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Generalization bounds for uniformly stable algorithms

Generalization bounds for uniformly stable algorithms. Vitaly Feldman Brain. with Jan Vondrak. Uniform stability. Domain (e.g. ) Dataset Learning algorithm Loss function. Uniform stability [ Bousquet,Elisseeff 02] : has uniform stability w.r.t. if

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Generalization bounds for uniformly stable algorithms

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  1. Generalization bounds for uniformly stable algorithms Vitaly Feldman Brain with Jan Vondrak

  2. Uniform stability • Domain (e.g. ) • Dataset • Learning algorithm • Loss function • Uniform stability [Bousquet,Elisseeff 02]: • has uniform stability w.r.t. if • For all neighboring

  3. Generalization error • Probability distribution over , • Population loss: • Empirical loss: • Generalization error/gap:

  4. Stochastic convex optimization • For all is convex -Lipschitz in • Minimize over • For all , being SGD with rate : • Uniform convergence error: • ERMmight have generalization error: • [Shalev-Shwartz,Shamir,Srebro,Sridharan ’09; Feldman 16]

  5. Stable optimization Strongly convex ERM [BE 02, SSSS 09] is uniformly stable and minimizes within Gradient descent on smooth losses [Hardt,Recht,Singer 16] : steps of GD on with is uniformly stable and minimizes within

  6. Generalization bounds For w/ range and uniform stability [Rogers,Wagner 78] [Bousquet,Elisseeff 02] Vacuous when NEW!

  7. Comparison [BE 02] This work Generalization error

  8. Second moment Previously [Devroye,Wagner 79; BE 02] • Chebyshev: TIGHT! NEW! [BE 02] This work Generalization error

  9. Implications For any loss , has uniform stability Stronger generalization bounds for DP prediction algorithms • Differentially-private prediction [Dwork,Feldman 18] • . A randomized algorithm has -DP prediction if for all

  10. Data-dependent functions Consider . E.g. • has uniform stability if • For all neighboring • Generalization error/gap:

  11. Generalization in expectation Goal: For all and ,

  12. Concentration via McDiarmid For all neighboring 1. 2. McDiarmid: where

  13. Proof technique Based on [Nissim,Stemmer 15; BNSSSU 16] Let for Need to bound • Let be a distribution over UNSTABLE!

  14. Stable max Exponential mechanism [McSherry,Talwar 07] : sample • Stable: - differentially private • Approximates max

  15. Game over Pick get • Let be a distribution over

  16. Conclusions • Better understanding of uniform stability • New technique • Open • Gap between upper and lower bounds • High probability generalization without strong uniformity

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