Maximilian Balandat

Datong Zhou, Maximilian Balandat, Claire Tomlin

The large scale deployment of Advanced Metering Infrastructure among residential energy customers has served as a boon for energy systems research relying on granular consumption data. Residential Demand Response aims to utilize the flexibility of consumers to reduce their energy usage during times when the grid is strained. Suitable incentive mechanisms to encourage customers to deviate from their usual behavior have to be implemented to correctly control the bids into the wholesale electricity market as a Demand Response provider. In this paper, we present a framework for short term load forecasting on an individual user level, and relate nonexperimental estimates of Demand Response efficacy, i.e. the estimated reduction of consumption during Demand Response events, to the variability of user consumption. We apply our framework on a data set from a residential Demand Response program in the Western United States. Our results suggest that users with more variable consumption patterns are more likely to reduce their consumption compared to users with a more regular consumption behavior.

Qie Hu, Frauke Oldewurtel, Maximilian Balandat et al.

The inter-temporal consumption flexibility of commercial buildings can be harnessed to improve the energy efficiency of buildings, or to provide ancillary service to the power grid. To do so, a predictive model of the building's thermal dynamics is required. In this paper, we identify a physics-based model of a multi-purpose commercial building including its heating, ventilation and air conditioning system during regular operation. We present our empirical results and show that large uncertainties in internal heat gains, due to occupancy and equipment, present several challenges in utilizing the building model for long-term prediction. In addition, we show that by learning these uncertain loads online and dynamically updating the building model, prediction accuracy is improved significantly.

Sebastian Ament, Samuel Daulton, David Eriksson et al.

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.

Hanxian Huang, Igor Fedorov, Andrey Gromov et al. · meta-ai, mila

Real-time AI experiences call for on-device large language models (OD-LLMs) optimized for efficient deployment on resource-constrained hardware. The most useful OD-LLMs produce near-real-time responses and exhibit broad hardware compatibility, maximizing user reach. We present a methodology for designing such models using hardware-in-the-loop architecture search under mobile latency constraints. This system is amenable to industry-scale deployment: it generates models deployable without custom kernels and compatible with standard mobile runtimes like Executorch. Our methodology avoids specialized attention mechanisms and instead uses attention skipping for long-context acceleration. Our approach jointly optimizes model architecture (layers, dimensions) and attention pattern. To efficiently evaluate candidates, we treat each as a pruned version of a pretrained backbone with inherited weights, thereby achieving high accuracy with minimal continued pretraining. We leverage the low cost of latency evaluation in a staged process: learning an accurate latency model first, then searching for the Pareto-frontier across latency and quality. This yields MobileLLM-Flash, a family of foundation models (350M, 650M, 1.4B) for efficient on-device use with strong capabilities, supporting up to 8k context length. MobileLLM-Flash delivers up to 1.8x and 1.6x faster prefill and decode on mobile CPUs with comparable or superior quality. Our analysis of Pareto-frontier design choices offers actionable principles for OD-LLM design.

Samuel Daulton, Xingchen Wan, David Eriksson et al.

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.

Aryan Deshwal, Sebastian Ament, Maximilian Balandat et al.

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.

Datong P. Zhou, Maximilian Balandat, Claire J. Tomlin

We evaluate the causal effect of hour-ahead price interventions on the reduction in residential electricity consumption using a data set from a large-scale experiment on 7,000 households in California. By estimating user-level counterfactuals using time-series prediction, we estimate an average treatment effect of ~0.10 kWh (11%) per intervention and household. Next, we leverage causal decision trees to detect treatment effect heterogeneity across users by incorporating census data. These decision trees depart from classification and regression trees, as we intend to estimate a causal effect between treated and control units rather than perform outcome regression. We compare the performance of causal decision trees with a simpler, yet more inaccurate k-means clustering approach that naively detects heterogeneity in the feature space, confirming the superiority of causal decision trees. Lastly, we comment on how our methods to detect heterogeneity can be used for targeting households to improve cost efficiency.

Poompol Buathong, Jiayue Wan, Raul Astudillo et al.

