A PTAS for Computing the Supremum of Gaussian

ProcessesRaghu Meka (IAS/DIMACS)

Gaussian Processes (GPs)

• Jointly Gaussian variables :• Any finite sum is Gaussian

Supremum of Gaussian Processes (GPs)

Given want to study

Why Gaussian Processes?

Stochastic Processes

Functional analysis

Convex Geometry

Machine LearningMany more!

Fundamental graph parameterEg:

Aldous-Fill 94: Compute cover time deterministically?

Cover times of Graphs

• KKLV00: approximation• Feige-Zeitouni’09: FPTAS for trees

Cover Times and GPsThm (Ding, Lee, Peres 10): O(1) det. poly.

time approximation for cover time.

Thm (DLP10): Winkler-Zuckerman “blanket-time” conjectures.• Transfer to GPs • Compute supremum of GP

Question (Lee10, Ding11): Given , compute a factor approx. to

Computing the Supremum

Question (Lee10, Ding11): PTAS for computing the supremum of GPs?

𝑣1

𝑣2

¿ 𝑋𝑡

0Random

Gaussian

• Covariance matrix• More intuitive

Question (Lee10, Ding11): Given , compute a factor approx. to

Computing the Supremum

• DLP10: O(1) factor approximation• Can’t beat O(1): Talagrand’s majorizing

measures

Main ResultThm: A PTAS for computing the

supremum of Gaussian processes.

Comparison inequalities from convex geometry

Thm: PTAS for computing cover time of bounded degree graphs.

Thm: Given , a det. algorithm to compute approx. to

Outline of Algorithm1. Dimension reduction

– Slepian’s Lemma, Johnson-Lindenstrauss

2. Optimal eps-nets in Gaussian space

– Kanter’s lemma, univariate to multivariate

Dimension Reduction

• , .

Idea: JL projection, solve in projected spaceUse deterministic JL – EIO02, S02.V

W

Analysis: Slepian’s Lemma

Problem: Relate supremum of projections

Analysis: Slepian’s Lemma

• Enough to solve for W• Enough to be exp. in

dimension

Outline of Algorithm1. Dimension reduction

– Slepian’s Lemma, Johnson-Lindenstrauss

2. Optimal eps-nets in Gaussian space

– Kanter’s lemma, univariate to multivariate

Nets in Gaussian Space• Goal: , in time approximate

• We solve the problem for all semi-norms

Nets in Gaussian space• Discrete approximations of

GaussianExplicit

• Integer rounding: (need granularity )• Dadusch-Vempala’12: Main thm: Explicit -net of size .

Optimal: Matching lowerbound

Construction of eps-net• Simplest possible: univariate to

multivariate

1. What resolution? Naïve: .2. How far out on the axes?

𝑘 𝑘

Even out mass in interval .

Construction of eps-net• Analyze ‘step-wise’ approximator

- 𝛿 3 𝛿2 𝛿 4 𝛿−3 𝛿−4 𝛿 −2 𝛿

1. What resolution? Naïve: .2. How far out on the axes?

- 𝛿 3 𝛿2 𝛿 4 𝛿−3 𝛿−4 𝛿 −2 𝛿

Construction of eps-net• Take univariate net and lift to

multivariate 𝑘 𝑘

What resolution enough?𝛾 𝛾𝑢

Main Lemma: Can take

- 𝛿 3 𝛿2 𝛿 4 𝛿−3 𝛿−4 𝛿 −2 𝛿𝛾 𝛾𝑢

Dimension Free Error Bounds

Lem: For , a norm,

• Proof by “sandwiching”• Exploit convexity critically

Analysis of Error

• Why interesting? For any norm,

Def: Sym. (less peaked), if sym. convex sets K

Sandwiching and Lifting Nets

Fact:

Proof:

- 𝛿 3 𝛿2 𝛿 4 𝛿−3 𝛿−4 𝛿 −2 𝛿

Spreading away from origin!

Sandwiching and Lifting Nets

Kanter’s Lemma(77): and unimodal,

Fact: By definition, Cor: By Kanter’s lemma,

Cor: Upper bound,

𝑘 𝑘

𝛾

Fact: Proof: For inward push compensates earlier spreading.

• Def: scaled down version of – , , pdf of .

Sandwiching and Lifting Nets

Push mass towards origin.

Sandwiching and Lifting Nets

Kanter’s Lemma(77): and unimodal,

Fact: By definition, Cor: By Kanter’s lemma, 𝑘 𝑘

Cor: Lower bound,

𝛾 𝛾 ℓ

Sandwiching and Lifting Nets

𝑘 𝑘

𝛾

𝑘

𝛾 ℓ

Combining both:

Outline of Algorithm1. Dimension reduction

– Slepian’s Lemma

2. Optimal eps-nets for Gaussians– Kanter’s lemma

PTAS for Supremum

Open Problems• FPTAS for computing supremum?

• Black-box algorithms?– JL step looks at points

• PTAS for cover time on all graphs?– Conjecture of Ding, Lee, Peres 10

Thank you