Published as a conference paper at ICLR 2025T OWARDSH IERARCHICAL R ECTIFIED F LOW Yichi Zhang 1, Yici Y an 1, Alex Schwing 1, Zhizhen Zhao 1 1 University of Illinois Urbana-Champaign A BSTRACT We formulate a hierarchical rectified flow to model data distributions. It hierarchically couples multiple ordinary differential equations (ODEs) and defines a time-differentiable stochastic process that generates a data distribution from a known source distribution. Each ODE resembles the ODE that is solved in a classic rectified flow, but differs in its domain, i.e., location, velocity, acceleration, etc. Unlike the classic rectified flow formulation, which formulates a single ODE in the location domain and only captures the expected velocity field (sufficient to capture a multi-modal data distribution), the hierarchical rectified flow formulation models the multi-modal random velocity field, acceleration field, etc., in their entirety. This more faithful modeling of the random velocity field enables integration paths to intersect when the underlying ODE is solved during data generation. Intersecting paths in turn lead to integration trajectories that are more straight than those obtained in the classic rectified flow formulation, where integration paths cannot intersect. This leads to modeling of data distributions with fewer neural function evaluations. We empirically verify this on synthetic 1D and 2D data as well as MNIST, CIFAR-10, and ImageNet-32 data. Our code is available at: https://riccizz.github.io/HRF/ . 1 I NTRODUCTION Diffusion models (Ho et al., 2020; Song et al., 2021a;b) and particularly also flow matching (Liu et al., 2023; Lipman et al., 2023; Albergo & V anden-Eijnden, 2023; Albergo et al., 2023) have gained significant attention recently. This is partly due to impressive results that have been reported across domains from computer vision (Ho et al., 2020) and medical imaging (Song et al., 2022) to robotics (Kapelyukh et al., 2023) and computational biology (Guo et al., 2024). Beyond impressive results, flow matching was also reported to faithfully model multimodal data distributions. In addition, sampling is reasonably straightforward: it requires to solve an ordinary differential equation (ODE) via forward integration of a set of source distribution points along an estimated velocity field from time zero to time one. The source distribution points are sampled from a simple and known source distribution, e.g., a standard Gaussian. The velocity field is obtained by matching velocities from a constructed "ground-truth" integration path with a parametric deep net using a mean squared error (MSE) objective. See Figure 1(a) for the "ground-truth" integration paths of classic rectified flow. Studying the "ground-truth" velocity distribution at a distinct location and time for rectified flow reveals a multimodal distribution.

Recently, machine learning (ML) models have gained prominence in predicting asset returns, selecting portfolios, and estimating stochastic discount factors, with significant success in these areas. ML techniques, by capturing complex and nonlinear relationships in financial data, are particularly well-suited for enhancing portfolio management decisions. For example, within the mean-variance portfolio framework, ML methods are increasingly used to estimate expected returns and (co)variances, often leading to more effective portfolio allocations. The literature consistently demonstrates the effectiveness of machine learning in these and other applications (e.g., Gu, Kelly, and Xiu (2020); Bianchi, B uchner, and Tamoni (2021); Cong, Tang, Wang, and Zhang (2021); Kelly, Malamud, and Zhou (2021); Patton and Weller (2022); Didisheim, Ke, Kelly, and Malamud (2023); Filipovic and Schneider (2024)). Despite the success of machine learning in asset pricing, existing literature typically treats ML predictions as point estimates and conducts asset pricing analyses as if they were true values, overlooking the associated uncertainty. This is surprising, given that uncertainty about input parameters is widely acknowledged as critical in portfolio selection (e.g., DeMiguel, Garlappi, and Uppal (2009)), and Garlappi, Uppal, and Wang (2007) show that incorporating forecast uncertainty in mean-variance portfolio allocation leads to distinct economic insights. However, quantifying prediction uncertainty in ML forecasts, particularly with neural networks, remains a complex challenge, limiting their broader application in asset pricing.

