[Paper Review] Generating Pareto records

James Allen Fill, Daniel Q. Naiman|arXiv (Cornell University)|Jan 17, 2019

Algorithms and Data Compression7 references4 citations

TL;DR

This paper presents an efficient algorithm for simulating multivariate Pareto records using dynamic maintenance of record-setting region generators—minima of the region where new records can occur. The key contribution is an asymptotic analysis showing the expected number of generators grows as $(\log n)^{d-1}$ in dimension $d$, enabling dramatic speedups over naive simulation, with precise bounds and expansions derived for algorithmic complexity.

ABSTRACT

We present, (partially) analyze, and apply an efficient algorithm for the simulation of multivariate Pareto records. A key role is played by minima of the record-setting region (we call these generators) each time a new record is generated, and two highlights of our work are (i) efficient dynamic maintenance of the set of generators and (ii) asymptotic analysis of the expected number of generators at each time.

Motivation & Objective

• To develop an efficient algorithm for simulating multivariate Pareto records, which are otherwise extremely rare under naive i.i.d. sampling.
• To dynamically maintain the set of minima (generators) of the record-setting region, which define the boundary where new records can occur.
• To analyze the expected number of generators at each time step, providing theoretical bounds and asymptotic expansions for algorithmic complexity.
• To enable large-scale empirical studies of multivariate record behavior by overcoming the prohibitive computational cost of naive simulation.
• To establish theoretical foundations for the number of generators, including exact expressions and asymptotic expansions in terms of $n$ and $d$.

Proposed method

• The algorithm uses importance sampling to efficiently generate Pareto records by focusing on the record-setting region defined by current records.
• Generators are maintained dynamically as the minima of the record-setting region, ensuring only the most relevant points are tracked.
• The method relies on coordinatewise maxima and orthant decomposition to efficiently update the generator set after each new record.
• Theoretical analysis uses order statistics and logarithmic asymptotics to derive bounds on the expected number of generators, $\mathcal{G}_{d,n}$.
• Key equations include the asymptotic expansion $\mathcal{G}_{d,n} = (\log n)^{d-1} \sum_{j=0}^{d-1} a_{d,j} (\log n)^{-j} + O(n^{-1} (\log n)^{d-1})$, with coefficients $a_{d,j}$ derived from gamma function derivatives.
• The algorithm's efficiency is demonstrated by showing it reduces the number of required observations from $10^{61}$ to just $10^4$ steps for 10,000 records in the bivariate case.

Experimental results

Research questions

• RQ1 How can multivariate Pareto records be simulated efficiently, given their extremely low probability under naive i.i.d. sampling?
• RQ2 What is the expected number of generators (minima of the record-setting region) at time $n$ in dimension $d$, and how does it grow with $n$?
• RQ3 Can tight bounds be established for the number of generators in terms of the number of current records?
• RQ4 What is the asymptotic behavior of the expected number of generators as $n \to \infty$ in fixed dimension $d$?
• RQ5 How does the number of generators relate to the number of records generated, and can this relationship be used to analyze algorithmic cost?

Key findings

• The expected number of generators at time $n$ in dimension $d$ grows asymptotically as $(\log n)^{d-1}$, with leading term $1 \cdot (\log n)^{d-1}$.
• An exact expression for $\mathcal{G}_{d,n}$ is derived using the sum of gamma function derivatives and binomial coefficients.
• An asymptotic expansion for $\mathcal{G}_{d,n}$ is established: $\mathcal{G}_{d,n} = (\log n)^{d-1} \sum_{j=0}^{d-1} a_{d,j} (\log n)^{-j} + O(n^{-1} (\log n)^{d-1})$, where $a_{d,j}$ are explicitly defined coefficients.
• For $d=2$, the number of generators satisfies $\tilde{\gamma}_m = \tilde{\rho}_m + 1$ almost surely, with $\tilde{\gamma}_m = O(m^{1/2})$, matching the conjectured growth rate.
• For $d \geq 3$, the number of generators is bounded by $\tilde{\gamma}_m \leq (1+o(1)) \frac{\tilde{\rho}_m^{d-1}}{(d-1)!}$, with tighter bounds for $d=3$ and $d=4$.
• The algorithm reduces the number of required observations from $10^{61}$ to $10^4$ steps for 10,000 records in the bivariate case, demonstrating a dramatic efficiency gain.