We evaluate TPSR and several baseline methods on the following four standard benchmark datasets: Feynman, Black-box, and Strogatz from SRBench [42], and In-domain Synthetic Data generated based on [18]. More details on each of these datasets are given below. The regression input points (x, y) from these equations are provided in Penn Machine Learning Benchmark (PMLB) [42, 43] and have been studied in SRBench [42] for the symbolic regression task. The input dimension is limited to d 10 and the true underlying function of points is known. We split the dataset into B bags of 200 input points (when N is larger than 200) since the transformer SR model is pretrained on N 200 input points as per [18]. The input points for these problems are included in PMLB [43] and have been examined in SRBench [42] for symbolic regression. The input dimension for these problems is restricted to d = 2 and the true underlying functions are provided. The aim of SR study on these black-box datasets is to find an interpretable model expression that fits the data effectively.

Symbolic regression (SR) is a challenging task in machine learning that involves finding a mathematical expression for a function based on its values. Recent advancements in SR have demonstrated the effectiveness of pre-trained transformer models in generating equations as sequences, leveraging large-scale pre-training on synthetic datasets and offering notable advantages in terms of inference time over classical Genetic Programming (GP) methods. However, these models primarily rely on supervised pre-training objectives borrowed from text generation and overlook equation discovery goals like accuracy and complexity. To address this, we propose TPSR, a Transformer-based Planning strategy for Symbolic Regression that incorporates Monte Carlo Tree Search planning algorithm into the transformer decoding process. Unlike conventional decoding strategies, TPSR enables the integration of non-differentiable equation verification feedback, such as fitting accuracy and complexity, as external sources of knowledge into the transformer equation generation process. Extensive experiments on various datasets show that our approach outperforms state-of-the-art methods, enhancing the model's fitting-complexity trade-off, extrapolation abilities, and robustness to noise

Owing to advancements in image synthesis techniques, stylization methodologies for large models have garnered remarkable outcomes. However, when it comes to processing facial images, the outcomes frequently fall short of expectations. Facial stylization is predominantly challenged by two significant hurdles. Firstly, obtaining a large dataset of high-quality stylized images is difficult. The scarcity and diversity of artistic styles make it impractical to compile comprehensive datasets for each style.

Forecasting future world events is a challenging but valuable task. Forecasts of climate, geopolitical conflict, pandemics and economic indicators help shape policy and decision making. In these domains, the judgment of expert humans contributes to the best forecasts. Given advances in language modeling, can these forecasts be automated? To this end, we introduce Autocast, a dataset containing thousands of forecasting questions and an accompanying news corpus. Questions are taken from forecasting tournaments, ensuring high quality, real-world importance, and diversity. The news corpus is organized by date, allowing us to precisely simulate the conditions under which humans made past forecasts (avoiding leakage from the future).

Machine learning over graphs has recently attracted growing attention due to its ability to analyze and learn complex relations within critical interconnected systems. However, the disparate impact that is amplified by the use of biased graph structures in these algorithms has raised significant concerns for their deployment in realworld decision systems. In addition, while synthetic graph generation has become pivotal for privacy and scalability considerations, the impact of generative learning algorithms on structural bias has not yet been investigated. Motivated by this, this work focuses on the analysis and mitigation of structural bias for both real and synthetic graphs. Specifically, we first theoretically analyze the sources of structural bias that result in disparity for the predictions of dyadic relations. To alleviate the identified bias factors, we design a novel fairness regularizer that offers a versatile use. Faced with the bias amplification in graph generation models brought to light in this work, we further propose a fair graph generation framework, FairWire, by leveraging our fair regularizer design in a generative model. Experimental results on real-world networks validate that the proposed tools herein deliver effective structural bias mitigation for both real and synthetic graphs.

One of the roadblocks for training generalist robotic models today is heterogeneity. Previous robot learning methods often collect data to train with one specific embodiment for one task, which is expensive and prone to overfitting. This work studies the problem of learning policy representations through heterogeneous pretraining on robot data across different embodiments and tasks at scale. We propose Heterogeneous Pre-trained Transformers (HPT), which pre-train a large, shareable trunk of a policy neural network to learn a task and embodiment agnostic shared representation. This general architecture aligns the specific proprioception and vision inputs from distinct embodiments to a short sequence of tokens and then processes such tokens to map to control robots for different tasks. Leveraging the recent large-scale multi-embodiment real-world robotic datasets as well as simulation, deployed robots, and human video datasets, we investigate pre-training policies across heterogeneity. We conduct experiments to investigate the scaling behaviors of training objectives, to the extent of 52 datasets. HPTs outperform several baselines and enhance the fine-tuned policy performance by over 20% on unseen tasks in multiple simulator benchmarks and real-world settings.

A.1 Algorithm for Learning Categorical Data The over all training algorithm for learning categorical generative models is similar to the other cases. A.2 Practical Algorithm We give a detailed practical algorithm. N(0,I) is a standard Gaussian noise. A simplified loss Similar to Song and Ermon [41], Ho et al. [18], we use a stochastic version of loss (15), in which we only uniformly sample temporal snapshots to compute the loss. Apply gradient descent to update .

We propose a family of First Hitting Diffusion Models (FHDM), deep generative models that generate data with a diffusion process that terminates at a random first hitting time. This yields an extension of the standard fixed-time diffusion models that terminate at a pre-specified deterministic time. Although standard diffusion models are designed for continuous unconstrained data, FHDM is naturally designed to learn distributions on continuous as well as a range of discrete and structure domains. Moreover, FHDM enables instance-dependent terminate time and accelerates the diffusion process to sample higher quality data with fewer diffusion steps. Technically, we train FHDM by maximum likelihood estimation on diffusion trajectories augmented from observed data with conditional first hitting processes (i.e., bridge) derived based on Doob's h-transform, deviating from the commonly used time-reversal mechanism. We apply FHDM to generate data in various domains such as point cloud (general continuous distribution), climate and geographical events on earth (continuous distribution on the sphere), unweighted graphs (distribution of binary matrices), and segmentation maps of 2D images (high-dimensional categorical distribution). We observe considerable improvement compared with the state-of-the-art approaches in both quality and speed.