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Optimization · Bayesian & Black-box Optimization

Haowei Wang, Jingyi Wang, Zhongxiang Dai, Nai-Yuan Chiang, Szu Hui Ng, Cosmin Petra

Constrained Bayesian optimization (CBO) methods have seen significant success in black-box optimization with constraints. One of the most commonly used CBO methods is the constrained expected improvement (CEI) algorithm. CEI is a natural extension of expected improvement (EI) when constraints are incorporated. However, the theoretical convergence rate of CEI has not been established. In this work, we study the convergence rate of CEI by analyzing its simple regret upper bound. First, we show that when the objective function $f$ and constraint function $c$ are assumed to each lie in a reproducing kernel Hilbert space (RKHS), CEI achieves the convergence rates of $\mathcal{O} \left(t^{-\frac{1}{2}}\log^{\frac{d+1}{2}}(t) \right) \ \text{and }\ \mathcal{O}\left(t^{\frac{-\nu}{2\nu+d}} \log^{\frac{\nu}{2\nu+d}}(t)\right)$ for the commonly used squared exponential and Matérn kernels, respectively. Second, we show that when $f$ and $c$ are assumed to be sampled from Gaussian processes (GPs), CEI achieves the same convergence rates with a high probability. Numerical experiments are performed to validate the theoretical analysis.

Deep Learning · Large Language Models

Lennart Schneider, Martin Wistuba, Aaron Klein, Jacek Golebiowski, Giovanni Zappella, Felice Antonio Merra

Optimal prompt selection is crucial for maximizing large language model (LLM) performance on downstream tasks, especially in black-box settings where models are only accessible via APIs.Black-box prompt selection is challenging due to potentially large, combinatorial search spaces, absence of gradient information, and high evaluation cost of prompts on a validation set.We propose HbBoPs, a novel method that combines a structural-aware deep kernel Gaussian Process with Hyperband as a multi-fidelity scheduler to efficiently select prompts.HbBoPs uses embeddings of instructions and few-shot exemplars, treating them as modular components within prompts.This enhances the surrogate model's ability to predict which prompt to evaluate next in a sample-efficient manner.Hyperband improves query-efficiency by adaptively allocating resources across different fidelity levels, reducing the number of validation instances required for evaluating prompts.Extensive experiments across ten diverse benchmarks and three LLMs demonstrate that HbBoPs outperforms state-of-the-art methods in both performance and efficiency.

Probabilistic Methods

Taeyoung Yun, Kiyoung Om, Jaewoo Lee, Sujin Yun, Jinkyoo Park

Optimizing high-dimensional and complex black-box functions is crucial in numerous scientific applications.While Bayesian optimization (BO) is a powerful method for sample-efficient optimization, it struggles with the curse of dimensionality and scaling to thousands of evaluations. Recently, leveraging generative models to solve black-box optimization problems has emerged as a promising framework.However, those methods often underperform compared to BO methods due to limited expressivity and difficulty of uncertainty estimation in high-dimensional spaces.To overcome these issues, we introduce \textbf{DiBO}, a novel framework for solving high-dimensional black-box optimization problems.Our method iterates two stages. First, we train a diffusion model to capture the data distribution and deep ensembles to predict function values with uncertainty quantification.Second, we cast the candidate selection as a posterior inference problem to balance exploration and exploitation in high-dimensional spaces. Concretely, we fine-tune diffusion models to amortize posterior inference.Extensive experiments demonstrate that our method outperforms state-of-the-art baselines across synthetic and real-world tasks. Our code is publicly available \href{https://github.com/umkiyoung/DiBO}{here}.

Optimization · Zero-order and Black-box Optimization

Toby Boyne, Jose Pablo Folch, Robert Lee, Behrang Shafei, Ruth Misener

We perform Bayesian optimization using a Gaussian process perspective on Bayesian Additive Regression Trees (BART). Our BART Kernel (BARK) uses tree agreement to define a posterior over piecewise-constant functions, and we explore the space of tree kernels using a Markov chain Monte Carlo approach. Where BART only samples functions, the resulting BARK model obtains samples of Gaussian processes defining distributions over functions, which allow us to build acquisition functions for Bayesian optimization. Our tree-based approach enables global optimization over the surrogate, even for mixed-feature spaces. Moreover, where many previous tree-based kernels provide uncertainty quantification over function values, our sampling scheme captures uncertainty over the tree structure itself. Our experiments show the strong performance of BARK on both synthetic and applied benchmarks, due to the combination of our fully Bayesian surrogate and the optimization procedure.

