Strongly Polynomial Time Complexity of Policy Iteration for $L_\infty$ Robust MDPs

Markov decision processes (MDPs) are a fundamental model in sequential decision making. Robust MDPs (RMDPs) extend this framework by allowing uncertainty in transition probabilities and optimizing against the worst-case realization of that uncertainty. In particular, $(s, a)$-rectangular RMDPs with $L_\infty$ uncertainty sets form a fundamental and expressive model: they subsume classical MDPs and turn-based stochastic games. We consider this model with discounted payoffs. The existence of polynomial and strongly-polynomial time algorithms is a fundamental problem for these optimization models. For MDPs, linear programming yields polynomial-time algorithms for any arbitrary discount factor, and the seminal work of Ye established strongly--polynomial time for a fixed discount factor. The generalization of such results to RMDPs has remained an important open problem. In this work, we show that a robust policy iteration algorithm runs in strongly-polynomial time for $(s, a)$-rectangular $L_\infty$ RMDPs with a constant (fixed) discount factor, resolving an important algorithmic question.

Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems

We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.