On a Gradient Approach to Chebyshev Center Problems with Applications to Function Learning

We introduce $\textsf{gradOL}$, the first gradient-based optimization framework for solving Chebyshev center problems, a fundamental challenge in optimal function learning and geometric optimization. $\textsf{gradOL}$ hinges on reformulating the semi-infinite problem as a finitary max-min optimization, making it amenable to gradient-based techniques. By leveraging automatic differentiation for precise numerical gradient computation, $\textsf{gradOL}$ ensures numerical stability and scalability, making it suitable for large-scale settings. Under strong convexity of the ambient norm, $\textsf{gradOL}$ provably recovers optimal Chebyshev centers while directly computing the associated radius. This addresses a key bottleneck in constructing stable optimal interpolants. Empirically, $\textsf{gradOL}$ achieves significant improvements in accuracy and efficiency on 34 benchmark Chebyshev center problems from a benchmark $\textsf{CSIP}$ library. Moreover, we extend $\textsf{gradOL}$ to general convex semi-infinite programming (CSIP), attaining up to $4000\times$ speedups over the state-of-the-art $\texttt{SIPAMPL}$ solver tested on the indicated $\textsf{CSIP}$ library containing 67 benchmark problems. Furthermore, we provide the first theoretical foundation for applying gradient-based methods to Chebyshev center problems, bridging rigorous analysis with practical algorithms. $\textsf{gradOL}$ thus offers a unified solution framework for Chebyshev centers and broader CSIPs.

GGNNs : Generalizing GNNs using Residual Connections and Weighted Message Passing

Many real-world phenomena can be modeled as a graph, making them extremely valuable due to their ubiquitous presence. GNNs excel at capturing those relationships and patterns within these graphs, enabling effective learning and prediction tasks. GNNs are constructed using Multi-Layer Perceptrons (MLPs) and incorporate additional layers for message passing to facilitate the flow of features among nodes. It is commonly believed that the generalizing power of GNNs is attributed to the message-passing mechanism between layers, where nodes exchange information with their neighbors, enabling them to effectively capture and propagate information across the nodes of a graph. Our technique builds on these results, modifying the message-passing mechanism further: one by weighing the messages before accumulating at each node and another by adding Residual connections. These two mechanisms show significant improvements in learning and faster convergence

Agree To Disagree

How frequently do individuals thoroughly review terms and conditions before proceeding to register for a service, install software, or access a website? The majority of internet users do not engage in this practice. This trend is not surprising, given that terms and conditions typically consist of lengthy documents replete with intricate legal terminology and convoluted sentences. In this paper, we introduce a Machine Learning-powered approach designed to automatically parse and summarize critical information in a user-friendly manner. This technology focuses on distilling the pertinent details that users should contemplate before committing to an agreement.