Fast T2T: Optimization Consistency Speeds Up Diffusion-Based Training-to-Testing Solving for Combinatorial Optimization

TL;DR

This work proposes to learn direct mappings from different noise levels to the optimal solution for a given instance, facilitating high-quality generation with minimal shots through an optimization consistency training protocol, and refers to this model as Fast T2T.

Abstract

Diffusion models have recently advanced Combinatorial Optimization (CO) as a powerful backbone for neural solvers. However, their iterative sampling process requiring denoising across multiple noise levels incurs substantial overhead. We propose to learn direct mappings from different noise levels to the optimal solution for a given instance, facilitating high-quality generation with minimal shots. This is achieved through an optimization consistency training protocol, which, for a given instance, minimizes the difference among samples originating from varying generative trajectories and time steps relative to the optimal solution. The proposed model enables fast single-step solution generation while retaining the option of multi-step sampling to trade for sampling quality, which offers a more effective and efficient alternative backbone for neural solvers. In addition, within the training-to-testing (T2T) framework, to bridge the gap between training on historical instances and solving new instances, we introduce a novel consistency-based gradient search scheme during the test stage, enabling more effective exploration of the solution space learned during training. It is achieved by updating the latent solution probabilities under objective gradient guidance during the alternation of noise injection and denoising steps. We refer to this model as Fast T2T. Extensive experiments on two popular tasks, the Traveling Salesman Problem (TSP) and Maximal Independent Set (MIS), demonstrate the superiority of Fast T2T regarding both solution quality and efficiency, even outperforming LKH given limited time budgets. Notably, Fast T2T with merely one-step generation and one-step gradient search can mostly outperform the SOTA diffusion-based counterparts that require hundreds of steps, while achieving tens of times speedup.

Figures (11)

• Figure 1: Optimization consistency models for CO solving where the model learns how to map from varying levels of noise to the solution distribution, conditioned on the problem graph instance.
• Figure 2: Vannila consistency models are trained to map points on any trajectory to its origin. Optimization consistency enforces that all trajectories conditioned on $G$ consistently map to the same initial point, i.e., the optimal solution of $G$.
• Figure 3: Results on TSP-500 and TSP-1000. AS: Active Search, S: Sampling Decoding, BS: Beam Search. $^*$ denotes results that are quoted from previous works li2023t2tqiu2022dimes.
• Figure 4: Effect of runtime to optimality drop for Fast T2T and LKH3 on TSP-100.
• Figure 5: Effect of runtime to optimality drop for Fast T2T and LKH3 on TSP-1000.

Theorems & Definitions (2)

• Definition 4.1: Optimization Consistency
• Definition 4.2