Zhanpeng Zhou

Tongcheng Zhang, Zhanpeng Zhou, Mingze Wang et al. · oxford

One crucial factor behind the success of deep learning lies in the implicit bias induced by noise inherent in gradient-based training algorithms. Motivated by empirical observations that training with noisy labels improves model generalization, we delve into the underlying mechanisms behind stochastic gradient descent (SGD) with label noise. Focusing on a two-layer over-parameterized linear network, we analyze the learning dynamics of label noise SGD, unveiling a two-phase learning behavior. In \emph{Phase I}, the magnitudes of model weights progressively diminish, and the model escapes the lazy regime; enters the rich regime. In \emph{Phase II}, the alignment between model weights and the ground-truth interpolator increases, and the model eventually converges. Our analysis highlights the critical role of label noise in driving the transition from the lazy to the rich regime and minimally explains its empirical success. Furthermore, we extend these insights to Sharpness-Aware Minimization (SAM), showing that the principles governing label noise SGD also apply to broader optimization algorithms. Extensive experiments, conducted under both synthetic and real-world setups, strongly support our theory. Our code is released at https://github.com/a-usually/Label-Noise-SGD.

Zhanpeng Zhou, Yongyi Yang, Xiaojiang Yang et al.

Recent work has revealed many intriguing empirical phenomena in neural network training, despite the poorly understood and highly complex loss landscapes and training dynamics. One of these phenomena, Linear Mode Connectivity (LMC), has gained considerable attention due to the intriguing observation that different solutions can be connected by a linear path in the parameter space while maintaining near-constant training and test losses. In this work, we introduce a stronger notion of linear connectivity, Layerwise Linear Feature Connectivity (LLFC), which says that the feature maps of every layer in different trained networks are also linearly connected. We provide comprehensive empirical evidence for LLFC across a wide range of settings, demonstrating that whenever two trained networks satisfy LMC (via either spawning or permutation methods), they also satisfy LLFC in nearly all the layers. Furthermore, we delve deeper into the underlying factors contributing to LLFC, which reveal new insights into the spawning and permutation approaches. The study of LLFC transcends and advances our understanding of LMC by adopting a feature-learning perspective.

Ling Tang, Wen Shen, Zhanpeng Zhou et al.

In this paper, we prove the representation defects of a cascaded convolutional decoder network, considering the capacity of representing different frequency components of an input sample. We conduct the discrete Fourier transform on each channel of the feature map in an intermediate layer of the decoder network. Then, we extend the 2D circular convolution theorem to represent the forward and backward propagations through convolutional layers in the frequency domain. Based on this, we prove three defects in representing feature spectrums. First, we prove that the convolution operation, the zero-padding operation, and a set of other settings all make a convolutional decoder network more likely to weaken high-frequency components. Second, we prove that the upsampling operation generates a feature spectrum, in which strong signals repetitively appear at certain frequencies. Third, we prove that if the frequency components in the input sample and frequency components in the target output for regression have a small shift, then the decoder usually cannot be effectively learned.

Zhanpeng Zhou, Wen Shen, Huixin Chen et al.

In this paper, we prove the effects of the BN operation on the back-propagation of the first and second derivatives of the loss. When we do the Taylor series expansion of the loss function, we prove that the BN operation will block the influence of the first-order term and most influence of the second-order term of the loss. We also find that such a problem is caused by the standardization phase of the BN operation. Experimental results have verified our theoretical conclusions, and we have found that the BN operation significantly affects feature representations in specific tasks, where losses of different samples share similar analytic formulas.

MiniMax, Aili Chen, Aonian Li et al.

We introduce the MiniMax-M2 series, a family of Mixture-of-Experts language models built around the principle that mini activations can unleash maximum real-world intelligence. The flagship M2 contains 229.9B total parameters with only 9.8B activated per token. Designed end-to-end for agentic deployment, the M2 series rests on three components: (i) agent-driven data pipelines producing large-scale, verifiable trajectories across agentic coding and agentic cowork, each grounded in an executable workspace and an artifact-aligned reward; (ii) Forge, a scalable agent-native RL system that adapts to long-horizon agent trajectories, paired with windowed-FIFO scheduling, prefix-tree merging, inference optimization, and a clean training-inference-agent decoupling that supports both white-box and black-box agents; (iii) the latest M2.7 checkpoint takes an early step toward self-evolution -- autonomously debugging training runs and modifying its own scaffold. Across M2 through M2.7, this combination translates a mini-activation footprint into frontier-tier performance on agentic coding, deep search, office-task, and reasoning benchmarks.

