Hessian Matrix

The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function, commonly used in optimization problems to describe the local curvature of the loss function with respect to model parameters. For a function f(x), the Hessian H is defined as H = ∂²f/∂x², where x represents the vector of parameters. The eigenvalues of the Hessian provide critical information about the nature of stationary points: positive eigenvalues indicate a local minimum, negative eigenvalues indicate a local maximum, and mixed signs indicate a saddle point. In the context of machine learning, the Hessian matrix is instrumental in second-order optimization methods, where it aids in determining the step direction and size during parameter updates. Its computational complexity, however, can be a limiting factor in high-dimensional spaces, necessitating approximations or alternative methods.