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.

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.

We propose a pair of completely data-driven algorithms for unsupervised classification and dimension reduction, and we empirically study their performance on a number of data sets, both simulated data in three-dimensions and images from the COIL-20 data set. The algorithms take as input a set of points sampled from a uniform distribution supported on a metric space, the latter embedded in an ambient metric space, and they output a clustering or reduction of dimension of the data. They work by constructing a natural family of graphs from the data and selecting the graph which maximizes the relative von Neumann entropy of certain normalized heat operators constructed from the graphs. Once the appropriate graph is selected, the eigenvectors of the graph Laplacian may be used to reduce the dimension of the data, and clusters in the data may be identified with the kernel of the associated graph Laplacian. Notably, these algorithms do not require information about the size of a neighborhood or the desired number of clusters as input, in contrast to popular algorithms such as $k$-means, and even more modern spectral methods such as Laplacian eigenmaps, among others. In our computational experiments, our clustering algorithm outperforms $k$-means clustering on data sets with non-trivial geometry and topology, in particular data whose clusters are not concentrated around a specific point, and our dimension reduction algorithm is shown to work well in several simple examples.