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Luigi Nardi

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16 papers

2 author rows

ICML Conference 2025 Conference Paper

A Unified Framework for Entropy Search and Expected Improvement in Bayesian Optimization

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.

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UAI Conference 2025 Conference Paper

Exploring Exploration in Bayesian Optimization

Leonard Papenmeier
Nuojin Cheng
Stephen Becker
Luigi Nardi

A well-balanced exploration-exploitation trade-off is crucial for successful acquisition functions in Bayesian optimization. However, there is a lack of quantitative measures for exploration, making it difficult to analyze and compare different acquisition functions. This work introduces two novel approaches -observation traveling salesman distance and observation entropy- to quantify the exploration characteristics of acquisition functions based on their selected observations. Using these measures, we examine the explorative nature of several well-known acquisition functions across a diverse set of black-box problems, uncover links between exploration and empirical performance, and reveal new relationships among existing acquisition functions. Beyond enabling a deeper understanding of acquisition functions, these measures also provide a foundation for guiding their design in a more principled and systematic manner.

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ICML Conference 2025 Conference Paper

Understanding High-Dimensional Bayesian 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.

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ICLR Conference 2024 Conference Paper

A General Framework for User-Guided Bayesian Optimization

Carl Hvarfner
Frank Hutter
Luigi Nardi

The optimization of expensive-to-evaluate black-box functions is prevalent in various scientific disciplines. Bayesian optimization is an automatic, general and sample-efficient method to solve these problems with minimal knowledge of the the underlying function dynamics. However, the ability of Bayesian optimization to incorporate prior knowledge or beliefs about the function at hand in order to accelerate the optimization is limited, which reduces its appeal for knowledgeable practitioners with tight budgets. To allow domain experts to customize the optimization routine, we propose ColaBO, the first Bayesian-principled framework for incorporating prior beliefs beyond the typical kernel structure, such as the likely location of the optimizer or the optimal value. The generality of ColaBO makes it applicable across different Monte Carlo acquisition functions and types of user beliefs. We empirically demonstrate ColaBO's ability to substantially accelerate optimization when the prior information is accurate, and to retain approximately default performance when it is misleading.

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ICML Conference 2024 Conference Paper

Vanilla Bayesian Optimization Performs Great in High Dimensions

Carl Hvarfner
Erik Hellsten
Luigi Nardi

High-dimensional optimization problems have long been considered the Achilles’ heel of Bayesian optimization algorithms. Spurred by the curse of dimensionality, a large collection of algorithms aim to make BO more performant in this setting, commonly by imposing various simplifying assumptions on the objective, thereby decreasing its presumed complexity. In this paper, we identify the degeneracies that make vanilla BO poorly suited to high-dimensional tasks, and further show how existing algorithms address these degeneracies through the lens of model complexity. Motivated by the model complexity measure, we derive an enhancement to the prior assumptions that are typical of the vanilla BO algorithm, which reduces the complexity to manageable levels without imposing structural restrictions on the objective. Our modification - a simple scaling of the Gaussian process lengthscale prior in the dimensionality - reveals that standard BO works drastically better than previously thought in high dimensions. Our insights are supplemented by substantial out-performance of existing state-of-the-art on multiple commonly considered real-world high-dimensional tasks.

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NeurIPS Conference 2023 Conference Paper

Bounce: Reliable High-Dimensional Bayesian Optimization for Combinatorial and Mixed Spaces

Leonard Papenmeier
Luigi Nardi
Matthias Poloczek

Impactful applications such as materials discovery, hardware design, neural architecture search, or portfolio optimization require optimizing high-dimensional black-box functions with mixed and combinatorial input spaces. While Bayesian optimization has recently made significant progress in solving such problems, an in-depth analysis reveals that the current state-of-the-art methods are not reliable. Their performances degrade substantially when the unknown optima of the function do not have a certain structure. To fill the need for a reliable algorithm for combinatorial and mixed spaces, this paper proposes Bounce that relies on a novel map of various variable types into nested embeddings of increasing dimensionality. Comprehensive experiments show that Bounce reliably achieves and often even improves upon state-of-the-art performance on a variety of high-dimensional problems.