Xuan Cui, Huiyue Li, Run Zeng et al.

As large language models (LLMs) scale to billions of parameters, full-parameter fine-tuning becomes compute- and memory-prohibitive. Parameter-efficient fine-tuning (PEFT) mitigates this issue by updating only a small set of task-specific parameters while keeping the base model frozen. Among PEFT approaches, low-rank adaptation (LoRA) is widely adopted; however, it enforces a uniform rank across layers despite substantial variation in layer importance, motivating {layerwise} rank allocation. Recent adaptive-rank variants (e.g., AdaLoRA) allocate ranks based on importance scores, yet typically rely on instantaneous gradients that capture only local sensitivity, overlooking non-local, pathwise effects within the same layer, which yields unstable and biased scores. To address this limitation, we introduce IGU-LoRA, an adaptive-rank LoRA that (i) computes within-layer Integrated Gradients (IG) sensitivities and aggregates them into a layer-level score for rank allocation, and (ii) applies an uncertainty-aware scheme using exponential moving averages with deviation tracking to suppress noisy updates and calibrate rank selection. Theoretically, we prove an upper bound on the composite trapezoidal rule approximation error for parameter-space IG under a pathwise Hessian-Lipschitz condition, which informs the quadrature budget. Across diverse tasks and architectures, IGU-LoRA consistently outperforms strong PEFT baselines at matched parameter budgets, improving downstream accuracy and robustness. Ablations confirm the contributions of pathwise within-layer sensitivity estimates and uncertainty-aware selection to effective rank allocation. Our code is publicly available at https://github.com/withyou12/igulora.git

Jianhao Huang, Zhanpeng Zhou, Renqiu Xia et al.

While next-token prediction (NTP) has been the standard objective for training language models, it often struggles to capture global structure in reasoning tasks. Multi-token prediction (MTP) has recently emerged as a promising alternative, yet its underlying mechanisms remain poorly understood. In this paper, we study how MTP facilitates reasoning, with a focus on planning. Empirically, we show that MTP consistently outperforms NTP on both synthetic graph path-finding tasks and more realistic reasoning benchmarks, such as Countdown and boolean satisfiability problems. Theoretically, we analyze a simplified two-layer Transformer on a star graph task. We prove that MTP induces a two-stage reverse reasoning process: the model first attends to the end node and then reconstructs the path by tracing intermediate nodes backward. This behavior arises from a gradient decoupling property of MTP, which provides a cleaner training signal compared to NTP. Ultimately, our results highlight how multi-token objectives inherently bias optimization toward robust and interpretable reasoning circuits.

Yiting Chen, Zhanpeng Zhou, Junchi Yan

The behavior of neural networks still remains opaque, and a recently widely noted phenomenon is that networks often achieve similar performance when initialized with different random parameters. This phenomenon has attracted significant attention in measuring the similarity between features learned by distinct networks. However, feature similarity could be vague in describing the same feature since equivalent features hardly exist. In this paper, we expand the concept of equivalent feature and provide the definition of what we call functionally equivalent features. These features produce equivalent output under certain transformations. Using this definition, we aim to derive a more intrinsic metric for the so-called feature complexity regarding the redundancy of features learned by a neural network at each layer. We offer a formal interpretation of our approach through the lens of category theory, a well-developed area in mathematics. To quantify the feature complexity, we further propose an efficient algorithm named Iterative Feature Merging. Our experimental results validate our ideas and theories from various perspectives. We empirically demonstrate that the functionally equivalence widely exists among different features learned by the same neural network and we could reduce the number of parameters of the network without affecting the performance.The IFM shows great potential as a data-agnostic model prune method. We have also drawn several interesting empirical findings regarding the defined feature complexity.

