Numerical Optimization with Adaptive Accuracy and Applications to Machine Learning
A large variety of real-life problems can be modeled as continuous optimization problems. Moreover, the technological advances of recent years have highlighted not only the need of storing huge amounts of data, but also of analyzing them to extract useful information. Thus, continuous numerical optimization plays a fundamental role in the solution of large-scale problems arising in many fields, such as machine learning, in all the applications of everyday life.
The high dimensionality of the problems requires ad-hoc numerical methods, able to deal with huge datasets and based either on approximation techniques, such as inexact evaluations or regularizations of functions in a deterministic framework, or on stochastic models, exploiting data redundancy and integral or flow-based representations. These methods share the idea that inexact solvers must be used to tackle difficulties such as high dimensionality, uncertainty, ill posedness, ill conditioning, multiple scales, sparsity, and thus they include deterministic and stochastic methods, hybrid methods (combining deterministic, stochastic and heuristic approaches), multiscale methods, etc., all using adaptive accuracy.
The project has the ambition of proposing, in a unifying framework, numerical optimization algorithms with adaptive accuracy for efficiently solving a variety of problems in the curse of dimensionality, non-convexity and/or non-smoothness, which share the aforementioned difficulties.
