Zhengyang Geng

Shaojie Bai, Zhengyang Geng, Yash Savani et al. · cmu, stanford

Many recent state-of-the-art (SOTA) optical flow models use finite-step recurrent update operations to emulate traditional algorithms by encouraging iterative refinements toward a stable flow estimation. However, these RNNs impose large computation and memory overheads, and are not directly trained to model such stable estimation. They can converge poorly and thereby suffer from performance degradation. To combat these drawbacks, we propose deep equilibrium (DEQ) flow estimators, an approach that directly solves for the flow as the infinite-level fixed point of an implicit layer (using any black-box solver), and differentiates through this fixed point analytically (thus requiring $O(1)$ training memory). This implicit-depth approach is not predicated on any specific model, and thus can be applied to a wide range of SOTA flow estimation model designs. The use of these DEQ flow estimators allows us to compute the flow faster using, e.g., fixed-point reuse and inexact gradients, consumes $4\sim6\times$ times less training memory than the recurrent counterpart, and achieves better results with the same computation budget. In addition, we propose a novel, sparse fixed-point correction scheme to stabilize our DEQ flow estimators, which addresses a longstanding challenge for DEQ models in general. We test our approach in various realistic settings and show that it improves SOTA methods on Sintel and KITTI datasets with substantially better computational and memory efficiency.

Zhengyang Geng, J. Zico Kolter

Deep Equilibrium (DEQ) Models, an emerging class of implicit models that maps inputs to fixed points of neural networks, are of growing interest in the deep learning community. However, training and applying DEQ models is currently done in an ad-hoc fashion, with various techniques spread across the literature. In this work, we systematically revisit DEQs and present TorchDEQ, an out-of-the-box PyTorch-based library that allows users to define, train, and infer using DEQs over multiple domains with minimal code and best practices. Using TorchDEQ, we build a ``DEQ Zoo'' that supports six published implicit models across different domains. By developing a joint framework that incorporates the best practices across all models, we have substantially improved the performance, training stability, and efficiency of DEQs on ten datasets across all six projects in the DEQ Zoo. TorchDEQ and DEQ Zoo are released as \href{https://github.com/locuslab/torchdeq}{open source}.

Zekun Li, Zhengyang Geng, Zhao Kang et al.

Reference-based line-art colorization is a challenging task in computer vision. The color, texture, and shading are rendered based on an abstract sketch, which heavily relies on the precise long-range dependency modeling between the sketch and reference. Popular techniques to bridge the cross-modal information and model the long-range dependency employ the attention mechanism. However, in the context of reference-based line-art colorization, several techniques would intensify the existing training difficulty of attention, for instance, self-supervised training protocol and GAN-based losses. To understand the instability in training, we detect the gradient flow of attention and observe gradient conflict among attention branches. This phenomenon motivates us to alleviate the gradient issue by preserving the dominant gradient branch while removing the conflict ones. We propose a novel attention mechanism using this training strategy, Stop-Gradient Attention (SGA), outperforming the attention baseline by a large margin with better training stability. Compared with state-of-the-art modules in line-art colorization, our approach demonstrates significant improvements in Fréchet Inception Distance (FID, up to 27.21%) and structural similarity index measure (SSIM, up to 25.67%) on several benchmarks. The code of SGA is available at https://github.com/kunkun0w0/SGA .

