Uniform Sampling of Polytopesand Concentration of Measure

Sam Power

Cambridge Centre for AnalysisCantab Capital Institute for the Mathematics of Information

sp825@cam.ac.uk

June 7, 2018

Sam Power (CCA) Geometric MCMC June 7, 2018 1 / 25

Overview

1 Problem Statement

2 Basic Methods of Solution

3 Markov Chain Approaches

4 New Insights and Algorithms

5 Conclusion and Future Directions

Sam Power (CCA) Geometric MCMC June 7, 2018 2 / 25

Task

Uniform Sampling from Polytopes

K = {x ∈ Rd : 〈ai, x〉 6 bi for 1 6 i 6 f}

Sam Power (CCA) Geometric MCMC June 7, 2018 3 / 25

Halfspace

Figure: A halfspace in 2d

Sam Power (CCA) Geometric MCMC June 7, 2018 4 / 25

Polytope via Halfspaces

Figure: Building a polygon by intersecting halfspaces

Sam Power (CCA) Geometric MCMC June 7, 2018 5 / 25

Low-Dimensional Approach

Figure: Uniform points in a circle: generate points in a square (easy), throw awaythe points that miss the circle (even easier!)

Sam Power (CCA) Geometric MCMC June 7, 2018 6 / 25

Ball Walk (Lovasz, Simonovits, ...)

Figure: At your current location ...

Sam Power (CCA) Geometric MCMC June 7, 2018 7 / 25

Ball Walk (Lovasz, Simonovits, ...)

Figure: Draw a ball of radius r around you and move uniformly within that ball.

Sam Power (CCA) Geometric MCMC June 7, 2018 8 / 25

Barrier Walks (Chen, Dwivedi, et al.)

Figure: Propose from a locally-informed ellipsoid instead

Sam Power (CCA) Geometric MCMC June 7, 2018 9 / 25

Hit-and-Run (Smith, Belisle, Romeijn)

Figure: Pick a direction uniformly at random, move uniformly along that line.

Sam Power (CCA) Geometric MCMC June 7, 2018 10 / 25

Hit-and-Run

Sam Power (CCA) Geometric MCMC June 7, 2018 11 / 25

Generalising Hit-and-Run

No ‘parameters’ to tune

· · · still subtle freedom in the algorithm

How to ‘hit’

How to ‘run’

Some work on ‘hit’ proposals (Kaufman & Smith, Polyak & Gryazina)

Almost no work on ‘run’ proposals.

Sam Power (CCA) Geometric MCMC June 7, 2018 12 / 25

Generalising Hit-and-Run

No ‘parameters’ to tune

· · · still subtle freedom in the algorithm

How to ‘hit’

How to ‘run’

Some work on ‘hit’ proposals (Kaufman & Smith, Polyak & Gryazina)

Almost no work on ‘run’ proposals.

Sam Power (CCA) Geometric MCMC June 7, 2018 12 / 25

Generalising Hit-and-Run

No ‘parameters’ to tune

· · · still subtle freedom in the algorithm

How to ‘hit’

How to ‘run’

Some work on ‘hit’ proposals (Kaufman & Smith, Polyak & Gryazina)

Almost no work on ‘run’ proposals.

Sam Power (CCA) Geometric MCMC June 7, 2018 12 / 25

Generalising Hit-and-Run

No ‘parameters’ to tune

· · · still subtle freedom in the algorithm

How to ‘hit’

How to ‘run’

Some work on ‘hit’ proposals (Kaufman & Smith, Polyak & Gryazina)

Almost no work on ‘run’ proposals.

Sam Power (CCA) Geometric MCMC June 7, 2018 12 / 25

Generalising Hit-and-Run

No ‘parameters’ to tune

· · · still subtle freedom in the algorithm

How to ‘hit’

How to ‘run’

Some work on ‘hit’ proposals (Kaufman & Smith, Polyak & Gryazina)

Almost no work on ‘run’ proposals.

Sam Power (CCA) Geometric MCMC June 7, 2018 12 / 25

Generalising Hit-and-Run

No ‘parameters’ to tune

· · · still subtle freedom in the algorithm

How to ‘hit’

How to ‘run’

Some work on ‘hit’ proposals (Kaufman & Smith, Polyak & Gryazina)

Almost no work on ‘run’ proposals.

Sam Power (CCA) Geometric MCMC June 7, 2018 12 / 25

General Setup

If we are

At x,

Sample direction u ∼ Hx, and

Move a distance t ∼ Rx,u in that direction,

we have transition kernel

fH,R(x→ y) =H(u) ·R(t)‖y − x‖d−1

where y = x+ tu.

