DDPS: Discrete Diffusion Posterior Sampling for Paths in Layered Graphs

TL;DR

DDPS tackles the problem of generating valid paths in layered graphs with diffusion models by introducing the padded adjacency-list matrix (PALM), a structured discrete representation that guarantees path feasibility. It adapts the D3PM framework for PALM-based training and sampling, and derives a discrete posterior-guided sampling procedure that uses the gradient of expected rewards to steer path generation toward preferred edges without retraining. Key contributions include the PALM representation, efficient training/inference for PALM-based diffusion, and a practical discrete guidance mechanism that improves reward attainment while preserving distributional fidelity. The results show DDPS outperforms naïve continuous-relaxation baselines in generating valid paths and reveals a trade-off between reward optimization and adherence to the learned diffusion prior, with a tunable guidance strength achieving near-optimal rewards in practice.

Abstract

Diffusion models form an important class of generative models today, accounting for much of the state of the art in cutting edge AI research. While numerous extensions beyond image and video generation exist, few of such approaches address the issue of explicit constraints in the samples generated. In this paper, we study the problem of generating paths in a layered graph (a variant of a directed acyclic graph) using discrete diffusion models, while guaranteeing that our generated samples are indeed paths. Our approach utilizes a simple yet effective representation for paths which we call the padded adjacency-list matrix (PALM). In addition, we show how to effectively perform classifier guidance, which helps steer the sampled paths to specific preferred edges without any retraining of the diffusion model. Our preliminary results show that empirically, our method outperforms alternatives which do not explicitly account for path constraints.

Figures (6)

• Figure 1: Example of a layered graph with $L=4$ layers. The first layer $\mathcal{V}^1$ is a singleton ($|\mathcal{V}^1|=1$), while subsequent layers have $|\mathcal{V}^l|=3$ for $l=2,3,4$. Edges only exist between adjacent layers ($l \to l+1$). A specific path (A, C, G, H) is highlighted in blue.
• Figure 2: The representation (transposed) for the path highlighted in figure \ref{['fig:lg_example']}. Columns correspond to vertices: blue indicates on-path vertices (A, C, G, H) and their specific one-hot vectors; gray indicates off-path vertices. Each gray column vector is an arbitrary one-hot selection if the vertex has outgoing edges. Faded entries are padding zeros.
• Figure 3: Valid rates for baseline and our method under different guidance strengths.
• Figure 4: Average reward values under various guidance scales for different reward configurations. Error bars denote standard deviations.
• Figure 5: Metrics in path distributions for the small-size graph Toy. The error bars represent standard deviations across different reward function instances within the reward configuration class (20 instances per configuration). The sample size is 65536.

Theorems & Definitions (6)

• Definition 1: Layered Graph, adapted from cerny2024layered
• Definition 2: Path
• Definition 3: Padded Adjacency--List Matrix (PALM)
• Definition 4: Layer Imitation Scores (IS-L)
• Definition 5: Spearman's Footrule Distance (SFD), adapted from diaconis1977spearman
• Remark 1