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A General Framework for Multi-fidelity Bayesian Optimization with Gaussian Processes
Jialin Song, Yuxin Chen, Yisong Yue
TL;DR
Multi-fidelity optimization seeks to use information sources with different costs and accuracies to optimize an expensive unknown function, while prior approaches may impose restrictive fidelity assumptions or lack theoretical guarantees. MF-MI-Greedy models dependent fidelities with Gaussian processes and selects actions using cost-sensitive mutual information. The paper proves low regret and reports evaluations on synthetic and real-world datasets.
Problem
Multi-fidelity Bayesian optimization must use differently priced information sources to optimize an expensive unknown function, while prior methods may rely on restrictive fidelity assumptions or heuristics with limited theoretical guarantees.
Source: §1 · pp. 1–2Method
Results
The paper proves low regret with a cost-aware regret notion and reports empirical performance on synthetic and real-world datasets.
Source: §1 · p. 2Takeaways & Limitations
The framework can lift off-the-shelf single-fidelity GP optimization algorithms to the multi-fidelity setting while preserving their original regret bounds.
Source: §7 · p. 13The stated model assumes fidelities are mutually dependent through a fixed, possibly unknown, joint distribution.
Source: §3 · p. 5
Abstract
from arXiv · showhide
How can we efficiently gather information to optimize an unknown function, when presented with multiple, mutually dependent information sources with different costs? For example, when optimizing a robotic system, intelligently trading off computer simulations and real robot testings can lead to significant savings. Existing methods, such as multi-fidelity GP-UCB or Entropy Search-based approaches, either make simplistic assumptions on the interaction among different fidelities or use simple heuristics that lack theoretical guarantees. In this paper, we study multi-fidelity Bayesian optimization with complex structural dependencies among multiple outputs, and propose MF-MI-Greedy, a principled algorithmic framework for addressing this problem. In particular, we model different fidelities using additive Gaussian processes based on shared latent structures with the target function. Then we use cost-sensitive mutual information gain for efficient Bayesian global optimization. We propose a simple notion of regret which incorporates the cost of different fidelities, and prove that MF-MI-Greedy achieves low regret. We demonstrate the strong empirical performance of our algorithm on both synthetic and real-world datasets.
1. Introduction
Expensive real-world evaluations make it useful to gather information from cheaper, less accurate sources. MF-MI-Greedy uses cross-fidelity mutual information to guide these choices and is presented with theoretical and empirical evaluation.
1. Introduction
- Cheaper, less accurate simulations can provide an alternative source of information when real experiments are labor-intensive and time-consuming.The introduction gives robotic simulation as an example.Source: §1 · pp. 1–2
- Existing multi-fidelity approaches may rely on strict relationships between fidelity quality and cost or two-stage query selection, potentially limiting practical use and producing sub-optimal selections.Source: §1 · p. 2
- MF-MI-Greedy maximizes mutual information across fidelity levels and connects them for joint posterior updates and hyperparameter optimization.The framework addresses multi-fidelity Bayesian optimization with complex structural dependencies among outputs.Source: §1 · p. 2
- The paper proves low regret using a notion that incorporates the costs of different fidelities.Source: §1 · p. 2
- The authors report empirical evaluation of MF-MI-Greedy on synthetic and real-world datasets.Source: §1 · p. 2
2. Background and Related Work
The section reviews Gaussian-process modeling and Bayesian optimization, then places multi-fidelity methods in that context. It highlights that prior multi-fidelity approaches use particular dependency assumptions or heuristics, and that theoretical performance is not well understood for some entropy-search methods.
