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To make Sophia decoupled, we can separate the Hessian estimation from the main optimizer. This will allow users to plug in different Hessian estimators without modifying the core optimizer code. Here's the research analysis, algorithmic pseudocode, and Python implementation for a decoupled Sophia optimizer.

Architectural Analysis
Create a base Hessian estimator class that defines the interface for all Hessian estimators.
Implement specific Hessian estimators (e.g., Hutchinson, Gauss-Newton-Bartlett) as subclasses of the base Hessian estimator class.
Modify the Sophia optimizer to accept a Hessian estimator object during initialization.
Update the optimizer's step method to use the provided Hessian estimator object for Hessian estimation.
Algorithm Pseudocode
Base Hessian Estimator
Define an abstract method estimate that takes the parameter θ and gradient as input and returns the Hessian estimate.
Hutchinson Estimator
Inherit from the base Hessian estimator class.
Implement the estimate method using the Hutchinson algorithm.
Gauss-Newton-Bartlett Estimator
Inherit from the base Hessian estimator class.
Implement the estimate method using the Gauss-Newton-Bartlett algorithm.
Decoupled Sophia Optimizer
Modify the Sophia optimizer to accept a Hessian estimator object during initialization.
Update the optimizer's step method to use the provided Hessian estimator object for Hessian estimation.