Speaker
Description
A wide range of applications in imaging, data science, and machine learning can be formulated as discrete inverse problems of the form
\begin{equation}
y = Kx + \varepsilon,
\end{equation}
where $K$ is a linear operator, $x \in \mathbb{R}^d$ is the unknown variable, and $y$ represents noisy observations.
Due to ill-conditioning and the increasingly large scale of modern problems, direct inversion is unstable or computationally infeasible.
A common approach is to adopt a variational formulation, in which the solution is obtained by minimizing a functional composed of a data fidelity term and a regularization term.
Deterministic first-order optimization methods are classical tools for solving such problems, but their computational cost can be prohibitive when the number of measurements is very large.
Stochastic optimization methods offer an effective alternative by reducing the cost per iteration through randomized sampling strategies.
These methods exploit the structure of the problem by operating on subsets of the data, while still ensuring convergence under appropriate assumptions.
In this work, we investigate stochastic first-order methods for regularized inverse problems, focusing on adaptive strategies for step-size selection and variance control.
We analyze the role of mini-batch sampling and dynamically tuned hyperparameters in improving convergence speed and robustness.
The proposed approach builds upon recent advances in stochastic gradient methods with adaptive learning rates and variance reduction, extending their use beyond traditional machine learning applications.
Numerical experiments on representative large-scale problems demonstrate the effectiveness and scalability of the proposed stochastic framework.