This paper studies streaming optimization problems with objectives given by a sum of locally coupled cost terms of the form $f(vx_{t-1},vx_t)$. In particular, we are interested in how the solution $\hat{\boldsymbol{x}}_{t|T}$ 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{\boldsymbol{x}}_{t|T}$ converges to a limit point $\boldsymbol{x}^*_{t}$ 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 per-iteration 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 inhomogeneous Poisson intensity estimation, support the theoretical results and show how the algorithm can be used in practice.

中文翻译：

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时变优化问题的流式解决方案

本文研究流式优化问题，其目标由以下形式的局部耦合成本项的总和给出$f(vx_{t-1},vx_t)$. 我们特别感兴趣的是如何解决$\hat{\boldsymbol{x}}_{t|T}$为了$t$变量的第帧变化为$T$增加。递增时$T$并且添加一个新的函数和一组新的变量通常会改变任何地方的解决方案，我们给出条件$ \hat{\boldsymbol{x}}_{t|T}$收敛到极限点$\boldsymbol{x}^*_{t}$以线性速率为$T\rightarrow \infty$. 因此，我们能够为内存有限的算法推导出理论保证，这表明将解决方案更新限制在过去的少量帧中几乎不会牺牲任何准确性。我们还提出了一种新的高效牛顿在线算法（NOA），受这些结果的启发，它以固定的每次迭代复杂度更新解决方案$ \mathcal {O}(3Bn^{3})$， 独立于$T$， 在哪里$B$对应于变量在过去的更新时间，并且$n$是单个块向量的大小。两个流优化示例，非均匀样本的在线重建和非均匀泊松强度估计，支持理论结果并展示了该算法如何在实践中使用。