Upload
erik
View
60
Download
0
Tags:
Embed Size (px)
DESCRIPTION
A PTAS for Computing the Supremum of Gaussian Processes. Raghu 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 - PowerPoint PPT Presentation
Citation preview
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