In Bayesian inference, computing the posterior distribution from the data is typically a non-trivial problem, which usually requires approximations such as mean-field approaches or numerical methods, like the Monte Carlo Markov Chain. Being a high-dimensional distribution over a set of correlated variables, the posterior distribution can undergo the notorious replica symmetry breaking transition. When it happens, several mean-field methods and virtually every Monte Carlo scheme can not provide a reasonable approximation to the posterior and its marginals. Replica symmetry is believed to be guaranteed whenever the data is generated with known prior and likelihood distributions, namely under the so-called Nishimori conditions. In this paper, we break this belief, by providing a counter-example showing that, under the Nishimori conditions, replica symmetry breaking arises. Introducing a simple, geometrical model that can be thought of as a patient zero retrieval problem in a highly infectious regime of the epidemic Susceptible-Infectious model, we show that under the Nishimori conditions, there is evidence of replica symmetry breaking. We achieve this result by computing the instability of the replica symmetric cavity method toward the one step replica symmetry broken phase. The origin of this phenomenon -- replica symmetry breaking under the Nishimori conditions -- is likely due to the correlated disorder appearing in the epidemic models.

ABSTRACT Natural gas (NG) is relatively a clean source of energy, particularly compared to fossil fuels, and worldwide consumption of NG has been increasing almost linearly in the last two decades. A similar trend can also be seen in Turkey, while another similarity is the high dependence on impor ts for the continuous NG supply. It is crucial to accurately forecast future NG demand (NGD) in Turkey, especially, for import contracts; in this respect, forecasts of monthly NGD for the following year are of utmost importance. In the current study, the h istorical monthly NG consumption data between 2014 and 2024 provided by SOCAR, the local residential NG distribution company for two cities in Turkey, Bursa and Kayseri, was used to determine out - of - sample monthly NGD forecasts for a period of one year and nine months using various time series models, including SARIMA and ETS models, and a novel proposed machine learning method. The proposed method, named Just - in - Time - Learning - Gaussia n Process Regression (JITL - GPR), uses a novel feature representation for t he past NG demand values; instead of using past demand values as column - wise separate features, they are placed on a two - dimensional (2 - D) grid of year - month values. For each test point, a kernel function, tailored for the NGD predictions, is used in GPR t o predict the query point. Since a model is constructed separately for each test point, the proposed method is, indeed, an example of JITL. The JITL - GPR method is easy to use and optimize, and offers a reduction in forecast errors compared to traditional t ime series methods and a state - of - the - art combinat ion model; therefore, it is a promising tool for NGD forecasting in similar settings. INTRODUCTION In the last few decades, there has been a shift in energy sources from fossil fuels to cleaner energy sources, such as wind and solar energy, mainly due to environmental concerns and related government regulations . However, these latter sources are depend ent on w eather conditions and require integration with grid technologies for continuous power generation. Natural gas (NG), typically, consists of (up to) ~95% of methane and 2 - 2.5% ethane - hexane+, with the remain der consist ing of nitrogen, CO NG p ower plants are easy to build and highly reliable, mak ing them invaluable for "clean" energy production. On the other hand, m ost countries depend on imports to maintain t heir NG supplies, and there is a delicate balance between import s and domestic demand . S toring excess import ed gas above actual demand is difficult and would result in economic losses, while import ing less than actual demand could result in a nationwide sh ortage.

Despite the growing interest in collaborative AI, designing systems that seamlessly integrate human input remains a major challenge. In this study, we developed a task to systematically examine human preferences for collaborative agents. We created and evaluated five collaborative AI agents with strategies that differ in the manner and degree they adapt to human actions. Participants interacted with a subset of these agents, evaluated their perceived traits, and selected their preferred agent. We used a Bayesian model to understand how agents' strategies influence the Human-AI team performance, AI's perceived traits, and the factors shaping human-preferences in pairwise agent comparisons. Our results show that agents who are more considerate of human actions are preferred over purely performance-maximizing agents. Moreover, we show that such human-centric design can improve the likability of AI collaborators without reducing performance. We find evidence for inequality-aversion effects being a driver of human choices, suggesting that people prefer collaborative agents which allow them to meaningfully contribute to the team. Taken together, these findings demonstrate how collaboration with AI can benefit from development efforts which include both subjective and objective metrics.

