arXiv subjects

Soroosh Shafiee

Publications and source records attributed to Soroosh Shafiee.

17 recordsLinked to original sources

arXiv2026

Estimate then Predict: Convex Formulation for Travel Demand Forecasting

Travel demand forecasting is essential for evaluating large-scale infrastructure projects, yet the traditional sequential four-step process can produce inconsistencies across trip distribution, mode choice, and traffic assignment. Although combined models address these inconsistencies, their practical use has been limited by simplified behavioral assumptions, computational burden, and the lack of a unified parameter-estimation framework. We propose a convex programming approach that integrates destination, mode, and route choices within a hierarchical extended logit model. The optimal primal solution characterizes the joint travel-demand equilibrium, while the optimal dual variables recover taste coefficients, alternative-specific constants, and destination- and mode-level scale parameters. The model captures mode correlations through nested logit and route overlap through path-size logit. Its convex structure provides global optimality guarantees and enables efficient solution using off-the-shelf conic solvers. Observed travel patterns are incorporated through moment and conditional-entropy constraints, while the route-level dispersion parameter is calibrated separately using partial link counts. Numerical experiments on eight benchmark networks demonstrate the scalability and computational efficiency of the formulation.

cs.GT↗

arXiv2026

Risk-Averse Wasserstein Distributionally Robust Online Learning

We study distributionally robust online learning, where a risk-averse learner updates decisions sequentially to guard against worst-case distributions drawn from a Wasserstein ambiguity set centered at past observations. While this paradigm is well understood in the offline setting through Wasserstein Distributionally Robust Optimization (DRO), its online extension poses significant challenges in convergence. In this paper, we formulate the problem as an online saddle-point stochastic game between a decision maker and an adversary selecting worst-case distributions, and propose a general framework that converges to a robust Nash equilibrium coinciding with the solution of the corresponding offline Wasserstein DRO problem.

cs.LG↗

arXiv2026

Oracle-Based Distributionally Robust Optimization under Optimal Transport Ambiguity Sets

Distributionally robust optimization (DRO) with optimal transport ambiguity sets is traditionally solved by reformulating the minimax problem into a single-level convex program. While theoretically tractable, these reformulations introduce numerous auxiliary variables and demanding conic constraints that scale poorly in practice. In this paper, we address this challenge by reducing the inner worst-case expectation problem exactly to a scalar budget allocation task. This structural insight yields an efficient algorithm that bypasses large lifted reformulations, alongside a fast post-processing scheme to recover an optimal worst-case distribution supported on at most $N+1$ points, where $N$ denotes the sample size. We embed this procedure within an oracle-based distributional best-response framework to directly compute an approximate primal-dual solution to the overall DRO problem. Furthermore, we extend our analysis to the dual DRO formulation, proving the existence of a least-favorable distribution supported on at most $\min\{N+n+1, KN\}$ atoms, where $n$ and $K$ denote the decision dimension and number of loss components, respectively, and provide an efficient convex programming reduction to extract it from the solution of the primal DRO. Numerical experiments demonstrate that the proposed approach significantly outperforms state-of-the-art reformulation-based solvers.

math.OC↗

arXiv2026

Auto-Conditioned Frank-Wolfe Algorithms

Frank-Wolfe methods are projection-free algorithms for constrained optimization whose practical performance often depends critically on the choice of step size. Classical closed-loop step-size rules typically require prior knowledge of a global smoothness constant, while line-search variants avoid this requirement at the cost of additional function evaluations and implementation overhead. In this paper, we develop a fully auto-conditioned framework for Frank-Wolfe-type methods. The framework replaces the global Lipschitz constant in closed-loop step sizes with a local Lipschitz estimator computed from first-order information along the iterates. We show that this abstraction captures several important projection-free subroutines, including standard Frank-Wolfe, Matching Pursuit, pairwise Frank-Wolfe, and away-step Frank-Wolfe. For the resulting general class of methods, we establish convergence to stationary points in the nonconvex setting and recover the standard sublinear convergence guarantees in the convex setting, without requiring prior knowledge of a global smoothness constant. We further show that, when specialized to particular Frank-Wolfe variants and combined with additional structural assumptions, the same auto-conditioned framework yields accelerated convergence rates. Numerical experiments demonstrate that the proposed methods provide substantial practical improvements over line-search-based alternatives, highlighting the benefits of adapting to local curvature while retaining the simplicity of closed-loop step-size rules.

