Fixed-point neural samplers on discrete spaces
Sampling from discrete, unnormalized distributions without curated data is a fundamental bottleneck in physics, chemistry, and combinatorics, because the state space is exponential and Markov chains mix too slowly. This paper introduces DIG-IN (Discrete Gibbs Iterative Neural Sampler), which trains masked diffusion models via fixed-point iteration by using Gibbs soft labels—local single-site conditionals derived from energy differences—as rich supervision instead of one-hot reconstruction targets. The method tempers importance weights by effective sample size to stabilize the bootstrap, and it scales to high-dimensional systems such as $D=2500$ lattice spin models and alloy phase diagrams.
The paper presents a strong synthesis: Gibbs soft labels lower the variance of the training signal compared to one-hot reconstruction (Proposition 3.1), ESS-tempered importance weighting stabilizes the self-generated bootstrap with idealized convergence guarantees (Theorem 3.2), and the construction extends elegantly beyond masked diffusion to arbitrary discrete interpolants via a minimum kinetic energy local flux (Equations 23–26). Empirically, DIG-IN is the only method that simultaneously avoids mode collapse in the ordered phase and yields accurate $\ln Z$ estimates on Ising and Potts models, and it convincingly outperforms FrOGS on alloy phase diagrams. Nevertheless, the theoretical guarantees are intentionally narrow—they assume exact population fitting and exact sampling—and do not cover the neural-network training actually used.
The most compelling contribution is the Gibbs soft-label construction. By training the masked denoiser on normalized single-site conditionals $h^d(c \mid x^{-d})$ rather than on one-hot labels, the authors replace a brittle reconstruction objective with a strong, local gradient-like signal that is provably lower variance and empirically prevents mode collapse. The extension to general discrete interpolants is equally elegant: a symmetric parameterization of forward and backward generators that minimizes probability-weighted kinetic energy (Equation 25) shows the core ideas are not artificially wedded to masking.
The convergence guarantee in Theorem 3.2 is explicitly idealized: it assumes exact sampling, exact population minimization, and a finite unrestricted table of predictors. The authors frankly acknowledge that it does not cover the practical setting in our experiments at all, leaving a substantial gap between theory and the neural-network training actually performed.
A second issue is computational cost: Gibbs labels require evaluating single-site energy differences, which costs $D(N-1)+1$ energy calls per configuration for a non-local black-box energy. The paper notes this as a limitation but does not quantify how much it slows training relative to baselines, nor does it validate proposed subsampling heuristics. Finally, the method introduces a large hyperparameter space—including buffer size $B$, refill size $b$, inner steps $K$, ESS fraction $\xi$, and damping $\eta$—without systematic ablation of their individual effects in the main experiments.
The empirical comparison is clean in the lattice-model regime: all methods receive the same training budget and evaluation protocol. DIG-IN dominates MDNS, PDNS, and DASBS on effective sample size, $\ln Z$ error, and mode total-variation distance (Table 1), and it handily outperforms FrOGS on alloy phase diagrams. However, LEAPS is ruled out by unstable training rather than inherent representational failure, and the paper does not benchmark against autoregressive baselines or recent GFlowNet training improvements outside the CTMC family. The amortized scaling to $D=2500$ is impressive, but it is conditioned on a 2-D parameter grid rather than demonstrating raw dimensionality scaling beyond prior work.
The algorithm is described in enough detail—Algorithm 1, Figure 1, and extensive appendices—to permit reimplementation, but no code, data, or pretrained checkpoints are released in the provided text. A barrier to reproduction is that the authors disclose using LLMs to implement parts of the experiment code and the figure and table scripts, which means the exact codebase may not be available or fully auditable without release. Reproducing the alloy experiments additionally requires the DFT-fitted cluster expansion evaluators, and the importance-weight buffer logic with EMA damping and adaptive tempering is intricate enough that independent replication would be nontrivial without source code.
Sampling from discrete, unnormalized distributions without access to data is a challenging problem. Neural samplers offer a promising approach by training generative models from density evaluations directly. Despite recent progress, existing discrete neural samplers are prone to mode collapse, come without convergence guarantees when trained via fixed-point iterations, and are often tied to a specific reference process such as masked or uniform diffusion. In this work, we introduce Discrete Gibbs Iterative Neural Sampler, a fixed-point neural sampler that addresses these limitations, enabling efficient, scalable learning, substantially reducing mode collapse in practice. Our framework builds on masked diffusion and also extends to transport between pairs of distributions. We demonstrate that the resulting method scales effectively to high-dimensional systems, supports amortized sampling across different conditions, and enables accurate estimation of alloy phase diagrams.
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