Continuous Algorithms: Optimization and Sampling

Spring 2020. TR: 9:30-10:45. Arch.(East) 207.

Santosh Vempala, OH: Tue 11-12, Klaus 2222.
TA: He Jia, OH: Thu 2-4, Arthita Ghosh: OH: Mon 11-1.

The design of algorithms is traditionally a discrete endeavor. However, many advances have come from a continuous viewpoint. Typically, a continuous process, deterministic or randomized is designed (or shown) to have desirable properties, such as approaching an optimal solution or a desired distribution, and an algorithm is derived from this by appropriate discretization. We will use this viewpoint to develop several general techniques and build up to the state-of-the-art in polynomial algorithms for optimization and sampling.

The course is offered simultaneously at UW by Yin Tat Lee.

Prerequisite: basic knowledge of algorithms, probability, linear algebra.


Textbook (in progress)


Grading:
6550: Biweekly HW, including one longer HW towards the end. You are encouraged to collaborate on HW, but must write your own solutions. Submit via gradescope (link on canvas).
Send comments on lecture notes each week, either in dropbox or by email.
8803: HW optional. Comment on lecture notes each week.
Bonus: suggesting simpler proofs, exercises, figures.


Schedule (tentative):
  1. Introduction
    Jan 7. Course overview.
    Jan 9. Gradient descent. Notes.
    HW1 has been posted on gradescope, due Jan 20.
    Jan 14. Gradient descent (contd.) Notes.
  2. Elimination
    Jan 16. Cutting Plane method; Ellipsoid. Notes (and from a while ago)
    Jan 21. Center-of-Gravity. Notes
    Jan 23. Ball method; lower bounds.
  3. Reduction
    Jan 28, 30. Duality (LP duality, SDP duality, ...) and Equivalences (Optimization, Membership, Separation; OPT, Integration->Sampling; Maxflow, Bipartite Matching).
  4. Geometrization I
    Feb 4. Mirror Descent, Frank-Wolfe
    Feb 6. Newton method
    Feb 11. Interior Point Method
    Feb 13. IPM for LP.
  5. Sparsification
    Feb 18. Regression and subspace embeddings.
    Feb 20. Matrix approximation.
    Feb 25. Coordinate Descent.
    Feb 27. Stochastic Gradient Descent.
  6. Acceleration
    Mar 3. Conjugate Gradient and Chebychev Expansion.
    Mar 5. Accelerated Gradient Descent.
  7. Decomposition
    Mar 10.
    Mar 12.
    Mar 17,19 (Spring break)
  8. Sampling
    Mar 24. Langevin Dynamics and SDE.
    Mar 26. Conductance, Ball walk
  9. Geometrization II
    Apr 7. Mixing of ball walk, Isoperimetry, Isotropy.
    Apr 9. Hit-and-Run, Dikin walk
    Apr 14. (Riemannian) HMC
    Apr 16. ODE. Complex analysis.