[Paper Review] Unsupervised Learning for Combinatorial Optimization with Principled Objective Relaxation

Haoyu Wang, Nan Wu+3 more|arXiv (Cornell University)|Jul 13, 2022

Ferroelectric and Negative Capacitance Devices4 citations

TL;DR

This paper proposes an unsupervised learning framework for combinatorial optimization (CO) using principled entry-wise concave relaxation of objective functions, enabling end-to-end training via back-propagation. The key contribution is a theoretical guarantee that low optimization loss in the relaxed, concave objective ensures high-quality integral solutions after rounding, significantly improving performance over baselines in proxy-based CO problems like circuit design and approximate computing.

ABSTRACT

Using machine learning to solve combinatorial optimization (CO) problems is challenging, especially when the data is unlabeled. This work proposes an unsupervised learning framework for CO problems. Our framework follows a standard relaxation-plus-rounding approach and adopts neural networks to parameterize the relaxed solutions so that simple back-propagation can train the model end-to-end. Our key contribution is the observation that if the relaxed objective satisfies entry-wise concavity, a low optimization loss guarantees the quality of the final integral solutions. This observation significantly broadens the applicability of the previous framework inspired by Erdos' probabilistic method. In particular, this observation can guide the design of objective models in applications where the objectives are not given explicitly while requiring being modeled in prior. We evaluate our framework by solving a synthetic graph optimization problem, and two real-world applications including resource allocation in circuit design and approximate computing. Our framework largely outperforms the baselines based on naïve relaxation, reinforcement learning, and Gumbel-softmax tricks.

Motivation & Objective

• To address the challenge of solving combinatorial optimization problems without labeled data, especially in proxy-based CO settings where objectives are expensive or implicit to evaluate.
• To develop a general unsupervised learning framework that avoids reliance on reinforcement learning or costly labeling while maintaining solution quality.
• To establish a theoretical foundation linking low relaxed loss to high-quality discrete solutions through entry-wise concave relaxation.
• To enable end-to-end training of neural networks on unlabeled data by parameterizing relaxed solutions with differentiable objectives.
• To demonstrate the framework’s effectiveness on synthetic and real-world CO problems, including circuit resource allocation and approximate computing.

Proposed method

• The framework uses a relaxation-plus-rounding pipeline: relax the discrete CO objective into a continuous, entry-wise concave function parameterized by a neural network.
• The relaxed objective is optimized via standard back-propagation on unlabeled data, enabling end-to-end training.
• A key theoretical contribution is proving that if the relaxed objective is entry-wise concave, a low-cost soft solution guarantees a high-quality integral solution after deterministic sequential rounding.
• The method incorporates constraints (e.g., resource limits) via normalized penalty terms, such as $ g_r(\bar{X};C) = \frac{n - \sum \bar{X}_i}{n-t} $, ensuring feasibility.
• The model is trained using a loss function $ l_r(\bar{X};C) = f_r(\bar{X};C) + \beta \cdot g_r(\bar{X};C) $, with $ \beta $ chosen to dominate the objective.
• During inference, the model takes a threshold $ t $ as input to dynamically adapt to different constraint levels, enabling a single model to handle multiple configurations.

Experimental results

Research questions

• RQ1 Can entry-wise concave relaxation of the objective function ensure high-quality discrete solutions in unsupervised combinatorial optimization?
• RQ2 How can unsupervised learning be systematically applied to proxy-based CO problems where objectives are not explicitly defined?
• RQ3 Does a low loss in a differentiable, concave relaxation translate to strong performance in the final discrete solution after rounding?
• RQ4 Can this framework outperform reinforcement learning and Gumbel-softmax-based methods in terms of solution quality and training stability?
• RQ5 To what extent can a single neural network model generalize across different constraint thresholds in resource-constrained optimization?

Key findings

• The proposed framework significantly outperforms baselines based on naïve relaxation, reinforcement learning, and Gumbel-softmax in solving synthetic and real-world CO problems.
• The entry-wise concave relaxation ensures that a low-cost soft solution leads to a high-quality integral solution after deterministic rounding, providing a theoretical performance guarantee.
• In the circuit resource allocation task, the model achieved higher optimum-recovery probabilities across varying thresholds compared to other methods.
• The framework enables a single model to generalize across multiple constraint levels (e.g., different A×C unit usage thresholds) by conditioning on the threshold $ t $ as input.
• The method demonstrated superior training speed, generalization, and scalability on large-scale problems compared to supervised and RL-based alternatives.
• The code and datasets are publicly available, supporting reproducibility and broader adoption in proxy-based CO applications.