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NeurIPS Conference 2023 Conference Paper

PriorBand: Practical Hyperparameter Optimization in the Age of Deep Learning

Neeratyoy Mallik
Edward Bergman
Carl Hvarfner
Danny Stoll
Maciej Janowski
Marius Lindauer
Luigi Nardi
Frank Hutter

Hyperparameters of Deep Learning (DL) pipelines are crucial for their downstream performance. While a large number of methods for Hyperparameter Optimization (HPO) have been developed, their incurred costs are often untenable for modern DL. Consequently, manual experimentation is still the most prevalent approach to optimize hyperparameters, relying on the researcher's intuition, domain knowledge, and cheap preliminary explorations. To resolve this misalignment between HPO algorithms and DL researchers, we propose PriorBand, an HPO algorithm tailored to DL, able to utilize both expert beliefs and cheap proxy tasks. Empirically, we demonstrate PriorBand's efficiency across a range of DL benchmarks and show its gains under informative expert input and robustness against poor expert beliefs.

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NeurIPS Conference 2023 Conference Paper

Self-Correcting Bayesian Optimization through Bayesian Active Learning

Carl Hvarfner
Erik Hellsten
Frank Hutter
Luigi Nardi

Gaussian processes are the model of choice in Bayesian optimization and active learning. Yet, they are highly dependent on cleverly chosen hyperparameters to reach their full potential, and little effort is devoted to finding good hyperparameters in the literature. We demonstrate the impact of selecting good hyperparameters for GPs and present two acquisition functions that explicitly prioritize hyperparameter learning. Statistical distance-based Active Learning (SAL) considers the average disagreement between samples from the posterior, as measured by a statistical distance. SAL outperforms the state-of-the-art in Bayesian active learning on several test functions. We then introduce Self-Correcting Bayesian Optimization (SCoreBO), which extends SAL to perform Bayesian optimization and active learning simultaneously. SCoreBO learns the model hyperparameters at improved rates compared to vanilla BO, while outperforming the latest Bayesian optimization methods on traditional benchmarks. Moreover, we demonstrate the importance of self-correction on atypical Bayesian optimization tasks.

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ICLR Conference 2022 Conference Paper

$\pi$BO: Augmenting Acquisition Functions with User Beliefs for Bayesian Optimization

Carl Hvarfner
Danny Stoll
Artur L. F. Souza
Marius Lindauer
Frank Hutter
Luigi Nardi

Bayesian optimization (BO) has become an established framework and popular tool for hyperparameter optimization (HPO) of machine learning (ML) algorithms. While known for its sample-efficiency, vanilla BO can not utilize readily available prior beliefs the practitioner has on the potential location of the optimum. Thus, BO disregards a valuable source of information, reducing its appeal to ML practitioners. To address this issue, we propose $\pi$BO, an acquisition function generalization which incorporates prior beliefs about the location of the optimum in the form of a probability distribution, provided by the user. In contrast to previous approaches, $\pi$BO is conceptually simple and can easily be integrated with existing libraries and many acquisition functions. We provide regret bounds when $\pi$BO is applied to the common Expected Improvement acquisition function and prove convergence at regular rates independently of the prior. Further, our experiments show that $\pi$BO outperforms competing approaches across a wide suite of benchmarks and prior characteristics. We also demonstrate that $\pi$BO improves on the state-of-the-art performance for a popular deep learning task, with a $12.5\times$ time-to-accuracy speedup over prominent BO approaches.

