The last few years have witnessed a surge in methods for symbolic regression, from advances in traditional evolutionary approaches to novel deep learning-based systems. Individual works typically focus on advancing the state-of-the-art for one particular class of solution strategies, and there have been few attempts to investigate the benefits of hybridizing or integrating multiple strategies. In this work, we identify five classes of symbolic regression solution strategies---recursive problem simplification, neural-guided search, large-scale pre-training, genetic programming, and linear models---and propose a strategy to hybridize them into a single modular, unified symbolic regression framework. Based on empirical evaluation using SRBench, a new community tool for benchmarking symbolic regression methods, our unified framework achieves state-of-the-art performance in its ability to (1) symbolically recover analytical expressions, (2) fit datasets with high accuracy, and (3) balance accuracy-complexity trade-offs, across 252 ground-truth and black-box benchmark problems, in both noiseless settings and across various noise levels. Finally, we provide practical use case-based guidance for constructing hybrid symbolic regression algorithms, supported by extensive, combinatorial ablation studies.

Test-time scaling (TTS) has been shown to improve the performance of large language models (LLMs) by sampling and aggregating diverse reasoning paths. However, existing research has overlooked a critical issue: selection bias of reasoning strategies during scaling. Specifically, when generating reasoning processes, LLMs tend to follow certain strategies (e.g., algebraic solutions for math problems) while neglecting other valid alternatives (e.g., geometric solutions), resulting in insufficient exploration of the solution space. To further understand the impact of this bias, we present a theoretical analysis that reveals when it undermines the effectiveness of test-time scaling. Motivated by this theoretical insight, we introduce TTS-Uniform, a framework designed to mitigate the selection bias of reasoning strategies. It (i) identifies potential strategies, (ii) uniformly allocates the sampling budget across them, and (iii) filters out unstable strategies prior to aggregation. Experimental results show that TTS-Uniform significantly enhances scaling effectiveness across multiple mainstream LLMs and benchmark datasets. Code is available at https://github.com/zongqianwu/Uniform-TTS. Chain-of-thought (CoT) (Wei et al., 2022; Kojima et al., 2022) enhances the reasoning capabilities of large language models (LLMs) by explicitly unfolding intermediate steps (i.e., reasoning paths) before arriving at the final answer. Building on CoT, test-time scaling (TTS) (Zhang et al., 2025; Ji et al., 2025) further improves performance by sampling and aggregating diverse paths. However, existing TTS research (Wang et al., 2022; Snell et al., 2024) overlooks a critical limitation of CoT, which in turn constrains the effectiveness of scaling. Specifically, when tackling a problem, CoT reasoning tends to follow certain strategies while neglecting other valid alternatives.

Individual works typically focus on advancing the state-of-the-art for one particular class of solution strategies, and there have been few attempts to investigate the benefits of hybridizing or integrating multiple strategies.

The last few years have witnessed a surge in methods for symbolic regression, from advances in traditional evolutionary approaches to novel deep learning-based systems. Individual works typically focus on advancing the state-of-the-art for one particular class of solution strategies, and there have been few attempts to investigate the benefits of hybridizing or integrating multiple strategies. In this work, we identify five classes of symbolic regression solution strategies---recursive problem simplification, neural-guided search, large-scale pre-training, genetic programming, and linear models---and propose a strategy to hybridize them into a single modular, unified symbolic regression framework. Based on empirical evaluation using SRBench, a new community tool for benchmarking symbolic regression methods, our unified framework achieves state-of-the-art performance in its ability to (1) symbolically recover analytical expressions, (2) fit datasets with high accuracy, and (3) balance accuracy-complexity trade-offs, across 252 ground-truth and black-box benchmark problems, in both noiseless settings and across various noise levels. Finally, we provide practical use case-based guidance for constructing hybrid symbolic regression algorithms, supported by extensive, combinatorial ablation studies.

This paper addresses an interesting and new problem to augment training data in a learnable and principled manner. Modern machine learning systems are known for their'hunger for data' and until now state-of-the-art approaches have relied mainly on heuristics to augment labeled training data. This paper tries to reduce the tedious task of finding a good combination of data augmentation strategies with best parameters by learning a sequence of best data augmentation strategies in a generative adversarial framework while working with unsupervised data. The motivation behind the problem is to reduce human labor without compromising the final discriminative classification performance. The problem formulation is pretty clear from the text.

