Source-linked AI summary
High probability generalization bounds for uniformly stable algorithms with nearly optimal rate
Vitaly Feldman, Jan Vondrak
TL;DR
The paper addresses the limited high-probability guarantees provided by classical stability-based generalization bounds. It proves a logarithmic-overhead bound for uniformly stable algorithms and applies it to stochastic gradient descent and privacy-preserving prediction.
Problem
Most stability notions yield bounds only in expectation or for the second moment, leaving high-probability generalization guarantees as a central challenge.
Method
The paper develops a high-probability analysis for uniformly stable algorithms, using a stability-preserving, data-dependent shift in the clamping range.
Results
A logarithmic-overhead high-probability generalization bound is obtained for any γ-uniformly stable algorithm, improving the dependence on the tail parameter over prior work.
Takeaways & Limitations
The result yields high-probability generalization bounds for multi-pass stochastic gradient descent and agnostically learned threshold functions with differentially private prediction.
Takeaways & Limitations
For random-shuffling gradient descent, little is known about the empirical error, and existing results apply only in more restrictive settings or to simple function classes.
Abstract
from arXiv · showhide
Algorithmic stability is a classical approach to understanding and analysis of the generalization error of learning algorithms. A notable weakness of most stability-based generalization bounds is that they hold only in expectation. Generalization with high probability has been established in a landmark paper of Bousquet and Elisseeff (2002) albeit at the expense of an additional $\sqrt{n}$ factor in the bound. Specifically, their bound on the estimation error of any $γ$-uniformly stable learning algorithm on $n$ samples and range in $[0,1]$ is $O(γ\sqrt{n \log(1/δ)} + \sqrt{\log(1/δ)/n})$ with probability $\geq 1-δ$. The $\sqrt{n}$ overhead makes the bound vacuous in the common settings where $γ\geq 1/\sqrt{n}$. A stronger bound was recently proved by the authors (Feldman and Vondrak, 2018) that reduces the overhead to at most $O(n^{1/4})$. Still, both of these results give optimal generalization bounds only when $γ= O(1/n)$. We prove a nearly tight bound of $O(γ\log(n)\log(n/δ) + \sqrt{\log(1/δ)/n})$ on the estimation error of any $γ$-uniformly stable algorithm. It implies that for algorithms that are uniformly stable with $γ= O(1/\sqrt{n})$, estimation error is essentially the same as the sampling error. Our result leads to the first high-probability generalization bounds for multi-pass stochastic gradient descent and regularized ERM for stochastic convex problems with nearly optimal rate --- resolving open problems in prior work. Our proof technique is new and we introduce several analysis tools that might find additional applications.
1 Introduction
The paper studies high-probability estimation error for uniformly stable algorithms, addressing the limitations of expectation-only bounds and prior high-probability bounds with large stability overheads. It proves a logarithmic-overhead bound and applies it to stable learning algorithms, including ERM and gradient descent.
- Problem setting: Uniform stability measures how much an algorithm’s output function changes when one dataset element is replaced, and estimation error compares its population and empirical means.The framework covers data-dependent functions with range in [0,1] and i.i.d. samples.
- Motivation: High-probability bounds are important because expectation-only guarantees are insufficient when an algorithm is used repeatedly or inside another algorithm.The paper contrasts stability-based expectation bounds with exponentially decaying tails from uniform convergence.
- Prior limitations: Prior stability bounds incurred substantial overheads, becoming vacuous in settings where γ is at least 1/√n and leaving high-probability guarantees for key applications unresolved.The introduction identifies open problems for strongly regularized ERM and stochastic gradient descent.
- Contribution: The main result gives any γ-uniformly stable algorithm a high-probability generalization bound with only logarithmic overhead.Theorem 1.1 applies to arbitrary distributions over Z and algorithms with range [0,1].
- Contribution: For γ = O(1/√n), stable algorithms achieve essentially the same estimation-error guarantees as algorithms that output a fixed function.The bound is optimal up to constant factors when γ is sufficiently small, whereas earlier results required γ = O(1/n) for comparable guarantees.
- Applications: The result yields improved high-probability guarantees for strongly convex ERM, regularized stochastic convex optimization, and deterministic or stochastic gradient descent.The paper also applies the framework to privacy-preserving prediction and reports nearly optimal rates for several applications.
2 Preliminaries
The paper formalizes estimation error for data-dependent functions and defines uniform stability as bounded sensitivity to replacing one dataset element. It reduces generalization analysis to unbiased leave-one-out quantities and a tail-bounding function.
