The proposed asymptotic theory for stochastic gradient descent (SGD) with dropout regularization can be extended to more general non-linear models by leveraging the foundational principles established in the linear regression context. One approach is to consider generalized additive models (GAMs) or non-linear neural networks, where the response variable is modeled as a non-linear function of the predictors.
To achieve this, the following steps can be taken:
Non-linear Function Approximation: Replace the linear regression model with a non-linear function ( f(x; \\beta) ) that captures the relationship between the predictors ( x ) and the response variable. This function can be parameterized by ( \\beta ), similar to how linear regression uses ( \\beta ) for linear combinations.
Dropout Regularization Adaptation: Adapt the dropout mechanism to the non-linear context by applying dropout to the activations of the hidden layers in neural networks or to the basis functions in GAMs. This ensures that the model retains the stochasticity introduced by dropout while learning non-linear relationships.
Geometric-Moment Contraction (GMC) Extension: Extend the GMC property to accommodate the non-linear transformations. This may involve proving that the iterates of the non-linear model still satisfy the contraction property under certain conditions, such as Lipschitz continuity of the non-linear function.
Central Limit Theorems (CLTs): Derive quenched CLTs for the non-linear iterates, similar to those established for linear models. This would require careful handling of the non-linearities and ensuring that the necessary moment conditions are satisfied.
Numerical Simulations: Conduct numerical experiments to validate the theoretical results in the context of non-linear models, ensuring that the proposed confidence intervals and asymptotic properties hold.
By following these steps, the insights gained from the linear regression analysis can be effectively translated to more complex non-linear models, thereby broadening the applicability of dropout regularization in stochastic optimization.