Diffusion Posterior Sampling (DPS) provides a principled Bayesian approach to inverse problems by sampling from $p(x_0 \mid y)$. However, in practice, the goal of inverse problem solving is not to cover the posterior but to recover the most accurate reconstruction, where optimization-based diffusion solvers often excel despite lacking a clear probabilistic foundation. We introduce Local MAP Sampling (LMAPS), a new inference framework that iteratively solving local MAP subproblems along the diffusion trajectory. This perspective clarifies their connection to global MAP estimation and DPS, offering a unified probabilistic interpretation for optimization-based methods. Building on this foundation, we develop practical algorithms with a probabilistically interpretable covariance approximation, a reformulated objective for stability and interpretability, and a gradient approximation for non-differentiable operators. Across a broad set of image restoration and scientific tasks, LMAPS achieves state-of-the-art performance, including $\geq 2$ dB gains on motion deblurring, JPEG restoration, and quantization, and $>1.5$ dB improvements on inverse scattering benchmarks.

Abstract: We introduce novel communication strategies in synchronous distributed Deep Learning consisting of decentralized gradient reduction orchestration and computational graph-aware grouping of gradient tensors. These new techniques produce an optimal overlap between computation and communication and result in near-linear scaling (0.93) of distributed training up to 27,600 NVIDIA V100 GPUs on the Summit Supercomputer. We demonstrate our gradient reduction techniques in the context of training a Fully Convolutional Neural Network to approximate the solution of a longstanding scientific inverse problem in materials imaging. The efficient distributed training on a dataset size of 0.5 PB, produces a model capable of an atomically-accurate reconstruction of materials, and in the process reaching a peak performance of 2.15(4) EFLOPS$_{16}$.