Autoformalization aims to translate natural-language mathematical statements into a formal language. While LLMs have accelerated progress in this area, existing methods still suffer from low accuracy. We identify two key abilities for effective autoformalization: comprehensive mastery of formal-language domain knowledge, and reasoning capability of natural language problem understanding and informal-formal alignment. Without the former, a model cannot identify the correct formal objects; without the latter, it struggles to interpret real-world contexts and map them precisely into formal expressions. To address these gaps, we introduce ThinkingF, a data synthesis and training pipeline that improves both abilities. First, we construct two datasets: one by distilling and selecting large-scale examples rich in formal knowledge, and another by generating informal-to-formal reasoning trajectories guided by expert-designed templates. We then apply SFT and RLVR with these datasets to further fuse and refine the two abilities. The resulting 7B and 32B models exhibit both comprehensive formal knowledge and strong informal-to-formal reasoning. Notably, StepFun-Formalizer-32B achieves SOTA BEq@1 scores of 40.5% on FormalMATH-Lite and 26.7% on ProverBench, surpassing all prior general-purpose and specialized models.

Regularization that incorporates the linear combination of empirical loss and explicit regularization terms as the loss function has been frequently used for many machine learning tasks. The explicit regularization term is designed in different types, depending on its applications. While regularized learning often boost the performance with higher accuracy and faster convergence, the regularization would sometimes hurt the empirical loss minimization and lead to poor performance. To deal with such issues in this work, we propose a novel strategy, namely Gr adients O rthogonal D ecomposition ( GrOD ), that improves the training procedure of regularized deep learning. Instead of linearly combining gradients of the two terms, GrOD re-estimates a new direction for iteration that does not hurt the empirical loss minimization while preserving the regularization affects, through orthogonal decomposition. We have performed extensive experiments to use GrOD improving the commonly used algorithms of transfer learning [ 2 ], knowledge distillation [ 3 ], and adversarial learning [ 4 ]. The experiment results based on large datasets, including Caltech 256 [ 5 ], MIT indoor 67 [ 6 ], CIFAR-10 [ 7 ], and ImageNet [ 8 ], show significant improvement made by GrOD for all three algorithms in all cases.

In this paper, we comprehensively reveal the learning dynamics of normalized neural network using Stochastic Gradient Descent (with momentum) and Weight Decay (WD), named as Spherical Motion Dynamics (SMD). Most related works focus on studying behavior of effective learning rate" inequilibrium" state, i.e. assuming weight norm remains unchanged. However, their discussion on why this equilibrium can be reached is either absent or less convincing. Our work directly explores the cause of equilibrium, as a special state of SMD. Specifically, 1) we introduce the assumptions that can lead to equilibrium state in SMD, and prove equilibrium can be reached in a linear rate regime under given assumptions; 2) we propose ``angular update" as a substitute for effective learning rate to depict the state of SMD, and derive the theoretical value of angular update in equilibrium state; 3) we verify our assumptions and theoretical results on various large-scale computer vision tasks including ImageNet and MSCOCO with standard settings. Experiment results show our theoretical findings agree well with empirical observations. We also show that the behavior of angular update in SMD can produce interesting effect to the optimization of neural network in practice.

Transfer learning have been frequently used to improve deep neural network training through incorporating weights of pre-trained networks as the starting-point of optimization for regularization. While deep transfer learning can usually boost the performance with better accuracy and faster convergence, transferring weights from inappropriate networks hurts training procedure and may lead to even lower accuracy. In this paper, we consider deep transfer learning as minimizing a linear combination of empirical loss and regularizer based on pre-trained weights, where the regularizer would restrict the training procedure from lowering the empirical loss, with conflicted descent directions (e.g., derivatives). Following the view, we propose a novel strategy making regularization-based Deep Transfer learning Never Hurt (DTNH) that, for each iteration of training procedure, computes the derivatives of the two terms separately, then re-estimates a new descent direction that does not hurt the empirical loss minimization while preserving the regularization affects from the pre-trained weights. Extensive experiments have been done using common transfer learning regularizers, such as L2-SP and knowledge distillation, on top of a wide range of deep transfer learning benchmarks including Caltech, MIT indoor 67, CIFAR-10 and ImageNet. The empirical results show that the proposed descent direction estimation strategy DTNH can always improve the performance of deep transfer learning tasks based on all above regularizers, even when transferring pre-trained weights from inappropriate networks. All in all, DTNH strategy can improve state-of-the-art regularizers in all cases with 0.1%-7% higher accuracy in all experiments.