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Pareto Set Identification With Posterior Sampling

Cyrille Kone, Marc Jourdan, Emilie Kaufmann

TL;DR

This paper proposes the PSIPS algorithm that deals simultaneously with structure and correlation without paying the computational cost of existing oracle-based algorithms, and demonstrates its good empirical performance in real-world and synthetic instances.

Abstract

The problem of identifying the best answer among a collection of items having real-valued distribution is well-understood. Despite its practical relevance for many applications, fewer works have studied its extension when multiple and potentially conflicting metrics are available to assess an item's quality. Pareto set identification (PSI) aims to identify the set of answers whose means are not uniformly worse than another. This paper studies PSI in the transductive linear setting with potentially correlated objectives. Building on posterior sampling in both the stopping and the sampling rules, we propose the PSIPS algorithm that deals simultaneously with structure and correlation without paying the computational cost of existing oracle-based algorithms. Both from a frequentist and Bayesian perspective, PSIPS is asymptotically optimal. We demonstrate its good empirical performance in real-world and synthetic instances.

Pareto Set Identification With Posterior Sampling

TL;DR

This paper proposes the PSIPS algorithm that deals simultaneously with structure and correlation without paying the computational cost of existing oracle-based algorithms, and demonstrates its good empirical performance in real-world and synthetic instances.

Abstract

The problem of identifying the best answer among a collection of items having real-valued distribution is well-understood. Despite its practical relevance for many applications, fewer works have studied its extension when multiple and potentially conflicting metrics are available to assess an item's quality. Pareto set identification (PSI) aims to identify the set of answers whose means are not uniformly worse than another. This paper studies PSI in the transductive linear setting with potentially correlated objectives. Building on posterior sampling in both the stopping and the sampling rules, we propose the PSIPS algorithm that deals simultaneously with structure and correlation without paying the computational cost of existing oracle-based algorithms. Both from a frequentist and Bayesian perspective, PSIPS is asymptotically optimal. We demonstrate its good empirical performance in real-world and synthetic instances.

Paper Structure

This paper contains 82 sections, 39 theorems, 302 equations, 10 figures, 4 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Pareto Set Identification
  3. Identification Strategy
  4. Fixed-confidence PSI
  5. Notation
  6. Lower Bound
  7. Algorithms
  8. Stopping Rules
  9. Sampling Rules
  10. Contributions
  11. Related Work
  12. PSIPS ALGORITHM
  13. The Posterior Sampling (PS) Stopping Rule
  14. Computational Cost
  15. Correctness
  16. ...and 67 more sections

Key Result

Lemma 1

An algorithm which is $\delta$-correct on all problems in $\mathcal{D}^K$ satisfies that, for all $\bm{\nu} \in \mathcal{D}^K$ with regression matrix $\bm \theta \in \Theta$, $\mathbb{E}_{\bm\nu}[\tau_{\delta}] \ge T^\star(\bm \theta) \log(1/(2.4\delta))$ where $T^\star(\bm \theta)$ is a characteris and its maximizer set of optimal allocations is $w^\star(\bm \theta)$.

Figures (10)

  • Figure 1: Empirical stopping times on the covid19 experiment with $\delta=0.01$ (left) and $\delta=0.001$ (right).
  • Figure 2: Impact of the correlation $\rho$ on the stopping time.
  • Figure 3: Empirical stopping time on the NoC experiment.
  • Figure 4: Empirical stopping time on random Gaussian (left) and Bernoulli (right) instances.
  • Figure 5: Average runtime for the first $T$ iterations in the covid19 experiment.
  • ...and 5 more figures

Theorems & Definitions (64)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 54 more