Asymptotic Convergence Rate and Statistical Inference for Stochastic Sequential Quadratic Programming
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We apply a stochastic sequential quadratic programming (StoSQP) algorithm to solve constrained nonlinear optimization problems, where the objective is stochastic and the constraints are deterministic. We study a fully stochastic setup, where only a single sample is available in each iteration for estimating the gradient and Hessian of the objective. We allow StoSQP to select a random stepsize α̅_t adaptively, such that β_t≤α̅_t ≤β_t+χ_t, where β_t, χ_t=o(β_t) are prespecified deterministic sequences. We also allow StoSQP to solve Newton system inexactly via randomized iterative solvers, e.g., with the sketch-and-project method; and we do not require the approximation error of inexact Newton direction to vanish. For this general StoSQP framework, we establish the asymptotic convergence rate for its last iterate, with the worst-case iteration complexity as a byproduct; and we perform statistical inference. In particular, with proper decaying β_t,χ_t, we show that: (i) the StoSQP scheme can take at most O(1/ϵ^4) iterations to achieve ϵ-stationarity; (ii) asymptotically and almost surely, (x_t -x^⋆, λ_t - λ^⋆) = O(√(β_tlog(1/β_t)))+O(χ_t/β_t), where (x_t,λ_t) is the primal-dual StoSQP iterate; (iii) the sequence 1/√(β_t)· (x_t -x^⋆, λ_t - λ^⋆) converges to a mean zero Gaussian distribution with a nontrivial covariance matrix. Moreover, we establish the Berry-Esseen bound for (x_t, λ_t) to measure quantitatively the convergence of its distribution function. We also provide a practical estimator for the covariance matrix, from which the confidence intervals of (x^⋆, λ^⋆) can be constructed using iterates {(x_t,λ_t)}_t. Our theorems are validated using nonlinear problems in CUTEst test set.
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