Mathematics of Data: From Theory to Computation (Fall 2014)



Prof. Volkan Cevher



Convex optimization offers a unified framework in obtaining numerical solutions to data analytics problems with provable statistical guarantees of correctness at well-understood computational costs.

To this end, this course reviews recent advances in convex optimization and statistical analysis in the wake of Big Data. We provide an overview of the emerging convex data models and their statistical guarantees, describe scalable numerical solution techniques such as first-order methods and their randomized versions, and illustrate the essential role of parallel and distributed computation.

Throughout the course, we put the mathematical concepts into action with large-scale applications from machine learning, signal processing, and statistics.


Learning outcomes

By the end of the course, the students are expected to understand the so-called time-data tradeoffs in data analytics. In particular, the students must be able to

  1. Choose an appropriate convex formulation for a data analytics problem at hand
  2. Estimate the underlying data size requirements for the correctness of its solution
  3. Implement an appropriate convex optimization algorithm based on the available computational platform
  4. Decide on a meaningful level of optimization accuracy for stopping the algorithm
  5. Characterize the time required for their algorithm to obtain a numerical solution with the chosen accuracy



Previous coursework in calculus, linear algebra, and probability is required. Familiarity with optimization is useful. 



The course consists of the following topics

Lecture 1 “Objects in Space”: Definitions of norms, inner products, and metrics for vector, matrix and tensor objects.
  Basics of complexity theory.
Lecture 2 Maximum likelihood principle as a motivation for convex optimization. 
  Fundamental structures in convex analysis, such as cones, smoothness, and conjugation.
Lecture 3 Unconstrained, smooth minimization techniques.
  Gradient methods.
  Variable metric algorithms.
  Time-data tradeoffs in ML estimation.
Lecture 4 Convex geometry of linear inverse problems. 
  Structured data models (e.g., sparse and low-rank) and convex gauge functions and formulations that encourage these structures.
  Computational aspects of gauge functions.
Lecture 5
Composite convex minimization. Regularized M-estimators. 
  Time-data tradeoffs in linear inverse problems.
Lecture 6 Convex demixing.
  Statistical dimension.
  Phase transitions in convex minimization.
  Smoothing approaches for non-smooth convex minimization.
Lecture 7 Constrained convex minimization-I.
  Introduction to convex duality.
  Classical solution methods (the augmented Lagrangian method, alternating minimization algorithm, alternating direction method of multipliers, and the Frank-Wolfe method) and their deficiencie.
Lecture 8 Constrained convex minimization-II. 
  Variational gap characterizations and dual smoothing.
  Scalable, black-box optimization techniques.
  Time data-tradeoffs for linear inverse problems.
Lecture 9 Classical black-box convex optimization techniques. 
  Linear programming, semidefinite programming, and the interior point method (IPM).
  Hierarchies of classical formulations.
  Time and space complexity of the IPM.
Lecture 10 Time-data tradeoffs in machine learning.
Lecture 11 Convex methods for Big Data I: Randomized coordinate descent methods.
  The Page Rank problem and Nesterov’s solution.
  Composite formulations.
Lecture 12 Convex methods for Big Data II: Stochastic gradient descent methods.
  Least squares: conjugate gradients vs. a simple stochastic gradient method.
  Dual and gradient averaging schemes.
  Stochastic mirror descent.
Lecture 13 Randomized linear algebra routines for scalable convex optimization.
  Probabilistic algorithms for constructing approximate low-rank matrix decompositions.
  Subset selection approaches.
  Theoretical approximation guarantees.
Lecture 14 Role of parallel and distributed computing.  
  How to avoid communication bottlenecks and synchronization.
  Consensus methods.
  Memory lock-free, decentralized, and asynchronous algorithms.