## Constrained and Localized Nonparametric Estimation and Optimization

### With John Lafferty (U of Chicago)

# Constrained and Localized Nonparametric Estimation and Optimization

We present work on two nonstandard frameworks for minimax analysis.

For the first problem, imagine that I estimate a statistical model

from data, and then want to share my model with you. But we are

communicating over a resource constrained channel. By sending lots of

bits, I can communicate my model accurately, with little loss in

statistical risk. Sending a small number of bits will incur some

excess risk. What can we say about the tradeoff between statistical

risk and the communication constraints? This is a type of rate

distortion and constrained minimax problem, for which we provide a

sharp analysis in certain nonparametric settings.

The second problem starts with the question “how difficult is it to

minimize a specific convex function?” This is tricky to

formalize traditional complexity analysis is expressed in terms of

the worst case over a large class of instances. We extend the

classical minimax analysis of stochastic convex optimization by

introducing a localized form of minimax complexity for individual

functions. This uses a computational analogue of the modulus of

continuity that is central to statistical minimax analysis, which

serves as a computational analogue of Fisher information.

Joint work with Sabyasachi Chatterjee, John Duchi, and Yuancheng Zhu.

- Speaker: John Lafferty (U of Chicago)
- Friday 04 November 2016, 16:00–17:00
- Venue: MR12, Centre for Mathematical Sciences, Wilberforce Road, Cambridge..
- Series: Statistics; organiser: Quentin Berthet.