## 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.

- Reduction
- Elimination
- Conditioning
- Geometrization
- Expansion
- Sparsification
- Acceleration
- Decomposition
- Discretization

The course is offered simultaneously at UW by

Yin Tat Lee.

**Prerequisite**: Basic knowledge of algorithms, probability, linear algebra.

**Textbook (updated May 2023)** (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):

- Introduction

Jan 7. Course overview.

Jan 9. Gradient descent. Notes.

Jan 14. Gradient descent (contd.) Notes.
- 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. Notes

- Reduction

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.

- Sparsification

Mar 3. Regression and subspace embeddings.Notes

Mar 5. Matrix approximation.

- Acceleration

Mar 10. Linear Regression and Chebychev Polynomials. Notes

Mar 12. Conjugate Gradient. Notes

Mar 17,19 (Spring break)
- Sampling

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.