arXiv subjects

Andrew E. B. Lim

Publications and source records attributed to Andrew E. B. Lim.

8 recordsLinked to original sources

arXiv2026

Posterior and Likelihood Sensitivity in Bayesian Distributionally Robust Optimization

We introduce the notion of worst-case posterior and worst-case likelihood sensitivity. These measure, respectively, the sensitivity of the expected cost to worst-case perturbations of the posterior distribution and worst-case perturbations of the likelihood of a Bayesian model. Each defines a quantitative measure of robustness. A decision maker concerned about the sensitivity of the out-of-sample expected cost to deviations from her assumptions will want a decision for which both sensitivities are small. We derive posterior and likelihood sensitivities for uncertainty sets defined in terms of deviation measures. Posterior sensitivity vanishes when the posterior variance shrinks to zero, which occurs when parameter uncertainty is eliminated from learning. Parameter learning does not eliminate likelihood sensitivity. A distributionally robust formulation of a Bayesian optimization problem makes a near-Pareto-optimal tradeoff between performance (expected cost) and robustness (posterior and likelihood sensitivity).

math.OC↗

arXiv2026

Robustness Measures in Distributionally Robust Optimization

Distributionally Robust Optimization (DRO) is a worst-case approach to decision making when there is model uncertainty. It is also well known that for certain uncertainty sets, DRO is approximated by a regularized nominal problem. We show that the regularizer is not just a penalty function but the worst-case sensitivity (WCS) of the expected cost with respect to deviations from the nominal model, giving it the interpretation of a robustness measure. This has substantial consequences for robust modeling. It shows that DRO is fundamentally a tradeoff between performance and robustness, where the robustness measure is determined by the uncertainty set. The robustness measure reveals properties of a cost distribution that affect sensitivity to misspecification. This leads to a systematic approach to selecting uncertainty sets. The family of DRO solutions obtained by varying the size of the uncertainty set traces a near Pareto-optimal performance--robustness frontier that can be used to select its size. The frontier identifies problem instances where the price of robustness is high and provides insight into effective ways of redesigning the system to reduce this cost. We derive WCS for a collection of commonly used uncertainty sets, and illustrate these ideas in a number of applications.

math.OC↗

arXiv2025

Dynamic Factor Models with Forward-Looking Views

Prediction models calibrated using historical data may forecast poorly if the dynamics of the present and future differ from observations in the past. For this reason, predictions can be improved if information like forward looking views about the state of the system are used to refine the forecast. We develop an approach for combining a dynamic factor model for risky asset prices calibrated on historical data, and noisy expert views of future values of the factors/covariates in the model, and study the implications for dynamic portfolio choice. By exploiting the graphical structure linking factors, asset prices, and views, we derive closed-form expressions for the dynamics of the factor and price processes after conditioning on the views. For linear factor models, the price process becomes a time-inhomogeneous affine process with a new covariate formed from the views. We establish a novel theoretical connection between the conditional factor process and a process we call a Mean-Reverting Bridge (MrB), an extension of the classical Brownian bridge. We derive the investor's optimal portfolio strategy and show that views influence both the myopic mean-variance term and the intertemporal hedge. The optimal dynamic portfolio when the long-run mean of the expected return is uncertain and learned online from data is also derived. More generally, our framework offers a generalizable approach for embedding forward-looking information about covariates in a dynamic factor model.

math.OC↗

arXiv2024

Partial Backorder Inventory System: Asymptotic Optimality and Demand Learning

We develop a stochastic inventory system which accounts for the limited patience of backlogged customers. While limited patience is a feature that is closer to the nature of unmet demand, our model also unifies the classic backlogging and lost-sales inventory systems which are special cases of the one we propose. We establish the uniform (asymptotic) optimality of the base-stock policy when both demand and patience distributions are known. When the backlogged demands become unobservable, we introduce a novel policy family that operates without backlogged demands information, and prove that it can approach the cost efficiency of the optimal policy in the system when the demand and patience distributions are known. Finally, we consider an online inventory control problem in which backlogged demand is unobservable and demand and patience distributions are also not known, and develop a UCB-type algorithm that yields a near-optimal policy. The regret bounds given by the algorithm are provably tight within the planning horizon, and are comparable to the state-of-the-art results in the literature, even in the face of partial and biased observations and weaker system ergodicity.

