Streaming Solutions for Time-Varying Optimization Problems

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