Optimization · Zero-order and Black-box Optimization

Leonard Papenmeier, Matthias Poloczek, Luigi Nardi

Recent work reported that simple Bayesian optimization (BO) methods perform well for high-dimensional real-world tasks, seemingly contradicting prior work and tribal knowledge. This paper investigates why. We identify underlying challenges that arise in high-dimensional BO and explain why recent methods succeed. Our empirical analysis shows that vanishing gradients caused by Gaussian process (GP) initialization schemes play a major role in the failures of high-dimensional Bayesian optimization (HDBO) and that methods that promote local search behaviors are better suited for the task. We find that maximum likelihood estimation (MLE) of GP length scales suffices for state-of-the-art performance. Based on this, we propose a simple variant of MLE called MSR that leverages these findings to achieve state-of-the-art performance on a comprehensive set of real-world applications. We present targeted experiments to illustrate and confirm our findings.

Probabilistic Methods · Bayesian Models and Methods

Nuojin Cheng, Leonard Papenmeier, Stephen Becker, Luigi Nardi

Bayesian optimization is a widely used method for optimizing expensive black-box functions, with Expected Improvement being one of the most commonly used acquisition functions. In contrast, information-theoretic acquisition functions aim to reduce uncertainty about the function’s optimum and are often considered fundamentally distinct from EI. In this work, we challenge this prevailing perspective by introducing a unified theoretical framework, Variational Entropy Search, which reveals that EI and information-theoretic acquisition functions are more closely related than previously recognized. We demonstrate that EI can be interpreted as a variational inference approximation of the popular information-theoretic acquisition function, named Max-value Entropy Search. Building on this insight, we propose VES-Gamma, a novel acquisition function that balances the strengths of EI and MES. Extensive empirical evaluations across both low- and high-dimensional synthetic and real-world benchmarks demonstrate that VES-Gamma is competitive with state-of-the-art acquisition functions and in many cases outperforms EI and MES.

Nuojin Cheng, Leonard Papenmeier, Stephen Becker, Luigi Nardi

Bayesian optimization is a widely used method for optimizing expensive black-box functions, with Expected Improvement being one of the most commonly used acquisition functions. In contrast, information-theoretic acquisition functions aim to reduce uncertainty about the function’s optimum and are often considered fundamentally distinct from EI. In this work, we challenge this prevailing perspective by introducing a unified theoretical framework, Variational Entropy Search, which reveals that EI and information-theoretic acquisition functions are more closely related than previously recognized. We demonstrate that EI can be interpreted as a variational inference approximation of the popular information-theoretic acquisition function, named Max-value Entropy Search. Building on this insight, we propose VES-Gamma, a novel acquisition function that balances the strengths of EI and MES. Extensive empirical evaluations across both low- and high-dimensional synthetic and real-world benchmarks demonstrate that VES-Gamma is competitive with state-of-the-art acquisition functions and in many cases outperforms EI and MES.

Optimization · Zero-order and Black-box Optimization

Jungtaek Kim

Bayesian optimization has attracted huge attention from diverse research areas in science and engineering, since it is capable of efficiently finding a global optimum of an expensive-to-evaluate black-box function. In general, a probabilistic regression model is widely used as a surrogate function to model an explicit distribution over function evaluations given an input to estimate and a training dataset. Beyond the probabilistic regression-based methods, density ratio estimation-based Bayesian optimization has been suggested in order to estimate a density ratio of the groups relatively close and relatively far to a global optimum. Developing this line of research further, supervised classifiers are employed to estimate a class probability for the two groups instead of a density ratio. However, the supervised classifiers used in this strategy are prone to be overconfident for known knowledge on global solution candidates. Supposing that we have access to unlabeled points, e.g., predefined fixed-size pools, we propose density ratio estimation-based Bayesian optimization with semi-supervised learning to solve this challenge. Finally, we show the empirical results of our methods and several baseline methods in two distinct scenarios with unlabeled point sampling and a fixed-size pool, and analyze the validity of our methods in diverse experiments.

