Information Routing across Batch Boundaries: Memory--Batch Tradeoffs in Lipschitz Bandits

• URL: http://arxiv.org/abs/2608.07922v1
• Date: Sat, 08 Aug 2026 04:57:24 GMT
• Title: Information Routing across Batch Boundaries: Memory--Batch Tradeoffs in Lipschitz Bandits
• Abstract summary: Adaptive learning needs both a state that preserves what observations imply and opportunities to act on that state.<n>We study this width--depth tradeoff in Lipschitz bandits.<n>For $Wgtrsim_dlog(eT)$, we characterize minimax expected pseudo-regret up to logarithmic factors.
• Score: 20.064604732340037
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Adaptive learning needs both a state that preserves what observations imply and opportunities to act on that state. We study this width--depth tradeoff in stochastic Lipschitz bandits. After each pull, the learner retains at most $W$ bits of live reward-dependent state and organizes its pulls into at most $B$ committed batches. For $W\gtrsim_d\log(eT)$, we characterize minimax expected pseudo-regret up to logarithmic factors; the lower bounds hold for every $W$. Besides the classical sequential and unrestricted-memory batch costs, the frontier contains the new penalty \[ T^{\frac{d+2}{d+3}} \bigl(1+(B-1)W\bigr)^{-\frac1{d(d+3)}}, \] proving that state width and update depth are not interchangeable. The interaction is an information-routing constraint: at regional scale $s$, low regret forces the committed action transcript to encode $Θ_d(s^{-d})$ regional decisions, while the collected boundary states carry at most $(B-1)W$ bits of entropy. Matching policies stream and erase verification statistics while retaining a mask of a safe active set, either in memory or fragment by fragment. The theorem recovers the full-dimensional worst-case batch-only frontier and logarithmic-memory achievability in the fully sequential specialization; static batch boundaries match predictable adaptive ones.

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