用内在时间精确追踪学习过程中的信息变化，揭示了在线学习的深层规律。
Adaptive Bayes exactly tracks information over intrinsic time
- 基于内在时间构建信息会计框架，逐轮计算损失与信息距离变化。
- 累积误差可分解为即时代价和信息距离减少，且在低噪声下自动收紧。
- 适用于多种学习场景，如强化学习、在线优化和博弈，具有统一建模能力。
贝叶斯更新与乘法权重更新通过序列反馈重加权专家、模型或动作。我们证明，任意此类更新的后悔值均满足一个精确的信息会计恒等式：每轮中，学习者相对于任一参照物的超额损失，等于本轮暴露的不确定性带来的即时代价，以及当前权重与参照物间信息距离的减少之和。累积代价定义了一条路径相关的不确定性时钟，即所实现序列的内在时间。将单步平衡求和，得到两种精确的累积后悔分解方式，对应于更新在轮次间的两种自然组合方式。由于分解是精确的，有利的随机或低噪声情形表现为所实现内在时间的自约束性质。该会计体系还确定了一个学习率——其反比于内在时间的平方根——该调度在选定在线学习场景中可媲美自适应基线。相同的微积分框架涵盖Hedge、乐观与侧信息变体、连续先验、提升算法、在线凸优化、上下文老虎机及重复博弈：所有情形下的路径相关会计形式一致。
原文摘要 · Abstract (English)
Bayesian and multiplicative-weights updates reweight experts, models, or actions from sequential feedback. We show that the regret of any such update obeys an exact information-accounting identity. On each round, the learner's excess loss to any chosen comparator is the sum of an immediate cost for the uncertainty exposed by the round and a reduction in the information distance from the learner's current weights to the comparator. The cumulative cost defines a pathwise uncertainty clock, the intrinsic time of the realized sequence. Summing one-step balances yields two exact adaptive decompositions of cumulative regret, one for each natural way of composing the update across rounds. Because the decompositions are exact, favorable stochastic or low-noise regimes appear as self-bounding properties of the realized intrinsic time. The accounting also fixes a learning rate, inverse in the square root of intrinsic time. That schedule is competitive with adaptive baselines in selected online-learning settings. The same calculus covers Hedge, optimistic and side-information variants, continuous priors, boosting, online convex optimization, contextual bandits, and repeated games: the pathwise account is the same in every case.
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