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Motivated by applications in emergency response and experimental design, we consider smooth stochastic optimization problems over probability measures supported on compact subsets of the Euclidean space. With the influence function as the variational object, we construct a deterministic Frank-Wolfe (dFW) recursion for probability spaces, made especially possible by a lemma that identifies a ``closed-form'' solution to the infinite-dimensional Frank-Wolfe sub-problem. Each iterate in dFW is expressed as a convex combination of the incumbent iterate and a Dirac measure concentrating on the minimum of the influence function at the incumbent iterate. To address common application contexts that have access only to Monte Carlo observations of the objective and influence function, we construct a stochastic Frank-Wolfe (sFW) variation that generates a random sequence of probability measures constructed using minima of increasingly accurate estimates of the influence function. We demonstrate that sFW's optimality gap sequence exhibits $O(k^{-1})$ iteration complexity almost surely and in expectation for smooth convex objectives, and $O(k^{-1/2})$ (in Frank-Wolfe gap) for smooth non-convex objectives. Furthermore, we show that an easy-to-implement fixed-step, fixed-sample version of (sFW) exhibits exponential convergence to $\varepsilon$-optimality. We end with a central limit theorem on the observed objective values at the sequence of generated random measures. To further intuition, we include several illustrative examples with exact influence function calculations.

Model update (MU) and candidate evaluation (CE) are classical steps incorporated inside many stochastic trust-region (TR) algorithms. The sampling effort exerted within these steps, often decided with the aim of controlling model error, largely determines a stochastic TR algorithm's sample complexity. Given that MU and CE are amenable to variance reduction, we investigate the effect of incorporating common random numbers (CRN) within MU and CE on complexity. Using ASTRO and ASTRO-DF as prototype first-order and zeroth-order families of algorithms, we demonstrate that CRN's effectiveness leads to a range of complexities depending on sample-path regularity and the oracle order. For instance, we find that in first-order oracle settings with smooth sample paths, CRN's effect is pronounced -- ASTRO with CRN achieves $\tilde{O}(\epsilon^{-2})$ a.s. sample complexity compared to $\tilde{O}(\epsilon^{-6})$ a.s. in the generic no-CRN setting. By contrast, CRN's effect is muted when the sample paths are not Lipschitz, with the sample complexity improving from $\tilde{O}(\epsilon^{-6})$ a.s. to $\tilde{O}(\epsilon^{-5})$ and $\tilde{O}(\epsilon^{-4})$ a.s. in the zeroth- and first-order settings, respectively. Since our results imply that improvements in complexity are largely inherited from generic aspects of variance reduction, e.g., finite-differencing for zeroth-order settings and sample-path smoothness for first-order settings within MU, we anticipate similar trends in other contexts.

We present batching as an omnibus device for uncertainty quantification using simulation output. We consider the classical context of a simulationist performing uncertainty quantification on an estimator $\theta_n$ (of an unknown fixed quantity $\theta$) using only the output data $(Y_1,Y_2,\ldots,Y_n)$ gathered from a simulation. By uncertainty quantification, we mean approximating the sampling distribution of the error $\theta_n-\theta$ toward: (A) estimating an assessment functional $\psi$, e.g., bias, variance, or quantile; or (B) constructing a $(1-\alpha)$-confidence region on $\theta$. We argue that batching is a remarkably simple and effective device for this purpose, and is especially suited for handling dependent output data such as what one frequently encounters in simulation contexts. We demonstrate that if the number of batches and the extent of their overlap are chosen appropriately, batching retains bootstrap's attractive theoretical properties of strong consistency and higher-order accuracy. For constructing confidence regions, we characterize two limiting distributions associated with a Studentized statistic. Our extensive numerical experience confirms theoretical insight, especially about the effects of batch size and batch overlap.

