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.
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.
Jan 7. Course overview.
Jan 9. Gradient descent. Notes.
Jan 14. Gradient descent (contd.) Notes.
Jan 16. Cutting Plane method; Ellipsoid. Notes (and from a while ago)
Jan 21. Center-of-Gravity. Notes
Jan 23. Ball method; lower bounds. Notes
Jan 28. Duality (LP duality, SDP duality, ...). Notes
Jan 30, 4. Equivalences (Optimization, Membership, Separation; Gradient, Evaluation). Notes
- Geometrization I
Feb 4. Mirror Descent. Notes
Feb 6. Frank-Wolfe. Notes
Feb 11. Newton method
Feb 13, 18. Interior Point Method for LP.Notes
Feb 20, 25. Self-concordance and IPM for convex optimization.
Mar 3. Regression and subspace embeddings.Notes
Mar 5. Matrix approximation.
Mar 10. Linear Regression and Chebychev Polynomials. Notes
Mar 12. Conjugate Gradient. Notes
Mar 17,19 (Spring break)
Mar 24. Langevin Dynamics and SDE.
Mar 26. Conductance and Convergence
- Geometrization II
Apr 7. Mixing of the ball walk, Isotropy.
Apr 9. Isoperimetry, KLS
Apr 14. Volume Computation
Apr 16. Gaussian Cooling
Additional topics: (Riemannian) HMC, ODE, Complex analysis.