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Stability and Accuracy Trade-offs in Statistical Estimation

Abhinav Chakraborty, Yuetian Luo, Rina Foygel Barber

TL;DR

This work formalizes stability as a constraint in statistical estimation and analyzes the resulting stability–accuracy trade-offs through stability-constrained minimax risks. It develops general lower bounds for worst-case and average-case stability, and constructs optimal stable estimators for four canonical problems: bounded-mean, heavy-tailed-mean, sparse-mean, and nonparametric regression. A key finding is that average-case stability typically imposes a qualitatively weaker restriction than worst-case stability, with phase transitions that vary by problem; several tasks exhibit sharp phase transitions under average-case stability, while others show gradual transitions. The results also illuminate connections to differential privacy, showing how stability constraints translate into privacy guarantees and vice versa, and they outline directions for future work on other stability notions and broader estimation settings.

Abstract

Algorithmic stability is a central concept in statistics and learning theory that measures how sensitive an algorithm's output is to small changes in the training data. Stability plays a crucial role in understanding generalization, robustness, and replicability, and a variety of stability notions have been proposed in different learning settings. However, while stability entails desirable properties, it is typically not sufficient on its own for statistical learning -- and indeed, it may be at odds with accuracy, since an algorithm that always outputs a constant function is perfectly stable but statistically meaningless. Thus, it is essential to understand the potential statistical cost of stability. In this work, we address this question by adopting a statistical decision-theoretic perspective, treating stability as a constraint in estimation. Focusing on two representative notions-worst-case stability and average-case stability-we first establish general lower bounds on the achievable estimation accuracy under each type of stability constraint. We then develop optimal stable estimators for four canonical estimation problems, including several mean estimation and regression settings. Together, these results characterize the optimal trade-offs between stability and accuracy across these tasks. Our findings formalize the intuition that average-case stability imposes a qualitatively weaker restriction than worst-case stability, and they further reveal that the gap between these two can vary substantially across different estimation problems.

Stability and Accuracy Trade-offs in Statistical Estimation

TL;DR

This work formalizes stability as a constraint in statistical estimation and analyzes the resulting stability–accuracy trade-offs through stability-constrained minimax risks. It develops general lower bounds for worst-case and average-case stability, and constructs optimal stable estimators for four canonical problems: bounded-mean, heavy-tailed-mean, sparse-mean, and nonparametric regression. A key finding is that average-case stability typically imposes a qualitatively weaker restriction than worst-case stability, with phase transitions that vary by problem; several tasks exhibit sharp phase transitions under average-case stability, while others show gradual transitions. The results also illuminate connections to differential privacy, showing how stability constraints translate into privacy guarantees and vice versa, and they outline directions for future work on other stability notions and broader estimation settings.

Abstract

Algorithmic stability is a central concept in statistics and learning theory that measures how sensitive an algorithm's output is to small changes in the training data. Stability plays a crucial role in understanding generalization, robustness, and replicability, and a variety of stability notions have been proposed in different learning settings. However, while stability entails desirable properties, it is typically not sufficient on its own for statistical learning -- and indeed, it may be at odds with accuracy, since an algorithm that always outputs a constant function is perfectly stable but statistically meaningless. Thus, it is essential to understand the potential statistical cost of stability. In this work, we address this question by adopting a statistical decision-theoretic perspective, treating stability as a constraint in estimation. Focusing on two representative notions-worst-case stability and average-case stability-we first establish general lower bounds on the achievable estimation accuracy under each type of stability constraint. We then develop optimal stable estimators for four canonical estimation problems, including several mean estimation and regression settings. Together, these results characterize the optimal trade-offs between stability and accuracy across these tasks. Our findings formalize the intuition that average-case stability imposes a qualitatively weaker restriction than worst-case stability, and they further reveal that the gap between these two can vary substantially across different estimation problems.
Paper Structure (108 sections, 33 theorems, 266 equations, 2 figures, 1 table)

This paper contains 108 sections, 33 theorems, 266 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Trade-offs and constraints in statistical decision theory.
  3. Our Contributions
  4. Paper Organization
  5. Notation
  6. Background and Problem Formulation
  7. Classical Minimax Framework
  8. Worst-case and Average-case Stability
  9. Stability-Constrained Minimax Risks
  10. A Glimpse of the Result
  11. General Tools for Lower Bounding $R_{n,p}(\theta(\mathcal{P}), \beta_n)$
  12. Mean Estimation with Bounded Support
  13. Refined Characterization When $d = 1$
  14. Heavy-tailed Mean Estimation
  15. Sparse Mean Estimation
  16. ...and 93 more sections

Key Result

Theorem 1

For the statistical estimation setting considered in Section sec:class-minimax under the worst-case stability constraint, we have where in the expected value, the joint distribution of $(\mathcal{D}_n, \mathcal{D}_n')$ could be any coupling between $P_1^{\otimes n}$ and $P_2^{\otimes n}$.

Figures (2)

  • Figure 1: Stability-vs.-accuracy trade-off curves for mean estimation under worst-case stability and average-case stability. Figure (a) is for the $1$-d mean estimation over a bounded distribution class; figure (b) is for the $1$-d mean estimation over a heavy-tailed distribution class with bounded $k$-th moment $(k \geq 2)$. Here, $r$ denotes the radius of the parameter space, and the vertical dotted lines mark the transition location of the curves.
  • Figure 2: A pictorial illustration on the shrinkage factor of our estimator $\widehat{\theta}$\ref{['eq:ave-est']}, and the naive estimator $\widetilde{\theta}$\ref{['eq:naive-estimator']} relative to the sample mean, given average-case stability $\beta_n = \frac{r}{(1+\delta_n)n}$ for some $\delta_n \in [0,1]$.

Theorems & Definitions (52)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: Sharp vs. gradual phase transition
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 42 more