We propose a general purpose confidence interval procedure (CIP) for statistical functionals constructed using data from a stationary time series. The procedures we propose are based on derived distribution-free analogues of the $\chi^2$ and Student's $t$ random variables for the statistical functional context, and hence apply in a wide variety of settings including quantile estimation, gradient estimation, M-estimation, CVAR-estimation, and arrival process rate estimation, apart from more traditional statistical settings. Like the method of subsampling, we use overlapping batches of time series data to estimate the underlying variance parameter; unlike subsampling and the bootstrap, however, we assume that the implied point estimator of the statistical functional obeys a central limit theorem (CLT) to help identify the weak asymptotics (called OB-x limits, x=I,II,III) of batched Studentized statistics. The OB-x limits, certain functionals of the Wiener process parameterized by the size of the batches and the extent of their overlap, form the essential machinery for characterizing dependence, and consequently the correctness of the proposed CIPs. The message from extensive numerical experimentation is that in settings where a functional CLT on the point estimator is in effect, using \emph{large overlapping batches} alongside OB-x critical values yields confidence intervals that are often of significantly higher quality than those obtained from more generic methods like subsampling or the bootstrap. We illustrate using examples from CVaR estimation, ARMA parameter estimation, and NHPP rate estimation; R and MATLAB code for OB-x critical values is available at~\texttt{web.ics.purdue.edu/~pasupath/}., Comment: 43 pages, 4 figures

Stochastic Gradient (SG) is the defacto iterative technique to solve stochastic optimization (SO) problems with a smooth (non-convex) objective $f$ and a stochastic first-order oracle. SG's attractiveness is due in part to its simplicity of executing a single step along the negative subsampled gradient direction to update the incumbent iterate. In this paper, we question SG's choice of executing a single step as opposed to multiple steps between subsample updates. Our investigation leads naturally to generalizing SG into Retrospective Approximation (RA) where, during each iteration, a "deterministic solver" executes possibly multiple steps on a subsampled deterministic problem and stops when further solving is deemed unnecessary from the standpoint of statistical efficiency. RA thus rigorizes what is appealing for implementation -- during each iteration, "plug in" a solver, e.g., L-BFGS line search or Newton-CG, as is, and solve only to the extent necessary. We develop a complete theory using relative error of the observed gradients as the principal object, demonstrating that almost sure and $L_1$ consistency of RA are preserved under especially weak conditions when sample sizes are increased at appropriate rates. We also characterize the iteration and oracle complexity (for linear and sub-linear solvers) of RA, and identify a practical termination criterion leading to optimal complexity rates. To subsume non-convex $f$, we present a certain "random central limit theorem" that incorporates the effect of curvature across all first-order critical points, demonstrating that the asymptotic behavior is described by a certain mixture of normals. The message from our numerical experiments is that the ability of RA to incorporate existing second-order deterministic solvers in a strategic manner might be important from the standpoint of dispensing with hyper-parameter tuning.

We present adaptive sequential SAA (sample average approximation) algorithms to solve large-scale two-stage stochastic linear programs. The iterative algorithm framework we propose is organized into \emph{outer} and \emph{inner} iterations as follows: during each outer iteration, a sample-path problem is implicitly generated using a sample of observations or ``scenarios," and solved only \emph{imprecisely}, to within a tolerance that is chosen \emph{adaptively}, by balancing the estimated statistical error against solution error. The solutions from prior iterations serve as \emph{warm starts} to aid efficient solution of the (piecewise linear convex) sample-path optimization problems generated on subsequent iterations. The generated scenarios can be independent and identically distributed (iid), or dependent, as in Monte Carlo generation using Latin-hypercube sampling, antithetic variates, or randomized quasi-Monte Carlo. We first characterize the almost-sure convergence (and convergence in mean) of the optimality gap and the distance of the generated stochastic iterates to the true solution set. We then characterize the corresponding iteration complexity and work complexity rates as a function of the sample size schedule, demonstrating that the best achievable work complexity rate is Monte Carlo canonical and analogous to the generic $\mathcal{O}(\epsilon^{-2})$ optimal complexity for non-smooth convex optimization. We report extensive numerical tests that indicate favorable performance, due primarily to the use of a sequential framework with an optimal sample size schedule, and the use of warm starts. The proposed algorithm can be stopped in finite-time to return a solution endowed with a probabilistic guarantee on quality.

