Xiaoan Xu, Jesus A. De Loera, Ethan X. Fang, Junwei Lu, Luze Xu
原文摘要
Classical optimization and learning-guided search offer complementary approaches to open problems in discrete mathematics, but their outputs are difficult to compare when search spaces, validity checks, and stopping rules differ. We present \emph{RadonBench}, a benchmark for constructing maximum-cardinality Radon-free lattice configurations: finite subsets of $\mathbb{Z}^d$ admitting no integral Radon partition. It remains open whether Radon-free configurations with $11$ points in $\mathbb{Z}^3$ or $21$ points in $\mathbb{Z}^4$ are possible. If they exist, they would improve the corresponding lower bounds on the integer Radon numbers. We compare exhaustive extension, greedy local search, PatternBoost, AlphaZero, and AlphaEvolve. All outputs use one success predicate: a convex target-sized configuration succeeds exactly when its number of integral Radon points is zero. PatternBoost and AlphaZero construct verified Radon-free configurations of sizes $10$ and $20$, one point short of the respective targets; AlphaZero's best convex target terminals have two integral Radon points in both dimensions. Exhaustive extension over fixed seeds and declared finite search boxes likewise reaches a minimum of two integral Radon points. Greedy local search and AlphaEvolve each produce target-sized near misses with one integral Radon point. No audited run reaches zero. These results establish reproducible near-miss baselines over explicitly scoped search domains.
Personalized reward models require user-specific preference comparisons, but collecting such feedback is costly and provides only limited supervision for each user. At the same time, large pools of responses can be generated without user identities or preference labels, yet existing personalized reward models leave this data unused. Directly learning from these responses is difficult because pairwise objectives constrain relative scores within a prompt, whereas pseudo-labeling individual responses requires scores that are comparable across prompts for the same user. We propose \textbf{PerSPL} (Personalized Semi-supervised Preference Learning), a backbone-agnostic framework that uses pair-mean score centering to stabilize this cross-prompt score coordinate and iteratively turns unlabeled responses into user-specific training signals. Against seven representative baselines across four benchmarks---PersonalLLM, HelpSteer2, Reddit TLDR, and Chatbot Arena---PerSPL achieves the highest reported mean accuracy among the evaluated methods for both seen and unseen users. The two latter benchmarks provide real-user preference comparisons in summarization and open-domain chatbot interaction, respectively. These results indicate that unlabeled responses can provide additional supervision for personalized reward modeling when per-user preference labels are scarce.