We derive generalization bounds for learning algorithms based on their robustness: the property that if a testing sample is "similar" to a training sample, then the testing error is close to the training error. This provides a novel approach, different from complexity or stability arguments, to study generalization of learning algorithms. One advantage of the robustness approach, compared to previous methods, is the geometric intuition it conveys. Consequently, robustness-based analysis is easy to extend to learning in non-standard setups such as Markovian samples or quantile loss. We further show that a weak notion of robustness is both sufficient and necessary for generalizability, which implies that robustness is a fundamental property that is required for learning algorithms to work.

Robustness and Generalization.

In this paper, we consider a model called CHARME (Conditional Heteroscedastic Autoregressive Mixture of Experts), a class of generalized mixture of nonlinear nonparametric AR-ARCH time series. Under certain Lipschitz-type conditions on the autoregressive and volatility functions, we prove that this model is stationary, ergodic and $\tau$-weakly dependent. These conditions are much weaker than those presented in the literature that treats this model. Moreover, this result forms the theoretical basis for deriving an asymptotic theory of the underlying (non)parametric estimation, which we present for this model. As an application, from the universal approximation property of neural networks (NN), we develop a learning theory for the NN-based autoregressive functions of the model, where the strong consistency and asymptotic normality of the considered estimator of the NN weights and biases are guaranteed under weak conditions.

José Gómez García

Papers

Learning charme models with neural networks