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Active Flow Matching

Yashvir S. Grewal, Daniel M. Steinberg, Thang D. Bui, Cheng Soon Ong, Edwin V. Bonilla

TL;DR

Active Flow Matching (AFM) is introduced, which reformulates variational objectives to operate on conditional endpoint distributions along the flow, enabling gradient-based steering of flow models toward high-fitness regions while preserving the rigour of VSD and CbAS.

Abstract

Discrete diffusion and flow matching models capture complex, non-additive and non-autoregressive structure in high-dimensional objective landscapes through parallel, iterative refinement. However, their implicit generative nature precludes direct integration with principled variational frameworks for online black-box optimisation, such as variational search distributions (VSD) and conditioning by adaptive sampling (CbAS). We introduce Active Flow Matching (AFM), which reformulates variational objectives to operate on conditional endpoint distributions along the flow, enabling gradient-based steering of flow models toward high-fitness regions while preserving the rigour of VSD and CbAS. We derive forward and reverse Kullback-Leibler (KL) variants using self-normalised importance sampling. Across a suite of online protein and small molecule design tasks, forward-KL AFM consistently performs competitively compared to state-of-the-art baselines, demonstrating effective exploration-exploitation under tight experimental budgets.

Active Flow Matching

TL;DR

Active Flow Matching (AFM) is introduced, which reformulates variational objectives to operate on conditional endpoint distributions along the flow, enabling gradient-based steering of flow models toward high-fitness regions while preserving the rigour of VSD and CbAS.

Abstract

Discrete diffusion and flow matching models capture complex, non-additive and non-autoregressive structure in high-dimensional objective landscapes through parallel, iterative refinement. However, their implicit generative nature precludes direct integration with principled variational frameworks for online black-box optimisation, such as variational search distributions (VSD) and conditioning by adaptive sampling (CbAS). We introduce Active Flow Matching (AFM), which reformulates variational objectives to operate on conditional endpoint distributions along the flow, enabling gradient-based steering of flow models toward high-fitness regions while preserving the rigour of VSD and CbAS. We derive forward and reverse Kullback-Leibler (KL) variants using self-normalised importance sampling. Across a suite of online protein and small molecule design tasks, forward-KL AFM consistently performs competitively compared to state-of-the-art baselines, demonstrating effective exploration-exploitation under tight experimental budgets.
Paper Structure (51 sections, 1 theorem, 27 equations, 2 figures, 4 tables, 1 algorithm)

This paper contains 51 sections, 1 theorem, 27 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Active Flow Matching (AFM).
  3. Background
  4. Active Generation
  5. Discrete Flow Matching
  6. Setup.
  7. Probability paths.
  8. Continuous-time Markov chain dynamics.
  9. Generating velocity from posteriors.
  10. Learning the posterior.
  11. Sampling algorithm.
  12. Extension to sequences.
  13. The implicit generator problem.
  14. Active Flow Matching
  15. Problem Setup
  16. ...and 36 more sections

Key Result

Theorem 3.1

Let $p^*({\mathbf{x}}) \propto p_{\mathrm{prior}}({\mathbf{x}}) w({\mathbf{x}})$ be the target distribution, where $w({\mathbf{x}}) = p(y \ge \tau \mid {\mathbf{x}})$ and $p_{\mathrm{prior}}$ is the uniform distribution. Under standard DFM assumptions with masked-source coupling, convex interpolant

Figures (2)

  • Figure 1: Budget-constrained optimisation on Ehrlich synthetic landscapes and the AAV capsid design task. We report the simple regret
  • Figure 2: Structure-based protein design with FoldX oracles. We plot the best score found vs. oracle calls;

Theorems & Definitions (4)

  • Theorem 3.1: Consistency of Forward-KL AFM
  • proof : Proof Sketch
  • Remark 3.2
  • proof