Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory (1004.2316v2)

Abstract: In regular statistical models, the leave-one-out cross-validation is asymptotically equivalent to the Akaike information criterion. However, since many learning machines are singular statistical models, the asymptotic behavior of the cross-validation remains unknown. In previous studies, we established the singular learning theory and proposed a widely applicable information criterion, the expectation value of which is asymptotically equal to the average Bayes generalization loss. In the present paper, we theoretically compare the Bayes cross-validation loss and the widely applicable information criterion and prove two theorems. First, the Bayes cross-validation loss is asymptotically equivalent to the widely applicable information criterion as a random variable. Therefore, model selection and hyperparameter optimization using these two values are asymptotically equivalent. Second, the sum of the Bayes generalization error and the Bayes cross-validation error is asymptotically equal to $2\lambda/n$, where $\lambda$ is the real log canonical threshold and $n$ is the number of training samples. Therefore the relation between the cross-validation error and the generalization error is determined by the algebraic geometrical structure of a learning machine. We also clarify that the deviance information criteria are different from the Bayes cross-validation and the widely applicable information criterion.

• The paper demonstrates that Bayes cross-validation loss and WAIC are asymptotically equivalent, offering robust tools for model selection in singular models.
• It proves that the sum of Bayes generalization and cross-validation errors equals 2λ/n, linking performance metrics to the model’s algebraic structure.
• The study differentiates WAIC and cross-validation from DIC, underscoring enhanced reliability in evaluating complex, non-regular statistical models.

Overview: Asymptotic Equivalence of Cross-Validation and Information Criteria in Singular Learning Theory

The paper "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory" by Sumio Watanabe explores the intricacies of model selection and evaluation in the context of singular statistical models. Singular models, which often arise in contemporary machine learning methods, notably differ from regular models, leading to unique theoretical challenges and insights. This paper primarily focuses on understanding the behavior of various model selection criteria in these singular learning models.

Key Highlights and Theoretical Contributions

1. Singular vs. Regular Models:
   • Regular models have a well-defined Fisher information matrix, leading to predictable asymptotic behaviors. For these models, leave-one-out cross-validation (LOOCV) is asymptotically equivalent to the Akaike Information Criterion (AIC).
   • Singular models, including neural networks, normal mixtures, hidden Markov models, and Bayesian networks, do not conform to these regularity conditions, requiring more nuanced approaches.
2. Foundations and Development of Singular Learning Theory:
   • Singular learning models have hierarchies, hidden variables, or grammatical constraints. Conventional techniques such as the maximum likelihood method often fail in these contexts, necessitating Bayesian estimation techniques.
   • The paper builds on the established concepts of Bayesian generalization error, Bayes cross-validation loss, and the widely applicable information criterion (WAIC).
3. Main Theoretical Results:
   • Theorem 1: The paper establishes that the Bayes cross-validation loss is asymptotically equivalent to the WAIC as a random variable. This finding posits that model selection and hyperparameter optimization using these two values are nearly identical in the limit for large sample sizes.
   • Theorem 2: It is shown that the sum of the Bayes generalization error and the Bayes cross-validation error asymptotically equals $2\lambda/n$, where $\lambda$ denotes the real log canonical threshold and $n$ is the number of training samples. This significant result ties the generalization properties to the algebraic geometrical structure of the learning machine, highlighting the importance of $\lambda$ as a determining factor.
4. Comparison with Deviance Information Criteria (DIC):
   • The paper differentiates DIC from cross-validation and WAIC, showing that DIC does not always align with the asymptotic behavior of the generalization error, especially in singular models.
   • The asymptotic properties and variances of the DIC, WAIC, and cross-validation errors are intricately dissected, providing detailed insight into their relative performance and reliability.

Practical and Theoretical Implications

The convergence properties of model evaluation criteria in singular learning models carry profound implications for both theoretical research and practical applications in machine learning:

1. Model Selection and Hyperparameter Tuning:
   • The findings advocate for the use of WAIC and Bayes cross-validation as robust tools for model selection in singular models, given their asymptotic equivalence.
   • The implications extend to improved reliability in the estimation of the generalization error, fostering more precise model validation and selection procedures.
2. Future Research Directions:
   • Refining Singular Learning Theory: The foundational results presented call for further exploration of the algebraic geometrical properties, particularly the real log canonical threshold $\lambda$, in diverse model structures and learning scenarios.
   • Extensions to More Complex Models: Future research should investigate the applicability and extensions of these theoretical results in more complex and higher-dimensional singular models, potentially incorporating newer deep learning architectures.
3. Computational Approaches and Efficiency:
   • While the theoretical underpinnings are robust, practical implementations, especially concerning the computational burden of Bayesian methods, remain a critical area of focus. Efficient algorithms and approximate methods for posterior distributions can significantly enhance the practical usability of WAIC and Bayes cross-validation.

Conclusion

This paper provides a rigorous and insightful exploration into the asymptotic behavior of cross-validation and information criteria within the field of singular learning models. By proving the asymptotic equivalence of Bayes cross-validation loss and WAIC, and elucidating their relationship with the algebraic geometrical structures of learning machines, Watanabe's work significantly advances our understanding and methods of model evaluation in complex, non-regular statistical models. The implications are far-reaching, impacting both theoretical advancements and practical methodologies in statistical machine learning.