128 research outputs found
Self-concordant Smoothing for Convex Composite Optimization
We introduce the notion of self-concordant smoothing for minimizing the sum
of two convex functions: the first is smooth and the second may be nonsmooth.
Our framework results naturally from the smoothing approximation technique
referred to as partial smoothing in which only a part of the nonsmooth function
is smoothed. The key highlight of our approach is in a natural property of the
resulting problem's structure which provides us with a variable-metric
selection method and a step-length selection rule particularly suitable for
proximal Newton-type algorithms. In addition, we efficiently handle specific
structures promoted by the nonsmooth function, such as -regularization
and group-lasso penalties. We prove local quadratic convergence rates for two
resulting algorithms: Prox-N-SCORE, a proximal Newton algorithm and
Prox-GGN-SCORE, a proximal generalized Gauss-Newton (GGN) algorithm. The
Prox-GGN-SCORE algorithm highlights an important approximation procedure which
helps to significantly reduce most of the computational overhead associated
with the inverse Hessian. This approximation is essentially useful for
overparameterized machine learning models and in the mini-batch settings.
Numerical examples on both synthetic and real datasets demonstrate the
efficiency of our approach and its superiority over existing approaches.Comment: 37 pages, 7 figures, 3 table
Convex Optimization for Big Data
This article reviews recent advances in convex optimization algorithms for
Big Data, which aim to reduce the computational, storage, and communications
bottlenecks. We provide an overview of this emerging field, describe
contemporary approximation techniques like first-order methods and
randomization for scalability, and survey the important role of parallel and
distributed computation. The new Big Data algorithms are based on surprisingly
simple principles and attain staggering accelerations even on classical
problems.Comment: 23 pages, 4 figurs, 8 algorithm
Rigorous optimization recipes for sparse and low rank inverse problems with applications in data sciences
Many natural and man-made signals can be described as having a few degrees of freedom relative to their size due to natural parameterizations or constraints; examples include bandlimited signals, collections of signals observed from multiple viewpoints in a network-of-sensors, and per-flow traffic measurements of the Internet. Low-dimensional models (LDMs) mathematically capture the inherent structure of such signals via combinatorial and geometric data models, such as sparsity, unions-of-subspaces, low-rankness, manifolds, and mixtures of factor analyzers, and are emerging to revolutionize the way we treat inverse problems (e.g., signal recovery, parameter estimation, or structure learning) from dimensionality-reduced or incomplete data. Assuming our problem resides in a LDM space, in this thesis we investigate how to integrate such models in convex and non-convex optimization algorithms for significant gains in computational complexity. We mostly focus on two LDMs: sparsity and low-rankness. We study trade-offs and their implications to develop efficient and provable optimization algorithms, and--more importantly--to exploit convex and combinatorial optimization that can enable cross-pollination of decades of research in both
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