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Controlling Continuous Relaxation for Combinatorial Optimization

Yuma Ichikawa

TL;DR

Experimental results demonstrate that CRA significantly enhances the performance of UL-based solvers, outperforming existing UL-based solvers and greedy algorithms in complex CO problems, and effectively eliminates artificial rounding and accelerates the learning process.

Abstract

Unsupervised learning (UL)-based solvers for combinatorial optimization (CO) train a neural network that generates a soft solution by directly optimizing the CO objective using a continuous relaxation strategy. These solvers offer several advantages over traditional methods and other learning-based methods, particularly for large-scale CO problems. However, UL-based solvers face two practical issues: (I) an optimization issue, where UL-based solvers are easily trapped at local optima, and (II) a rounding issue, where UL-based solvers require artificial post-learning rounding from the continuous space back to the original discrete space, undermining the robustness of the results. This study proposes a Continuous Relaxation Annealing (CRA) strategy, an effective rounding-free learning method for UL-based solvers. CRA introduces a penalty term that dynamically shifts from prioritizing continuous solutions, effectively smoothing the non-convexity of the objective function, to enforcing discreteness, eliminating artificial rounding. Experimental results demonstrate that CRA significantly enhances the performance of UL-based solvers, outperforming existing UL-based solvers and greedy algorithms in complex CO problems. Additionally, CRA effectively eliminates artificial rounding and accelerates the learning process.

Controlling Continuous Relaxation for Combinatorial Optimization

TL;DR

Experimental results demonstrate that CRA significantly enhances the performance of UL-based solvers, outperforming existing UL-based solvers and greedy algorithms in complex CO problems, and effectively eliminates artificial rounding and accelerates the learning process.

Abstract

Unsupervised learning (UL)-based solvers for combinatorial optimization (CO) train a neural network that generates a soft solution by directly optimizing the CO objective using a continuous relaxation strategy. These solvers offer several advantages over traditional methods and other learning-based methods, particularly for large-scale CO problems. However, UL-based solvers face two practical issues: (I) an optimization issue, where UL-based solvers are easily trapped at local optima, and (II) a rounding issue, where UL-based solvers require artificial post-learning rounding from the continuous space back to the original discrete space, undermining the robustness of the results. This study proposes a Continuous Relaxation Annealing (CRA) strategy, an effective rounding-free learning method for UL-based solvers. CRA introduces a penalty term that dynamically shifts from prioritizing continuous solutions, effectively smoothing the non-convexity of the objective function, to enforcing discreteness, eliminating artificial rounding. Experimental results demonstrate that CRA significantly enhances the performance of UL-based solvers, outperforming existing UL-based solvers and greedy algorithms in complex CO problems. Additionally, CRA effectively eliminates artificial rounding and accelerates the learning process.
Paper Structure (53 sections, 6 theorems, 21 equations, 8 figures, 8 tables)

This paper contains 53 sections, 6 theorems, 21 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Background
  4. Combinatorial optimization
  5. Example: MIS problem
  6. Unsupervised learning based solvers
  7. (Type I) Learning generalized heuristics from history/data
  8. (Type II) Learning effective heuristics on a specific single instance
  9. Continuous relaxation annealing for UL-based solvers
  10. Motivation: practical issues of UL-based solvers
  11. (I) Ambiguity in rounding method after learning
  12. (II) Difficulty in optimizing NNs
  13. Continuous relaxation annealing
  14. Penalty term to control discreteness and continuity
  15. Annealing penalty term
  16. ...and 38 more sections

Key Result

Theorem 3.1

Assuming the objective function $\hat{l}({\bm p};C)$ is bounded within the domain $[0, 1]^{N}$, as $\gamma \to +\infty$, the relaxed solutions ${\bm p}^{\ast} \in \mathrm{argmin}_{{\bm p}} \hat{r}({\bm p}; C, {\bm \lambda}, \gamma)$ converge to the original solutions ${\bm x}^{\ast} \in \mathrm{argm

Figures (8)

  • Figure 1: Annealing strategy. When $\gamma < 0$, it facilitates exploration by reducing the non-convexity of the objective function. As $\gamma$ increases, it promotes optimal discrete solutions by smoothing away suboptimal continuous ones.
  • Figure 2: Independent set density of the MIS problem on $d$-RRG. Results for graphs with $N=10{,}000$ nodes (Left) and $N=20{,}000$ nodes (Right). the dashed lines represent the theoretical results barbier2013hard.
  • Figure 3: Cut ratio of the MaxCut problem on $d$-RRG as a function of the degree $d$ Results for $N=10{,}000$ (Left) and $N=20{,}000$ (Right). The dashed lines represents the theoretical upper bounds parisi1980sequence.
  • Figure 4: The dynamics of cost function for MIS problems on RRGs with $N=10{,}000$ nodes varying degrees $d$ as a function of the number of parameters updates $N_{\mathrm{EPOCH}}$.
  • Figure 5: ApR on DBM problems.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Theorem 3.1
  • Lemma B.1
  • proof
  • Lemma B.2
  • proof
  • Lemma B.3
  • proof
  • Theorem B.4
  • proof
  • Corollary B.5