Non-convex Optimization for Machine Learning

By Prateek Jain, Purushottam Kar. Found. Trends Mach. Learn., 2017

550 citations recorded by the source corpus.

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. We hope that an insight into the inner workings of these methods will allow the reader to appreciate the unique marriage of task structure and generative models that allow these heuristic techniques to (provably) succeed. The monograph will lead the reader through several widely used non-convex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.

Subfields: Artificial Intelligence, Neurology

View Paper: https://www.semanticscholar.org/paper/43d1fe40167c5f2ed010c8e06c8e008c774fd22b

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Citations in the corpus

A Sufficient Condition for Convergences of Adam and RMSProp (2018, background): “Large-scale non-convex stochastic optimization [5], covering a slew of applications in statistics and machine learning [13, 5] such as learning a latent variable from massive data whose probability density distribution is unknown, takes the following generic formulation:”
A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation (2018, methodology): “In addition to such asymptotic techniques, the modern literature on first-order stochastic optimization also focuses heavily on nonasymptotic analysis (Bubeck 2015, Jain and Kar 2017, Bottou et al. 2018).”
Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview (2018, methodology): “Roughly speaking, [219] concentrates on the introduction of generic optimization algorithms, with low-rank factorization (and other learning problems) used as special examples to illustrate the generic results.”
Minimization Over the Nonconvex Sparsity Constraint Using A Hybrid First-order method (2021, methodology): “The gradient projection method described in [27] for nonconvexly constrained problems can be applied for solving ( P ).”
Statistically Efficient Off-Policy Policy Gradients (2020, unknown): “, Athey & Wager (2017); Kallus (2018); Kallus & Zhou (2018); Swaminathan & Joachims (2015) for H = 0 and Chen et al.”
Reinforcement Learning: Theory and Algorithms (2019, background): “We refer the reader to Jiang et al. [2017] for detailed examples, as well as connections of Bellman rank with other rank type notions in the RL literature to measure problem complexity.”
A Survey of Optimization Methods From a Machine Learning Perspective (2019, unknown): “There is some work on summarizing the optimization methods for solving nonconvex functions from the perspective of machine learning [133].”
First-order Stochastic Algorithms for Escaping From Saddle Points in Almost Linear Time (2017, background): “We consider a non-convex optimization problem with f(x; ξ) = ∑d i=1 ξi(x 4 i − 4x(2)i ) where ξi are a normal random variables with mean of 1 so that the saddle points of the expected function are known [10].”