# EE 227C (Spring 2018)

Convex Optimization and Approximation

University of California, Berkeley

Time: **TuTh 12:30PM - 1:59PM**, Location: **Etcheverry 3106**

Instructor: Moritz Hardt (Email: hardt+ee227c@berk…edu)

Graduate Instructor: Max Simchowitz (Email: msimchow@berk…edu).

Office hours: Max on Mon 3-4pm, Soda 310 (starting 1/29), Moritz on Fri 9–9:50a, SDH 722

## Summary

This course will explore theory and algorithms for nonlinear optimization. We will focus on problems that arise in machine learning and modern data analysis, paying attention to concerns about complexity, robustness, and implementation in these domains. We will also see how tools from convex optimization can help tackle non-convex optimization problems common in practice.

## Assignments

Assignments will be posted on
Piazza. If you haven’t
already, sign up here.
Homeworks will be assigned roughly every two weeks, and 2–3 problems will be selected for grading (we will not tell you which ones in advance). Assignments should be submitted through GradeScope; the course is listed as EE227C, which you may join with entry code 9P5NDV. *All homeworks should be latexed.* Students will be permitted two unexcused late assignments (up to a week late). Students requesting additional extensions should email Max.

## Grading

Grading policy: 50% homeworks, 10% scribing, 20% midterm exam, 20% final exam.

## Course notes

Course notes will be publicly available. Participants will collaboratively create and maintain notes over the course of the semester using git. See this repository for source files. Most lectures will have an accompanying Jupyter notebook containing plots and illustrative examples.

### Sign up for scribing here

All three scribes should collaborate to provide a *single* tex file as seen
here.
Students are required to closely follow these instructions.

We suggest that each scribe takes down notes, and then all three meet after class to consolidate.

### All available lecture notes (pdf)

See individual lectures below. These notes likely contain several mistakes. If you spot any please send an email or pull request.

## Schedule

# | Date | Topic | ipynb | |
---|---|---|---|---|

Part I: Basic gradient methods |
||||

1 | 1/16 | Convexity | ipynb | |

2 | 1/18 | Gradient method (non-smooth and smooth) | — | |

3 | 1/23 | Gradient method (strongly convex) | — | |

4 | 1/25 | Some applications of gradient methods | ipynb | |

5 | 1/30 | Conditional gradient (Frank-Wolfe algorithm) | ipynb | |

Part II: Krylov methods and acceleration |
||||

6 | 2/1 | Discovering acceleration with Chebyshev polynomials | ipynb | |

7 | 2/6 | Nesterov’s accelerated gradient descent | — | |

8 | 2/8 | Eigenvalue intermezzo | — | |

9 | 2/13 | Lower bounds, robustness vs acceleration | py | |

Part III: Stochastic optimization |
||||

10 | 2/15 | Stochastc optimization | — | |

11 | 2/20 | Learning, regularization, and generalization | — | |

12 | 2/22 | Coordinate Descent (guest lecture by Max Simchowitz) | — | |

Part IV: Dual methods |
||||

13 | 2/27 | Duality theory | — | |

14 | 3/1 | Dual decomposition, method of multipliers | — | |

15 | 3/6 | Stochastic Dual Coordinate Ascent | — | |

16 | 3/8 | Backpropagation and adjoints | — | |

Part V: Non-convex problems |
||||

17 | 3/13 | Non-convex problems | — | |

18 | 3/15 | Saddle points | ||

19 | 3/20 | Alternating minimization and expectaction maximization | — | ipynb |

20 | 3/22 | Derivative-free optimization, policy gradient, controls | — | ipynb |

– | 3/27 | No class (Spring break) | ||

– | 3/29 | No class (Spring break) | ||

21 | 4/3 | Guest lecture by Ludwig Schmidt on non-convex constraints | ||

22 | 4/5 | Guest lecture by Ludwig Schmidt on non-convex constraints | ipynb | |

Part VI: Higher-order and interior point methods |
||||

23 | 4/10 | Newton method | ||

24 | 4/12 | Second-order methods code | — | ipynb |

25 | 4/17 | Barrier methods | ||

26 | 4/19 | Primal-dual interior point methods | ||

27 | 4/24 | Sum of squares | ||

28 | 4/26 | Last lecture | ||

29 | 5/1 | Reading, review, recitation | ||

30 | 5/3 | Reading, review, recitation |

## Background

The prerequisites are previous coursework in linear algebra, multivariate calculus, probability and statistics. Coursework or background in optimization theory as covered in EE227BT is highly recommended. The class will involve some basic programming. Students are encouraged to use either Julia or Python. We discourage the use of MATLAB.

## Material

### Textbooks

- Nonlinear Programming (3rd edition). D. Bertsekas, Athena Scientific.
- Numerical Optimization. J. Nocedal and S. J. Wright, Springer Series in Operations Research, Springer-Verlag, New York, 2006 (2nd edition).
- Convex Optimization. S. Boyd and L. Vandenberghe. Cambridge University Press, Cambridge, 2003. PDF available here
- Introductory Lectures on Convex Optimization: A Basic Course. Y. Nesterov. Kluwer, 2004.
- Convex Optimization: Algorithms and Complexity. S. Bubeck. PDF available here
- Nonlinear Programming D. P. Bertsekas. Athena Scientific, Belmont, Massachusetts. (2nd edition). 1999.
- Participants will furthermore have access to a yet unpublished optimization text
called
*Nonlinear Optimization for Machine Learning: New Shit Has Come to Light*.

### Lecture notes

- Efficient Methods in Convex Programming. A. Nemirovski. Lecture Notes as PDF available here.

### Blog posts

- The Zen of Gradient Descent Moritz Hardt
- Why Momentum Really Works Gabriel Goh.
- Robustness versus Acceleration Moritz Hardt