We also study the more general case of an additional entropy regularizer.

These assumptions imply that an optimal solutionx?

Appendix E provides a worked example of this.

Appendix E provides a worked example of this.

We also study the more general case of an additional entropy regularizer.

A hallmark of modern machine learning is its ability to deal with high dimensional problems by exploiting structural assumptions that limit the degrees of freedom in the underlying model. A deep understanding of the capabilities and limits of high dimensional learning methods under specific assumptions such as sparsity, group sparsity, and low rank has been attsined. Efforts [1,2] are now underway to distill this valuable experience by proposing general unified frameworks that can achieve the twio goals of summarizing previous analyses and enabling their application to notions of structure hitherto unexplored. Inspired by these developments, we propose and analyze a general computational scheme based on a greedy strategy to solve convex optimization problems that arise when dealing with structurally constrained high-dimensional problems. Our framework not only unifies existing greedy algorithms by recovering them as special cases but also yields novel ones.

Modem statistical estimators developed over the past decade have statistical or sample complexity that depends only weakly on the number of parameters when there is some structore to the problem, such as sparsity. A central question is whether similar advances can be made in their computational complexity as well. In this paper, we propose strategies that indicate that such advances can indeed be made. In particular, we investigate the greedy coordinate descent algorithm, and note that performing the greedy step efficiently weakens the costly dependence on the problem size provided the solution is sparse. We then propose a snite of methods that perform these greedy steps efficiently by a reduction to nearest neighbor search. We also devise a more amenable form of greedy descent for composite non-smooth objectives; as well as several approximate variants of such greedy descent. We develop a practical implementation of our algorithm that combines greedy coordinate descent with locality sensitive hashing. Without tuning the latter data structore, we are not only able to significantly speed up the vanilla greedy method, hot also outperform cyclic descent when the problem size becomes large. Our resnlts indicate the effectiveness of our nearest neighbor strategies, and also point to many open questions regarding the development of computational geometric techniques tailored towards first-order optimization methods.

Conic programming has well-documented merits in a gamut of signal processing and machine learning tasks. This contribution revisits a recently developed first-order conic descent (CD) solver, and advances it in three aspects: intuition, theory, and algorithmic implementation. It is found that CD can afford an intuitive geometric derivation that originates from the dual problem. This opens the door to novel algorithmic designs, with a momentum variant of CD, momentum conic descent (MOCO) exemplified. Diving deeper into the dual behavior CD and MOCO reveals: i) an analytically justified stopping criterion; and, ii) the potential to design preconditioners to speed up dual convergence. Lastly, to scale semidefinite programming (SDP) especially for low-rank solutions, a memory efficient MOCO variant is developed and numerically validated.

Graph-based algorithms have demonstrated state-of-the-art performance in the nearest neighbor search (NN-Search) problem. These empirical successes urge the need for theoretical results that guarantee the search quality and efficiency of these algorithms. However, there exists a practice-to-theory gap in the graph-based NN-Search algorithms. Current theoretical literature focuses on greedy search on exact near neighbor graph while practitioners use approximate near neighbor graph (ANN-Graph) to reduce the preprocessing time. This work bridges this gap by presenting the theoretical guarantees of solving NN-Search via greedy search on ANN-Graph for low dimensional and dense vectors. To build this bridge, we leverage several novel tools from computational geometry. Our results provide quantification of the trade-offs associated with the approximation while building a near neighbor graph. We hope our results will open the door for more provable efficient graph-based NN-Search algorithms.