Duality Theory for Non-Markovian Linear Gaussian Models

TL;DR

This paper introduces a duality theory for non-Markovian linear Gaussian models, leading to a transformer-like linear predictor with linear computational complexity, improving over classical methods.

Contribution

It develops a dual control system and establishes a duality principle linking optimal control to filtering in non-Markovian Gaussian models, with explicit formulas for the dual filter.

Findings

Dual control system formulated as a backward difference equation.

Duality principle linking control problem to filtering.

Explicit optimal control formula for a linear predictor with linear complexity.

Abstract

This work develops a duality theory for partially observed linear Gaussian models in discrete time. The state process evolves according to a causal but non-Markovian (or higher-order Gauss-Markov) structure, captured by a lower-triangular transition operator, which is related to transformer, with $T$ as the context length. The main contributions are: (i) a dual control system for the linear Gaussian model, formulated as a backward difference equation (B $\Delta$ E); (ii) a duality principle establishing that a specific linear-quadratic optimal control problem for the B $\Delta$ E is dual to the filtering problem for the partially observed model; and (iii) an explicit optimal control formula yielding a novel (transformer-like) linear predictor, referred to as the dual filter, whose computational complexity scales linearly in the time horizon $T$, in contrast to the $O(T^3)$ cost of classical smoothing and Wiener-Hopf approaches.