Zhanpeng Zhou, Zijun Chen, Yilan Chen et al.

The pretraining-finetuning paradigm has become the prevailing trend in modern deep learning. In this work, we discover an intriguing linear phenomenon in models that are initialized from a common pretrained checkpoint and finetuned on different tasks, termed as Cross-Task Linearity (CTL). Specifically, we show that if we linearly interpolate the weights of two finetuned models, the features in the weight-interpolated model are often approximately equal to the linear interpolation of features in two finetuned models at each layer. We provide comprehensive empirical evidence supporting that CTL consistently occurs for finetuned models that start from the same pretrained checkpoint. We conjecture that in the pretraining-finetuning paradigm, neural networks approximately function as linear maps, mapping from the parameter space to the feature space. Based on this viewpoint, our study unveils novel insights into explaining model merging/editing, particularly by translating operations from the parameter space to the feature space. Furthermore, we delve deeper into the root cause for the emergence of CTL, highlighting the role of pretraining.

Zhanpeng Zhou, Mingze Wang, Yuchen Mao et al.

Sharpness-Aware Minimization (SAM) has substantially improved the generalization of neural networks under various settings. Despite the success, its effectiveness remains poorly understood. In this work, we discover an intriguing phenomenon in the training dynamics of SAM, shedding light on understanding its implicit bias towards flatter minima over Stochastic Gradient Descent (SGD). Specifically, we find that SAM efficiently selects flatter minima late in training. Remarkably, even a few epochs of SAM applied at the end of training yield nearly the same generalization and solution sharpness as full SAM training. Subsequently, we delve deeper into the underlying mechanism behind this phenomenon. Theoretically, we identify two phases in the learning dynamics after applying SAM late in training: i) SAM first escapes the minimum found by SGD exponentially fast; and ii) then rapidly converges to a flatter minimum within the same valley. Furthermore, we empirically investigate the role of SAM during the early training phase. We conjecture that the optimization method chosen in the late phase is more crucial in shaping the final solution's properties. Based on this viewpoint, we extend our findings from SAM to Adversarial Training.

Jinbo Wang, Mingze Wang, Zhanpeng Zhou et al.

Transformers consist of diverse building blocks, such as embedding layers, normalization layers, self-attention mechanisms, and point-wise feedforward networks. Thus, understanding the differences and interactions among these blocks is important. In this paper, we uncover a clear Sharpness Disparity across these blocks, which emerges early in training and intriguingly persists throughout the training process. Motivated by this finding, we propose Blockwise Learning Rate (LR), a strategy that tailors the LR to each block's sharpness, accelerating large language model (LLM) pre-training. By integrating Blockwise LR into AdamW, we consistently achieve lower terminal loss and nearly $2\times$ speedup compared to vanilla AdamW. We demonstrate this acceleration across GPT-2 and LLaMA, with model sizes ranging from 0.12B to 2B and datasets of OpenWebText, MiniPile, and C4. Finally, we incorporate Blockwise LR into Adam-mini (Zhang et al., 2024), a recently proposed memory-efficient variant of Adam, achieving a combined $2\times$ speedup and $2\times$ memory saving. These results underscore the potential of exploiting the sharpness disparity to improve LLM training.

Andi Han, Wei Huang, Zhanpeng Zhou et al.

Deep learning with noisy labels presents significant challenges. In this work, we theoretically characterize the role of label noise from a feature learning perspective. Specifically, we consider a signal-noise data distribution, where each sample comprises a label-dependent signal and label-independent noise, and rigorously analyze the training dynamics of a two-layer convolutional neural network under this data setup, along with the presence of label noise. Our analysis identifies two key stages. In Stage I, the model perfectly fits all the clean samples (i.e., samples without label noise) while ignoring the noisy ones (i.e., samples with noisy labels). During this stage, the model learns the signal from the clean samples, which generalizes well on unseen data. In Stage II, as the training loss converges, the gradient in the direction of noise surpasses that of the signal, leading to overfitting on noisy samples. Eventually, the model memorizes the noise present in the noisy samples and degrades its generalization ability. Furthermore, our analysis provides a theoretical basis for two widely used techniques for tackling label noise: early stopping and sample selection. Experiments on both synthetic and real-world setups validate our theory.

