Semidefinite Programming with Applications in Statistical Learning

Objective

Interior point algorithms and a dramatic growth in computing power have revolutionized optimization in
the last two decades. Highly nonlinear problems which were previously thought intractable are now
routinely solved at reasonable scales. Semidefinite programs (i.e. linear programs on the cone of positive
semidefinite matrices) are a perfect example of this trend: reasonably large, highly nonlinear but convex
eigenvalue optimization problems are now solved efficiently by reliable numerical packages. This in turn
means that a wide array of new applications for semidefinite programming have been discovered,
mimicking the early development of linear programming. To cite only a few examples, semidefinite
programs have been used to solve collaborative filtering problems (e.g. make personalized movie
recommendations), approximate the solution of combinatorial programs, optimize the mixing rate of
Markov chains over networks, infer dependence patterns from multivariate time series or produce optimal
kernels in classification problems.
These new applications also come with radically different algorithmic requirements. While interior point
methods solve relatively small problems with a high precision, most recent applications of semidefinite
programming in statistical learning for example form very large-scale problems with comparatively low
precision targets, programs for which current algorithms cannot form even a single iteration. This
proposal seeks to break this limit on problem size by deriving reliable first-order algorithms for solving
large-scale semidefinite programs with a significantly lower cost per iteration, using for example
subsampling techniques to considerably reduce the cost of forming gradients.
Beyond these algorithmic challenges, the proposed research will focus heavily on applications of convex
programming to statistical learning and signal processing theory where optimization and duality results
quantify the statistical performance of coding or variable selection algorithms for example. Finally,
another central goal of this work will be to produce efficient, customized algorithms for some key
problems arising in machine learning and statistics.

Fields of science

Funding Scheme

ERC-SG - ERC Starting Grant

Host institution

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS

EU contribution

€ 1 148 460,00

Address

RUE MICHEL ANGE 3
75794 Paris
France

Region

Ile-de-France Ile-de-France Paris

Activity type

Research Organisations

Administrative Contact

Julie Zittel (Ms.)

Principal investigator

Alexandre Werner Geoffroy Gobert D'aspremont Lynden (Prof.)

Links

Contact the organisation Website

Total cost

No data

Beneficiaries (1)