Stochastic Damped L-BFGS with Controlled Norm of the Hessian Approximation
This work addresses the challenge of balancing Hessian approximation quality and conditioning for practitioners using L-BFGS in deep learning, particularly for highly nonconvex and ill-conditioned problems.
The paper introduces VARCHEN, a new stochastic variance-reduced damped L-BFGS algorithm that controls the norm of the Hessian approximation using eigenvalue estimates. It demonstrates improved robustness over SdLBFGS-VR and SVRG on a modified DavidNet problem, while showing comparable performance on logistic regression and nonconvex SVM.
We propose a new stochastic variance-reduced damped L-BFGS algorithm, where we leverage estimates of bounds on the largest and smallest eigenvalues of the Hessian approximation to balance its quality and conditioning. Our algorithm, VARCHEN, draws from previous work that proposed a novel stochastic damped L-BFGS algorithm called SdLBFGS. We establish almost sure convergence to a stationary point and a complexity bound. We empirically demonstrate that VARCHEN is more robust than SdLBFGS-VR and SVRG on a modified DavidNet problem -- a highly nonconvex and ill-conditioned problem that arises in the context of deep learning, and their performance is comparable on a logistic regression problem and a nonconvex support-vector machine problem.