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NeurIPS Conference 2022 Conference Paper

Increasing the Scope as You Learn: Adaptive Bayesian Optimization in Nested Subspaces

Leonard Papenmeier
Luigi Nardi
Matthias Poloczek

Recent advances have extended the scope of Bayesian optimization (BO) to expensive-to-evaluate black-box functions with dozens of dimensions, aspiring to unlock impactful applications, for example, in the life sciences, neural architecture search, and robotics. However, a closer examination reveals that the state-of-the-art methods for high-dimensional Bayesian optimization (HDBO) suffer from degrading performance as the number of dimensions increases, or even risk failure if certain unverifiable assumptions are not met. This paper proposes BAxUS that leverages a novel family of nested random subspaces to adapt the space it optimizes over to the problem. This ensures high performance while removing the risk of failure, which we assert via theoretical guarantees. A comprehensive evaluation demonstrates that BAxUS achieves better results than the state-of-the-art methods for a broad set of applications.

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NeurIPS Conference 2022 Conference Paper

Joint Entropy Search For Maximally-Informed Bayesian Optimization

Carl Hvarfner
Frank Hutter
Luigi Nardi

Information-theoretic Bayesian optimization techniques have become popular for optimizing expensive-to-evaluate black-box functions due to their non-myopic qualities. Entropy Search and Predictive Entropy Search both consider the entropy over the optimum in the input space, while the recent Max-value Entropy Search considers the entropy over the optimal value in the output space. We propose Joint Entropy Search (JES), a novel information-theoretic acquisition function that considers an entirely new quantity, namely the entropy over the joint optimal probability density over both input and output space. To incorporate this information, we consider the reduction in entropy from conditioning on fantasized optimal input/output pairs. The resulting approach primarily relies on standard GP machinery and removes complex approximations typically associated with information-theoretic methods. With minimal computational overhead, JES shows superior decision-making, and yields state-of-the-art performance for information-theoretic approaches across a wide suite of tasks. As a light-weight approach with superior results, JES provides a new go-to acquisition function for Bayesian optimization.

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IROS Conference 2021 Conference Paper

Learning of Parameters in Behavior Trees for Movement Skills

Matthias Mayr
Konstantinos I. Chatzilygeroudis
Faseeh Ahmad
Luigi Nardi
Volker Krüger

Reinforcement Learning (RL) is a powerful mathematical framework that allows robots to learn complex skills by trial-and-error. Despite numerous successes in many applications, RL algorithms still require thousands of trials to converge to high-performing policies, can produce dangerous behaviors while learning, and the optimized policies (usually modeled as neural networks) give almost zero explanation when they fail to perform the task. For these reasons, the adoption of RL in industrial settings is not common. Behavior Trees (BTs), on the other hand, can provide a policy representation that a) supports modular and composable skills, b) allows for easy interpretation of the robot actions, and c) provides an advantageous low-dimensional parameter space. In this paper, we present a novel algorithm that can learn the parameters of a BT policy in simulation and then generalize to the physical robot without any additional training. We leverage a physical simulator with a digital twin of our workstation, and optimize the relevant parameters with a black-box optimizer. We showcase the efficacy of our method with a 7-DOF KUKAiiwa manipulator in a task that includes obstacle avoidance and a contact-rich insertion (peg-in-hole), in which our method outperforms the baselines.

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ICRA Conference 2018 Conference Paper

SLAMBench2: Multi-Objective Head-to-Head Benchmarking for Visual SLAM

Bruno Bodin
Harry Wagstaff
Sajad Saeedi 0001
Luigi Nardi
Emanuele Vespa
John Mawer
Andy Nisbet
Mikel Luján

SLAM is becoming a key component of robotics and augmented reality (AR) systems. While a large number of SLAM algorithms have been presented, there has been little effort to unify the interface of such algorithms, or to perform a holistic comparison of their capabilities. This is a problem since different SLAM applications can have different functional and non-functional requirements. For example, a mobile phone-based AR application has a tight energy budget, while a UAV navigation system usually requires high accuracy. SLAMBench2 is a benchmarking framework to evaluate existing and future SLAM systems, both open and close source, over an extensible list of datasets, while using a comparable and clearly specified list of performance metrics. A wide variety of existing SLAM algorithms and datasets is supported, e. g. ElasticFusion, InfiniTAM, ORB-SLAM2, OKVIS, and integrating new ones is straightforward and clearly specified by the framework. SLAMBench2 is a publicly-available software framework which represents a starting point for quantitative, comparable and val-idatable experimental research to investigate trade-offs across SLAM systems.

