from a holistic perspective, spanning Data, Algorithms, and System Hardware. We characterize the carbon footprint of AI computing by examining the model development cycle across industry-scale machine learning use cases and, at the same time, considering the life cycle of system hardware. Taking a step further, we capture the operational and manufacturing carbon footprint of AI computing and present an end-to-end analysis for what and how hardware-software design and at-scale optimization can help reduce the overall carbon footprint of AI. Based on the industry experience and lessons learned, we share the key challenges and chart out important development directions across the many dimensions of AI. We hope the key messages and insights presented in this paper can inspire the community to advance the field of AI in an environmentally-responsible manner.

Multi-Objective Bayesian Optimization over High-Dimensional Search Spaces

The ability to optimize multiple competing objective functions with high sample efficiency is imperative in many applied problems across science and industry. Multi-objective Bayesian optimization (BO) achieves strong empirical performance on such problems, but even with recent methodological advances, it has been restricted to simple, low-dimensional domains. Most existing BO methods exhibit poor performance on search spaces with more than a few dozen parameters. In this work we propose MORBO, a method for multi-objective Bayesian optimization over high-dimensional search spaces. MORBO performs local Bayesian optimization within multiple trust regions simultaneously, allowing it to explore and identify diverse solutions even when the objective functions are difficult to model globally. We show that MORBO significantly advances the state-of-the-art in sample-efficiency for several high-dimensional synthetic and real-world multi-objective problems, including a vehicle design problem with 222 parameters, demonstrating that MORBO is a practical approach for challenging and important problems that were previously out of reach for BO methods.