Bayesian optimization is a powerful framework for optimizing functions that are expensive or time-consuming to evaluate. Recent work has considered Bayesian optimization of function networks (BOFN), where the objective function is given by a network of functions, each taking as input the output of previous nodes in the network as well as additional parameters. Leveraging this network structure has been shown to yield significant performance improvements. Existing BOFN algorithms for general-purpose networks evaluate the full network at each iteration. However, many real-world applications allow for evaluating nodes individually. To exploit this, we propose a novel knowledge gradient acquisition function that chooses which node and corresponding inputs to evaluate in a cost-aware manner, thereby reducing query costs by evaluating only on a part of the network at each step. We provide an efficient approach to optimizing our acquisition function and show that 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.

Natalie Maus, Zhiyuan Jerry Lin, Maximilian Balandat et al.

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.

Samuel Daulton, David Eriksson, Maximilian Balandat et al.

Bayesian optimization (BO) is a popular technique for sample-efficient optimization of black-box functions. In many applications, the parameters being tuned come with a carefully engineered default configuration, and practitioners only want to deviate from this default when necessary. Standard BO, however, does not aim to minimize deviation from the default and, in practice, often pushes weakly relevant parameters to the boundary of the search space. This makes it difficult to distinguish between important and spurious changes and increases the burden of vetting recommendations when the optimization objective omits relevant operational considerations. We introduce BONSAI, a default-aware BO policy that prunes low-impact deviations from a default configuration while explicitly controlling the loss in acquisition value. BONSAI is compatible with a variety of acquisition functions, including expected improvement and upper confidence bound (GP-UCB). We theoretically bound the regret incurred by BONSAI, showing that, under certain conditions, it enjoys the same no-regret property as vanilla GP-UCB. Across many real-world applications, we empirically find that BONSAI substantially reduces the number of non-default parameters in recommended configurations while maintaining competitive optimization performance, with little effect on wall time.

Jihao Andreas Lin, Sebastian Ament, Louis C. Tiao et al.

Gaussian processes (GPs) are powerful and widely used probabilistic regression models, but their effectiveness in practice is often limited by the choice of kernel function. This kernel function is typically handcrafted from a small set of standard functions, a process that requires expert knowledge, results in limited adaptivity to data, and imposes strong assumptions on the hypothesis space. We study Empirical GPs, a principled framework for constructing flexible, data-driven GP priors that overcome these limitations. Rather than relying on standard parametric kernels, we estimate the mean and covariance functions empirically from a corpus of historical observations, enabling the prior to reflect rich, non-trivial covariance structures present in the data. Theoretically, we show that the resulting model converges to the GP that is closest (in KL-divergence sense) to the real data generating process. Practically, we formulate the problem of learning the GP prior from independent datasets as likelihood estimation and derive an Expectation-Maximization algorithm with closed-form updates, allowing the model handle heterogeneous observation locations across datasets. We demonstrate that Empirical GPs achieve competitive performance on learning curve extrapolation and time series forecasting benchmarks.

Zhiyuan Jerry Lin, Benjamin Letham, Samuel Dooley et al.

System prompts are a central control mechanism in modern AI systems, shaping behavior across conversations, tasks, and user populations. Yet they are difficult to tune when feedback is available only as aggregate metrics rather than per-example labels, failures, or critiques. We study this aggregate feedback setting as sample-constrained black-box optimization over discrete, variable-length text. We introduce ReElicit, a Bayesian optimization framework based on \emph{embedding by elicitation}. Given a task description, previously evaluated prompts, and scalar scores, an LLM elicits a compact, interpretable feature space and maps prompts into it. Leveraging a probabilistic Gaussian process surrogate, an acquisition function then selects target feature vectors, which the LLM realizes and refines into deployable system prompts. Re-eliciting the feature space as new evaluations arrive lets the representation adapt to the observed prompt-score history. We evaluate the setting using offline benchmark accuracy as a controlled aggregate proxy: the optimizer observes one scalar score per prompt and no per-example labels, errors, or critiques. Across ten system prompt optimization tasks with a 30 total evaluation budget, ReElicit achieves the strongest aggregate performance profile among representative aggregate-only prompt-optimization baselines. These results suggest that LLMs can serve as adaptive semantic representation builders, not only prompt generators, for Bayesian optimization over natural-language artifacts.

