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Stability, Complexity and Data-Dependent Worst-Case Generalization Bounds

Mario Tuci, Lennart Bastian, Benjamin Dupuis, Nassir Navab, Tolga Birdal, Umut Şimşekli

TL;DR

This paper tackles the challenge of generalization guarantees for stochastic optimization by introducing random-set stability for data-dependent random sets $\mathcal{W}_{S,U}$ and bounding the worst-case generalization gap $G_S(w)$ along the entire trajectory. The authors derive an expected bound that hinges on a stability parameter $\beta_n$ and a data-/algorithm-dependent complexity term, avoiding intractable mutual-information terms. They show how to recover IT-free topological/fractal bounds, including $\mathbf{E}^{\alpha}(\mathcal{W}_{S,U})$ and $\mathbf{PMag}(\cdot)$, within this stability framework and validate the approach with experiments on ViT and GraphSAGE that reveal a meaningful interplay between stability and topological complexity. Overall, the framework provides computable, interpretable bounds that connect optimization dynamics, geometry of training trajectories, and generalization performance in data-dependent settings.

Abstract

Providing generalization guarantees for stochastic optimization algorithms remains a key challenge in learning theory. Recently, numerous works demonstrated the impact of the geometric properties of optimization trajectories on generalization performance. These works propose worst-case generalization bounds in terms of various notions of intrinsic dimension and/or topological complexity, which were found to empirically correlate with the generalization error. However, most of these approaches involve intractable mutual information terms, which limit a full understanding of the bounds. In contrast, some authors built on algorithmic stability to obtain worst-case bounds involving geometric quantities of a combinatorial nature, which are impractical to compute. In this paper, we address these limitations by combining empirically relevant complexity measures with a framework that avoids intractable quantities. To this end, we introduce the concept of \emph{random set stability}, tailored for the data-dependent random sets produced by stochastic optimization algorithms. Within this framework, we show that the worst-case generalization error can be bounded in terms of (i) the random set stability parameter and (ii) empirically relevant, data- and algorithm-dependent complexity measures of the random set. Moreover, our framework improves existing topological generalization bounds by recovering previous complexity notions without relying on mutual information terms. Through a series of experiments in practically relevant settings, we validate our theory by evaluating the tightness of our bounds and the interplay between topological complexity and stability.

Stability, Complexity and Data-Dependent Worst-Case Generalization Bounds

TL;DR

This paper tackles the challenge of generalization guarantees for stochastic optimization by introducing random-set stability for data-dependent random sets and bounding the worst-case generalization gap along the entire trajectory. The authors derive an expected bound that hinges on a stability parameter and a data-/algorithm-dependent complexity term, avoiding intractable mutual-information terms. They show how to recover IT-free topological/fractal bounds, including and , within this stability framework and validate the approach with experiments on ViT and GraphSAGE that reveal a meaningful interplay between stability and topological complexity. Overall, the framework provides computable, interpretable bounds that connect optimization dynamics, geometry of training trajectories, and generalization performance in data-dependent settings.

Abstract

Providing generalization guarantees for stochastic optimization algorithms remains a key challenge in learning theory. Recently, numerous works demonstrated the impact of the geometric properties of optimization trajectories on generalization performance. These works propose worst-case generalization bounds in terms of various notions of intrinsic dimension and/or topological complexity, which were found to empirically correlate with the generalization error. However, most of these approaches involve intractable mutual information terms, which limit a full understanding of the bounds. In contrast, some authors built on algorithmic stability to obtain worst-case bounds involving geometric quantities of a combinatorial nature, which are impractical to compute. In this paper, we address these limitations by combining empirically relevant complexity measures with a framework that avoids intractable quantities. To this end, we introduce the concept of \emph{random set stability}, tailored for the data-dependent random sets produced by stochastic optimization algorithms. Within this framework, we show that the worst-case generalization error can be bounded in terms of (i) the random set stability parameter and (ii) empirically relevant, data- and algorithm-dependent complexity measures of the random set. Moreover, our framework improves existing topological generalization bounds by recovering previous complexity notions without relying on mutual information terms. Through a series of experiments in practically relevant settings, we validate our theory by evaluating the tightness of our bounds and the interplay between topological complexity and stability.

Paper Structure

This paper contains 50 sections, 16 theorems, 77 equations, 13 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Technical Background
  4. Main Theoretical Results
  5. Random set stability
  6. Applicability of random set stability
  7. Projected SGD
  8. Worst-case generalization bound for data-dependent stable random sets
  9. Applications
  10. Assumptions
  11. Intrinsic dimensions and topological bounds
  12. Empirical Evaluations
  13. Experimental Design
  14. Quantity Estimation
  15. Results and Discussion
  16. ...and 35 more sections

Key Result

Lemma 3.1

Consider a fixed $K \in \mathbb{N}^\star$, $1\leq k \leq K$, and algorithms$\mathcal{A}_k$ is an algorithm with values in $\mathbb{R}^d$, i.e., its output is the $k$-th iterate of the optimization algorithm.$\mathcal{A}_k(S,U) := w_k$. Let $\mathcal{A}(S,U) = \mathcal{W}_{S,U} = \{\mathcal{A}_k(S,U

Figures (13)

  • Figure 1: (Left) Evolution of the loss function $\ell(\cdot, Z)$ for a fixed sample $Z \in \mathcal{Z}$ over $T=1000$ iterations, for two neighboring datasets $S,S' \in \mathcal{Z}^n$. While the classical notion of algorithmic stability measures the error at a specific iteration $t$, our stability notion extends this perspective to the entire training trajectory. (Right) Numerical estimation of our new random set stability parameter ($\beta_n$) as $n$ increases. The experiments demonstrate that $\beta_n$ decreases with larger $n$.
  • Figure 2: Variation of $\mathbf{E}^1$ with sample size $n$ for the model ViT. Pearson correlation coefficients $r$ are reported for each subgroup.
  • Figure 3: Variation of $\mathbf{E}^1$ with sample size $n$ for the model GraphSage. Pearson correlation coefficients $r$ are reported for each subgroup.
  • Figure 4: Variation of $\mathbf{PMag}(\sqrt[3]{100} \cdot \mathcal{W}_{S,U})$ with sample size $n$ for the model GraphSage. Pearson correlation coefficients $r$ are reported for each subgroup.
  • Figure 5: Variation of $\mathbf{PMag}(\sqrt[3]{10000} \cdot \mathcal{W}_{S,U})$ with sample size $n$ for the model GraphSage. Pearson correlation coefficients $r$ are reported for each subgroup.
  • ...and 8 more figures

Theorems & Definitions (45)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.1
  • Corollary 3.1
  • Lemma 3.1
  • ...and 35 more