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We consider statistical inference of equality-constrained stochasticnonlinear optimization problems. We develop a fully online stochasticsequential quadratic programming (StoSQP) method to solve the problems, whichcan be regarded as applying Newton's method to the first-order optimalityconditions (i.e., the KKT conditions). Motivated by recent designs of numericalsecond-order methods, we allow StoSQP to adaptively select any random stepsize$\bar{\alpha}_t$, as long as $\beta_t\leq \bar{\alpha}_t \leq \beta_t+\chi_t$,for some control sequences $\beta_t$ and $\chi_t=o(\beta_t)$. To reduce thedominant computational cost of second-order methods, we additionally allowStoSQP to inexactly solve quadratic programs via efficient randomized iterativesolvers that utilize sketching techniques. Notably, we do not require theapproximation error to diminish as iteration proceeds. For the developedmethod, we show that under mild assumptions (i) computationally, it can take atmost $O(1/\epsilon^4)$ iterations (same as samples) to attain$\epsilon$-stationarity; (ii) statistically, its primal-dual sequence$1/\sqrt{\beta_t}\cdot (x_t - x^\star, \lambda_t - \lambda^\star)$ converges toa mean-zero Gaussian distribution with a nontrivial covariance matrix dependingon the underlying sketching distribution. Additionally, we establish thealmost-sure convergence rate of the iterate $(x_t, \lambda_t)$ along with theBerry-Esseen bound; the latter quantitatively measures the convergence rate ofthe distribution function. We analyze a plug-in limiting covariance matrixestimator, and demonstrate the performance of the method both on benchmarknonlinear problems in CUTEst test set and on linearly/nonlinearly constrainedregression problems.

Statistical Inference of Constrained Stochastic Optimization via Sketched Sequential Quadratic Programming