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Pareto-Optimality, Smoothness, and Stochasticity in Learning-Augmented One-Max-Search

Ziyad Benomar, Lorenzo Croissant, Vianney Perchet, Spyros Angelopoulos

TL;DR

The paper tackles the online One-Max Search with a prediction of the maximum price, addressing the need for algorithms that are simultaneously Pareto-optimal between consistency and robustness and smoothly responsive to prediction error. It introduces a deterministic, Pareto-optimal, and smooth family of threshold policies $\{\mathsf{A}_r^{\rho}\}$ with thresholds $\Phi_r^{\rho}$, showing that smoothing is optimally achieved at $\rho=1$ and establishing a lower bound on any Pareto-optimal algorithm’s smoothness. It extends the deterministic results to stochastic settings by coupling prices and predictions and employing optimal transport to bound performance under dependent predictions, with explicit instantiations for stochastic predictions, independent marginals, and OT-based dependence. The work is validated by synthetic and real-data experiments (Bitcoin), demonstrating the practical benefit of smoothness in handling prediction uncertainty, and it outlines future extensions to multi-unit variants and broader financial-optimization contexts.

Abstract

One-max search is a classic problem in online decision-making, in which a trader acts on a sequence of revealed prices and accepts one of them irrevocably to maximise its profit. The problem has been studied both in probabilistic and in worst-case settings, notably through competitive analysis, and more recently in learning-augmented settings in which the trader has access to a prediction on the sequence. However, existing approaches either lack smoothness, or do not achieve optimal worst-case guarantees: they do not attain the best possible trade-off between the consistency and the robustness of the algorithm. We close this gap by presenting the first algorithm that simultaneously achieves both of these important objectives. Furthermore, we show how to leverage the obtained smoothness to provide an analysis of one-max search in stochastic learning-augmented settings which capture randomness in both the observed prices and the prediction.

Pareto-Optimality, Smoothness, and Stochasticity in Learning-Augmented One-Max-Search

TL;DR

The paper tackles the online One-Max Search with a prediction of the maximum price, addressing the need for algorithms that are simultaneously Pareto-optimal between consistency and robustness and smoothly responsive to prediction error. It introduces a deterministic, Pareto-optimal, and smooth family of threshold policies with thresholds , showing that smoothing is optimally achieved at and establishing a lower bound on any Pareto-optimal algorithm’s smoothness. It extends the deterministic results to stochastic settings by coupling prices and predictions and employing optimal transport to bound performance under dependent predictions, with explicit instantiations for stochastic predictions, independent marginals, and OT-based dependence. The work is validated by synthetic and real-data experiments (Bitcoin), demonstrating the practical benefit of smoothness in handling prediction uncertainty, and it outlines future extensions to multi-unit variants and broader financial-optimization contexts.

Abstract

One-max search is a classic problem in online decision-making, in which a trader acts on a sequence of revealed prices and accepts one of them irrevocably to maximise its profit. The problem has been studied both in probabilistic and in worst-case settings, notably through competitive analysis, and more recently in learning-augmented settings in which the trader has access to a prediction on the sequence. However, existing approaches either lack smoothness, or do not achieve optimal worst-case guarantees: they do not attain the best possible trade-off between the consistency and the robustness of the algorithm. We close this gap by presenting the first algorithm that simultaneously achieves both of these important objectives. Furthermore, we show how to leverage the obtained smoothness to provide an analysis of one-max search in stochastic learning-augmented settings which capture randomness in both the observed prices and the prediction.

Paper Structure

This paper contains 46 sections, 28 theorems, 131 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Related work
  4. Learning-augmented algorithms.
  5. Pareto-optimal algorithms.
  6. One-Max Search.
  7. Preliminaries
  8. Pareto-Optimal and Smooth Algorithms
  9. Multiplicative error
  10. Extension to the additive error
  11. Stochastic One-Max Search
  12. Competitive analysis in the stochastic framework
  13. Instantiations of Lemma \ref{['thm: [Stochastic bounds] bound in expectation using true coupling']}
  14. Stochastic predictions, deterministic prices.
  15. Stochastic independent predictions and prices
  16. ...and 31 more sections

Key Result

Theorem 3.1

For any fixed of robustness $r$, the set of all thresholds $\Phi: [1,\theta] \to [1,\theta]$ such that $\hbox{A}_\Phi$ has robustness $r$ and consistency $1/r\theta$ is

Figures (8)

  • Figure 1: The set $\mathcal{P}_r$ of Theorem \ref{['thm:thresh-pareto-optimal']} is depicted shaded. A threshold $\Phi$ is $r$-robust and $1/r\theta$-consistent if and only if its graph lies in this shaded area.
  • Figure 2: The threshold functions $\Phi^\rho_r$ of $\hbox{A}_{r}^{\rho}$, as defined by Eq. \ref{['eq:rho.family']}. Note that all are equal on $[1,r^{-1})$.
  • Figure 3: Performance of $\hbox{A}^\rho_r$ with $\rho \in \{0,0.5,1\}$.
  • Figure 4: Comparison of $\hbox{A}^1_r$ and $\hbox{A}^0_r$ on the Bitcoin price dataset.
  • Figure 5: Numerical quadrature of \ref{['prop: [independent marginals] uniform marginals case']} for $c_1=1$, $c_2=\theta$ (for $\theta\in[1,10]$) as a function of $s\in[1,5]$.
  • ...and 3 more figures

Theorems & Definitions (48)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 4.1
  • proof
  • Corollary 4.1
  • Example 4.1
  • Corollary 4.1
  • Corollary 4.1
  • ...and 38 more