Ashwini Pokle, Zhengyang Geng, Zico Kolter

Diffusion-based generative models are extremely effective in generating high-quality images, with generated samples often surpassing the quality of those produced by other models under several metrics. One distinguishing feature of these models, however, is that they typically require long sampling chains to produce high-fidelity images. This presents a challenge not only from the lenses of sampling time, but also from the inherent difficulty in backpropagating through these chains in order to accomplish tasks such as model inversion, i.e. approximately finding latent states that generate known images. In this paper, we look at diffusion models through a different perspective, that of a (deep) equilibrium (DEQ) fixed point model. Specifically, we extend the recent denoising diffusion implicit model (DDIM; Song et al. 2020), and model the entire sampling chain as a joint, multivariate fixed point system. This setup provides an elegant unification of diffusion and equilibrium models, and shows benefits in 1) single image sampling, as it replaces the fully-serial typical sampling process with a parallel one; and 2) model inversion, where we can leverage fast gradients in the DEQ setting to much more quickly find the noise that generates a given image. The approach is also orthogonal and thus complementary to other methods used to reduce the sampling time, or improve model inversion. We demonstrate our method's strong performance across several datasets, including CIFAR10, CelebA, and LSUN Bedrooms and Churches.

Benhao Huang, Zhengyang Geng, Zico Kolter

Scaling test-time compute by iteratively updating a latent state has emerged as a powerful paradigm for reasoning. Yet the internal mechanisms that enable these iterative models to generalize beyond memorized patterns remain unclear. We hypothesize that generalizable reasoning arises from learning task-conditioned attractors: latent dynamical systems whose stable fixed points correspond to valid solutions. We formalize this process through Equilibrium Reasoners (EqR), which enable test-time scaling without external verifiers or task-specific priors. EqR scales internal dynamics along two axes: depth, by running more iterations, and breadth, by aggregating stochastic trajectories from multiple initializations. Empirically, gains from test-time scaling are tightly coupled with stronger convergence toward solution-aligned attractors. This attractor perspective allows neural networks to adaptively allocate test-time compute based on task difficulty. While simple cases converge within 1 to 5 iteration steps, harder cases benefit from massive test-time scaling. By unrolling up to the equivalent of 40,000 layers, scalable latent reasoning boosts accuracy from 2.6% for feedforward models to over 99% on Sudoku-Extreme. These results suggest that learned attractor landscapes provide a useful mechanistic lens for understanding scalable reasoning in iterative latent models.

Zhengyang Geng, Yiyang Lu, Zongze Wu et al.

MeanFlow (MF) has recently been established as a framework for one-step generative modeling. However, its ``fastforward'' nature introduces key challenges in both the training objective and the guidance mechanism. First, the original MF's training target depends not only on the underlying ground-truth fields but also on the network itself. To address this issue, we recast the objective as a loss on the instantaneous velocity $v$, re-parameterized by a network that predicts the average velocity $u$. Our reformulation yields a more standard regression problem and improves the training stability. Second, the original MF fixes the classifier-free guidance scale during training, which sacrifices flexibility. We tackle this issue by formulating guidance as explicit conditioning variables, thereby retaining flexibility at test time. The diverse conditions are processed through in-context conditioning, which reduces model size and benefits performance. Overall, our $\textbf{improved MeanFlow}$ ($\textbf{iMF}$) method, trained entirely from scratch, achieves $\textbf{1.72}$ FID with a single function evaluation (1-NFE) on ImageNet 256$\times$256. iMF substantially outperforms prior methods of this kind and closes the gap with multi-step methods while using no distillation. We hope our work will further advance fastforward generative modeling as a stand-alone paradigm.

Weijian Luo, Colin Zhang, Debing Zhang et al.

We propose Diff-Instruct* (DI*), a data-efficient post-training approach for one-step text-to-image generative models to improve its human preferences without requiring image data. Our method frames alignment as online reinforcement learning from human feedback (RLHF), which optimizes the one-step model to maximize human reward functions while being regularized to be kept close to a reference diffusion process. Unlike traditional RLHF approaches, which rely on the Kullback-Leibler divergence as the regularization, we introduce a novel general score-based divergence regularization that substantially improves performance as well as post-training stability. Although the general score-based RLHF objective is intractable to optimize, we derive a strictly equivalent tractable loss function in theory that can efficiently compute its \emph{gradient} for optimizations. We introduce \emph{DI*-SDXL-1step}, which is a 2.6B one-step text-to-image model at a resolution of $1024\times 1024$, post-trained from DMD2 w.r.t SDXL. \textbf{Our 2.6B \emph{DI*-SDXL-1step} model outperforms the 50-step 12B FLUX-dev model} in ImageReward, PickScore, and CLIP score on the Parti prompts benchmark while using only 1.88\% of the inference time. This result clearly shows that with proper post-training, the small one-step model is capable of beating huge multi-step diffusion models. Our model is open-sourced at this link: https://github.com/pkulwj1994/diff_instruct_star. We hope our findings can contribute to human-centric machine learning techniques.