math.OC↗

arXiv2026

A Saddle Point Algorithm for Robust Data-Driven Factor Model Problems

We study the factor model problem, which aims to uncover low-dimensional structures in high-dimensional datasets. Adopting a robust data-driven approach, we formulate the problem as a saddle-point optimization. Our primary contribution is a first-order algorithm that solves this reformulation by leveraging a linear minimization oracle (LMO). We further develop semi-closed form solutions (up to a scalar) for three specific LMOs, corresponding to the Frobenius norm, Kullback-Leibler divergence, and Gelbrich (aka Wasserstein) distance. The analysis includes explicit quantification of these LMOs' regularity conditions, notably the Lipschitz constants of the dual function, which govern the algorithm's convergence performance. Numerical experiments confirm our method's effectiveness in high-dimensional settings, outperforming standard off-the-shelf optimization solvers.

math.OC↗

arXiv2026

Projection-Free Algorithms for Minimax Problems

This paper addresses constrained smooth saddle-point problems in settings where projection onto the feasible sets is computationally expensive. We bridge the gap between projection-based and projection-free optimization by introducing a unified dual dynamic smoothing framework that enables the design of efficient single-loop algorithms. Within this framework, we establish convergence results for nonconvex-concave and nonconvex-strongly concave settings. Furthermore, we show that this framework is naturally applicable to convex-concave problems. We propose and analyze three algorithmic variants based on the application of a linear minimization oracle over the minimization variable, the maximization variable, or both. Notably, our analysis yields anytime convergence guarantees without requiring a pre-specified iteration horizon. These results significantly narrow the performance gap between projection-free and projection-based methods for minimax optimization.

math.OC↗

arXiv2026

GPU-friendly and Linearly Convergent First-order Methods for Certifying Optimal $k$-sparse GLMs

We investigate the problem of certifying optimality for sparse generalized linear models (GLMs), where sparsity is enforced through a cardinality constraint. While Branch-and-Bound (BnB) frameworks can certify optimality using perspective relaxations, existing methods for solving these relaxations are computationally intensive, limiting their scalability. To address this challenge, we reformulate the relaxations as composite optimization problems and develop a unified proximal framework that is both linearly convergent and computationally efficient. Under specific geometric regularity conditions, our analysis links primal quadratic growth to dual quadratic decay, yielding error bounds that make the Fenchel duality gap a sharp proxy for progress towards the solution set. This leads to a duality gap-based restart scheme that upgrades a broad class of sublinear proximal methods to provably linearly convergent methods, and applies beyond the sparse GLM setting. For the implicit perspective regularizer, we further derive specialized routines to evaluate the regularizer and its proximal operator exactly in log-linear time, avoiding costly generic conic solvers. The resulting iterations are dominated by matrix--vector multiplications, which enables GPU acceleration. Experiments on synthetic and real-world datasets show orders-of-magnitude faster dual-bound computations and substantially improved BnB scalability on large instances.

math.OC↗

arXiv2025

Conditional Gradient Methods with Standard LMO for Stochastic Simple Bilevel Optimization

We propose efficient methods for solving stochastic simple bilevel optimization problems with convex inner levels, where the goal is to minimize an outer stochastic objective function subject to the solution set of an inner stochastic optimization problem. Existing methods often rely on costly projection or linear optimization oracles over complex sets, limiting their scalability. To overcome this, we propose an iteratively regularized conditional gradient approach that leverages linear optimization oracles exclusively over the base feasible set. Our proposed methods employ a vanishing regularization sequence that progressively emphasizes the inner problem while biasing towards desirable minimal outer objective solutions. In the one-sample stochastic setting and under standard convexity assumptions, we establish non-asymptotic convergence rates of $O(t^{-1/4})$ for both the outer and inner objectives. In the finite-sum setting with a mini-batch scheme, the corresponding rates become $O(t^{-1/2})$. When the outer objective is nonconvex, we prove non-asymptotic convergence rates of $O(t^{-1/7})$ for both the outer and inner objectives in the one-sample stochastic setting, and $O(t^{-1/4})$ in the finite-sum setting. Experimental results on over-parametrized regression and dictionary learning tasks demonstrate the practical advantages of our approach over existing methods, confirming our theoretical findings.

