Asymptotics of Discrete MDL for Online Prediction

Abstract

Abstract-Minimum description length (MDL) is an important principle for induction and prediction, with strong relations to optimal Bayesian learning. This paper deals with learning processes which are independent and identically distributed (i.i.d.) by means of two-part MDL, where the underlying model class is countable. We consider the online learning framework, i.e., observations come in one by one, and the predictor is allowed to update its state of mind after each time step. We identify two ways of predicting by MDL for this setup, namely, a static and a dynamic one. (A third variant, hybrid MDL, will turn out inferior.) We will prove that under the only assumption that the data is generated by a distribution contained in the model class, the MDL predictions converge to the true values almost surely. This is accomplished by proving finite bounds on the quadratic, the Hellinger, and the Kullback-Leibler loss of the MDL learner, which are, however, exponentially worse than for Bayesian prediction. We demonstrate that these bounds are sharp, even for model classes containing only Bernoulli distributions. We show how these bounds imply regret bounds for arbitrary loss functions. Our results apply to a wide range of setups, namely, sequence prediction, pattern classification, regression, and universal induction in the sense of algorithmic information theory among others.

Catching up Faster by Switching Sooner: A Predictive Approach to Adaptive Estimation with an Application to the AIC–BIC Dilemma

“…Now, if the AIC–BIC dilemma is interpreted as a conflict between consistency and optimal sequential prediction, then cumulative risk is a natural and often‐considered performance criterion (Haussler and Opper, 1997; Rissanen et al. , 1992; Barron, 1998a; Yang and Barron, 1999; Poland and Hutter, 2005), and we can reasonably claim that our results solve the dilemma. However, it can also be interpreted as a dichotomy between model selection for truth finding and model selection‐based (non‐sequential) estimation.…”

“…Minimax cumulative risk has previously been studied by, among others, Haussler and Opper (1997), Rissanen et al. (1992), Barron (1998a), Yang and Barron (1999) and Poland and Hutter (2005).…”

Offline to Online Conversion

“…Several estimation procedures do not only provide q n on X n , but measures on X ∞ or equivalently for each n separately a TC q n : X * → [0;1] (see Bayes and crude MDL below). While this opens further options forq, e.g.q(x n+1 |x 1:n ):=q n (x 1:n+1 )/q n (x 1:n ) with some (weak) results for MDL [PH05], it does not solve our main problem.…”

“…Crude MDL simply selects q n := argmax ν∈M {ν(x 1:n ) w(ν)} at time n, which is a probability measure on X ∞ . While this opens additional options for defining q, they also can perform poorly in the worst case [PH05]. Note that most versions of MDL perform often very well in practice, comparable to Bayes; robustness and proving guarantees are the open problems.…”

Fully General Online Imitation Learning

“…where 13 follows from Equations 7 and 8. Our ρ i , ρ norm i , and ρ stat i are closely inspired by Poland and Hutter (2005), who constructed (in our notation) ρ 1 , ρ norm 1 , and ρ stat 1 . Our first lemma bounds the deviation from ρ i being a measure.…”