math.OC↗

arXiv2023

A data-driven approach to beating SAA out-of-sample

While solutions of Distributionally Robust Optimization (DRO) problems can sometimes have a higher out-of-sample expected reward than the Sample Average Approximation (SAA), there is no guarantee. In this paper, we introduce a class of Distributionally Optimistic Optimization (DOO) models, and show that it is always possible to ``beat" SAA out-of-sample if we consider not just worst-case (DRO) models but also best-case (DOO) ones. We also show, however, that this comes at a cost: Optimistic solutions are more sensitive to model error than either worst-case or SAA optimizers, and hence are less robust and calibrating the worst- or best-case model to outperform SAA may be difficult when data is limited.

math.OC↗

arXiv2020

Worst-case sensitivity

We introduce the notion of Worst-Case Sensitivity, defined as the worst-case rate of increase in the expected cost of a Distributionally Robust Optimization (DRO) model when the size of the uncertainty set vanishes. We show that worst-case sensitivity is a Generalized Measure of Deviation and that a large class of DRO models are essentially mean-(worst-case) sensitivity problems when uncertainty sets are small, unifying recent results on the relationship between DRO and regularized empirical optimization with worst-case sensitivity playing the role of the regularizer. More generally, DRO solutions can be sensitive to the family and size of the uncertainty set, and reflect the properties of its worst-case sensitivity. We derive closed-form expressions of worst-case sensitivity for well known uncertainty sets including smooth $ϕ$-divergence, total variation, "budgeted" uncertainty sets, uncertainty sets corresponding to a convex combination of expected value and CVaR, and the Wasserstein metric. These can be used to select the uncertainty set and its size for a given application.

econ.EM↗

arXiv2020

Calibration of Distributionally Robust Empirical Optimization Models

We study the out-of-sample properties of robust empirical optimization problems with smooth $ϕ$-divergence penalties and smooth concave objective functions, and develop a theory for data-driven calibration of the non-negative "robustness parameter" $δ$ that controls the size of the deviations from the nominal model. Building on the intuition that robust optimization reduces the sensitivity of the expected reward to errors in the model by controlling the spread of the reward distribution, we show that the first-order benefit of ``little bit of robustness" (i.e., $δ$ small, positive) is a significant reduction in the variance of the out-of-sample reward while the corresponding impact on the mean is almost an order of magnitude smaller. One implication is that substantial variance (sensitivity) reduction is possible at little cost if the robustness parameter is properly calibrated. To this end, we introduce the notion of a robust mean-variance frontier to select the robustness parameter and show that it can be approximated using resampling methods like the bootstrap. Our examples show that robust solutions resulting from "open loop" calibration methods (e.g., selecting a $90\%$ confidence level regardless of the data and objective function) can be very conservative out-of-sample, while those corresponding to the robustness parameter that optimizes an estimate of the out-of-sample expected reward (e.g., via the bootstrap) with no regard for the variance are often insufficiently robust.

stat.ML↗

arXiv2012

Performance-based regularization in mean-CVaR portfolio optimization

We introduce performance-based regularization (PBR), a new approach to addressing estimation risk in data-driven optimization, to mean-CVaR portfolio optimization. We assume the available log-return data is iid, and detail the approach for two cases: nonparametric and parametric (the log-return distribution belongs in the elliptical family). The nonparametric PBR method penalizes portfolios with large variability in mean and CVaR estimations. The parametric PBR method solves the empirical Markowitz problem instead of the empirical mean-CVaR problem, as the solutions of the Markowitz and mean-CVaR problems are equivalent when the log-return distribution is elliptical. We derive the asymptotic behavior of the nonparametric PBR solution, which leads to insight into the effect of penalization, and justification of the parametric PBR method. We also show via simulations that the PBR methods produce efficient frontiers that are, on average, closer to the population efficient frontier than the empirical approach to the mean-CVaR problem, with less variability.

q-fin.PM↗