  We study the convergence rate of the famous Symmetric Rank-1 (SR1) algorithm
which has wide applications in different scenarios. Although it has been
extensively investigated, SR1 still lacks a non-asymptotic superlinear rate
compared with other quasi-Newton methods such as DFP and BFGS. In this paper we
address this problem. Inspired by the recent work on explicit convergence
analysis of quasi-Newton methods, we obtain the first explicit non-asymptotic
rates of superlinear convergence for the vanilla SR1 methods with correction
strategy to achieve the numerical stability. Specifically, the vanilla SR1 with
the correction strategy achieves the rates of the form
$\left(\frac{4n\ln(e\kappa) }{k}\right)^{k/2}$ for general smooth
strongly-convex functions where $k$ is the iteration counter, $\kappa$ is the
condition number of the objective function and $n$ is the dimension of the
problem. For the quadratic function, the vanilla SR1 algorithm can find the
optima of the objective function at most $n$ steps.

Optimization and Control