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ICRA Conference 2017 Conference Paper

Application-oriented design space exploration for SLAM algorithms

Sajad Saeedi 0001
Luigi Nardi
Edward Johns
Bruno Bodin
Paul H. J. Kelly
Andrew J. Davison

In visual SLAM, there are many software and hardware parameters, such as algorithmic thresholds and GPU frequency, that need to be tuned; however, this tuning should also take into account the structure and motion of the camera. In this paper, we determine the complexity of the structure and motion with a few parameters calculated using information theory. Depending on this complexity and the desired performance metrics, suitable parameters are explored and determined. Additionally, based on the proposed structure and motion parameters, several applications are presented, including a novel active SLAM approach which guides the camera in such a way that the SLAM algorithm achieves the desired performance metrics. Real-world and simulated experimental results demonstrate the effectiveness of the proposed design space and its applications.

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ICRA Conference 2016 Conference Paper

Comparative design space exploration of dense and semi-dense SLAM

M. Zeeshan Zia
Luigi Nardi
Andrew Jack
Emanuele Vespa
Bruno Bodin
Paul H. J. Kelly
Andrew J. Davison

SLAM has matured significantly over the past few years, and is beginning to appear in serious commercial products. While new SLAM systems are being proposed at every conference, evaluation is often restricted to qualitative visualizations or accuracy estimation against a ground truth. This is due to the lack of benchmarking methodologies which can holistically and quantitatively evaluate these systems. Further investigation at the level of individual kernels and parameter spaces of SLAM pipelines is non-existent, which is absolutely essential for systems research and integration. We extend the recently introduced SLAMBench framework to allow comparing two state-of-the-art SLAM pipelines, namely KinectFusion and LSD-SLAM, along the metrics of accuracy, energy consumption, and processing frame rate on two different hardware platforms, namely a desktop and an embedded device. We also analyze the pipelines at the level of individual kernels and explore their algorithmic and hardware design spaces for the first time, yielding valuable insights.

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ICRA Conference 2015 Conference Paper

Introducing SLAMBench, a performance and accuracy benchmarking methodology for SLAM

Luigi Nardi
Bruno Bodin
M. Zeeshan Zia
John Mawer
Andy Nisbet
Paul H. J. Kelly
Andrew J. Davison
Mikel Luján

Real-time dense computer vision and SLAM offer great potential for a new level of scene modelling, tracking and real environmental interaction for many types of robot, but their high computational requirements mean that use on mass market embedded platforms is challenging. Meanwhile, trends in low-cost, low-power processing are towards massive parallelism and heterogeneity, making it difficult for robotics and vision researchers to implement their algorithms in a performance-portable way. In this paper we introduce SLAMBench, a publicly-available software framework which represents a starting point for quantitative, comparable and validatable experimental research to investigate trade-offs in performance, accuracy and energy consumption of a dense RGB-D SLAM system. SLAMBench provides a KinectFusion implementation in C++, OpenMP, OpenCL and CUDA, and harnesses the ICL-NUIM dataset of synthetic RGB-D sequences with trajectory and scene ground truth for reliable accuracy comparison of different implementation and algorithms. We present an analysis and breakdown of the constituent algorithmic elements of KinectFusion, and experimentally investigate their execution time on a variety of multicore and GPU-accelerated platforms. For a popular embedded platform, we also present an analysis of energy efficiency for different configuration alternatives.

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