We propose a two-stage deep learning framework for the inverse design of rectangular patch antennas. Our approach leverages generative modeling to learn a latent representation of antenna frequency response curves and conditions a subsequent generative model on these responses to produce feasible antenna geometries. We further demonstrate that leveraging search and optimization techniques at test-time improves the accuracy of the generated designs and enables consideration of auxiliary objectives such as manufacturability. Our approach generalizes naturally to different design criteria, and can be easily adapted to more complex geometric design spaces.

Stochastic smooth nonconvex minimax problems are prevalent in machine learning, e.g., GAN training, fair classification, and distributionally robust learning. Stochastic gradient descent ascent (GDA)-type methods are popular in practice due to their simplicity and single-loop nature. However, there is a significant gap between the theory and practice regarding high-probability complexity guarantees for these methods on stochastic nonconvex minimax problems. Existing high-probability bounds for GDA-type single-loop methods only apply to convex/concave minimax problems and to particular non-monotone variational inequality problems under some restrictive assumptions. In this work, we address this gap by providing the first high-probability complexity guarantees for nonconvex/PL minimax problems corresponding to a smooth function that satisfies the PL-condition in the dual variable.

The Column Subset Selection (CSS) problem has been widely studied in dimensionality reduction and feature selection. The goal of the CSS problem is to output a submatrix S, consisting of k columns from an n d input matrix A that minimizes the residual error ‖A-SS \dagger A‖_F 2, where S \dagger is the Moore-Penrose inverse matrix of S. Many previous approximation algorithms have non-linear running times in both n and d, while the existing linear-time algorithms have a relatively larger approximation ratios. Additionally, the local search algorithms in existing results for solving the CSS problem are heuristic. To achieve linear running time while maintaining better approximation using a local search strategy, we propose a local search-based approximation algorithm for the CSS problem with exactly k columns selected.

We consider the robust routing problem with uncertain travel times under the min-max regret criterion, which represents an extended and robust version of the classic traveling salesman problem (TSP) and vehicle routing problem (VRP). The general budget uncertainty set is employed to capture the uncertainty, which provides the capability to control the conservatism of obtained solutions and covers the commonly used interval uncertainty set as a special case. The goal is to obtain a robust solution that minimizes the maximum deviation from the optimal routing time in the worst-case scenario. Given the significant advancements and broad applications of neural combinatorial optimization methods in recent years, we present our initial attempt to combine neural approaches for solving this problem. We propose a dual multi-head cross attention mechanism to extract problem features represented by the inputted uncertainty sets.

Learning the unique directed acyclic graph corresponding to an unknown causal model is a challenging task. Methods based on functional causal models can identify a unique graph, but either suffer from the curse of dimensionality or impose strong parametric assumptions. To address these challenges, we propose a novel hybrid approach for global causal discovery in observational data that leverages local causal substructures. We first present a topological sorting algorithm that leverages ancestral relationships in linear structural causal models to establish a compact top-down hierarchical ordering, encoding more causal information than linear orderings produced by existing methods. We demonstrate that this approach generalizes to nonlinear settings with arbitrary noise.

We introduce a simple but general online learning framework in which a learner plays against an adversary in a vector-valued game that changes every round. Even though the learner's objective is not convex-concave (and so the minimax theorem does not apply), we give a simple algorithm that can compete with the setting in which the adversary must announce their action first, with optimally diminishing regret. We demonstrate the power of our framework by using it to (re)derive optimal bounds and efficient algorithms across a variety of domains, ranging from multicalibration to a large set of no-regret algorithms, to a variant of Blackwell's approachability theorem for polytopes with fast convergence rates. As a new application, we show how to (multi)calibeat'' an arbitrary collection of forecasters --- achieving an exponentially improved dependence on the number of models we are competing against, compared to prior work.

Local Bayesian optimization is a promising practical approach to solve the high dimensional black-box function optimization problem. Among them is the approximated gradient class of methods, which implements a strategy similar to gradient descent. These methods have achieved good experimental results and theoretical guarantees. However, given the distributional properties of the Gaussian processes applied on these methods, there may be potential to further exploit the information of the Gaussian processes to facilitate the BO search. In this work, we develop the relationship between the steps of the gradient descent method and one that minimizes the Upper Confidence Bound (UCB), and show that the latter can be a better strategy than direct gradient descent when a Gaussian process is applied as a surrogate.

We propose a novel piecewise stationary linear bandit (PSLB) model, where the environment randomly samples a context from an unknown probability distribution at each changepoint, and the quality of an arm is measured by its return averaged over all contexts. The contexts and their distribution, as well as the changepoints are unknown to the agent.We design Piecewise-Stationary \varepsilon -Best Arm Identification (PS \varepsilon BAI), an algorithm that is guaranteed to identify an \varepsilon -optimal arm with probability \ge 1-\delta and with a minimal number of samples.PS \varepsilon BAI consists of two subroutines, PS \varepsilon BAI and Naïve \varepsilon -BAI (N \varepsilon BAI), which are executed in parallel. PS \varepsilon BAI actively detects changepoints and aligns contexts to facilitate the arm identification process.When PS \varepsilon BAI and N \varepsilon BAI are utilized judiciously in parallel, PS \varepsilon BAI is shown to have a finite expected sample complexity. By proving a lower bound, we show the expected sample complexity of PS \varepsilon BAI is optimal up to a logarithmic factor.We compare PS \varepsilon BAI to baseline algorithms using numerical experiments which demonstrate its efficiency.Both our analytical and numerical results corroborate that the efficacy of PS \varepsilon BAI is due to the delicate change detection and context alignment procedures embedded in PS \varepsilon BAI.

Graph neural networks (GNNs) have been widely used to predict properties and heuristics of mixed-integer linear programs (MILPs) and hence accelerate MILP solvers. This paper investigates the capacity of GNNs to represent strong branching (SB), the most effective yet computationally expensive heuristic employed in the branch-and-bound algorithm. In the literature, message-passing GNN (MP-GNN), as the simplest GNN structure, is frequently used as a fast approximation of SB and we find that not all MILPs's SB can be represented with MP-GNN. We precisely define a class of "MP-tractable" MILPs for which MP-GNNs can accurately approximate SB scores. Particularly, we establish a universal approximation theorem: for any data distribution over the MP-tractable class, there always exists an MP-GNN that can approximate the SB score with arbitrarily high accuracy and arbitrarily high probability, which lays a theoretical foundation of the existing works on imitating SB with MP-GNN.

Combinatorial optimization problems are widespread but inherently challenging due to their discrete nature. The primary limitation of existing methods is that they can only access a small fraction of the solution space at each iteration, resulting in limited efficiency for searching the global optimal. To overcome this challenge, diverging from conventional efforts of expanding the solver's search scope, we focus on enabling information to actively propagate to the solver through heat diffusion. Utilizing heat diffusion, we propose a framework for solving general combinatorial optimization problems. The proposed methodology demonstrates superior performance across a range of the most challenging and widely encountered combinatorial optimizations.