Zhanpeng Zhou, Yongyi Yang, Mahito Sugiyama et al.

Understanding how deep neural networks learn remains a fundamental challenge in modern machine learning. A growing body of evidence suggests that training dynamics undergo a distinct phase transition, yet our understanding of this transition is still incomplete. In this paper, we introduce an interval-wise perspective that compares network states across a time window, revealing two new phenomena that illuminate the two-phase nature of deep learning. i) \textbf{The Chaos Effect.} By injecting an imperceptibly small parameter perturbation at various stages, we show that the response of the network to the perturbation exhibits a transition from chaotic to stable, suggesting there is an early critical period where the network is highly sensitive to initial conditions; ii) \textbf{The Cone Effect.} Tracking the evolution of the empirical Neural Tangent Kernel (eNTK), we find that after this transition point the model's functional trajectory is confined to a narrow cone-shaped subset: while the kernel continues to change, it gets trapped into a tight angular region. Together, these effects provide a structural, dynamical view of how deep networks transition from sensitive exploration to stable refinement during training.

Zhanpeng Zhou, Yongyi Yang, Jie Ren et al.

Understanding the learning dynamics of neural networks is a central topic in the deep learning community. In this paper, we take an empirical perspective to study the learning dynamics of neural networks in real-world settings. Specifically, we investigate the evolution process of the empirical Neural Tangent Kernel (eNTK) during training. Our key findings reveal a two-phase learning process: i) in Phase I, the eNTK evolves significantly, signaling the rich regime, and ii) in Phase II, the eNTK keeps evolving but is constrained in a narrow space, a phenomenon we term the cone effect. This two-phase framework builds on the hypothesis proposed by Fort et al. (2020), but we uniquely identify the cone effect in Phase II, demonstrating its significant performance advantages over fully linearized training.

Jinbo Wang, Binghui Li, Zhanpeng Zhou et al.

Batch size scheduling (BSS) plays a critical role in large-scale deep learning training, influencing both optimization dynamics and computational efficiency. Yet, its theoretical foundations remain poorly understood. In this work, we show that the functional scaling law (FSL) framework introduced in Li et al. (2025a) provides a principled lens for analyzing BSS. Specifically, we characterize the optimal BSS under a fixed data budget and show that its structure depends sharply on task difficulty. For easy tasks, optimal schedules keep increasing batch size throughout. In contrast, for hard tasks, the optimal schedule maintains small batch sizes for most of training and switches to large batches only in a late stage. To explain the emergence of late switching, we uncover a dynamical mechanism -- the fast catch-up effect -- which also manifests in large language model (LLM) pretraining. After switching from small to large batches, the loss rapidly aligns with the constant large-batch trajectory. Using FSL, we show that this effect stems from rapid forgetting of accumulated gradient noise, with the catch-up speed determined by task difficulty. Crucially, this effect implies that large batches can be safely deferred to late training without sacrificing performance, while substantially reducing data consumption. Finally, extensive LLM pretraining experiments -- covering both Dense and MoE architectures with up to 1.1B parameters and 1T tokens -- validate our theoretical predictions. Across all settings, late-switch schedules consistently outperform constant-batch and early-switch baselines.

Bingrui Li, Jiaxin Wen, Zhanpeng Zhou et al.

As hyperparameter tuning becomes increasingly costly at scale, efficient tuning methods are essential. Yet principles for guiding hyperparameter tuning remain limited. In this work, we seek to establish such principles by considering a broad range of hyperparameters, including batch size, learning rate, and weight decay. We identify a phenomenon we call trajectory invariance, where pre-training loss curves, gradient noise, and gradient norm exhibit invariance--closely overlapping--with respect to a quantity that combines learning rate and weight decay. This phenomenon effectively reduces the original two-dimensional hyperparameter space to one dimension, yielding an efficient tuning rule: follow the salient direction revealed by trajectory invariance. Furthermore, we refine previous scaling laws and challenge several existing viewpoints. Overall, our work proposes new principles for efficient tuning and inspires future research on scaling laws.