Optimization · Zero-order and Black-box Optimization

Mujin Cheon, Jay Lee, Dong-Yeun Koh, Calvin Tsay

To avoid myopic behavior, multi-step lookahead Bayesian optimization (BO) algorithms consider the sequential nature of BO and have demonstrated promising results in recent years. However, owing to the curse of dimensionality, most of these methods make significant approximations or suffer scalability issues. This paper presents a novel reinforcement learning (RL)-based framework for multi-step lookahead BO in high-dimensional black-box optimization problems. The proposed method enhances the scalability and decision-making quality of multi-step lookahead BO by efficiently solving the sequential dynamic program of the BO process in a near-optimal manner using RL. We first introduce an Attention-DeepSets encoder to represent the state of knowledge to the RL agent and subsequently propose a multi-task, fine-tuning procedure based on end-to-end (encoder-RL) on-policy learning. We evaluate the proposed method, EARL-BO (Encoder Augmented RL for BO), on synthetic benchmark functions and hyperparameter tuning problems, finding significantly improved performance compared to existing multi-step lookahead and high-dimensional BO methods.

Probabilistic Methods · Bayesian Models and Methods

Melis Ilayda Bal, Pier Giuseppe Sessa, Mojmir Mutny, Andreas Krause

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.

Optimization · Zero-order and Black-box Optimization

Taiwo Adebiyi, Bach Do, Ruda Zhang

Bayesian optimization devolves the global optimization of a costly objective function to the global optimization of a sequence of acquisition functions. This inner-loop optimization can be catastrophically difficult if it involves posterior sample paths, especially in higher dimensions. We introduce an efficient global optimization strategy for posterior samples based on global rootfinding. It provides gradient-based optimizers with two sets of judiciously selected starting points, designed to combine exploration and exploitation. The number of starting points can be kept small without sacrificing optimization quality. Remarkably, even with just one point from each set, the global optimum is discovered most of the time. The algorithm scales practically linearly to high dimensions, breaking the curse of dimensionality. For Gaussian process Thompson sampling (GP-TS), we demonstrate remarkable improvement in both inner- and outer-loop optimization, surprisingly outperforming alternatives like EI and GP-UCB in most cases. Our approach also improves the performance of other posterior sample-based acquisition functions, such as variants of entropy search. Furthermore, we propose a sample-average formulation of GP-TS, which has a parameter to explicitly control exploitation and can be computed at the cost of one posterior sample.

Reinforcement Learning · Deep RL

Yu Heng Hung, Kai-Jie Lin, Yu-Heng Lin, Chien-Yi Wang, Cheng Sun, Ping-Chun Hsieh

Bayesian optimization (BO) offers an efficient pipeline for optimizing black-box functions with the help of a Gaussian process prior and an acquisition function (AF). Recently, in the context of single-objective BO, learning-based AFs witnessed promising empirical results given its favorable non-myopic nature. Despite this, the direct extension of these approaches to multi-objective Bayesian optimization (MOBO) suffer from the hypervolume identifiability issue, which results from the non-Markovian nature of MOBO problems. To tackle this, inspired by the non-Markovian RL literature and the success of Transformers in language modeling, we present a generalized deep Q-learning framework and propose BOFormer, which substantiates this framework for MOBO via sequence modeling. Through extensive evaluation, we demonstrate that BOFormer constantly achieves better performance than the benchmark rule-based and learning-based algorithms in various synthetic MOBO and real-world multi-objective hyperparameter optimization problems.

Theory · Probabilistic Methods

Xinyu Zhang, Daolang Huang, Samuel Kaski, Julien Martinelli

Preferential Bayesian Optimization (PBO) is a sample-efficient method to learn latent user utilities from preferential feedback over a pair of designs. It relies on a statistical surrogate model for the latent function, usually a Gaussian process, and an acquisition strategy to select the next candidate pair to get user feedback on. Due to the non-conjugacy of the associated likelihood, every PBO step requires a significant amount of computations with various approximate inference techniques. This computational overhead is incompatible with the way humans interact with computers, hindering the use of PBO in real-world cases. Building on the recent advances of amortized BO, we propose to circumvent this issue by fully amortizing PBO, meta-learning both the surrogate and the acquisition function. Our method comprises a novel transformer neural process architecture, trained using reinforcement learning and tailored auxiliary losses.On a benchmark composed of synthetic and real-world datasets, our method is several orders of magnitude faster than the usual Gaussian process-based strategies and often outperforms them in accuracy.