We consider a single stage stochastic program without recourse with a strictly convex loss function. We assume a compact decision space and grid it with a finite set of points. In addition, we assume that the decision maker can generate samples of the stochastic variable independently at each grid point and form a sample average approximation (SAA) of the stochastic program. Our objective in this paper is to characterize an asymptotically optimal linear sample allocation rule, given a fixed sampling budget, which maximizes the decay rate of probability of making false decision.

We quantify the large deviations of Gaussian extreme value statistics on closed convex sets in d-dimensional Euclidean space. The asymptotics imply that the extreme value distribution exhibits a rate function that is a simple quadratic function of a unique "dominating point" located on the boundary of the convex set. Furthermore, the dominating point is identified as the optimizer of a certain convex quadratic programming problem, indicating a "collusion" between the dependence structure of the Gaussian random vectors and the geometry of the convex set in determining the asymptotics. We specialize our main result to polyhedral sets which appear frequently in other contexts involving logarithmic asymptotics. We also extend the main result to characterize the large deviations of Gaussian-mixture extreme value statistics on general convex sets. Our results have implications to contexts arising in rare-event probability estimation and stochastic optimization, since the nature of the dominating point and the rate function suggest importance sampling measures.

We study the optimal placement problem of a stock trader who wishes to clear his/her inventory by a predetermined time horizon t, by using a limit order or a market order. For a diffusive market, we characterize the optimal limit order placement policy and analyze its behavior under different market conditions. In particular, we show that, in the presence of a negative drift, there exists a critical time t0>0 such that, for any time horizon t>t0, there exists an optimal placement, which, contrary to earlier work, is different from one that is placed "infinitesimally" close to the best ask, such as the best bid and second best bid. We also propose a simple method to approximate the critical time t0 and the optimal order placement.

We consider unconstrained optimization problems where only "stochastic" estimates of the objective function are observable as replicates from a Monte Carlo oracle. The Monte Carlo oracle is assumed to provide no direct observations of the function gradient. We present ASTRO-DF --- a class of derivative-free trust-region algorithms, where a stochastic local interpolation model is constructed, optimized, and updated iteratively. Function estimation and model construction within ASTRO-DF is adaptive in the sense that the extent of Monte Carlo sampling is determined by continuously monitoring and balancing metrics of sampling error (or variance) and structural error (or model bias) within ASTRO-DF. Such balancing of errors is designed to ensure that Monte Carlo effort within ASTRO-DF is sensitive to algorithm trajectory, sampling more whenever an iterate is inferred to be close to a critical point and less when far away. We demonstrate the almost-sure convergence of ASTRO-DF's iterates to a first-order critical point when using linear or quadratic stochastic interpolation models. The question of using more complicated models, e.g., regression or stochastic kriging, in combination with adaptive sampling is worth further investigation and will benefit from the methods of proof presented here. We speculate that ASTRO-DF's iterates achieve the canonical Monte Carlo convergence rate, although a proof remains elusive., Comment: 27 pages, no figures

We consider the question of efficient estimation in the tails of Gaussian copulas. Our special focus is estimating expectations over multi-dimensional constrained sets that have a small implied measure under the Gaussian copula. We propose three estimators, all of which rely on a simple idea: identify certain \emph{dominating} point(s) of the feasible set, and appropriately shift and scale an exponential distribution for subsequent use within an importance sampling measure. As we show, the efficiency of such estimators depends crucially on the local structure of the feasible set around the dominating points. The first of our proposed estimators $\estOpt$ is the "full-information" estimator that actively exploits such local structure to achieve bounded relative error in Gaussian settings. The second and third estimators $\estExp$, $\estLap$ are "partial-information" estimators, for use when complete information about the constraint set is not available, they do not exhibit bounded relative error but are shown to achieve polynomial efficiency. We provide sharp asymptotics for all three estimators. For the NORTA setting where no ready information about the dominating points or the feasible set structure is assumed, we construct a multinomial mixture of the partial-information estimator $\estLap$ resulting in a fourth estimator $\estNt$ with polynomial efficiency, and implementable through the ecoNORTA algorithm. Numerical results on various example problems are remarkable, and consistent with theory.