Shangda Yang, Vitaly Zankin, Maximilian Balandat et al.

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. Often these expectations must be computed by Monte Carlo (MC). The complexity rate of naive MC degrades for nested operations, whereas MLMC is capable of achieving the canonical MC convergence rate for this type of problem, independently of dimension and without any smoothness assumptions. Our theoretical study focuses on the approximation improvements for twoand three-step look-ahead acquisition functions, but, as we discuss, the approach is generalizable in various ways, including beyond the context of BO. Our findings are verified numerically and the benefits of MLMC for BO are illustrated on several benchmark examples. Code is available at https://github.com/Shangda-Yang/MLMCBO .

Sebastian Ament, Elizabeth Santorella, David Eriksson et al.

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.

Jihao Andreas Lin, Sebastian Ament, Maximilian Balandat et al.

A key task in AutoML is to model learning curves of machine learning models jointly as a function of model hyper-parameters and training progression. While Gaussian processes (GPs) are suitable for this task, naïve GPs require $\mathcal{O}(n^3m^3)$ time and $\mathcal{O}(n^2 m^2)$ space for $n$ hyper-parameter configurations and $\mathcal{O}(m)$ learning curve observations per hyper-parameter. Efficient inference via Kronecker structure is typically incompatible with early-stopping due to missing learning curve values. We impose $\textit{latent Kronecker structure}$ to leverage efficient product kernels while handling missing values. In particular, we interpret the joint covariance matrix of observed values as the projection of a latent Kronecker product. Combined with iterative linear solvers and structured matrix-vector multiplication, our method only requires $\mathcal{O}(n^3 + m^3)$ time and $\mathcal{O}(n^2 + m^2)$ space. We show that our GP model can match the performance of a Transformer on a learning curve prediction task.

Qing Feng, Samuel Daulton, Benjamin Letham et al.

Online experiments in internet systems, also known as A/B tests, are used for a wide range of system tuning problems, such as optimizing recommender system ranking policies and learning adaptive streaming controllers. Decision-makers generally wish to optimize for long-term treatment effects of the system changes, which often requires running experiments for a long time as short-term measurements can be misleading due to non-stationarity in treatment effects over time. The sequential experimentation strategies--which typically involve several iterations--can be prohibitively long in such cases. We describe a novel approach that combines fast experiments (e.g., biased experiments run only for a few hours or days) and/or offline proxies (e.g., off-policy evaluation) with long-running, slow experiments to perform sequential, Bayesian optimization over large action spaces in a short amount of time.

Katarzyna Kobalczyk, Zhiyuan Jerry Lin, Benjamin Letham et al.

For many real-world applications, feedback is essential in translating complex, nuanced, or subjective goals into quantifiable optimization objectives. We propose a language-in-the-loop framework that uses a large language model (LLM) to convert unstructured feedback in the form of natural language into scalar utilities to conduct BO over a numeric search space. Unlike preferential BO, which only accepts restricted feedback formats and requires customized models for each domain-specific problem, our approach leverages LLMs to turn varied types of textual feedback into consistent utility signals and to easily include flexible user priors without manual kernel design. At the same time, our method maintains the sample efficiency and principled uncertainty quantification of BO. We show that this hybrid method not only provides a more natural interface to the decision maker but also outperforms conventional BO baselines and LLM-only optimizers, particularly in feedback-limited regimes.

Samuel Daulton, Sait Cakmak, Maximilian Balandat et al.

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.

Raul Astudillo, Daniel R. Jiang, Maximilian Balandat et al.

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.

Carole-Jean Wu, Ramya Raghavendra, Udit Gupta et al.

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.

Samuel Daulton, David Eriksson, Maximilian Balandat et al.

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.

Wesley J. Maddox, Maximilian Balandat, Andrew Gordon Wilson et al.

Bayesian Optimization is a sample-efficient black-box optimization procedure that is typically applied to problems with 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 only model aggregate functions of the outcomes. 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 more than 65,000 outputs.

David Eriksson, Pierce I-Jen Chuang, Samuel Daulton et al.