Yiyang Lu, Susie Lu, Qiao Sun et al.

Modern diffusion/flow-based models for image generation typically exhibit two core characteristics: (i) using multi-step sampling, and (ii) operating in a latent space. Recent advances have made encouraging progress on each aspect individually, paving the way toward one-step diffusion/flow without latents. In this work, we take a further step towards this goal and propose "pixel MeanFlow" (pMF). Our core guideline is to formulate the network output space and the loss space separately. The network target is designed to be on a presumed low-dimensional image manifold (i.e., x-prediction), while the loss is defined via MeanFlow in the velocity space. We introduce a simple transformation between the image manifold and the average velocity field. In experiments, pMF achieves strong results for one-step latent-free generation on ImageNet at 256x256 resolution (2.22 FID) and 512x512 resolution (2.48 FID), filling a key missing piece in this regime. We hope that our study will further advance the boundaries of diffusion/flow-based generative models.

Zhengyang Geng, Ashwini Pokle, William Luo et al.

Consistency models (CMs) offer faster sampling than traditional diffusion models, but their training is resource-intensive. For example, as of 2024, training a state-of-the-art CM on CIFAR-10 takes one week on 8 GPUs. In this work, we propose an effective scheme for training CMs that largely improves the efficiency of building such models. Specifically, by expressing CM trajectories via a particular differential equation, we argue that diffusion models can be viewed as a special case of CMs. We can thus fine-tune a consistency model starting from a pretrained diffusion model and progressively approximate the full consistency condition to stronger degrees over the training process. Our resulting method, which we term Easy Consistency Tuning (ECT), achieves vastly reduced training times while improving upon the quality of previous methods: for example, ECT achieves a 2-step FID of 2.73 on CIFAR10 within 1 hour on a single A100 GPU, matching Consistency Distillation trained for hundreds of GPU hours. Owing to this computational efficiency, we investigate the scaling laws of CMs under ECT, showing that they obey the classic power law scaling, hinting at their ability to improve efficiency and performance at larger scales. Our code (https://github.com/locuslab/ect) is publicly available, making CMs more accessible to the broader community.

Zhengyang Geng, Mingyang Deng, Xingjian Bai et al.

We propose a principled and effective framework for one-step generative modeling. We introduce the notion of average velocity to characterize flow fields, in contrast to instantaneous velocity modeled by Flow Matching methods. A well-defined identity between average and instantaneous velocities is derived and used to guide neural network training. Our method, termed the MeanFlow model, is self-contained and requires no pre-training, distillation, or curriculum learning. MeanFlow demonstrates strong empirical performance: it achieves an FID of 3.43 with a single function evaluation (1-NFE) on ImageNet 256x256 trained from scratch, significantly outperforming previous state-of-the-art one-step diffusion/flow models. Our study substantially narrows the gap between one-step diffusion/flow models and their multi-step predecessors, and we hope it will motivate future research to revisit the foundations of these powerful models.

Jiawei Yang, Zhengyang Geng, Xuan Ju et al.

We show that Fréchet Distance (FD), long considered impractical as a training objective, can in fact be effectively optimized in the representation space. Our idea is simple: decouple the population size for FD estimation (e.g., 50k) from the batch size for gradient computation (e.g., 1024). We term this approach FD-loss. Optimizing FD-loss reveals several surprising findings. First, post-training a base generator with FD-loss in different representation spaces consistently improves visual quality. Under the Inception feature space, a one-step generator achieves0.72 FID on ImageNet 256x256. Second, the same FD-loss repurposes multi-step generators into strong one-step generators without teacher distillation, adversarial training or per-sample targets. Third, FID can misrank visual quality: modern representations can yield better samples despite worse Inception FID. This motivates FDr$^k$, a multi-representation metric. We hope this work will encourage further exploration of distributional distances in diverse representation spaces as both training objectives and evaluation metrics for generative models.

