On stochastic and deterministic quasi-Newton methods for non-Strongly convex optimization: Asymptotic convergence and rate analysis

Motivated by applications arising from large scale optimization and machine learning, we consider stochastic quasi-Newton (SQN) methods for solving unconstrained convex optimization problems. The convergence analysis of the SQN methods, both full and limited-memory variants, require the objective function to be strongly convex. However, this assumption is fairly restrictive and does not hold for applications such as minimizing the logistic regression loss function. To the best of our knowledge, no rate statements currently exist for SQN methods in the absence of such an assumption. Also, among the existing first-order methods for addressing stochastic optimization problems with merely convex objectives, those equipped with provable convergence rates employ averaging. However, this averaging technique has a detrimental impact on inducing sparsity. Motivated by these gaps, the main contributions of the paper are as follows: (i) Addressing large scale stochastic optimization problems, we develop an iteratively regularized stochastic limited-memory BFGS (IRS-LBFGS) algorithm, where the stepsize, regularization parameter, and the Hessian inverse approximation matrix are updated iteratively. We establish the convergence to an optimal solution of the original problem both in an almost-sure and mean senses. We derive the convergence rate in terms of the objective function's values and show that it is of the order $\mathcal{O}\left(k^{-\left(\frac{1}{3}-\epsilon\right)}\right)$, where $\epsilon$ is an arbitrary small positive scalar; (ii) In deterministic regime, we show that the regularized limited-memory BFGS algorithm displays a rate of the order $\mathcal{O}\left(\frac{1}{k^{1 -\epsilon'}}\right)$, where $\epsilon'$ is an arbitrary small positive scalar. We present our numerical experiments performed on a large scale text classification problem.

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