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Fast Rates for Empirical Risk Minimization of Strict Saddle Problems

Alon Gonen, Shai Shalev-Shwartz

TL;DR

This work analyzes empirical risk minimization for non-convex objectives that satisfy the strict saddle property, showing that efficient ERM methods (e.g., SGD) are statistically stable and generalize well. By relating strict saddle geometry to local strong convexity and leveraging average stability, the authors derive fast-rate generalization bounds that are dimension-free in the unconstrained case and provide explicit sample complexities for practical problems. They instantiate the theory in PCA and ICA via tensor decomposition, obtaining improved sample-size requirements that scale with spectral gaps or tensor dimensions rather than the full 1/ε^2 rate. The results unify and extend statistical analyses of several strict-saddle problems, offering a framework for fast-converging ERM analyses in structured non-convex settings.

Abstract

We derive bounds on the sample complexity of empirical risk minimization (ERM) in the context of minimizing non-convex risks that admit the strict saddle property. Recent progress in non-convex optimization has yielded efficient algorithms for minimizing such functions. Our results imply that these efficient algorithms are statistically stable and also generalize well. In particular, we derive fast rates which resemble the bounds that are often attained in the strongly convex setting. We specify our bounds to Principal Component Analysis and Independent Component Analysis. Our results and techniques may pave the way for statistical analyses of additional strict saddle problems.

Fast Rates for Empirical Risk Minimization of Strict Saddle Problems

TL;DR

This work analyzes empirical risk minimization for non-convex objectives that satisfy the strict saddle property, showing that efficient ERM methods (e.g., SGD) are statistically stable and generalize well. By relating strict saddle geometry to local strong convexity and leveraging average stability, the authors derive fast-rate generalization bounds that are dimension-free in the unconstrained case and provide explicit sample complexities for practical problems. They instantiate the theory in PCA and ICA via tensor decomposition, obtaining improved sample-size requirements that scale with spectral gaps or tensor dimensions rather than the full 1/ε^2 rate. The results unify and extend statistical analyses of several strict-saddle problems, offering a framework for fast-converging ERM analyses in structured non-convex settings.

Abstract

We derive bounds on the sample complexity of empirical risk minimization (ERM) in the context of minimizing non-convex risks that admit the strict saddle property. Recent progress in non-convex optimization has yielded efficient algorithms for minimizing such functions. Our results imply that these efficient algorithms are statistically stable and also generalize well. In particular, we derive fast rates which resemble the bounds that are often attained in the strongly convex setting. We specify our bounds to Principal Component Analysis and Independent Component Analysis. Our results and techniques may pave the way for statistical analyses of additional strict saddle problems.

Paper Structure

This paper contains 25 sections, 21 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Applications
  4. PCA
  5. ICA
  6. Our approach
  7. Preliminaries
  8. Stability and generalization error
  9. Strict saddle functions
  10. Stability Bounds for Strict Saddle Empirical Risks: Unconstrained Setting
  11. Stability Bounds for Strict Saddle Empirical Risks: Constrained Setting
  12. First and second-order conditions
  13. Strict saddle property in the constrained setting
  14. Analysis in the constrained setting
  15. Application to PCA
  16. ...and 10 more sections

Key Result

theorem 1

Let $\epsilon \in (0,1)$. Suppose that that the empirical risk is $(\alpha, \gamma, \tau)$-strict saddle with high probability (see Section sec:strictSaddle). Then the sample complexity of every ERM hypothesis is at most $\max \left \{ \frac{\beta_1}{\gamma}, \frac{\rho}{\tau}, \frac{2 \rho^2}{\alph

Theorems & Definitions (40)

  • theorem 1
  • theorem 2
  • remark 1
  • remark 1
  • theorem 3
  • theorem 4
  • definition 1
  • lemma 1
  • definition 2
  • remark 2
  • ...and 30 more