Zhengyang Geng, Ashwini Pokle, J. Zico Kolter · deepmind

Diffusion models excel at producing high-quality samples but naively require hundreds of iterations, prompting multiple attempts to distill the generation process into a faster network. However, many existing approaches suffer from a variety of challenges: the process for distillation training can be complex, often requiring multiple training stages, and the resulting models perform poorly when utilized in single-step generative applications. In this paper, we introduce a simple yet effective means of distilling diffusion models directly from initial noise to the resulting image. Of particular importance to our approach is to leverage a new Deep Equilibrium (DEQ) model as the distilled architecture: the Generative Equilibrium Transformer (GET). Our method enables fully offline training with just noise/image pairs from the diffusion model while achieving superior performance compared to existing one-step methods on comparable training budgets. We demonstrate that the DEQ architecture is crucial to this capability, as GET matches a $5\times$ larger ViT in terms of FID scores while striking a critical balance of computational cost and image quality. Code, checkpoints, and datasets are available.

Weijian Luo, Zemin Huang, Zhengyang Geng et al.

Despite their strong performances on many generative tasks, diffusion models require a large number of sampling steps in order to generate realistic samples. This has motivated the community to develop effective methods to distill pre-trained diffusion models into more efficient models, but these methods still typically require few-step inference or perform substantially worse than the underlying model. In this paper, we present Score Implicit Matching (SIM) a new approach to distilling pre-trained diffusion models into single-step generator models, while maintaining almost the same sample generation ability as the original model as well as being data-free with no need of training samples for distillation. The method rests upon the fact that, although the traditional score-based loss is intractable to minimize for generator models, under certain conditions we can efficiently compute the gradients for a wide class of score-based divergences between a diffusion model and a generator. SIM shows strong empirical performances for one-step generators: on the CIFAR10 dataset, it achieves an FID of 2.06 for unconditional generation and 1.96 for class-conditional generation. Moreover, by applying SIM to a leading transformer-based diffusion model, we distill a single-step generator for text-to-image (T2I) generation that attains an aesthetic score of 6.42 with no performance decline over the original multi-step counterpart, clearly outperforming the other one-step generators including SDXL-TURBO of 5.33, SDXL-LIGHTNING of 5.34 and HYPER-SDXL of 5.85. We will release this industry-ready one-step transformer-based T2I generator along with this paper.

Zemin Huang, Zhengyang Geng, Weijian Luo et al.

In the realm of Artificial Intelligence Generated Content (AIGC), flow-matching models have emerged as a powerhouse, achieving success due to their robust theoretical underpinnings and solid ability for large-scale generative modeling. These models have demonstrated state-of-the-art performance, but their brilliance comes at a cost. The process of sampling from these models is notoriously demanding on computational resources, as it necessitates the use of multi-step numerical ordinary differential equations (ODEs). Against this backdrop, this paper presents a novel solution with theoretical guarantees in the form of Flow Generator Matching (FGM), an innovative approach designed to accelerate the sampling of flow-matching models into a one-step generation, while maintaining the original performance. On the CIFAR10 unconditional generation benchmark, our one-step FGM model achieves a new record Fréchet Inception Distance (FID) score of 3.08 among few-step flow-matching-based models, outperforming original 50-step flow-matching models. Furthermore, we use the FGM to distill the Stable Diffusion 3, a leading text-to-image flow-matching model based on the MM-DiT architecture. The resulting MM-DiT-FGM one-step text-to-image model demonstrates outstanding industry-level performance. When evaluated on the GenEval benchmark, MM-DiT-FGM has delivered remarkable generating qualities, rivaling other multi-step models in light of the efficiency of a single generation step.