Optimization · Zero-order and Black-box Optimization

Dhruv Agarwal, Manoj Ghuhan Arivazhagan, Rajarshi Das, Sandesh Swamy, Sopan Khosla, Rashmi Gangadharaiah

Scaling test-time compute to search for optimal solutions is an important step towards building generally-capable language models that can reason. Recent work, however, shows that tasks of varying complexity require distinct search strategies to solve optimally, thus making it challenging to design a one-size-fits-all approach. Prior solutions either attempt to predict task difficulty to select the optimal search strategy, often infeasible in practice, or use a static, pre-defined strategy, e.g., repeated parallel sampling or greedy sequential search, which is sub-optimal. In this work, we argue for an alternative view using the probabilistic framework of Bayesian optimization (BO), where the search strategy is adapted dynamically based on the evolving uncertainty estimates of solutions as search progresses. To this end, we introduce Bayesian-OPRO (BOPRO)––a generalization of a recent method for in-context optimization, which iteratively samples from new proposal distributions by modifying the prompt to the LLM with a subset of its previous generations selected to explore or exploit different parts of the search space. We evaluate our method on word search, molecule optimization, and a joint hypothesis+program search task using a 1-D version of the challenging Abstraction and Reasoning Corpus (1D-ARC). Our results show that BOPRO outperforms all baselines in word search (≥10 points) and molecule optimization (higher quality and 17% fewer invalid molecules), but trails a best-k prompting strategy in program search. Our analysis reveals that despite the ability to balance exploration and exploitation using BOPRO, failure is likely due to the inability of code representation models in distinguishing sequences with low edit-distances.

Optimization · Zero-order and Black-box Optimization

Seunghun Lee, Jinyoung Park, Jaewon Chu, Minseo Yoon, Hyunwoo Kim

Bayesian Optimization (BO) has been recognized for its effectiveness in optimizing expensive and complex objective functions.Recent advancements in Latent Bayesian Optimization (LBO) have shown promise by integrating generative models such as variational autoencoders (VAEs) to manage the complexity of high-dimensional and structured data spaces.However, existing LBO approaches often suffer from the value discrepancy problem, which arises from the reconstruction gap between input and latent spaces.This value discrepancy problem propagates errors throughout the optimization process, leading to suboptimal outcomes.To address this issue, we propose a Normalizing Flow-based Bayesian Optimization (NF-BO), which utilizes normalizing flow as a generative model to establish one-to-one encoding function from the input space to the latent space, along with its left-inverse decoding function, eliminating the reconstruction gap. Specifically, we introduce SeqFlow, an autoregressive normalizing flow for sequence data.In addition, we develop a new candidate sampling strategy that dynamically adjusts the exploration probability for each token based on its importance.Through extensive experiments, our NF-BO method demonstrates superior performance in molecule generation tasks, significantly outperforming both traditional and recent LBO approaches.

Qiwen Zhu, Yanjie Wang, Shilv Cai, Liqun Chen, Jiahuan Zhou, Luxin Yan, Sheng Zhong 0001, Xu Zou 0002

In this paper, we introduce a novel approach to single-image super-resolution (SISR) that balances perceptual quality and distortion through multi-objective optimization (MOO). Traditional pixel-based distortion metrics like PSNR and SSIM often fail to align with human perceptual quality, resulting in blurry outputs despite high scores. To address this, we propose the Multi-Objective Bayesian Optimization Super-Resolution (MOBOSR) framework, which dynamically adjusts loss weights during training. This reduces the need for manual hyperparameter tuning and lessens computational demands compared to AutoML. Our method conceptualizes the relationship between loss weights and image quality assessment (IQA) metrics as black-box objective functions, optimized to achieve an optimal perception-distortion Pareto frontier. Extensive experiments demonstrate that MOBOSR surpasses current state-of-the-art methods in both perception and distortion, significantly advancing the perception-distortion Pareto frontier. Our work lays a foundation for future exploration of the balance between perceptual quality and fidelity in image restoration tasks. Source codes and pretrained models are available at: https://github.com/ZhuKeven/MOBOSR.

