Approximate Universal Prediction under Interaction

Richard Cornelius Suwandi

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…