Approximate Universal Prediction under Interaction

Abstract

Bayesian mixtures, and Solomonoff induction in particular, provide an idealized model of prediction learned by minimizing population log loss, and they underlie universal agents such as AIXI. An agent that plans with an approximation of such a predictor decides where that predictor is evaluated, and a planner can be drawn toward branches on which its model is optimistic. We study which approximation guarantees survive this feedback. For normalized finite-horizon causal mixtures, the transfer coefficient factors into a policy likelihood ratio and task evidence available before each prediction, so task rarity and action-branch rarity can compound. Within one fixed universal mixture, infinitely many computable environments admit excess prediction error of order $κη\,2^{K(μ)}$, where $η$ is the training KL budget and $κ$ measures policy coverage. This order is sharp, and for each target a computable predictor attains it at a suitable finite horizon, so computability alone does not remove the penalty. For bounded returns, value error is controlled by the probability of entering a poorly covered branch plus a square-root term in $η$. Both terms are necessary, and finite maximum coverage ratios are not required. Motivated by the mentor override of Golden Handcuffs, we derive margins for preserving deferral decisions and conditional KL budgets with an unavoidable inverse-history-probability penalty, and a counterexample separates agreement of deferral decisions from task-wise event avoidance. The guarantees apply to any predictor architecture that meets explicit population approximation and coverage conditions, which gives concrete targets for implementing idealized universal agents.