Sharpened Lazy Incremental Quasi-Newton Method
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We consider the finite sum minimization of n strongly convex and smooth functions with Lipschitz continuous Hessians in d dimensions. In many applications where such problems arise, including maximum likelihood estimation, empirical risk minimization, and unsupervised learning, the number of observations n is large, and it becomes necessary to use incremental or stochastic algorithms whose per-iteration complexity is independent of n. Of these, the incremental/stochastic variants of the Newton method exhibit superlinear convergence, but incur a per-iteration complexity of O(d^3), which may be prohibitive in large-scale settings. On the other hand, the incremental Quasi-Newton method incurs a per-iteration complexity of O(d^2) but its superlinear convergence rate has only been characterized asymptotically. This work puts forth the Sharpened Lazy Incremental Quasi-Newton (SLIQN) method that achieves the best of both worlds: an explicit superlinear convergence rate with a per-iteration complexity of O(d^2). Building upon the recently proposed Sharpened Quasi-Newton method, the proposed incremental variant incorporates a hybrid update strategy incorporating both classic and greedy BFGS updates. The proposed lazy update rule distributes the computational complexity between the iterations, so as to enable a per-iteration complexity of O(d^2). Numerical tests demonstrate the superiority of SLIQN over all other incremental and stochastic Quasi-Newton variants.
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