A Selection Premium Decomposition for the Expected Maximum of Random Walks

TL;DR

This work analyzes the upward bias that arises when $K$ models are evaluated on a shared validation sequence. It introduces the selection premium $\varphi_K$ and proves the exact per-step decomposition $\mathbb{E}[M_n]=\sum_{i=1}^n \mathbb{E}[\varphi_K(S_{i-1})]$, effectively a multi-arm analogue of Wald's equation. The authors develop a winner's curse decomposition for unequal means, and establish detailed decay properties: exact Gaussian decay, asymptotic behavior for finite-variance increments with a bias-concentration law, and nonasymptotic sub-Gaussian bounds. These results reveal that selection bias concentrates in the early competition phase and provide practical bounds and intuition for evaluation design and cross-validation in settings with multiple competing candidates.

Abstract

When $K$ models are evaluated on the same validation set of size $n$, the selected winner's apparent performance is biased upward. Suppose $K$ models are evaluated on a shared sequence of i.i.d. observations $X_1,\dots, X_n$, where model $k$ achieves response $f_k(X_i)$ with mean $μ_k = \mathbb E[f_k(X)]$. Writing $Y_{i,k} = f_k(X_i)-μ_k$ for the centered increment and $S_{n,k} = \sum_{i=1}^n Y_{i,k}$ for the centered cumulative score, the expected maximum satisfies $0\le\mathbb E\bigl[\max_k S_{n,k}\bigr] = \sum_{i=1}^n \mathbb E\bigl[\varphi_K(S_{i-1})\bigr]$ where $\varphi_K(u) = \mathbb{E}\bigl[\max_k(u_k + Y_k)\bigr] - \max_k u_k$, $u\in \mathbb R^K$, is the selection premium function. This formula corresponds to the null hypothesis case (all models are equal in the sense that they have the same mean), which clarifies that the bias arises from selection. While this decomposition follows from elementary conditioning and telescoping, we develop the analytical consequences in five directions. (i) structural properties of $\varphi_K$; (ii) extension to stopping times, recovering Wald's equation at $K=1$; (iii) a winner's curse decomposition for heterogeneous means; (iv) a universal bias concentration law showing that the first $α$-fraction of observations generates a $\sqrtα$-fraction of total bias.

Figures (2)

• Figure 1: Selection premium decay. Left: Normalized premium $\psi_K(i)$ for Gaussian arms with various $K$. Right: Distribution comparison for $K = 10$. The normalized curves are nearly identical across distributions, suggesting a universal decay law.
• Figure 2: Scaling in $n$ and $K$ for two uniform and Rademacher increment laws. Panel (a) verifies the $\sqrt{n}$ growth at fixed $K$; panel (b) verifies the $\sqrt{\log K}$ growth at fixed $n$.

Theorems & Definitions (26)

• Definition 2.1: Selection Premium Function
• Remark 2.2: Interpretation of $\varphi_K(u)$
• Theorem 3.1: Decomposition identity
• proof
• Remark 3.2: On the proof
• Proposition 3.3: Properties of $\varphi_K$
• proof
• Corollary 3.4
• proof
• Theorem 3.5: Stopping time extension