This paper presents a novel approach to multiobjective algorithms aimed at modeling the Pareto set using neural networks. Whereas previous methods mainly focused on identifying a finite number of solutions, our approach allows for the direct modeling of the entire Pareto set. Furthermore, we establish an equivalence between learning the complete Pareto set and maximizing the associated hypervolume, which enables the convergence analysis of hypervolume (as a new metric) for Pareto set learning. Specifically, our new analysis framework reveals the connection between the learned Pareto solution and its representation in a polar coordinate system. We evaluate our proposed approach on various benchmark problems and real-world problems, and the encouraging results make it a potentially viable alternative to existing multiobjective algorithms.

Practitioners often navigate LLM performance trade-offs by plotting Pareto frontiers of optimal accuracy-cost trade-offs. However, this approach offers no way to compare between LLMs with distinct strengths and weaknesses: for example, a cheap, error-prone model vs a pricey but accurate one. To address this gap, we propose economic evaluation of LLMs. Our framework quantifies the performance trade-off of an LLM as a single number based on the economic constraints of a concrete use case, all expressed in dollars: the cost of making a mistake, the cost of incremental latency, and the cost of abstaining from a query. We apply our economic evaluation framework to compare the performance of reasoning and non-reasoning models on difficult questions from the MATH benchmark, discovering that reasoning models offer better accuracy-cost tradeoffs as soon as the economic cost of a mistake exceeds \$0.01. In addition, we find that single large LLMs often outperform cascades when the cost of making a mistake is as low as \$0.1. Overall, our findings suggest that when automating meaningful human tasks with AI models, practitioners should typically use the most powerful available model, rather than attempt to minimize AI deployment costs, since deployment costs are likely dwarfed by the economic impact of AI errors.

In this paper, we introduce a simple yet effective reward dimension reduction method to tackle the scalability challenges of multi-objective reinforcement learning algorithms. While most existing approaches focus on optimizing two to four objectives, their abilities to scale to environments with more objectives remain uncertain. Our method uses a dimension reduction approach to enhance learning efficiency and policy performance in multi-objective settings. While most traditional dimension reduction methods are designed for static datasets, our approach is tailored for online learning and preserves Pareto-optimality after transformation. We propose a new training and evaluation framework for reward dimension reduction in multi-objective reinforcement learning and demonstrate the superiority of our method in environments including one with sixteen objectives, significantly outperforming existing online dimension reduction methods.

This paper compares Julia reduction and hyperbolic reduction with the aim of finding equivalent binary forms with minimal coefficients. We demonstrate that hyperbolic reduction generally outperforms Julia reduction, particularly in the cases of sextics and decimics, though neither method guarantees achieving the minimal form. We further propose an additional shift and scaling to approximate the minimal form more closely. Finally, we introduce a machine learning framework to identify optimal transformations that minimize the heights of binary forms. This study provides new insights into the geometry and algebra of binary forms and highlights the potential of AI in advancing symbolic computation and reduction techniques. The findings, supported by extensive computational experiments, lay the groundwork for hybrid approaches that integrate traditional reduction methods with data-driven techniques.

This project embarks on a journey to merge the abstract realm of Galois theory with the practical capabilities of machine learning This paper introduces a novel approach to understanding Galois (ML). Our goal is to harness ML's pattern recognition and prediction theory, one of the foundational areas of algebra, through the lens of abilities to address some of the most challenging aspects of Galois machine learning. By analyzing polynomial equations with machine theory, potentially revolutionizing our understanding and approach learning techniques, we aim to streamline the process of determining to polynomial solvability and related problems.

Feature extraction techniques are crucial in medical image classification; however, classical feature extractors in addition to traditional machine learning classifiers often exhibit significant limitations in providing sufficient discriminative information for complex image sets. While Convolutional Neural Networks (CNNs) and Vision Transformer (ViT) have shown promise in feature extraction, they are prone to overfitting due to the inherent characteristics of medical imaging data, including small sample sizes or high intra-class variance. In this work, the Medical Image Attention-based Feature Extractor (MIAFEx) is proposed, a novel method that employs a learnable refinement mechanism to enhance the classification token within the Transformer encoder architecture. This mechanism adjusts the token based on learned weights, improving the extraction of salient features and enhancing the model's adaptability to the challenges presented by medical imaging data. The MIAFEx output features quality is compared against classical feature extractors using traditional and hybrid classifiers. Also, the performance of these features is compared against modern CNN and ViT models in classification tasks, demonstrating its superiority in accuracy and robustness across multiple complex classification medical imaging datasets. This advantage is particularly pronounced in scenarios with limited training data, where traditional and modern models often struggle to generalize effectively. The source code of this proposal can be found at https://github.com/Oscar-RamosS/Medical-Image-Attention-based-Feature-Extractor-MIAFEx

We propose three completely data-driven methods for estimating the real cohomology groups $H^k (X ; \mathbb{R})$ of a compact metric-measure space $(X, d_X, \mu_X)$ embedded in a metric-measure space $(Y,d_Y,\mu_Y)$, given a finite set of points $S$ sampled from a uniform distrbution $\mu_X$ on $X$, possibly corrupted with noise from $Y$. We present the results of several computational experiments in the case that $X$ is embedded in $\mathbb{R}^n$, where two of the three algorithms performed well.