- Data-dependent functions: A data-dependent function maps a dataset and evaluation point to a value, while in supervised learning its estimation error is the true loss minus empirical loss.The function can represent the loss of a learned model on an example.
- Uniform stability: Uniform stability γ requires changing one dataset element to alter the function value by at most γ.The definition is equivalent to sensitivity at most γ for each fixed evaluation point.
- Randomized algorithms: Randomized algorithms can be analyzed through stability of their expected output or stability for fixed random bits, yielding different generalization guarantees.The expected-output approach is simpler, while fixed-randomness stability can support high-probability guarantees over algorithmic randomness.
- Concentration: The analysis uses McDiarmid’s inequality for functions with bounded sensitivity.This supplies concentration for functions of the dataset.
- Proof setup: The leave-one-out estimation error and tail-bounding function Dδ(n, R, γ) provide the main quantities for the subsequent proof.The unbiased transformation subtracts the population mean and changes the range to [-1,1] and stability to at most 2γ when M ranges in [0,1].
3 Proof of the Main Result
The proof recursively combines dataset-size reduction with range reduction while preserving unbiasedness and stability. A data-dependent clamping construction resolves the key bias-versus-stability obstacle and supports the main concentration bound.
- Proof strategy: The proof targets exponential-tail concentration by reducing D(n, R, γ) to smaller-dataset or smaller-range instances.The two operations are applied recursively, with dataset-size reduction performed by partitioning coordinates into blocks.
- Range reduction: McDiarmid-based centering and clamping reduce the effective range while accounting for sampling error and values clipped outside the range.The centered function subtracts φ(z), and the additional errors come from estimating φ and clamping tails.
- Dataset size reduction: Block partitioning views the estimation error as an average of blockwise errors, reducing dataset size from n to n′ without significantly changing the error.Conditioning on coordinates outside a block yields a smaller problem with dataset size n′.
- Inductive argument: Repeatedly combining the two reductions yields an inductive bound of approximately Õ(γ), with log n recursion levels producing the logarithmic overhead.The hardest regime is γ = R/√n; other regimes follow by applying one operation once.
- Technical obstacle: Simple fixed-range clamping can destroy unbiasedness or double the stability parameter, so the proof uses a dataset-dependent shift of the clamping interval.The shifted interval preserves both zero mean and uniform stability, but requires delicate analysis.
- Range reduction: The range-reduction construction produces a function that remains unbiased, differs from the original by a controlled amount, and retains uniform stability γ.The construction uses clamping around a dataset-dependent center b_s and preserves the zero-mean condition.
- Range reduction: The centered clamped function is bounded in a reduced range while preserving stability, enabling the recursive concentration argument.The resulting bound is expressed through R′ and the controlled clamping error.
- Main bound: The main theorem is obtained by combining the reduction lemmas through an inductive statement and treating the γ ≥ 1 case separately.The cited proof steps identify the reduction lemmas as the route to the main result.
4 Applications
The paper applies its high-probability stability bounds to stochastic convex optimization, regularized empirical risk minimization, deterministic and stochastic gradient descent, and differentially private prediction. These applications provide generalization guarantees for multiple passes, batch sizes, and practical optimization procedures.
- Regularized ERM: Theorem 4.1 establishes uniform stability for empirical minimization of 1-Lipschitz, strongly convex losses over a convex body.The result applies when the loss functions are λ-strongly convex and have range in [0, 1].
- Regularized ERM: Regularization yields an efficient algorithm with essentially the same stability and generalization bounds when the empirical objective is minimized within 1/n.The paper chooses a strongly convex regularizer with λ = 1/n2/3 in the cited application.
- Gradient descent: The bounds apply to gradient descent variants with different rates, arbitrary batch sizes, and multiple passes, without requiring the output to be close to the empirical minimum.This extends the analysis to algorithms used in practice rather than only iteration schedules chosen for convergence proofs.
- Gradient descent: For projected gradient descent on convex, 1-Lipschitz, σ-smooth losses, the paper gives a high-probability generalization application using η = 1/σ and T = ⌊σ√n/log n⌋.The stated setting uses losses with range in [0, 1] over a convex body.
- Stochastic gradient descent: For stochastic gradient descent, concentration of the stability parameter gives at most logarithmic overhead even with a constant number of passes and batch size 1.The overhead becomes relatively smaller as the number of passes grows, and the analysis covers randomized batch-selection strategies.
- Differential privacy: The paper also derives high-probability generalization bounds for differentially private prediction that are stronger than prior bounds in several parameter regimes.The application targets arbitrary loss functions with range in [0, 1] and compares favorably with the cited prior results in selected regimes.