March 2026

Coordinating multiple autonomous agents to explore and serve spatially heterogeneous demand requires jointly learning unknown spatial patterns and planning trajectories that maximize task performance. Pure model-based approaches provide structured uncertainty estimates but lack adaptive policy learning, while deep reinforcement learning often suffers from poor sample efficiency when spatial priors are absent. This paper presents a hybrid belief-reinforcement learning (HBRL) framework to address this gap. In the first phase, agents construct spatial beliefs using a Log-Gaussian Cox Process (LGCP) […]

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Shock! Shock! I learned yesterday that an open problem I’d been working on for several weeks had just been solved by Claude Opus 4.6 – Anthropic’s hybrid reasoning model that had been released three weeks earlier! It seems that I’ll have to revise my opinions about “generative AI” one of these days. What a joy it is to learn not only that my conjecture has a nice solution but also to celebrate this dramatic advance in automatic deduction […]

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Everyone believes that the digits of constants such as π or √2 cannot be distinguished from a sequence of random bits. The first few trillion successfully pass all tests of randomness. However, proving that they indeed behave perfectly randomly is arguably one of the oldest and most difficult unsolved math conjectures. So far, nobody succeeded in proving even the most basic facts for any of these constants, for instance: In the binary digits of π, is the proportion […]

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Frozen self-supervised representations often transfer well with only a few labels across many semantic tasks. We argue that a single geometric quantity, emph{directional} CDNV (decision-axis variance), sits at the core of two favorable behaviors: strong few-shot transfer within a task, and low interference across many tasks. We show that both emerge when variability emph{along} class-separating directions is small. First, we prove sharp non-asymptotic multiclass generalization bounds for downstream classification whose leading term is the directional CDNV. The bounds […]

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We study reinforcement learning in infinite-horizon discounted Markov decision processes with continuous state spaces, where data are generated online from a single trajectory under a Markovian behavior policy. To avoid maintaining an infinite-dimensional, function-valued estimate, we propose the novel Q-Measure-Learning, which learns a signed empirical measure supported on visited state-action pairs and reconstructs an action-value estimate via kernel integration. The method jointly estimates the stationary distribution of the behavior chain and the Q-measure through coupled stochastic approximation, leading […]

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