This objective is similar to Maximum Entropy Correlated Equilibrium (MECE) [48], and the proofs here are similar to the framework set out there. A drawback of MECE is that it is not easy to determine the minimum p permissible. If we choose p that does not permit a valid solution, then the parameters will diverge. We can circumvent this problem by optimizing the distance to a target ˆ p. And µis for balancing the linear objective.

In this section, we provide an open-ended discussion on how the prior distribution ( i.e., imbalanced

Again, we supposept,u1,,uq are obtained from the process in SectionB.3.3.

3D human meshes show a natural hierarchical structure (like torso-limbs-fingers). But existing video-based 3D human mesh recovery methods usually learn mesh features in Euclidean space. It's hard to catch this hierarchical structure accurately. So wrong human meshes are reconstructed. To solve this problem, we propose a hyperbolic space learning method leveraging temporal motion prior for recovering 3D human meshes from videos. First, we design a temporal motion prior extraction module. This module extracts the temporal motion features from the input 3D pose sequences and image feature sequences respectively. Then it combines them into the temporal motion prior. In this way, it can strengthen the ability to express features in the temporal motion dimension. Since data representation in non-Euclidean space has been proved to effectively capture hierarchical relationships in real-world datasets (especially in hyperbolic space), we further design a hyperbolic space optimization learning strategy. This strategy uses the temporal motion prior information to assist learning, and uses 3D pose and pose motion information respectively in the hyperbolic space to optimize and learn the mesh features. Then, we combine the optimized results to get an accurate and smooth human mesh. Besides, to make the optimization learning process of human meshes in hyperbolic space stable and effective, we propose a hyperbolic mesh optimization loss. Extensive experimental results on large publicly available datasets indicate superiority in comparison with most state-of-the-art.

Hyperbolic geometry has emerged as a powerful tool for modeling complex, structured data, particularly where hierarchical or tree-like relationships are present. By enabling embeddings with lower distortion, hyperbolic neural networks offer promising alternatives to Euclidean-based models for capturing intricate data structures. Despite these advantages, they often face performance challenges, particularly in computational efficiency and tasks requiring high precision. In this work, we address these limitations by simplifying key operations within hyperbolic neural networks, achieving notable improvements in both runtime and performance. Our findings demonstrate that streamlined hyperbolic operations can lead to substantial gains in computational speed and predictive accuracy, making hyperbolic neural networks a more viable choice for a broader range of applications. Recent datasets with complex geometric structures have highlighted the limitations of traditional Euclidean spaces in capturing hierarchical or tree-like data found in domains like complex networks (Krioukov et al., 2010), natural language processing (L opez et al., 2019; Zhu et al., 2020; L opez & Strube, 2020), and protein interactions (Zitnik et al., 2019). Hyperbolic neural networks (HNNs) (Ganea et al., 2018) and hyperbolic graph convolutional networks (HGCNs) (Chami et al., 2019) have demonstrated superior performance over Euclidean models in learning from hierarchical data.

Fuel moisture content (FMC) is a key predictor for wildfire rate of spread (ROS). Machine learning models of FMC are being used more in recent years, augmenting or replacing traditional physics-based approaches. Wildfire rate of spread (ROS) has a highly nonlinear relationship with FMC, where small differences in dry fuels lead to large differences in ROS. In this study, custom loss functions that place more weight on dry fuels were examined with a variety of machine learning models of FMC. The models were evaluated with a spatiotemporal cross-validation procedure to examine whether the custom loss functions led to more accurate forecasts of ROS. Results show that the custom loss functions improved accuracy for ROS forecasts by a small amount. Further research would be needed to establish whether the improvement in ROS forecasts leads to more accurate real-time wildfire simulations.

This paper aims to explore the application of machine learning in forecasting Chinese macroeconomic variables. Specifically, it employs various machine learning models to predict the quarterly real GDP growth of China, and analyzes the factors contributing to the performance differences among these models. Our findings indicate that the average forecast errors of machine learning models are generally lower than those of traditional econometric models or expert forecasts, particularly in periods of economic stability. However, during certain inflection points, although machine learning models still outperform traditional econometric models, expert forecasts may exhibit greater accuracy in some instances due to experts' more comprehensive understanding of the macroeconomic environment and real-time economic variables. In addition to macroeconomic forecasting, this paper employs interpretable machine learning methods to identify the key attributive variables from different machine learning models, aiming to enhance the understanding and evaluation of their contributions to macroeconomic fluctuations.

We introduce Hyperbolic Prototype Learning, a type of supervised learning, where class labels are represented by ideal points (points at infinity) in hyperbolic space. Learning is achieved by minimizing the 'penalized Busemann loss', a new loss function based on the Busemann function of hyperbolic geometry. We discuss several theoretical features of this setup. In particular, Hyperbolic Prototype Learning becomes equivalent to logistic regression in the one-dimensional case.

Linear two-timescale stochastic approximation (SA) scheme is an important class of algorithms which has become popular in reinforcement learning (RL), particularly for the policy evaluation problem. Recently, a number of works have been devoted to establishing the finite time analysis of the scheme, especially under the Markovian (non-i.i.d.) noise settings that are ubiquitous in practice. In this paper, we provide a finite-time analysis for linear two timescale SA. Our bounds show that there is no discrepancy in the convergence rate between Markovian and martingale noise, only the constants are affected by the mixing time of the Markov chain. With an appropriate step size schedule, the transient term in the expected error bound is o (1 /k c) and the steady-state term is O (1 /k), where c 1 and k is the iteration number. Furthermore, we present an asymptotic expansion of the expected error with a matching lower bound of Ω(1 /k). A simple numerical experiment is presented to support our theory. Keywords: stochastic approximation, reinforcement learning, GTD learning, Markovian noise 1. Introduction Since its introduction close to 70 years ago, the stochastic approximation (SA) scheme (Robbins and Monro, 1951) has been a powerful tool for root finding when only noisy samples are available. During the past two decades, considerable progresses in the practical and theoretical research of SA have been made, see (Bena ım, 1999; Kushner and Yin, 2003; Borkar, 2008) for an overview. Among others, linear SA schemes are popular in reinforcement learning (RL) as they lead to policy evaluation methods with linear function approximation, of particular importance is temporal difference (TD) learning (Sutton, 1988) for which finite time analysis has been reported in (Srikant and Ying, 2019; Lakshminarayanan and Szepesvari, 2018; Bhandari et al., 2018; Dalal et al., 2018a). The TD learning scheme based on classical (linear) SA is known to be inadequate for the off-policy learning paradigms in RL, where data samples are drawn from a behavior policy different from the policy being evaluated (Baird, 1995; Tsitsiklis and V an Roy, 1997). To circumvent this Authors listed in alphabetical order. These methods fall within the scope of linear two-timescale SA scheme introduced by Borkar (1997): θ k 1 θ k β k{null b 1( X k 1) null A 11(X k 1)θ k null A 12(X k 1) w k}, (1) w k 1 w k γ k{null b 2( X k 1) null A 21( X k 1)θ k null A 22(X k 1)w k}.