Entropy-SGD optimizes the prior of a PAC-Bayes bound: Generalization properties of Entropy-SGD and data-dependent priors (1712.09376v3)

Abstract: We show that Entropy-SGD (Chaudhari et al., 2017), when viewed as a learning algorithm, optimizes a PAC-Bayes bound on the risk of a Gibbs (posterior) classifier, i.e., a randomized classifier obtained by a risk-sensitive perturbation of the weights of a learned classifier. Entropy-SGD works by optimizing the bound's prior, violating the hypothesis of the PAC-Bayes theorem that the prior is chosen independently of the data. Indeed, available implementations of Entropy-SGD rapidly obtain zero training error on random labels and the same holds of the Gibbs posterior. In order to obtain a valid generalization bound, we rely on a result showing that data-dependent priors obtained by stochastic gradient Langevin dynamics (SGLD) yield valid PAC-Bayes bounds provided the target distribution of SGLD is {\epsilon}-differentially private. We observe that test error on MNIST and CIFAR10 falls within the (empirically nonvacuous) risk bounds computed under the assumption that SGLD reaches stationarity. In particular, Entropy-SGLD can be configured to yield relatively tight generalization bounds and still fit real labels, although these same settings do not obtain state-of-the-art performance.

• The paper links Entropy-SGD to PAC-Bayes bounds, showing it minimizes the bound on Gibbs posterior risk by optimizing the data-dependent prior.
• Entropy-SGLD can achieve nonvacuous risk bounds using data-dependent priors, provided the mechanism satisfies acceptable differential privacy (ε-differentially private).
• Empirical tests demonstrate Entropy-SGLD balances empirical risk and overfitting resistance, achieving valid bounds but requiring careful calibration for practical use.

An Examination of Entropy-SGD within the Framework of PAC-Bayes Bounds

The paper, titled "Entropy-SGD optimizes the prior of a PAC-Bayes bound: Generalization properties of Entropy-SGD and data-dependent priors," investigates the application of Entropy-SGD (Stochastic Gradient Descent) in the context of PAC-Bayes bounds. This work is conducted by Gintare Karolina Dziugaite and Daniel M. Roy, affiliated with the University of Cambridge and the University of Toronto, respectively. The central focus of this paper is to explore how Entropy-SGD can be interpreted as optimizing the PAC-Bayes bound's prior, thus exploring the generalization properties of this optimization algorithm under specific conditions.

Core Contributions

The authors offer a unique perspective by linking Entropy-SGD to statistical learning theory through PAC-Bayes bounds. They elucidate that Entropy-SGD seeks to minimize this bound on the risk of the Gibbs posterior classifier. This is achieved by optimizing the prior, traditionally considered independent of the data in PAC-Bayes settings. A significant finding here, reflecting on the generalization aspect, is that Entropy-SGD and its variants, such as Entropy-SGLD (incorporating Langevin Dynamics), can achieve nonvacuous risk bounds through data-dependent priors while still maintaining acceptable training error on datasets like MNIST and CIFAR10.

Implications and Theoretical Insights

By interpreting Entropy-SGD within this PAC-Bayes framework, the authors highlight potential overfitting issues. Entropy-SGD, operating outside the hypothesis that the prior is data-independent, needs adjustments to secure valid generalization bounds. The authors demonstrate that configuring Entropy-SGLD to incorporate data-dependent priors via stochastic gradient Langevin dynamics results in a valid PAC-Bayes bound, provided the mechanism is differentially private to an acceptable degree (ε-differentially private). This satisfies the theoretical requirements for convergence towards PAC-Bayesian bounds and holds significance for guiding future development and verification of machine learning models under rigorous theoretical frameworks.

Numerical Validation

The empirical evaluation explores the practical performance of the Entropy-SGLD algorithm on MNIST and CIFAR10 datasets. The results confirm that with suitable calibration (e.g., thermal noise adjustment and τ parameter tuning), Entropy-SGLD can offer a relatively tight generalization bound. Crucially, it manages to achieve meaningful trade-offs between empirical risk minimization and overfitting resistance, albeit not reaching state-of-the-art test-set performance. These findings underscore the importance of balancing differential privacy, data dependency of the priors, and computational tractability when extending the PAC-Bayes approach to real-world learning tasks.

Future Directions

The exploration of Entropy-SGD through the lens of PAC-Bayes theory lays the groundwork for further exploration into optimization-driven generalization properties. A possible direction for expanding this research lies in achieving finer control over SGD's privacy parameters and its gradient estimation stability, as these directly impact the generalization capability of algorithms in varying learning environments. Moreover, the discussion raises pertinent questions on applying weakened stability notions to develop even sharper PAC-Bayes bounds while retaining computational feasibility and robust empirical generalization.

This research offers a coherent synthesis of Entropy-SGD with PAC-Bayes theoretical grounds, providing valuable insights for advancing understanding of generalization performance in deep learning models. Future work might explore quantitative benchmarks where Entropy-SGD provides competitive advantages over traditional SGD variants, potentially broadening the application scope of this theoretically rich algorithm.