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We consider the use of a curvature-adaptive step size in gradient-basediterative methods, including quasi-Newton methods, for minimizingself-concordant functions, extending an approach first proposed for Newton'smethod by Nesterov. This step size has a simple expression that can be computedanalytically; hence, line searches are not needed. We show that using this stepsize in the BFGS method (and quasi-Newton methods in the Broyden convex classother than the DFP method) results in superlinear convergence for stronglyconvex self-concordant functions. We present numerical experiments comparinggradient descent and BFGS methods using the curvature-adaptive step size totraditional methods on deterministic logistic regression problems, and toversions of stochastic gradient descent on stochastic optimization problems.

Quasi-Newton Methods: Superlinear Convergence Without Line Searches for Self-Concordant Functions