Tongtian Zhu, Tianyu Zhang, Mingze Wang et al.

Decentralized learning provides a scalable alternative to parameter-server-based training, yet its performance is often hindered by limited peer-to-peer communication. In this paper, we study how communication should be scheduled over time to improve global generalization, including determining when and how frequently devices synchronize. Counterintuitive empirical results show that concentrating communication budgets in the later stages of decentralized training remarkably improves global generalization. Surprisingly, we uncover that fully connected communication at the final step, implemented by a single global merging, can significant improve the generalization performance of decentralized learning under serve high data heterogeneity. Our theoretical contributions, which explains these phenomena, are first to establish that the globally merged model of decentralized SGD can match the convergence rate of parallel SGD. Technically, we reinterpret part of the discrepancy among local models, which were previously considered as detrimental noise, as constructive components essential for matching this rate. This work provides promising results that decentralized learning is able to generalize under high data heterogeneity and limited communication, while offering broad new avenues for model merging research. The code will be made publicly available.

Zijun Chen, Zhanpeng Zhou, Bo Zhang et al.

Model merging has gained increasing attention due to its intriguing property: interpolating the parameters of different task-specific fine-tuned models leads to multi-task abilities. However, despite its empirical success, the underlying mechanisms of model merging remain poorly understood. In this work, we delve into the mechanism behind model merging from a representation perspective. Our analysis reveals that model merging achieves multi-task abilities through two key capabilities: i) distinguishing samples from different tasks, and ii) adapting to the corresponding expert model for each sample. These two capabilities allow the merged model to retain task-specific expertise, enabling efficient multi-task adaptation. Building on these insights, we propose \texttt{SE-Merging}, a self-enhanced model merging framework that leverages these two characteristics to dynamically identify the corresponding task for each sample and then adaptively rescales the merging coefficients to further enhance task-specific expertise in the merged model. Notably, \texttt{SE-Merging} achieves dynamic model merging without additional training. Extensive experiments demonstrate that \texttt{SE-Merging} achieves significant performance improvements while remaining compatible with existing model merging techniques.

Jie Ren, Die Zhang, Yisen Wang et al.

This paper provides a unified view to explain different adversarial attacks and defense methods, \emph{i.e.} the view of multi-order interactions between input variables of DNNs. Based on the multi-order interaction, we discover that adversarial attacks mainly affect high-order interactions to fool the DNN. Furthermore, we find that the robustness of adversarially trained DNNs comes from category-specific low-order interactions. Our findings provide a potential method to unify adversarial perturbations and robustness, which can explain the existing defense methods in a principle way. Besides, our findings also make a revision of previous inaccurate understanding of the shape bias of adversarially learned features.

Jie Ren, Zhanpeng Zhou, Qirui Chen et al.

Masking some input variables of a deep neural network (DNN) and computing output changes on the masked input sample represent a typical way to compute attributions of input variables in the sample. People usually mask an input variable using its baseline value. However, there is no theory to examine whether baseline value faithfully represents the absence of an input variable, \emph{i.e.,} removing all signals from the input variable. Fortunately, recent studies show that the inference score of a DNN can be strictly disentangled into a set of causal patterns (or concepts) encoded by the DNN. Therefore, we propose to use causal patterns to examine the faithfulness of baseline values. More crucially, it is proven that causal patterns can be explained as the elementary rationale of the Shapley value. Furthermore, we propose a method to learn optimal baseline values, and experimental results have demonstrated its effectiveness.

Jie Ren, Die Zhang, Yisen Wang et al.

This paper provides a unified view to explain different adversarial attacks and defense methods, i.e. the view of multi-order interactions between input variables of DNNs. Based on the multi-order interaction, we discover that adversarial attacks mainly affect high-order interactions to fool the DNN. Furthermore, we find that the robustness of adversarially trained DNNs comes from category-specific low-order interactions. Our findings provide a potential method to unify adversarial perturbations and robustness, which can explain the existing defense methods in a principle way. Besides, our findings also make a revision of previous inaccurate understanding of the shape bias of adversarially learned features.