A computational ethics framework is essential for AI and autonomous systems operating in complex, real-world environments. Existing approaches often lack the adaptability needed to integrate ethical principles into dynamic and ambiguous contexts, limiting their effectiveness across diverse scenarios. To address these challenges, we outline the necessary ingredients for building a holistic, meta-level framework that combines intermediate representations, probabilistic reasoning, and knowledge representation. The specifications therein emphasize scalability, supporting ethical reasoning at both individual decision-making levels and within the collective dynamics of multi-agent systems. By integrating theoretical principles with contextual factors, it facilitates structured and context-aware decision-making, ensuring alignment with overarching ethical standards. We further explore proposed theorems outlining how ethical reasoners should operate, offering a foundation for practical implementation. These constructs aim to support the development of robust and ethically reliable AI systems capable of navigating the complexities of real-world moral decision-making scenarios.

Abstraction is essential for reducing the complexity of systems across diverse fields, yet designing effective abstraction methodology for probabilistic models is inherently challenging due to stochastic behaviors and uncertainties. Current approaches often distill detailed probabilistic data into higher-level summaries to support tractable and interpretable analyses, though they typically struggle to fully represent the relational and probabilistic hierarchies through single-layered abstractions. We introduce a hierarchical probabilistic abstraction framework aimed at addressing these challenges by extending a measure-theoretic foundation for hierarchical abstraction. The framework enables modular problem-solving via layered mappings, facilitating both detailed layer-specific analysis and a cohesive system-wide understanding. This approach bridges high-level conceptualization with low-level perceptual data, enhancing interpretability and allowing layered analysis. Our framework provides a robust foundation for abstraction analysis across AI subfields, particularly in aligning System 1 and System 2 thinking, thereby supporting the development of diverse abstraction methodologies.

We introduce Spatial Reasoning Models (SRMs), a framework to perform reasoning over sets of continuous variables via denoising generative models. SRMs infer continuous representations on a set of unobserved variables, given observations on observed variables. Current generative models on spatial domains, such as diffusion and flow matching models, often collapse to hallucination in case of complex distributions. To measure this, we introduce a set of benchmark tasks that test the quality of complex reasoning in generative models and can quantify hallucination. The SRM framework allows to report key findings about importance of sequentialization in generation, the associated order, as well as the sampling strategies during training. It demonstrates, for the first time, that order of generation can successfully be predicted by the denoising network itself. Using these findings, we can increase the accuracy of specific reasoning tasks from <1% to >50%.

Diffusion models have recently driven significant breakthroughs in generative modeling. While state-of-the-art models produce high-quality samples on average, individual samples can still be low quality. Detecting such samples without human inspection remains a challenging task. To address this, we propose a Bayesian framework for estimating generative uncertainty of synthetic samples. We outline how to make Bayesian inference practical for large, modern generative models and introduce a new semantic likelihood (evaluated in the latent space of a feature extractor) to address the challenges posed by high-dimensional sample spaces. Through our experiments, we demonstrate that the proposed generative uncertainty effectively identifies poor-quality samples and significantly outperforms existing uncertainty-based methods. Notably, our Bayesian framework can be applied post-hoc to any pretrained diffusion or flow matching model (via the Laplace approximation), and we propose simple yet effective techniques to minimize its computational overhead during sampling.

The interplay of optimizers and architectures in neural networks is complicated and hard to understand why some optimizers work better on some specific architectures. In this paper, we find that the traditionally used sharpness metric does not fully explain the intricate interplay and introduces information-theoretic metrics called entropy gap to better help analyze. It is found that both sharpness and entropy gap affect the performance, including the optimization dynamic and generalization. We further use information-theoretic tools to understand a recently proposed optimizer called Lion and find ways to improve it.