Federal Project Grant Award Summary

The National Science Foundation (NSF) Division of Mathematical Sciences awarded $150,000 to the University of Colorado, Boulder, under the Mathematical and Physical Sciences (CFDA 47.049) program for a collaborative research project on gradient-free optimization of matrix functions. The award runs from July 1, 2026, through June 30, 2028. The project develops mathematical and computational tools to make gradient-free optimization (GFO)—optimization methods that do not rely on calculus-based gradients—substantially more efficient by exploiting hidden low-dimensional structure in matrix-valued gradients. This research addresses a critical barrier for small businesses, academic research groups, and public-sector organizations lacking large-scale computing infrastructure who need to fine-tune machine learning models or optimize complex simulations.

The project deliverables include novel algorithms for objective functions with gradients exhibiting low intrinsic dimensionality (sparsity, low rank, or sparsity-plus-low-rank structure), computational linear algebra techniques adapted to gradient-free settings, and integration of these methods into modern optimization algorithms. Additional outputs include the training of PhD students in interdisciplinary methods, openly available software tools, and instructional materials connecting linear algebra to contemporary deep learning applications. The work aims to establish mathematical and algorithmic foundations that reduce worst-case GFO barriers through advanced gradient estimation and matrix operation techniques in the gradient-free setting.