This paper deals with the problem of quantifying impact model misspecification when computing general expected values interest. The methodology that we propose is applicable in great generality; particular, provide examples involving path-dependent expectations stochastic processes. Our approach consists bounds for expectation interest regardless probability measure used, as long lies within a prescribed tolerance measured terms flexible class distances from suitable baseline model. These...

We show that several machine learning estimators, including square-root LASSO (Least Absolute Shrinkage and Selection) regularized logistic regression can be represented as solutions to distributionally robust optimization (DRO) problems. The associated uncertainty regions are based on suitably defined Wasserstein distances. Hence, our representations allow us view regularization a result of introducing an artificial adversary perturbs the empirical distribution account for out-of-sample...

Recently, (Blanchet, Kang, and Murhy 2016, Blanchet, Kang 2017) showed that several machine learning algorithms, such as square-root Lasso, Support Vector Machines, regularized logistic regression, among many others, can be represented exactly distributionally robust optimization (DRO) problems. The distributional uncertainty is defined a neighborhood centered at the empirical distribution. We propose methodology which learns in natural data-driven way. show rigorously our framework...

We consider optimal transport based distributionally robust optimization (DRO) problems with locally strongly convex cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about value policy, worst-case adversarial model. These expose a rich structure embedded in DRO problem (e.g. strong even if non-DRO was not convex, suitable scaling of Lagrangian for constraint, etc. which are crucial design...

Summary Estimators based on Wasserstein distributionally robust optimization are obtained as solutions of min-max problems in which the statistician selects a parameter minimizing worst-case loss among all probability models within certain distance from underlying empirical measure sense. While motivated by need to identify optimal model parameters or decision choices that misspecification, these estimators recover wide range regularized estimators, including square-root lasso and support...

We consider optimal transport-based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about value policy, worst-case adversarial model. These expose a rich structure embedded in DRO problem (e.g., strong even if non-DRO is not convex, suitable scaling of Lagrangian for constraint, etc., which are crucial...

This paper deals with the problem of quantifying impact model misspecification when computing general expected values interest. The methodology that we propose is applicable in great generality, particular, provide examples involving path-dependent expectations stochastic processes. Our approach consists bounds for expectation interest regardless probability measure used, as long lies within a prescribed tolerance measured terms flexible class distances from suitable baseline model. These...

We consider statistical methods which invoke a min-max distributionally robust formulation to extract good out-of-sample performance in data-driven optimization and learning problems. Acknowledging the distributional uncertainty from limited samples, formulations introduce an adversarial inner player explore unseen covariate data. The resulting Distributionally Robust Optimization (DRO) formulations, include Wasserstein DRO (our main focus), are specified using optimal transportation...

We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) level crossing and 3) within a regenerative cycle. Exponential twisting state-independent methods, which are effective in efficiently estimating these light-tailed increments not applicable when heavy-tailed. To address latter case, more complex...

Wasserstein distributionally robust optimization estimators are obtained as solutions of min-max problems in which the statistician selects a parameter minimizing worst-case loss among all probability models within certain distance (in sense) from underlying empirical measure. While motivated by need to identify optimal model parameters or decision choices that misspecification, these recover wide range regularized estimators, including square-root lasso and support vector machines, others,...

Abstract We present the first exact simulation method for multidimensional reflected Brownian motion (RBM). Exact in this setting is challenging because of presence correlated local-time-like terms definition RBM. apply recently developed so-called ε-strong techniques (also known as tolerance-enforced simulation) which allow us to provide a piecewise linear approximation RBM with ε (deterministic) error uniform norm. A novel conditional acceptance–rejection step then used eliminate error. In...

Scalable and efficient importance sampling for managing tail risks As the models employed in realm of risk analytics optimization become increasingly sophisticated, it is crucial that management tools, such as variance reduction techniques, are typically designed stylized on a case by basis evolve to scale well gain broader applicability. In paper titled “Achieving efficiency black-box simulation distribution tails with self-structuring samplers,” authors take step toward this goal...

Given data on choices made by consumers for different assortments, a key challenge is to develop parsimonious models that describe and predict consumer choice behavior. One such model the marginal distribution (MDM), which requires only specification of distributions random utilities alternatives explain data.

Motivated by the prominence of Conditional Value-at-Risk (CVaR) as a measure for tail risk in settings affected uncertainty, we develop new formula approximating CVaR based optimization objectives and their gradients from limited samples. Unlike state-of-the-art sample average approximations which require impractically large amounts data probability regions, proposed approximation scheme exploits self-similarity heavy-tailed distributions to extrapolate suitable lower quantiles. The...

This paper considers efficient Importance Sampling (IS) for the estimation of tail risks a loss defined in terms sophisticated object such as machine learning predictor or mixed-integer linear optimization formulation. Assuming only black-box access to and distribution underlying random vector, presents an IS algorithm estimating Value at Risk Conditional Risk. The key challenge any procedure, namely, identifying appropriate change-of-measure, is automated with self-structuring...

Efficient simulation of rare events involving sums heavy-tailed random variables has been an active research area in applied probability over the last fifteen years. These problems are viewed as challenging, since large deviations theory inspired and exponential twisting based importance samplin

We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d.regularly varying increments; namely, 1) the large deviation probabilities, 2) level crossing and 3) within a regenerative cycle.Exponential twisting state-independent methods, which are effective in efficiently estimating these light-tailed increments not applicable when heavy-tailed.To address latter case, more complex...