Innate resistance to anti-PD-1 immunotherapy remains a major clinical challenge in metastatic melanoma, with the underlying molecular networks being poorly understood. To address this, we constructed a dynamic Probabilistic Boolean Network model using transcriptomic data from patient tumor biopsies to elucidate the regulatory logic governing therapy response. We then employed a reinforcement learning agent to systematically discover optimal, multi-step therapeutic interventions and used explainable artificial intelligence to mechanistically interpret the agent's control policy. The analysis revealed that a precisely timed, 4-step temporary inhibition of the lysyl oxidase like 2 protein (LOXL2) was the most effective strategy. Our explainable analysis showed that this ''hit-and-run" intervention is sufficient to erase the molecular signature driving resistance, allowing the network to self-correct without requiring sustained intervention. This study presents a novel, time-dependent therapeutic hypothesis for overcoming immunotherapy resistance and provides a powerful computational framework for identifying non-obvious intervention protocols in complex biological systems.

Hit-and-Run is a coordinate-free Gibbs sampler, yet the quantitative advantages of its coordinate-free property remain largely unexplored beyond empirical studies. In this paper, we prove sharp estimates for the Wasserstein contraction of Hit-and-Run in Gaussian target measures via coupling methods and conclude mixing time bounds. Our results uncover ballistic and superdiffusive convergence rates in certain settings. Furthermore, we extend these insights to a coordinate-free variant of the randomized Kaczmarz algorithm, an iterative method for linear systems, and demonstrate analogous convergence rates. These findings offer new insights into the advantages and limitations of coordinate-free methods for both sampling and optimization.

Counting integer solutions of linear constraints has found interesting applications in various fields. It is equivalent to the problem of counting lattice points inside a polytope. However, state-of-the-art algorithms for this problem become too slow for even a modest number of variables. In this paper, we propose a new framework to approximate the lattice counts inside a polytope with a new random-walk sampling method. The counts computed by our approach has been proved approximately bounded by a $(\epsilon, \delta)$-bound. Experiments on extensive benchmarks show that our algorithm could solve polytopes with dozens of dimensions, which significantly outperforms state-of-the-art counters.