A PTAS for Computing the Supremum of Gaussian Processes
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A PTAS for Computing the Supremum of Gaussian
ProcessesRaghu Meka (IAS/DIMACS)
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Gaussian Processes (GPs)
• Jointly Gaussian variables :• Any finite sum is Gaussian
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Supremum of Gaussian Processes (GPs)
Given want to study
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Why Gaussian Processes?
Stochastic Processes
Functional analysis
Convex Geometry
Machine LearningMany more!
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Fundamental graph parameterEg:
Aldous-Fill 94: Compute cover time deterministically?
Cover times of Graphs
• KKLV00: approximation• Feige-Zeitouni’09: FPTAS for trees
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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
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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
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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
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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
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Outline of Algorithm1. Dimension reduction
– Slepian’s Lemma, Johnson-Lindenstrauss
2. Optimal eps-nets in Gaussian space
– Kanter’s lemma, univariate to multivariate
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Dimension Reduction
• , .
Idea: JL projection, solve in projected spaceUse deterministic JL – EIO02, S02.V
W
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Analysis: Slepian’s Lemma
Problem: Relate supremum of projections
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Analysis: Slepian’s Lemma
• Enough to solve for W• Enough to be exp. in
dimension
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Outline of Algorithm1. Dimension reduction
– Slepian’s Lemma, Johnson-Lindenstrauss
2. Optimal eps-nets in Gaussian space
– Kanter’s lemma, univariate to multivariate
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Nets in Gaussian Space• Goal: , in time approximate
• We solve the problem for all semi-norms
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Nets in Gaussian space• Discrete approximations of
GaussianExplicit
• Integer rounding: (need granularity )• Dadusch-Vempala’12: Main thm: Explicit -net of size .
Optimal: Matching lowerbound
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Construction of eps-net• Simplest possible: univariate to
multivariate
1. What resolution? Naïve: .2. How far out on the axes?
𝑘 𝑘
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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
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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
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- 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
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Analysis of Error
• Why interesting? For any norm,
Def: Sym. (less peaked), if sym. convex sets K
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Sandwiching and Lifting Nets
Fact:
Proof:
- 𝛿 3 𝛿2 𝛿 4 𝛿−3 𝛿−4 𝛿 −2 𝛿
Spreading away from origin!
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Sandwiching and Lifting Nets
Kanter’s Lemma(77): and unimodal,
Fact: By definition, Cor: By Kanter’s lemma,
Cor: Upper bound,
𝑘 𝑘
𝛾
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Fact: Proof: For inward push compensates earlier spreading.
• Def: scaled down version of – , , pdf of .
Sandwiching and Lifting Nets
Push mass towards origin.
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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
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Sandwiching and Lifting Nets
𝑘 𝑘
𝛾
𝑘
𝛾 ℓ
Combining both:
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Outline of Algorithm1. Dimension reduction
– Slepian’s Lemma
2. Optimal eps-nets for Gaussians– Kanter’s lemma
PTAS for Supremum
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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
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Thank you