使用原始递归超级预测器进行通用个体序列预测
Universal Individual-Sequence Prediction with a Primitive-Recursive Superpredictor
AI总结：
研究个体二进制序列顺序预测，构造可计算概率预测器，其相对于原始递归预测器有次线性遗憾界，证明PPM预测器是原始递归的，该预测器在特定源实现上达最优，还与其他预测器建立严格分离。
AI中文摘要：
我们研究在零一损失下个体二进制序列的顺序预测。没有可计算的主预测器能在每个序列上与所有全可计算预测器竞争。因此我们考虑有理值原始递归预测器，它包含有限状态、基于上下文和部分匹配预测（PPM）规则等。我们构造了一个可计算概率预测器，相对于每个原始递归预测器在每个个体序列上有明确的次线性遗憾界。还证明了PPM预测器是原始递归的。所以我们的预测器在每个可计算平稳遍历二进制源的每个马丁 - 洛夫随机实现上达到无限过去贝叶斯误差。这种最优性扩展到根据任意原始递归调度交错的有限多个独立此类源。最后，我们与有限状态预测和每个固定原始递归预测器建立了严格分离。
英文摘要：
We study sequential prediction of individual binary sequences under zero-one loss. No computable master can compete on every sequence with all total computable predictors. We therefore consider rational-valued primitive-recursive forecasters, a broad syntactically enumerable class containing finite-state, context-based, and Prediction by Partial Matching (PPM) rules. We construct a computable probabilistic predictor with an explicit sublinear regret bound relative to every primitive-recursive forecaster on every individual sequence. We further prove that the PPM predictor is primitive recursive. Consequently, our predictor attains the infinite-past Bayes error on every Martin-Löf random realization of every computable stationary ergodic binary source. This optimality extends to finitely many independent such sources interleaved according to an arbitrary primitive-recursive schedule. Finally, we establish strict separations from finite-state prediction and from every fixed primitive-recursive predictor. keywords: Universal prediction, individual sequences, prediction with expert advice, primitive recursive functions, Kolmogorov complexity, PPM.