Sam Power (CCA) Geometric MCMC June 7, 2018 13 / 25

Perfect Hit-and-Run

Observation: we can make this kernel independent of y:

Hx(u) ∝ a(x, u)d (1)Rx,u(t) ∝ td−1 · I[0 6 t 6 a(x, u)] (2)

Intuition: convex geometry

; perfect sampling!

· · · but, there’s a hitch

Sam Power (CCA) Geometric MCMC June 7, 2018 14 / 25

Perfect Hit-and-Run

Observation: we can make this kernel independent of y:

Hx(u) ∝ a(x, u)d (1)Rx,u(t) ∝ td−1 · I[0 6 t 6 a(x, u)] (2)

Intuition: convex geometry

; perfect sampling!

· · · but, there’s a hitch

Sam Power (CCA) Geometric MCMC June 7, 2018 14 / 25

Perfect Hit-and-Run

Observation: we can make this kernel independent of y:

Hx(u) ∝ a(x, u)d (1)Rx,u(t) ∝ td−1 · I[0 6 t 6 a(x, u)] (2)

Intuition: convex geometry

; perfect sampling!

· · · but, there’s a hitch

Sam Power (CCA) Geometric MCMC June 7, 2018 14 / 25

Perfect Hit-and-Run

Observation: we can make this kernel independent of y:

Hx(u) ∝ a(x, u)d (1)Rx,u(t) ∝ td−1 · I[0 6 t 6 a(x, u)] (2)

Intuition: convex geometry

; perfect sampling!

· · · but, there’s a hitch

Sam Power (CCA) Geometric MCMC June 7, 2018 14 / 25

Seeing a polygon from the inside

Sam Power (CCA) Geometric MCMC June 7, 2018 15 / 25

Seeing a polygon from the inside

Sam Power (CCA) Geometric MCMC June 7, 2018 16 / 25

Seeing a polygon from the inside

Sam Power (CCA) Geometric MCMC June 7, 2018 17 / 25

Seeing a polygon from the inside

Sam Power (CCA) Geometric MCMC June 7, 2018 18 / 25

Making it practical

Bottleneck is sampling H

Could try focus Markov chain methods on H, then t is easy

· · · but think about what H looks like

More tractable: use a finite direction set

Sam Power (CCA) Geometric MCMC June 7, 2018 19 / 25

Making it practical

Bottleneck is sampling H

Could try focus Markov chain methods on H, then t is easy

· · · but think about what H looks like

More tractable: use a finite direction set

Sam Power (CCA) Geometric MCMC June 7, 2018 19 / 25

Making it practical

Bottleneck is sampling H

Could try focus Markov chain methods on H, then t is easy

· · · but think about what H looks like

More tractable: use a finite direction set

Sam Power (CCA) Geometric MCMC June 7, 2018 19 / 25

Preferential Gibbs Sampler

Sam Power (CCA) Geometric MCMC June 7, 2018 20 / 25

Overcomplete Preferential Gibbs Sampler

Sam Power (CCA) Geometric MCMC June 7, 2018 21 / 25

Making it stochastic

Key difficulty is in picking good directions

Want algorithm to be insensitive to conditioning

So · · ·

Pick a random collection of directions at each step.

Sam Power (CCA) Geometric MCMC June 7, 2018 22 / 25

Making it stochastic

Key difficulty is in picking good directions

Want algorithm to be insensitive to conditioning

So · · ·

Pick a random collection of directions at each step.

Sam Power (CCA) Geometric MCMC June 7, 2018 22 / 25

Making it stochastic

Key difficulty is in picking good directions

Want algorithm to be insensitive to conditioning

So · · ·

Pick a random collection of directions at each step.

Sam Power (CCA) Geometric MCMC June 7, 2018 22 / 25

Making it stochastic

Key difficulty is in picking good directions

Want algorithm to be insensitive to conditioning

So · · ·

Pick a random collection of directions at each step.

Sam Power (CCA) Geometric MCMC June 7, 2018 22 / 25

Dynamic Compass

Sam Power (CCA) Geometric MCMC June 7, 2018 23 / 25

Future Directions

Understand tradeoffs in # of directions

Getting away from ‘bad’ points

Go adaptive?

Affine-invariant version

Sam Power (CCA) Geometric MCMC June 7, 2018 24 / 25

Thank you!

Sam Power (CCA) Geometric MCMC June 7, 2018 25 / 25

Problem StatementBasic Methods of SolutionMarkov Chain ApproachesNew Insights and AlgorithmsConclusion and Future Directions

fd@rm@0: fd@rm@1: fd@rm@2: fd@rm@3: fd@rm@4: fd@rm@5: fd@rm@6: fd@rm@7: