Majorizing measures for the optimizer

ConferencePreprintprobability
Sander Borst, Daniel Dadush, Neil Olver and Makrand Sinha,
Accepted to ICTS 2021

The theory of majorizing measures, extensively developed by Fernique, Talagrand and many others, provides one of the most general frameworks for controlling the behavior of stochastic processes. In particular, it can be applied to derive quantitative bounds on the expected suprema and the degree of continuity of sample paths for many processes.

One of the crowning achievements of the theory is Talagrand’s tight alternative characterization of the suprema of Gaussian processes in terms of majorizing measures. The proof of this theorem was difficult, and thus considerable effort was put into the task of developing both shorter and easier to understand proofs. A major reason for this difficulty was considered to be theory of majorizing measures itself, which had the reputation of being opaque and
mysterious. As a consequence, most recent treatments of the theory (including by Talagrand himself) have eschewed the use of majorizing measures in favor of a purely combinatorial approach (the generic chaining) where objects based on sequences of partitions provide roughly matching upper and lower bounds on the desired expected supremum.

In this paper, we return to majorizing measures as a primary object of study, and give a viewpoint that we think is natural and clarifying from an optimization perspective. As our main contribution, we give an algorithmic proof
of the majorizing measures theorem based on two parts:

  • We make the simple (but apparently new) observation that finding the best majorizing measure can be cast as a convex program.
    This also allows for efficiently computing the measure using off-the-shelf methods from convex optimization.
  • We obtain tree-based upper and lower bound certificates by rounding, in a series of steps, the primal and dual solutions to this convex program.

While duality has conceptually been part of the theory since its beginnings, as far as we are aware no explicit link to convex optimization has been previously made.

Performance of the smallest-variance-first rule in appointment sequencing

approximationJournalprobability
Madelon A. de Kemp, Michel Mandjes and Neil Olver
Accepted to Operations Research

A classical problem in appointment scheduling, with applications in health care, concerns the determination of the patients’ arrival times that minimize a cost function that is a weighted sum of mean waiting times and mean idle times. Part of this problem is the sequencing problem, which focuses on ordering the patients. We assess the performance of the smallest-variance-first (SVF) rule, which sequences patients in order of increasing variance of their service durations. While it was known that SVF is not always optimal, many papers have found that it performs well in practice and simulation. We give theoretical justification for these observations by proving quantitative worst-case bounds on the ratio between the cost incurred by the SVF rule and the minimum attainable cost, in a number of settings. We also show that under quite general conditions, this ratio approaches 1 as the number of patients grows large, showing that the SVF rule is asymptotically optimal. While this viewpoint in terms of approximation ratio is a standard approach in many algorithmic settings, our results appear to be the first of this type in the appointment scheduling literature.

Fast, Deterministic and Sparse Dimensionality Reduction

Conferenceprobability
Daniel Dadush, Cristóbal Guzman, Neil Olver
SODA 2018

Explosion and linear transit times in infinite trees

Journalprobability
Omid Amini, Luc Devroye, Simon Griffiths and Neil Olver
Probability Theory and Related Fields (accepted)

Let $T$ be an infinite rooted tree with weights $w_e$ assigned to its edges. Denote by $m_n(T)$ the minimum weight of a path from the root to a node of the $n$th generation. We consider the possible behaviour of $m_n(T)$ with focus on the two following cases: we say $T$ is explosive if \[ \lim_{n\to \infty}m_n(T) < \infty, \] and say that $T$ exhibits linear growth if \[ \liminf_{n\to \infty} \frac{m_n(T)}{n} > 0. \]
We consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost surely. We then apply this characterization to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres. As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.

Adaptive Rumor Spreading

Conferenceprobability
José Correa, Marcos Kiwi , Neil Olver and Alberto Vera
WINE 2015

Motivated by the recent emergence of the so-called opportunistic communication networks, we consider the issue of adaptivity in the most basic continuous time (asynchronous) rumor spreading process. In our setting a rumor has to be spread to a population; the service provider can push it at any time to any node in the network and has unit cost for doing this. On the other hand, as usual in rumor spreading, nodes share the rumor upon meeting and this imposes no cost on the service provider. Rather than fixing a budget on the number of pushes, we consider the cost version of the problem with a fixed deadline and ask for a minimum cost strategy that spreads the rumor to every node. A non-adaptive strategy can only intervene at the beginning and at the end, while an adaptive strategy has full knowledge and intervention capabilities. Our main result is that in the homogeneous case (where every pair of nodes randomly meet at the same rate) the benefit of adaptivity is bounded by a constant. This requires a subtle analysis of the underlying random process that is of interest in its own right.

Pipage Rounding, Pessimistic Estimators and Matrix Concentration

Conferenceprobability
N. Harvey, N. Olver
SODA 2014

Pipage rounding is a dependent random sampling technique that has several interesting properties and diverse applications.
One property that has been useful in applications is negative correlation of the resulting vector. There are some further properties that would be interesting to derive, but do not seem to follow from negative correlation. In particular, recent concentration results for sums of independent random matrices are not known to extend to a negatively dependent setting.

We introduce a simple but useful technique called concavity of pessimistic estimators. This technique allows us to show concentration of submodular functions and concentration of matrix sums under pipage rounding. The former result answers a question of Chekuri et al. (2009). To prove the latter result, we derive a new variant of Lieb’s celebrated concavity theorem in matrix analysis.

We provide numerous applications of these results. One is to spectrally-thin trees, a spectral analog of the thin trees that played a crucial role in the recent breakthrough on the asymmetric traveling salesman problem. We show a polynomial time algorithm that, given a graph where every edge has effective conductance at least $\kappa$, returns an $O(\kappa^{-1} \cdot \log n / \log \log n)$-spectrally-thin tree. There are further applications to rounding of semidefinite programs and to a geometric question of extracting a nearly-orthonormal basis from an isotropic distribution.

On Explosions in Heavy-tailed Branching Random Walks

Journalprobability
Omid Amini, Luc Devroye, Simon Griffiths and Neil Olver
Annals of Probability 41(3B):1864-1899, 2013

Consider a branching random walk on $\mathbb{R}$, with offspring distribution $Z$ and nonnegative displacement distribution W. We say that explosion occurs if an infinite number of particles may be found within a finite distance of the origin. In this paper, we investigate this phenomenon when the offspring distribution $Z$ is heavy-tailed. Under an appropriate condition, we are able to characterize the pairs $(Z, W)$ for which explosion occurs, by demonstrating the equivalence of explosion with a seemingly much weaker event: that the sum over generations of the minimum displacement in each generation is finite. Furthermore, we demonstrate that our condition on the tail is best possible for this equivalence to occur. We also investigate, under additional smoothness assumptions, the behavior of $M_n$, the position of the particle in generation $n$ closest to the origin, when explosion does not occur (and hence $\lim_{n\rightarrow\infty}M_n=\infty$).