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Make Optimization Once and for All with Fine-grained Guidance

Mingjia Shi, Ruihan Lin, Xuxi Chen, Yuhao Zhou, Zezhen Ding, Pingzhi Li, Tong Wang, Kai Wang, Zhangyang Wang, Jiheng Zhang, Tianlong Chen

TL;DR

Diff-L2O addresses the inefficiency and limited generalization of traditional L2O by modeling the solution space with diffusion processes and incorporating guidance from optimization meta-features. It presents a unified theoretical framework including ISHD ODE dynamics, SDEs, and diffusion-based sampling, plus a PAC-Bayesian style generalization bound linking sample diversity to performance. Empirically, Diff-L2O achieves rapid initialization and faster convergence on LASSO, Rastrigin, Ackley, and MNIST, with minute-level training and strong compatibility with conventional optimizers via a hybrid approach. The work highlights the practical impact of broad solution-space exploration for robust optimization across diverse tasks.

Abstract

Learning to Optimize (L2O) enhances optimization efficiency with integrated neural networks. L2O paradigms achieve great outcomes, e.g., refitting optimizer, generating unseen solutions iteratively or directly. However, conventional L2O methods require intricate design and rely on specific optimization processes, limiting scalability and generalization. Our analyses explore general framework for learning optimization, called Diff-L2O, focusing on augmenting sampled solutions from a wider view rather than local updates in real optimization process only. Meanwhile, we give the related generalization bound, showing that the sample diversity of Diff-L2O brings better performance. This bound can be simply applied to other fields, discussing diversity, mean-variance, and different tasks. Diff-L2O's strong compatibility is empirically verified with only minute-level training, comparing with other hour-levels.

Make Optimization Once and for All with Fine-grained Guidance

TL;DR

Diff-L2O addresses the inefficiency and limited generalization of traditional L2O by modeling the solution space with diffusion processes and incorporating guidance from optimization meta-features. It presents a unified theoretical framework including ISHD ODE dynamics, SDEs, and diffusion-based sampling, plus a PAC-Bayesian style generalization bound linking sample diversity to performance. Empirically, Diff-L2O achieves rapid initialization and faster convergence on LASSO, Rastrigin, Ackley, and MNIST, with minute-level training and strong compatibility with conventional optimizers via a hybrid approach. The work highlights the practical impact of broad solution-space exploration for robust optimization across diverse tasks.

Abstract

Learning to Optimize (L2O) enhances optimization efficiency with integrated neural networks. L2O paradigms achieve great outcomes, e.g., refitting optimizer, generating unseen solutions iteratively or directly. However, conventional L2O methods require intricate design and rely on specific optimization processes, limiting scalability and generalization. Our analyses explore general framework for learning optimization, called Diff-L2O, focusing on augmenting sampled solutions from a wider view rather than local updates in real optimization process only. Meanwhile, we give the related generalization bound, showing that the sample diversity of Diff-L2O brings better performance. This bound can be simply applied to other fields, discussing diversity, mean-variance, and different tasks. Diff-L2O's strong compatibility is empirically verified with only minute-level training, comparing with other hour-levels.

Paper Structure

This paper contains 64 sections, 3 theorems, 17 equations, 7 figures, 6 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Preliminary
  4. Optimization's general trajectory formulation.
  5. Discretization.
  6. Stochastic optimization's dynamics.
  7. Diffusion process.
  8. Discussion: modeling solutions is feasible
  9. Takeaways.
  10. Case: overparameterization.
  11. Case: underparameterization.
  12. Case: low performance.
  13. Closest doesn't mean best.
  14. The closest is yet informative.
  15. Diff-L2O: How to model solutions
  16. ...and 49 more sections

Key Result

Theorem 2.1

(General PAC-Bayesian on stochastic solution space.) In this general theorem, $\Delta$ requires only a non-negative general convex distance, and we do not restrict the optimization objective to the downstream tasks. With a initial prior process $p$, $\forall q$ (posterior) w/ $n$ #samples, we have t where $\mathcal{M}:=\mathbf{E}_{h\sim p}\exp\{n\Delta(h)\}$ is related to the optimization task, in

Figures (7)

  • Figure 1: Diff-L2O's intuitions: wider views and better sampling diversity with artificial sampling on solution spaces.
  • Figure 2: Comparison on optimizees across #dimension: LASSO, Rastrigin and Ackley.
  • Figure 3: Ablation: compatibility of Diff-L2O with conventional optimizers.
  • Figure 4: Visualization: learning surface. Fast convergence happens within several epochs.
  • Figure 5: Visualization of the learned and the ground-truth distribution (true). The distributions are generally matched.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3