Streaming Solutions for Time-Varying Optimization Problems

Abstract

This paper studies streaming optimization problems that have objectives of the form <inline-formula><tex-math notation="LaTeX">$ \sum _{t=1}^{T}f(x_{t-1},x_{t})$</tex-math></inline-formula>. In particular, we are interested in how the solution <inline-formula><tex-math notation="LaTeX">$\hat{x}_{t|T}$</tex-math></inline-formula> for the <inline-formula><tex-math notation="LaTeX">$t$</tex-math></inline-formula>th frame of variables changes as <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> increases. While incrementing <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> and adding a new functional and a new set of variables does in general change the solution everywhere, we give conditions under which <inline-formula><tex-math notation="LaTeX">$\hat{x}_{t|T}$</tex-math></inline-formula> converges to a limit point <inline-formula><tex-math notation="LaTeX">$x^*_{t}$</tex-math></inline-formula> at a linear rate as <inline-formula><tex-math notation="LaTeX">$T\rightarrow \infty$</tex-math></inline-formula>. As a consequence, we are able to derive theoretical guarantees for algorithms with limited memory, showing that limiting the solution updates to only a small number of frames in the past sacrifices almost nothing in accuracy. We also present a new efficient Newton online algorithm (NOA), inspired by these results, that updates the solution with fixed per-iteration complexity of <inline-formula><tex-math notation="LaTeX">$ \mathcal{O}(3Bn^{3})$</tex-math></inline-formula>, independent of <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$B$</tex-math></inline-formula> corresponds to how far in the past the variables are updated, and <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the size of a single block-vector. Two streaming optimization examples, online reconstruction from non-uniform samples and inhomogeneous Poisson intensity estimation, support the theoretical results and show how the algorithm can be used in practice.