math.OC↗

arXiv2025

Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization

We study optimal transport-based distributionally robust optimization problems where a fictitious adversary, often envisioned as nature, can choose the distribution of the uncertain problem parameters by reshaping a prescribed reference distribution at a finite transportation cost. In this framework, we show that robustification is intimately related to various forms of variation and Lipschitz regularization even if the transportation cost function fails to be (some power of) a metric. We also derive conditions for the existence and the computability of a Nash equilibrium between the decision-maker and nature, and we demonstrate numerically that nature's Nash strategy can be viewed as a distribution that is supported on remarkably deceptive adversarial samples. Finally, we identify practically relevant classes of optimal transport-based distributionally robust optimization problems that can be addressed with efficient gradient descent algorithms even if the loss function or the transportation cost function are nonconvex (but not both at the same time).

math.OC↗

arXiv2025

Mean-Covariance Robust Risk Measurement

We introduce a universal framework for mean-covariance robust risk measurement and portfolio optimization. We model uncertainty in terms of the Gelbrich distance on the mean-covariance space, along with prior structural information about the population distribution. Our approach is related to the theory of optimal transport and exhibits superior statistical and computational properties than existing models. We find that, for a large class of risk measures, mean-covariance robust portfolio optimization boils down to the Markowitz model, subject to a regularization term given in closed form. This includes the finance standards, value-at-risk and conditional value-at-risk, and can be solved highly efficiently.

q-fin.PM↗

arXiv2025

Scalable First-order Method for Certifying Optimal k-Sparse GLMs

This paper investigates the problem of certifying optimality for sparse generalized linear models (GLMs), where sparsity is enforced through an $\ell_0$ cardinality constraint. While branch-and-bound (BnB) frameworks can certify optimality by pruning nodes using dual bounds, existing methods for computing these bounds are either computationally intensive or exhibit slow convergence, limiting their scalability to large-scale problems. To address this challenge, we propose a first-order proximal gradient algorithm designed to solve the perspective relaxation of the problem within a BnB framework. Specifically, we formulate the relaxed problem as a composite optimization problem and demonstrate that the proximal operator of the non-smooth component can be computed exactly in log-linear time complexity, eliminating the need to solve a computationally expensive second-order cone program. Furthermore, we introduce a simple restart strategy that enhances convergence speed while maintaining low per-iteration complexity. Extensive experiments on synthetic and real-world datasets show that our approach significantly accelerates dual bound computations and is highly effective in providing optimality certificates for large-scale problems.

cs.LG↗

arXiv2025

Distributionally Robust Optimization

Distributionally robust optimization (DRO) studies decision problems under uncertainty where the probability distribution governing the uncertain problem parameters is itself uncertain. A key component of any DRO model is its ambiguity set, that is, a family of probability distributions consistent with any available structural or statistical information. DRO seeks decisions that perform best under the worst distribution in the ambiguity set. This worst case criterion is supported by findings in psychology and neuroscience, which indicate that many decision-makers have a low tolerance for distributional ambiguity. DRO is rooted in statistics, operations research and control theory, and recent research has uncovered its deep connections to regularization techniques and adversarial training in machine learning. This survey presents the key findings of the field in a unified and self-contained manner.

math.OC↗

arXiv2025

Wasserstein Distributionally Robust Estimation in High Dimensions: Performance Analysis and Optimal Hyperparameter Tuning

Distributionally robust optimization (DRO) has become a powerful framework for estimation under uncertainty, offering strong out-of-sample performance and principled regularization. In this paper, we propose a DRO-based method for linear regression and address a central question: how to optimally choose the robustness radius, which controls the trade-off between robustness and accuracy. Focusing on high-dimensional settings where the dimension and the number of samples are both large and comparable in size, we employ tools from high-dimensional asymptotic statistics to precisely characterize the estimation error of the resulting estimator. Remarkably, this error can be recovered by solving a simple convex-concave optimization problem involving only four scalar variables. This characterization enables efficient selection of the radius that minimizes the estimation error. In doing so, it achieves the same effect as cross-validation, but at a fraction of the computational cost. Numerical experiments confirm that our theoretical predictions closely match empirical performance and that the optimal radius selected through our method aligns with that chosen by cross-validation, highlighting both the accuracy and the practical benefits of our approach.