Meng-Hao Guo, Jiajun Xu, Yi Zhang et al.

Reasoning stands as a cornerstone of intelligence, enabling the synthesis of existing knowledge to solve complex problems. Despite remarkable progress, existing reasoning benchmarks often fail to rigorously evaluate the nuanced reasoning capabilities required for complex, real-world problemsolving, particularly in multi-disciplinary and multimodal contexts. In this paper, we introduce a graduate-level, multi-disciplinary, EnglishChinese benchmark, dubbed as Reasoning Bench (R-Bench), for assessing the reasoning capability of both language and multimodal models. RBench spans 1,094 questions across 108 subjects for language model evaluation and 665 questions across 83 subjects for multimodal model testing in both English and Chinese. These questions are meticulously curated to ensure rigorous difficulty calibration, subject balance, and crosslinguistic alignment, enabling the assessment to be an Olympiad-level multi-disciplinary benchmark. We evaluate widely used models, including OpenAI o1, GPT-4o, DeepSeek-R1, etc. Experimental results indicate that advanced models perform poorly on complex reasoning, especially multimodal reasoning. Even the top-performing model OpenAI o1 achieves only 53.2% accuracy on our multimodal evaluation. Data and code are made publicly available at here.

Fu-Yun Wang, Zhengyang Geng, Hongsheng Li

Diffusion models achieve superior generation quality but suffer from slow generation speed due to the iterative nature of denoising. In contrast, consistency models, a new generative family, achieve competitive performance with significantly faster sampling. These models are trained either through consistency distillation, which leverages pretrained diffusion models, or consistency training/tuning directly from raw data. In this work, we propose a novel framework for understanding consistency models by modeling the denoising process of the diffusion model as a Markov Decision Process (MDP) and framing consistency model training as the value estimation through Temporal Difference~(TD) Learning. More importantly, this framework allows us to analyze the limitations of current consistency training/tuning strategies. Built upon Easy Consistency Tuning (ECT), we propose Stable Consistency Tuning (SCT), which incorporates variance-reduced learning using the score identity. SCT leads to significant performance improvements on benchmarks such as CIFAR-10 and ImageNet-64. On ImageNet-64, SCT achieves 1-step FID 2.42 and 2-step FID 1.55, a new SoTA for consistency models.

Meng-Hao Guo, Xuanyu Chu, Qianrui Yang et al.

The rapid advancement of native multi-modal models and omni-models, exemplified by GPT-4o, Gemini, and o3, with their capability to process and generate content across modalities such as text and images, marks a significant milestone in the evolution of intelligence. Systematic evaluation of their multi-modal output capabilities in visual thinking processes (also known as multi-modal chain of thought, M-CoT) becomes critically important. However, existing benchmarks for evaluating multi-modal models primarily focus on assessing multi-modal inputs and text-only reasoning while neglecting the importance of reasoning through multi-modal outputs. In this paper, we present a benchmark, dubbed RBench-V, designed to assess models' vision-indispensable reasoning abilities. To construct RBench-V, we carefully hand-pick 803 questions covering math, physics, counting, and games. Unlike previous benchmarks that typically specify certain input modalities, RBench-V presents problems centered on multi-modal outputs, which require image manipulation such as generating novel images and constructing auxiliary lines to support the reasoning process. We evaluate numerous open- and closed-source models on RBench-V, including o3, Gemini 2.5 Pro, Qwen2.5-VL, etc. Even the best-performing model, o3, achieves only 25.8% accuracy on RBench-V, far below the human score of 82.3%, highlighting that current models struggle to leverage multi-modal reasoning. Data and code are available at https://evalmodels.github.io/rbenchv

Tianle Cai, Yuhong Li, Zhengyang Geng et al.

