In a previous paper, we began the study of sequence prediction algorithms adapted to stringological word complexity measures. One measure we considered was left-to-right (most-significant-digit-first) automaticity. Here, we show a statistically and computationally efficient algorithm adapted to the dual'' right-to-left (least-significant-digit-first) automaticity, which turns out to be substantially different for our purpose. We also demonstrate a prediction algorithm for a more expressive measure that we call arithmetic repetition complexity''. In particular, the latter can be used for predicting the so-called mix-automatic sequences.
In previous papers, we began the study of sequence prediction algorithms adapted to stringological word complexity measures. In particular, we defined a complexity measure called Arithmetic Repetition Complexity (ARC) which admits a polynomial-time prediction algorithm with a mistake bound quasilinear in the complexity. Here, we show a weaker complexity measure related to ARC that admits an especially efficient prediction algorithm: an algorithm that runs in quasilinear time and polylog space for appropriate highly-structured sequences. The complexity measure is defined via a restricted class of "zipline programs" (a variant of straight-line programs), which we call layered. We thus get a less expressive measure with a more efficient algorithm (compared to our results for ARC), demonstrating a possible tradeoff.
Sequence prediction methods for linear dynamical systems with long memory, i.e. marginally stable systems, typically achieve regret that grows linearly with the hidden dimension of the underlying generative model. While many methods have been developed to address this regime with varying success, we show that simply using the second-order Vovk-Azoury-Warmuth (VAW) algorithm to learn a short autoregressive-with-inputs (ARX) model achieves astoundingly strong results: for bounded sequential data from a marginally-stable linear dynamical system with spectra in the complex disk except for angular wedge of width δ around the negative real axis, this algorithm achieves dimension-free regret O(δ−4log2T). These bounds are state-of-the-art to our knowledge. The key components for our result come from 1) using the theory of ``Universal Sequence Preconditioning'' (USP) \cite{marsdenuniversal} to prove the existence of an optimal setting of autoregressive coefficients, 2) the application of VAW which takes better advantage of the memory compression provided by USP, and 3) the analysis of Faber polynomials on circular sectors to extend these results to systems with complex spectra.
We consider the problem of sequential prediction of an m-ary sequence, where at each epoch, (i) the environment selects an outcome from an m-ary alphabet, (ii) the learner selects a probability distribution over the same alphabet (unaware of the outcome generated by the environment), and finally, (iii) the learner incurs a cost that depends on the probability assigned to the outcome. The cost function we consider captures the complexity of predicting the outcome generated by the environment, in a scenario where the aforementioned prediction is performed via comparative queries to a lying oracle. We consider both stochastic and adversarial environments, propose algorithms for both settings, and establish logarithmic upper bounds on their regret.