2.1 Background on Gaussian Processes
- A Gaussian process models indexed random variables whose every finite subset has a multivariate Gaussian distribution.Source: §2.1 · p. 3
2.2 Bayesian Optimization via Gaussian Processes
- GP-UCB selects queries using an upper-confidence bound and has a cumulative-regret bound connected to the mutual information gained through queries.Source: §2.2 · p. 3
- Entropy-search methods select informative queries to learn the optimizer's location or value; max-value entropy search avoids dependence on search-space dimension and has regret analysis.The passage contrasts this analysis with earlier entropy-search methods.Source: §2.2 · pp. 3–4
- Random Fourier features are described as a dimension-reduction tool for speeding GP regression, and the proposed framework can use different single-fidelity optimization subroutines.Source: §2.2 · p. 4
- Multi-output Gaussian processes model correlated outputs through shared latent processes, while this paper uses an additive fidelity model in experiments for efficiency.Source: §2.2 · p. 4
- Prior multi-fidelity methods include autoregressive co-kriging and entropy-search approaches using co-kriging or convolved Gaussian processes.Source: §2.2 · p. 4
- For multi-fidelity entropy search and predictive entropy search heuristics, theoretical performance is little understood despite promising empirical results on some datasets.Source: §2.2 · p. 5
- Some prior methods impose fidelity structures tied to approximation error or select action and fidelity in separate stages; the paper instead jointly selects both.The authors state that separate-stage selection may choose target-fidelity queries even when a low-fidelity function differs only by a constant shift.Source: §2.2 · p. 5
3. Problem Statement
The problem is to maximize a target payoff function while using costly target queries and potentially informative auxiliary fidelities. The formal setup assigns query costs and a total budget to an adaptive policy.
3. Problem Statement
- The target function is observed through noisy queries, and auxiliary functions provide additional noisy observations as lower-fidelity versions.A physical-system reward and simulated payoff are given as an example.Source: §3 · p. 5
- The fidelities are mutually dependent under a fixed, possibly unknown joint distribution modeled as a multiple-output Gaussian process.Each fidelity has a GP marginal distribution.Source: §3 · p. 5
- Each query action has a fidelity-specific cost, while lower-fidelity actions receive minimal immediate reward even though they may inform future target-function decisions.Source: §3 · pp. 5–6
- A policy adaptively selects actions from observations, subject to a fixed total cost budget, to maximize expected cumulative reward.Source: §3 · p. 6
- The objective is to identify a target-fidelity action with performance close to the maximum of the unknown payoff function.Source: §3 · p. 6
- When a target-fidelity query is much more costly and the budget permits only one such query, the problem reduces to choosing informative lower-fidelity actions before that final query.The paper relates this case to the optimal value-of-information problem.Source: §3 · p. 6
4. The Multi-fidelity BO Framework
MF-MI-Greedy balances low-fidelity information gathering against target-fidelity optimization under a budget. Its exploration subroutine greedily selects cost-effective actions and uses stopping rules to reserve resources for target queries.
4. The Multi-fidelity BO Framework
- MF-MI-Greedy alternates low-fidelity exploration with target-fidelity optimization, balancing information per unit cost against the more expensive target actions.Source: §4 · p. 6
- Low-fidelity action quality is measured by information gain about the target payoff, and an exploration phase seeks actions that are informative, cost-effective relative to target actions, and not excessively exploratory.The information gain is described as entropy reduction in the target-payoff posterior.Source: §4 · pp. 6–7
- Explore-LF greedily selects low-fidelity actions by benefit-cost ratio and stops when the budget is exhausted, a target action has a better ratio, or cumulative benefit-cost ratio is small.Source: §4 · p. 7
- After exploration, the method updates the joint-GP posterior using all observations and selects a target-fidelity action with a single-fidelity optimization subroutine.Source: §4 · p. 7
11 Output: Selected set of items L from lower fidelities
MF-MI-Greedy uses a single-fidelity Bayesian optimization subroutine to select target-fidelity actions, then repeats the process until the budget is exhausted and estimates the target optimizer.
11 Output: Selected set of items L from lower fidelities
- At each round, the target-fidelity action comes from a single-fidelity Bayesian optimization subroutine, which may be an off-the-shelf method.MF-MI-Greedy repeats this process until the preset budget is exhausted, then outputs an estimator of the target function optimizer.Source: §11 · p. 8
5. Theoretical Analysis
The theoretical analysis introduces a cost-aware notion of multi-fidelity regret and bounds the cumulative regret of MF-MI-Greedy through mutual information. It also identifies conditions under which informative, lower-cost fidelities make the method more advantageous and supports no-regret guarantees for several optimization subroutines.