Large Language Models (LLMs) employ auto-regressive decoding that requires sequential computation, with each step reliant on the previous one's output. This creates a bottleneck as each step necessitates moving the full model parameters from High-Bandwidth Memory (HBM) to the accelerator's cache. While methods such as speculative decoding have been suggested to address this issue, their implementation is impeded by the challenges associated with acquiring and maintaining a separate draft model. In this paper, we present Medusa, an efficient method that augments LLM inference by adding extra decoding heads to predict multiple subsequent tokens in parallel. Using a tree-based attention mechanism, Medusa constructs multiple candidate continuations and verifies them simultaneously in each decoding step. By leveraging parallel processing, Medusa substantially reduces the number of decoding steps required. We present two levels of fine-tuning procedures for Medusa to meet the needs of different use cases: Medusa-1: Medusa is directly fine-tuned on top of a frozen backbone LLM, enabling lossless inference acceleration. Medusa-2: Medusa is fine-tuned together with the backbone LLM, enabling better prediction accuracy of Medusa heads and higher speedup but needing a special training recipe that preserves the backbone model's capabilities. Moreover, we propose several extensions that improve or expand the utility of Medusa, including a self-distillation to handle situations where no training data is available and a typical acceptance scheme to boost the acceptance rate while maintaining generation quality. We evaluate Medusa on models of various sizes and training procedures. Our experiments demonstrate that Medusa-1 can achieve over 2.2x speedup without compromising generation quality, while Medusa-2 further improves the speedup to 2.3-3.6x.

Zhengyang Geng, Xin-Yu Zhang, Shaojie Bai et al.

This paper focuses on training implicit models of infinite layers. Specifically, previous works employ implicit differentiation and solve the exact gradient for the backward propagation. However, is it necessary to compute such an exact but expensive gradient for training? In this work, we propose a novel gradient estimate for implicit models, named phantom gradient, that 1) forgoes the costly computation of the exact gradient; and 2) provides an update direction empirically preferable to the implicit model training. We theoretically analyze the condition under which an ascent direction of the loss landscape could be found, and provide two specific instantiations of the phantom gradient based on the damped unrolling and Neumann series. Experiments on large-scale tasks demonstrate that these lightweight phantom gradients significantly accelerate the backward passes in training implicit models by roughly 1.7 times, and even boost the performance over approaches based on the exact gradient on ImageNet.

Yifei Wang, Zhengyang Geng, Feng Jiang et al.

Multi-view methods learn representations by aligning multiple views of the same image and their performance largely depends on the choice of data augmentation. In this paper, we notice that some other useful augmentations, such as image rotation, are harmful for multi-view methods because they cause a semantic shift that is too large to be aligned well. This observation motivates us to relax the exact alignment objective to better cultivate stronger augmentations. Taking image rotation as a case study, we develop a generic approach, Pretext-aware Residual Relaxation (Prelax), that relaxes the exact alignment by allowing an adaptive residual vector between different views and encoding the semantic shift through pretext-aware learning. Extensive experiments on different backbones show that our method can not only improve multi-view methods with existing augmentations, but also benefit from stronger image augmentations like rotation.

Zhengyang Geng, Meng-Hao Guo, Hongxu Chen et al.

As an essential ingredient of modern deep learning, attention mechanism, especially self-attention, plays a vital role in the global correlation discovery. However, is hand-crafted attention irreplaceable when modeling the global context? Our intriguing finding is that self-attention is not better than the matrix decomposition (MD) model developed 20 years ago regarding the performance and computational cost for encoding the long-distance dependencies. We model the global context issue as a low-rank recovery problem and show that its optimization algorithms can help design global information blocks. This paper then proposes a series of Hamburgers, in which we employ the optimization algorithms for solving MDs to factorize the input representations into sub-matrices and reconstruct a low-rank embedding. Hamburgers with different MDs can perform favorably against the popular global context module self-attention when carefully coping with gradients back-propagated through MDs. Comprehensive experiments are conducted in the vision tasks where it is crucial to learn the global context, including semantic segmentation and image generation, demonstrating significant improvements over self-attention and its variants.