Using a similar reasoning as in the proof of Proposition 3.2, we can now substitute the linear

We will show that the optimal solutions to both these optimization problems are equal. Cp will each be of the form x avg ( x) . This fact will be useful later on. We will now verify that this assignment satisfies all the KKT conditions. Next, we verify primal feasibility.

Using a similar reasoning as in the proof of Proposition 3.2, we can now substitute the linear

Ensuring fairness in AI systems is critical, especially in high-stakes domains such as lending, hiring, and healthcare. This urgency is reflected in emerging global regulations that mandate fairness assessments and independent bias audits. However, procuring the necessary complete data for fairness testing remains a significant challenge. In industry settings, legal and privacy concerns restrict the collection of demographic data required to assess group disparities, and auditors face practical and cultural challenges in gaining access to data. In practice, data relevant for fairness testing is often split across separate sources: internal datasets held by institutions with predictive attributes, and external public datasets such as census data containing protected attributes, each providing only partial, marginal information. Our work seeks to leverage such available separate data to estimate model fairness when complete data is inaccessible. We propose utilising the available separate data to estimate a set of feasible joint distributions and then compute the set plausible fairness metrics. Through simulation and real experiments, we demonstrate that we can derive meaningful bounds on fairness metrics and obtain reliable estimates of the true metric. Our results demonstrate that this approach can serve as a practical and effective solution for fairness testing in real-world settings where access to complete data is restricted.

In this work, we explore online convex optimization (OCO) and introduce a new condition and analysis that provides fast rates by exploiting the curvature of feasible sets. In online linear optimization, it is known that if the average gradient of loss functions exceeds a certain threshold, the curvature of feasible sets can be exploited by the follow-the-leader (FTL) algorithm to achieve a logarithmic regret. This study reveals that algorithms adaptive to the curvature of loss functions can also leverage the curvature of feasible sets. In particular, we first prove that if an optimal decision is on the boundary of a feasible set and the gradient of an underlying loss function is non-zero, then the algorithm achieves a regret bound of O(\rho \log T) in stochastic environments. Here, \rho 0 is the radius of the smallest sphere that includes the optimal decision and encloses the feasible set.

We study a grouped bandit setting where each arm comprises multiple independent sub-arms referred to as attributes. Each attribute of each arm has an independent stochastic reward. We impose the constraint that for an arm to be deemed feasible, the mean reward of all its attributes should exceed a specified threshold. The goal is to find the arm with the highest mean reward averaged across attributes among the set of feasible arms in the fixed confidence setting. We first characterize a fundamental limit on the performance of any policy. Following this, we propose a near-optimal confidence interval-based policy to solve this problem and provide analytical guarantees for the policy. We compare the performance of the proposed policy with that of two suitably modified versions of action elimination via simulations.

We develop a class of mixed-integer formulations for disjunctive constraints intermediate to the big-M and convex hull formulations in terms of relaxation strength. The main idea is to capture the best of both the big-M and convex hull formulations: a computationally light formulation with a tight relaxation. The "P-split" formulations are based on a lifted transformation that splits convex additively separable constraints into P partitions and forms the convex hull of the linearized and partitioned disjunction. The "P-split" formulations are derived for disjunctive constraints with convex constraints within each disjuct, and we generalize the results for the case with nonconvex constraints within the disjuncts. We analyze the continuous relaxation of the P-split formulations and show that, under certain assumptions, the formulations form a hierarchy starting from a big-M equivalent and converging to the convex hull. The goal of the P-split formulations is to form strong approximations of the convex hull through a computationally simpler formulation. We computationally compare the P-split formulations against big-M and convex hull formulations on 344 test instances. The test problems include K-means clustering, semi-supervised clustering, P_ball problems, and optimization over trained ReLU neural networks. The computational results show promising potential of the P-split formulations. For many of the test problems, P-split formulations are solved with a similar number of explored nodes as the convex hull formulation, while reducing the solution time by an order of magnitude and outperforming big-M both in time and number of explored nodes.

Deep neural networks (DNNs) are emerging as a potential solution to solve NP-hard wireless resource allocation problems. However, in the presence of intricate constraints, e.g., users' quality-of-service (QoS) constraints, guaranteeing constraint satisfaction becomes a fundamental challenge. In this paper, we propose a novel unsupervised learning framework to solve the classical power control problem in a multi-user interference channel, where the objective is to maximize the network sumrate under users' minimum data rate or QoS requirements and power budget constraints. Utilizing a differentiable projection function, two novel deep learning (DL) solutions are pursued. The first is called Deep Implicit Projection Network (DIPNet), and the second is called Deep Explicit Projection Network (DEPNet). DIPNet utilizes a differentiable convex optimization layer to implicitly define a projection function. On the other hand, DEPNet uses an explicitly-defined projection function, which has an iterative nature and relies on a differentiable correction process. DIPNet requires convex constraints; whereas, the DEPNet does not require convexity and has a reduced computational complexity. To enhance the sum-rate performance of the proposed models even further, Frank-Wolfe algorithm (FW) has been applied to the output of the proposed models. Extensive simulations depict that the proposed DNN solutions not only improve the achievable data rate but also achieve zero constraint violation probability, compared to the existing DNNs. The proposed solutions outperform the classic optimization methods in terms of computation time complexity.

Probabilistic inference in pairwise Markov Random Fields (MRFs), i.e. computing the partition function or computing a MAP estimate of the variables, is a foundational problem in probabilistic graphical models. Semidefinite programming relaxations have long been a theoretically powerful tool for analyzing properties of probabilistic inference, but have not been practical owing to the high computational cost of typical solvers for solving the resulting SDPs. In this paper, we propose an efficient method for computing the partition function or MAP estimate in a pairwise MRF by instead exploiting a recently proposed coordinate-descent-based fast semidefinite solver. We also extend semidefinite relaxations from the typical binary MRF to the full multi-class setting, and develop a compact semidefinite relaxation that can again be solved efficiently using the solver. We show that the method substantially outperforms (both in terms of solution quality and speed) the existing state of the art in approximate inference, on benchmark problems drawn from previous work. We also show that our approach can scale to large MRF domains such as fully-connected pairwise CRF models used in computer vision.