Artificial intelligence models typically reduce to optimization problems: find the best solution according to a problem-specific metric. Sometimes, the nature of the problems means that the standard calculus-based tools cannot be applied. This setting is known as gradient-free optimization, and is particularly relevant for small businesses, academic research groups, and public-sector organizations that lack large-scale computing infrastructure yet still need to fine-tune machine learning models or optimize complex simulations. This project develops new mathematical and computational tools that make gradient-free optimization dramatically more efficient by exploiting hidden low-dimensional structure in these problems. This will lower the computational barrier to entry for a broad range of users. The project will train PhD students in these interdisciplinary methods, produce openly available software, and develop instructional materials connecting linear algebra to modern deep learning. Gradient-free optimization (GFO) has deep theoretical foundations, yet remains poorly understood in high dimensions. This project will establish mathematical and algorithmic tools that break worst-case GFO barriers by exploiting structure in matrix-valued gradients. Algorithms for objective functions whose gradients exhibit various kinds of low intrinsic dimensionality, such as sparsity, low rank, or sparsity-plus-low-rank will be developed. These gradient estimation techniques will be wrappe