We explored the addition bias, a cognitive tendency to prefer adding elements over removing them to alter an initial state or structure, by conducting four preregistered experiments examining the problem-solving behavior of both humans and OpenAl's GPT-4 large language model. The experiments involved 588 participants from the U.S. and 680 iterations of the GPT-4 model. The problem-solving task was either to create symmetry within a grid (Experiments 1 and 3) or to edit a summary (Experiments 2 and 4). As hypothesized, we found that overall, the addition bias was present. Solution efficiency (Experiments 1 and 2) and valence of the instruction (Experiments 3 and 4) played important roles. Human participants were less likely to use additive strategies when subtraction was relatively more efficient than when addition and subtraction were equally efficient. GPT-4 exhibited the opposite behavior, with a strong addition bias when subtraction was more efficient. In terms of instruction valence, GPT-4 was more likely to add words when asked to "improve" compared to "edit", whereas humans did not show this effect. When we looked at the addition bias under different conditions, we found more biased responses for GPT-4 compared to humans. Our findings highlight the importance of considering comparable and sometimes superior subtractive alternatives, as well as reevaluating one's own and particularly the language models' problem-solving behavior.

Split learning (SL) has been recently proposed as a way to enable resource-constrained devices to train multi-parameter neural networks (NNs) and participate in federated learning (FL). In a nutshell, SL splits the NN model into parts, and allows clients (devices) to offload the largest part as a processing task to a computationally powerful helper. In parallel SL, multiple helpers can process model parts of one or more clients, thus, considerably reducing the maximum training time over all clients (makespan). In this paper, we focus on orchestrating the workflow of this operation, which is critical in highly heterogeneous systems, as our experiments show. In particular, we formulate the joint problem of client-helper assignments and scheduling decisions with the goal of minimizing the training makespan, and we prove that it is NP-hard. We propose a solution method based on the decomposition of the problem by leveraging its inherent symmetry, and a second one that is fully scalable. A wealth of numerical evaluations using our testbed's measurements allow us to build a solution strategy comprising these methods. Moreover, we show that this strategy finds a near-optimal solution, and achieves a shorter makespan than the baseline scheme by up to 52.3%.

Machine-learning methods are gradually being adopted in a great variety of social, economic, and scientific contexts, yet they are notorious for struggling with exact mathematics. A typical example is computer algebra, which includes tasks like simplifying mathematical terms, calculating formal derivatives, or finding exact solutions of algebraic equations. Traditional software packages for these purposes are commonly based on a huge database of rules for how a specific operation (e.g., differentiation) transforms a certain term (e.g., sine function) into another one (e.g., cosine function). Thus far, these rules have usually needed to be discovered and subsequently programmed by humans. Focusing on the paradigmatic example of solving linear equations in symbolic form, we demonstrate how the process of finding elementary transformation rules and step-by-step solutions can be automated using reinforcement learning with deep neural networks.

In recent years, by utilizing optimization techniques to formulate the propagation of deep model, a variety of so-called Optimization-Derived Learning (ODL) approaches have been proposed to address diverse learning and vision tasks. Although having achieved relatively satisfying practical performance, there still exist fundamental issues in existing ODL methods. In particular, current ODL methods tend to consider model construction and learning as two separate phases, and thus fail to formulate their underlying coupling and depending relationship. In this work, we first establish a new framework, named Hierarchical ODL (HODL), to simultaneously investigate the intrinsic behaviors of optimization-derived model construction and its corresponding learning process. Then we rigorously prove the joint convergence of these two sub-tasks, from the perspectives of both approximation quality and stationary analysis. To our best knowledge, this is the first theoretical guarantee for these two coupled ODL components: optimization and learning. We further demonstrate the flexibility of our framework by applying HODL to challenging learning tasks, which have not been properly addressed by existing ODL methods. Finally, we conduct extensive experiments on both synthetic data and real applications in vision and other learning tasks to verify the theoretical properties and practical performance of HODL in various application scenarios.