Universal MMSE Filtering With Logarithmic Adaptive Regret

TL;DR

This paper introduces a linear-time algorithm for online real-valued signal estimation that minimizes mean-square-error, achieving logarithmic adaptive regret against the best linear filter in hindsight, thus advancing online filtering methods.

Contribution

The paper presents a novel linear-time algorithm for online MMSE filtering that guarantees logarithmic adaptive regret, resolving a key open problem in the field.

Findings

Achieves logarithmic adaptive regret against the best linear filter.

Runs in linear time relative to the number of filter coefficients.

Provides an asymptotically tight regret bound.

Abstract

We consider the problem of online estimation of a real-valued signal corrupted by oblivious zero-mean noise using linear estimators. The estimator is required to iteratively predict the underlying signal based on the current and several last noisy observations, and its performance is measured by the mean-square-error. We describe and analyze an algorithm for this task which: 1. Achieves logarithmic adaptive regret against the best linear filter in hindsight. This bound is assyptotically tight, and resolves the question of Moon and Weissman [1]. 2. Runs in linear time in terms of the number of filter coefficients. Previous constructions required at least quadratic time.