Prediction and dimension

Abstract

Given a set X of sequences over a finite alphabet, we investigate the following three quantities.(i) The feasible predictability of X is the highest success ratio that a polynomial-time randomized predictor can achieve on all sequences in X. (ii) The deterministic feasible predictability of X is the highest success ratio that a polynomial-time deterministic predictor can achieve on all sequences in X. (iii) The feasible dimension of X is the polynomial-time effectivization of the classical Hausdorff dimension ("fractal dimension") of X.Predictability is known to be stable in the sense that the feasible predictability of X ∪ Y is always the minimum of the feasible predictabilities of X and Y. We show that deterministic predictability also has this property if X and Y are computably presentable. We show that deterministic predictability coincides with predictability on singleton sets. Our main theorem states that the feasible dimension of X is bounded above by the maximum entropy of the predictability of X and bounded below by the segmented self-information of the predictability of X, and that these bounds are tight.

Effective fractal dimension: foundations and applications

“…This extends to resource-bounded dimension by imposing resource-bounds on the predictors to define resource-bounded unpredictabilities. This result explains and illuminates the relationships between prediction and Hausdorff dimension that were obtained by Fortnow and Lutz [15], Ryabko [49,50,51], and Staiger [57]. This section is based on [20].…”

“…They have been shown to be closely related to several measures of complexity in cluding Kolmogorov complexity [43,31,23], Boolean circuit-size complexity [30,23], and predictability [15]. Applications have also been given to the study of polynomial-time degrees [2] and approximate optimization [19].…”

SIGACT news complexity theory column 48

“…First, Hitchcock [26] proved that, for any of the above-mentioned resource bounds ∆, ∆-dimension is precisely unpredictability by ∆-computable predictors in the log-loss model of prediction, and it was shown in [4] that strong ∆-dimension admits a dual unpredictability characterization. When combined with results by Fortnow and Lutz [19], this characterization also yielded new relationships between log-loss prediction and linear-loss prediction.…”

Martingale Families and Dimension in P

“…Recently resource-bounded measure has been refined via effective dimension which is an effectivization of Hausdorff dimension, yielding applications in a variety of topics, including algorithmic information theory, computational complexity, prediction, and data compression [13,17,14,4,2,6].…”