When tuning the architecture and hyperparameters of large machine learning models for on-device deployment, it is desirable to understand the optimal trade-offs between on-device latency and model accuracy. In this work, we leverage recent methodological advances in Bayesian optimization over high-dimensional search spaces and multi-objective Bayesian optimization to efficiently explore these trade-offs for a production-scale on-device natural language understanding model at Facebook.

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, $q$NEHVI, that reduces computational complexity of parallel EHVI from exponential to polynomial with respect to the batch size. $q$NEHVI 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 $q$NEHVI 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.

Shali Jiang, Daniel R. Jiang, Maximilian Balandat et al.

Bayesian optimization is a sequential decision making framework for optimizing expensive-to-evaluate black-box functions. Computing a full lookahead policy amounts to solving a highly intractable stochastic dynamic program. Myopic approaches, such as expected improvement, are often adopted in practice, but they ignore the long-term impact of the immediate decision. Existing nonmyopic approaches are mostly heuristic and/or computationally expensive. In this paper, we provide the first efficient implementation of general multi-step lookahead Bayesian optimization, formulated as a sequence of nested optimization problems within a multi-step scenario tree. Instead of solving these problems in a nested way, we equivalently optimize all decision variables in the full tree jointly, in a ``one-shot'' fashion. Combining this with an efficient method for implementing multi-step Gaussian process ``fantasization,'' we demonstrate that multi-step expected improvement is computationally tractable and exhibits performance superior to existing methods on a wide range of benchmarks.

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.

Maximilian Balandat, Brian Karrer, Daniel R. Jiang et al.

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.

Maximilian Balandat, Walid Krichene, Claire Tomlin et al.

We study a general version of the adversarial online learning problem. We are given a decision set $\mathcal{X}$ in a reflexive Banach space $X$ and a sequence of reward vectors in the dual space of $X$. At each iteration, we choose an action from $\mathcal{X}$, based on the observed sequence of previous rewards. Our goal is to minimize regret, defined as the gap between the realized reward and the reward of the best fixed action in hindsight. Using results from infinite dimensional convex analysis, we generalize the method of Dual Averaging (or Follow the Regularized Leader) to our setting and obtain general upper bounds on the worst-case regret that subsume a wide range of results from the literature. Under the assumption of uniformly continuous rewards, we obtain explicit anytime regret bounds in a setting where the decision set is the set of probability distributions on a compact metric space $S$ whose Radon-Nikodym derivatives are elements of $L^p(S)$ for some $p > 1$. Importantly, we make no convexity assumptions on either the set $S$ or the reward functions. We also prove a general lower bound on the worst-case regret for any online algorithm. We then apply these results to the problem of learning in repeated continuous two-player zero-sum games, in which players' strategy sets are compact metric spaces. In doing so, we first prove that if both players play a Hannan-consistent strategy, then with probability 1 the empirical distributions of play weakly converge to the set of Nash equilibria of the game. We then show that, under mild assumptions, Dual Averaging on the (infinite-dimensional) space of probability distributions indeed achieves Hannan-consistency. Finally, we illustrate our results through numerical examples.

Datong Zhou, Maximilian Balandat, Claire Tomlin

The widespread deployment of Advanced Metering Infrastructure has made granular data of residential electricity consumption available on a large scale. Smart meters enable a two way communication between residential customers and utilities. One field of research that relies on such granular consumption data is Residential Demand Response, where individual users are incentivized to temporarily reduce their consumption during periods of high marginal cost of electricity. To quantify the economic potential of Residential Demand Response, it is important to estimate the reductions during Demand Response hours, taking into account the heterogeneity of electricity users. In this paper, we incorporate latent variables representing behavioral archetypes of electricity users into the process of short term load forecasting with Machine Learning methods, thereby differentiating between varying levels of energy consumption. The latent variables are constructed by fitting Conditional Mixture Models of Linear Regressions and Hidden Markov Models on smart meter readings of a Residential Demand Response program in the western United States. We observe a notable increase in the accuracy of short term load forecasts compared to the case without latent variables. We then estimate the reductions during Demand Response events conditional on the latent variables, and discover a higher DR reduction among users with automated smart home devices compared to those without.