Wouter M. Koolen

A closer look at adaptive regret

For the prediction with expert advice setting, we consider methods to construct algorithms that have low adaptive regret. The adaptive regret of an algorithm on a time interval [t1, t2] is the loss of the algorithm minus the loss of the best expert over that interval. Adaptive regret measures how well the algorithm approximates the best expert locally, and so is different from, although closely related to, both the classical regret, measured over an initial time interval [1, t], and the tracking regret, where the algorithm is compared to a good sequence of experts over [1, t]. We investigate two existing intuitive methods for deriving algorithms with low adaptive regret, one based on specialist experts and the other based on restarts. Quite surprisingly, we show that both methods lead to the same algorithm, namely Fixed Share, which is known for its tracking regret. We provide a thorough analysis of the adaptive regret of Fixed Share. We obtain the exact worst-case adaptive regret for Fixed Share, from which the classical tracking bounds follow. We prove that Fixed Share is optimal for adaptive regret: the worst-case adaptive regret of any algorithm is at least that of an instance of Fixed Share.

Some Mathematical Refinements Concerning Error Minimization in the Genetic Code

The genetic code is known to have a high level of error robustness and has been shown to be very error robust compared to randomly selected codes, but to be significantly less error robust than a certain code found by a heuristic algorithm. We formulate this optimization problem as a Quadratic Assignment Problem and use this to formally verify that the code found by the heuristic algorithm is the global optimum. We also argue that it is strongly misleading to compare the genetic code only with codes sampled from the fixed block model, because the real code space is orders of magnitude larger. We thus enlarge the space from which random codes can be sampled from approximately 2.433 × 1018 codes to approximately 5.908 × 1045 codes. We do this by leaving the fixed block model, and using the wobble rules to formulate the characteristics acceptable for a genetic code. By relaxing more constraints, three larger spaces are also constructed. Using a modified error function, the genetic code is found to be more error robust compared to a background of randomly generated codes with increasing space size. We point out that these results do not necessarily imply that the code was optimized during evolution for error minimization, but that other mechanisms could be the reason for this error robustness.

Universal Codes From Switching Strategies

We discuss algorithms for combining sequential prediction strategies, a task which can be viewed as a natural generalization of the concept of universal coding. We describe a graphical language based on hidden Markov models for defining prediction strategies, and we provide both existing and new models as examples. The models include efficient, parameterless models for switching between the input strategies over time, including a model for the case where switches tend to occur in clusters, and finally a new model for the scenario where the prediction strategies have a known relationship, and where jumps are typically between strongly related ones. This last model is relevant for coding time series data where parameter drift is expected. As theoretical contributions, we introduce an interpolation construction that is useful in the development and analysis of new algorithms, and we establish a new sophisticated lemma for analyzing the individual sequence regret of parameterized models.

Time-bounded incompressibility of compressible strings and sequences

For every total recursive time bound t, a constant fraction of all compressible (low Kolmogorov complexity) strings is t-bounded incompressible (high time-bounded Kolmogorov complexity); there are uncountably many infinite sequences of which every initial segment of length n is compressible to logn yet t-bounded incompressible below 14n-logn; and there is a countably infinite number of recursive infinite sequences of which every initial segment is similarly t-bounded incompressible. These results and their proofs are related to, but different from, Barzdinss lemma.