A Continuous Relaxation for Discrete Bayesian Optimization (2404.17452v1)

Abstract: To optimize efficiently over discrete data and with only few available target observations is a challenge in Bayesian optimization. We propose a continuous relaxation of the objective function and show that inference and optimization can be computationally tractable. We consider in particular the optimization domain where very few observations and strict budgets exist; motivated by optimizing protein sequences for expensive to evaluate bio-chemical properties. The advantages of our approach are two-fold: the problem is treated in the continuous setting, and available prior knowledge over sequences can be incorporated directly. More specifically, we utilize available and learned distributions over the problem domain for a weighting of the Hellinger distance which yields a covariance function. We show that the resulting acquisition function can be optimized with both continuous or discrete optimization algorithms and empirically assess our method on two bio-chemical sequence optimization tasks.

• The paper introduces a continuous relaxation method that transforms discrete Bayesian optimization for bio-chemical sequences into a tractable continuous problem.
• It adapts Gaussian Processes with a novel weighted Hellinger kernel to incorporate prior knowledge and efficiently explore vast discrete sequence spaces.
• Empirical validation on protein sequence tasks demonstrates enhanced optimization performance even in cold-start scenarios with minimal observations.

Continuous Relaxation for Discrete Bayesian Optimization in Bio-Chemical Sequence Tasks

Introduction to the Paper

The paper presents a novel approach to Bayesian Optimization (BO) for optimizing discrete sequences, particularly within the restrictive domain of expensive-to-evaluate bio-chemical properties such as protein sequences. The key innovation proposed is a method for continuous relaxation of the objective function in Bayesian Optimization. This adaptation transforms the optimization of inherently discrete sequence inputs, such as proteins, into a continuous domain while maintaining computationally tractable inference and optimization.

Key Challenges and Proposed Solutions

The primary challenge in discrete Bayesian Optimization addressed in the paper is the difficulty of managing high-dimensional and discrete sequence spaces with very few available observations and a strict evaluation budget. The proposed solution involves relaxing the optimization problem from discrete to continuous by mapping sequences to distributions over sequences. This continuous relaxation allows the utilization of Gaussian Processes (GPs) in the relaxed space and enables the inclusion of prior knowledge about sequence distributions directly into the optimization process.

Continuous Objective Relaxation

• Discrete sequence optimization is converted into a continuous problem by minimizing the expected value of the objective function over a space of probability distributions.
• A space of factorizable distributions, significantly reducing complexity, is considered instead of the full space of all possible probability distributions over sequence inputs.

Modeling via Gaussian Processes

• Gaussian Process (GP) models are adapted to handle the continuous relaxation of discrete sequences.
• A novel weighted Hellinger distance-based covariance function is introduced. This covariance function can effectively measure similarities between sequences in the transformed continuous space, incorporating prior knowledge from pre-trained models like Hidden Markov Models or Variational Autoencoders.

Optimization Strategies

• Various strategies to optimize the acquisition function in the continuous relaxed space are explored. These include using continuous parameterizations provided by latent variable models and employing manifold optimization techniques.
• The paper ensures that the optimization process can revert to choosing optimal discrete sequences efficiently from the continuous distribution space.

Empirical Validation

Two primary bio-chemical sequence optimization tasks are used to validate the method:

1. Optimization of protein sequences for properties crucial to protein functionality.
2. Comparative assessment with state-of-the-art methods under stringent conditions of limited observations and evaluations, showing promising enhancements over existing techniques.

Results Overview

• The method demonstrates empirical effectiveness in cold-start scenarios where only minimal observations are available initially, which is often the case in real-world biochemical optimization tasks.
• The proposed GP with a weighted Hellinger kernel adapts effectively to the underlying problem structure, allowing significant enhancements in optimization performance in limited data regimes.

Conclusions and Future Work

The paper concludes that the approach of continuous relaxation for discrete Bayesian Optimization using a novel weighted Hellinger distance-based kernel can significantly enhance the effectiveness of sequence optimization tasks, particularly in biochemistry. For future work, the authors suggest further exploration into optimizing and extending pre-trained models that inform the relaxation process and kernel design.

Implications

The method provides a robust framework for tackling high-dimensional, discrete optimization problems in domains where data acquisition is expensive and the available budget for evaluations is low. The continuous relaxation approach broadens the applicability of Bayesian Optimization to new problem domains, including those in computational biology and related fields, where traditional methods might fall short. Future developments based on this research could lead to more effective drug discovery processes and deeper understanding of protein functionalities through optimized sequence explorations.