What is the role of curvature in high-dimensional optimization landscapes?

Curvature determines how sensitive the cost function is to parameter changes in different directions.

In high-dimensional spaces, curvature is anisotropic—some directions are extremely flat while others are steep. This creates ill-conditioned optimization problems where gradient descent oscillates or slows down. Preconditioning methods like Adam or second-order approximations attempt to normalize curvature effects to improve convergence efficiency.