Taeyoung Yun, Kanghoon Lee, Sujin Yun, Ilmyung Kim, Won-Woo Jung, Min-Cheol Kwon, Kyujin Choi, Yoohyeon Lee, Jinkyoo Park

Complex urban road networks with high vehicle occupancy frequently face severe traffic congestion. Designing an effective strategy for managing multiple traffic lights plays a crucial role in managing congestion. However, most current traffic light management systems rely on human-crafted decisions, which may not adapt well to diverse traffic patterns. In this paper, we delve into two pivotal design components of the traffic light management system that can be dynamically adjusted to various traffic conditions: phase combination and phase time allocation. While numerous studies have sought an efficient strategy for managing traffic lights, most of these approaches consider a fixed traffic pattern and are limited to relatively small road networks. To overcome these limitations, we introduce a novel and practical framework to formulate the optimization of such design components using an offline meta black-box optimization. We then present a simple yet effective method to efficiently find a solution for the aforementioned problem. In our framework, we first collect an offline meta dataset consisting of pairs of design choices and corresponding congestion measures from various traffic patterns. After collecting the dataset, we employ the Attentive Neural Process (ANP) to predict the impact of the proposed design on congestion across various traffic patterns with well-calibrated uncertainty. Finally, Bayesian optimization, with ANP as a surrogate model, is utilized to find an optimal design for unseen traffic patterns through limited online simulations. Our experiment results show that our method outperforms state-of-the-art baselines on complex road networks in terms of the number of waiting vehicles. Surprisingly, the deployment of our method into a real-world traffic system was able to improve traffic throughput by 4.80% compared to the original strategy.

Abdoulatif Cissé, Xenophon Evangelopoulos, Sam Carruthers, Vladimir V. Gusev, Andrew I. Cooper

Robotics and automation offer massive acceleration for solving intractable, multivariate scientific problems such as materials discovery, but the available search spaces can be dauntingly large. Bayesian optimization has emerged as a popular sample-efficient optimization engine, thriving in tasks where no analytic form of the target function/property is known. Here, we exploit expert human knowledge in the form of hypotheses to direct Bayesian searches more quickly to promising regions of chemical space. Previous methods have used underlying distributions derived from existing experimental measurements, which is unfeasible for new, unexplored scientific tasks. Also, such distributions cannot capture intricate hypotheses. Our proposed method uses expert human hypotheses to generate improved seed samples. Unpromising seeds are automatically discounted, while promising seeds are used to augment the surrogate model data, thus achieving better-informed sampling. This process continues in a global versus local search fashion, organized in a bilevel optimization framework. We validate the performance of our method on a range of synthetic functions and demonstrate its practical utility on a real chemical design task where the use of expert hypotheses accelerates the search performance significantly.

Xudong Yin, Yao Yang

Explanatory Artificial Intelligence plays a vital role in machine learning, due to its widespread application in decision-making scenarios, e.g., credit lending. Counterfactual Explanation (CFE) is a new kind of explanatory method that involves asking “what if ”, i.e. what would have happened if model inputs slightly change. To answer the question, Counterfactual Explanation aims at finding a minimum perturbation in model inputs leading to a different model decision. Compared with model-agnostic approaches, model-specific CFE approaches designed only for specific type of models usually have better performance in finding optimal counterfactual perturbations, owing to access to the inner workings of models. To deal with this dilemma, this work first proposes CMAES-based Counterfactual Explanations (CMACE): an effective model-agnostic counterfactual generating approach based on Covariance Matrix Adaptation Evolution Strategy (CMA-ES) and a warm starting scheme that provides good initialization of the counterfactual's mean and covariance parameters for CMA-ES taking advantage of prior information of training samples. CMACE significantly outperforms another state-of-art (SOTA) model-agnostic approach (Bayesian Counterfactual Generator, BayCon) with various experimental settings. Extensive experiments also demonstrate that CMACE is superior to a SOTA model-specific approach (Flexible Optimizable Counterfactual Explanations for Tree Ensembles, FOCUS) that is designed for tree-based models using gradient-based optimization.

Alaleh Ahmadianshalchi, Syrine Belakaria, Janardhan Rao Doppa

We consider the problem of constrained multi-objective optimization over black-box objectives, with user-defined preferences, with a largely infeasible input space. Our goal is to approximate the optimal Pareto set from the small fraction of feasible inputs. The main challenges include huge design space, multiple objectives, numerous constraints, and rare feasible inputs identified only through expensive experiments. We present PAC-MOO, a novel preference-aware multi-objective Bayesian optimization algorithm to solve this problem. It leverages surrogate models for objectives and constraints to intelligently select the sequence of inputs for evaluation to achieve the target goal.