Streaming Solutions for Time-Varying Optimization Problems

This paper studies streaming optimization problems that have objectives of the form $\sum_{t=1}^Tf(\mathbf{x}_{t-1},\mathbf{x}_t)$. In particular, we are interested in how the solution $\hat{\mathbf{x} }_{t|T}$ for the $t$th frame of variables changes as $T$ increases. While incrementing $T$ and adding a new functional and a new set of variables does in general change the solution everywhere, we give conditions under which $\hat{\mathbf{x} }_{t|T}$ converges to a limit point $\mathbf{x}^*_t$ at a linear rate as $T\rightarrow\infty$. 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 complexity of $\mathcal{O}( {3Bn^3})$, independent of $T$, where $B$ corresponds to how far in the past the variables are updated, and $n$ is the size of a single block-vector. Two streaming optimization examples, online reconstruction from non-uniform samples and non-homogeneous Poisson intensity estimation, support the theoretical results and show how the algorithm can be used in practice.