5.1 Multi-fidelity Regret
- An episode consists of lower-fidelity actions followed by one target-fidelity action, and its regret uses the episode cost and target-fidelity reward.Cumulative regret compares a policy's cumulative reward with the best possible reward under the same budget.Source: §5.1 · pp. 8–9
5.2 Regret Analysis
- Theorem 4 bounds cumulative regret as a function of mutual information between the target fidelity and actions selected by the algorithm.The theorem's setup also gives a high-probability upper bound for the single-fidelity comparator in terms of budget and target-fidelity information gain.Source: §5.2 · p. 9
- When low-fidelity actions are informative about the target and cost much less, MF-MI-Greedy is more advantageous than single-fidelity optimization.Under this condition, the algorithm is less likely to focus on exploring the target fidelity.Source: §5.2 · p. 10
- Using GP-UCB, EST, or GP-MI as the optimization-phase subroutine makes MF-MI-Greedy no-regret.The analysis describes no-regret as cumulative regret per unit budget tending to zero as the budget grows.Source: §5.2 · p. 10
6. Experiments
The experiments evaluate MF-MI-Greedy on synthetic and real optimization tasks using an additive GP model, and compare it with MF-GP-UCB and GP-UCB. Results show general competitiveness on synthetic tasks, robustness to changed cost definitions in cosmological inference, and a better final nanophotonic design than either baseline.
6. Experiments
- The experiments cover three synthetic test-function tasks and two practical optimization problems, using a total budget of 100 target-function call costs.When the optimum is known, the evaluation compares simple regret; otherwise, it compares simple reward.Source: §6 · p. 10, §6.1 · p. 10
6.1 Experimental Setup
- The experimental setup models each lower-fidelity function as the target function plus an error process, with Gaussian processes for both components.Because the target function is shared across fidelities, observations at any fidelity update every fidelity's posterior.Source: §6.1 · p. 10
6.2 Compared Methods
- The comparisons include MF-GP-UCB and GP-UCB, with GP-UCB serving as the single-fidelity optimization subroutine in the proposed framework.Source: §6.2 · p. 10
6.3 Synthetic Examples
- MF-MI-Greedy is generally competitive with MF-GP-UCB on synthetic tasks, but its simple regrets tend to be larger early in optimization.The authors suggest that untuned termination parameters may cause over-exploration of regions that reveal little about the optimum.Source: §6.3 · p. 11, Figure 2 · p. 11
6.4 Real Experiments
- The real-task evaluation includes cosmological likelihood inference and nanophotonic structure design.Source: §6.4 · p. 11
6.4.1 Maximum Likelihood Inference for Cosmological Parameters
- MF-MI-Greedy outperforms both baselines under the original cosmological cost definition, maintains consistently high likelihood when costs change, and MF-GP-UCB degrades significantly under the changed costs.The authors interpret these results as evidence of robustness to cost estimates, which may be inaccurate in practical applications.Source: Figure 3 · p. 12, §6.4.1 · p. 12
6.4.2 Experimental Design for Optimizing Nanophotonic Structures
- After initial exploration, MF-MI-Greedy reaches a better final nanophotonic design than MF-GP-UCB and GP-UCB.The experiment uses simulations at three mesh sizes, with costs [1, 4, 9].Source: §6.4.2 · p. 13
7. Conclusion
The conclusion presents a general framework for multi-fidelity Bayesian optimization, with a cost-aware regret notion and theoretical guarantees. Experiments demonstrate its performance on synthetic and real datasets.
7. Conclusion
- The framework lifts popular off-the-shelf single-fidelity GP optimization algorithms to the multi-fidelity setting while preserving their original regret bounds.The paper also introduces a simple regret notion incorporating different fidelity costs and reports experiments on synthetic and real datasets.Source: §7 · p. 13
A.1 Proofs of Theorem 4
The proof decomposes cumulative regret into contributions from lower-fidelity exploration and target-fidelity optimization. It bounds the target-fidelity part using the assumed guarantee of the single-fidelity optimization subroutine.
A.1 Proofs of Theorem 4
- The cumulative regret separates regret from lower-fidelity exploration from regret on target-fidelity actions selected by SF-GP-OPT.Source: §A.1 · p. 16
- For the target-fidelity component, the proof relies on the regret guarantee assumed for the SF-GP-OPT subroutine.Source: §A.1 · pp. 16–17
A.2 Proof of Corollary 5
The corollary's no-regret claim follows by checking that the candidate single-fidelity optimizers meet Theorem 4's assumption.
A.2 Proof of Corollary 5
- GP-UCB, EST, and GP-MI yield no-regret MF-MI-Greedy instances when they satisfy Theorem 4's assumption for SF-GP-OPT.Source: §A.2 · p. 17