stat.ML↗

arXiv2024

Robust Distribution Learning with Local and Global Adversarial Corruptions

We consider learning in an adversarial environment, where an $\varepsilon$-fraction of samples from a distribution $P$ are arbitrarily modified (global corruptions) and the remaining perturbations have average magnitude bounded by $ρ$ (local corruptions). Given access to $n$ such corrupted samples, we seek a computationally efficient estimator $\hat{P}_n$ that minimizes the Wasserstein distance $\mathsf{W}_1(\hat{P}_n,P)$. In fact, we attack the fine-grained task of minimizing $\mathsf{W}_1(Π_\# \hat{P}_n, Π_\# P)$ for all orthogonal projections $Π\in \mathbb{R}^{d \times d}$, with performance scaling with $\mathrm{rank}(Π) = k$. This allows us to account simultaneously for mean estimation ($k=1$), distribution estimation ($k=d$), as well as the settings interpolating between these two extremes. We characterize the optimal population-limit risk for this task and then develop an efficient finite-sample algorithm with error bounded by $\sqrt{\varepsilon k} + ρ+ \tilde{O}(d\sqrt{k}n^{-1/(k \lor 2)})$ when $P$ has bounded covariance. This guarantee holds uniformly in $k$ and is minimax optimal up to the sub-optimality of the plug-in estimator when $ρ= \varepsilon = 0$. Our efficient procedure relies on a novel trace norm approximation of an ideal yet intractable 2-Wasserstein projection estimator. We apply this algorithm to robust stochastic optimization, and, in the process, uncover a new method for overcoming the curse of dimensionality in Wasserstein distributionally robust optimization.

cs.LG↗

arXiv2024

A Robust Optimization Approach to Network Control Using Local Information Exchange

Designing policies for a network of agents is typically done by formulating an optimization problem where each agent has access to state measurements of all the other agents in the network. Such policy designs with centralized information exchange result in optimization problems that are typically hard to solve, require establishing substantial communication links, and do not promote privacy since all information is shared among the agents. Designing policies based on arbitrary communication structures can lead to non-convex optimization problems which are typically NP-hard. In this work, we propose an optimization framework for decentralized policy designs. In contrast to the centralized information exchange, our approach requires only local communication exchange among the neighboring agents matching the physical coupling of the network. Thus, each agent only requires information from its direct neighbors, minimizing the need for excessive communication and promoting privacy amongst the agents. Using robust optimization techniques, we formulate a convex optimization problem with a loosely coupled structure that can be solved efficiently. We numerically demonstrate the efficacy of the proposed approach in energy management and supply chain applications. We show that the proposed approach leads to solutions that closely approximate those obtained by the centralized formulation only at a fraction of the computational effort.

math.OC↗

arXiv2023

Constrained Optimization of Rank-One Functions with Indicator Variables

Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled with indicator variables for identifying the support of the continuous variables. In this paper we investigate compact extended formulations for such problems through perspective reformulation techniques. In contrast to the majority of previous work that relies on support function arguments and disjunctive programming techniques to provide convex hull results, we propose a constructive approach that exploits a hidden conic structure induced by perspective functions. To this end, we first establish a convex hull result for a general conic mixed-binary set in which each conic constraint involves a linear function of independent continuous variables and a set of binary variables. We then demonstrate that extended representations of sets associated with epigraphs of rank-one convex functions over constraints modeling indicator relations naturally admit such a conic representation. This enables us to systematically give perspective formulations for the convex hull descriptions of these sets with nonlinear separable or non-separable objective functions, sign constraints on continuous variables, and combinatorial constraints on indicator variables. We illustrate the efficacy of our results on sparse nonnegative logistic regression problems.

math.OC↗

arXiv2023

Outlier-Robust Wasserstein DRO

Distributionally robust optimization (DRO) is an effective approach for data-driven decision-making in the presence of uncertainty. Geometric uncertainty due to sampling or localized perturbations of data points is captured by Wasserstein DRO (WDRO), which seeks to learn a model that performs uniformly well over a Wasserstein ball centered around the observed data distribution. However, WDRO fails to account for non-geometric perturbations such as adversarial outliers, which can greatly distort the Wasserstein distance measurement and impede the learned model. We address this gap by proposing a novel outlier-robust WDRO framework for decision-making under both geometric (Wasserstein) perturbations and non-geometric (total variation (TV)) contamination that allows an $\varepsilon$-fraction of data to be arbitrarily corrupted. We design an uncertainty set using a certain robust Wasserstein ball that accounts for both perturbation types and derive minimax optimal excess risk bounds for this procedure that explicitly capture the Wasserstein and TV risks. We prove a strong duality result that enables tractable convex reformulations and efficient computation of our outlier-robust WDRO problem. When the loss function depends only on low-dimensional features of the data, we eliminate certain dimension dependencies from the risk bounds that are unavoidable in the general setting. Finally, we present experiments validating our theory on standard regression and classification tasks.

stat.ML↗