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 Phase I, the magnitudes of model weights progressively diminish, and the model escapes the lazy regime; enters the rich regime. In 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.

The Adam optimizer is widely used for transformer optimization in practice, which makes understanding the underlying optimization mechanisms an important problem. However, due to the Adam's complexity, theoretical analysis of how it optimizes transformers remains a challenging task. Fortunately, Sign Gradient Descent (SignGD) serves as an effective surrogate for Adam. Despite its simplicity, theoretical understanding of how SignGD optimizes transformers still lags behind. In this work, we study how SignGD optimizes a two-layer transformer -- consisting of a softmax attention layer with trainable query-key parameterization followed by a linear layer -- on a linearly separable noisy dataset. We identify four stages in the training dynamics, each exhibiting intriguing behaviors. Based on the training dynamics, we prove the fast convergence but poor generalization of the learned transformer on the noisy dataset. We also show that Adam behaves similarly to SignGD in terms of both optimization and generalization in this setting. Additionally, we find that the poor generalization of SignGD is not solely due to data noise, suggesting that both SignGD and Adam requires high-quality data for real-world tasks. Finally, experiments on synthetic and real-world datasets empirically support our theoretical results.

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.

Transformers have become the cornerstone of modern AI. Unlike traditional architectures, transformers exhibit a distinctive characteristic: diverse types of building blocks, such as embedding layers, normalization layers, self-attention mechanisms, and point-wise feed-forward networks, work collaboratively. Understanding the disparities and interactions among these blocks is therefore important. In this paper, we uncover a clear **sharpness disparity** across these blocks, which intriguingly emerges early in training and persists throughout the training process. Building on this insight, we propose a novel **Blockwise Learning Rate (LR)** strategy to accelerate large language model (LLM) pre-training. Specifically, by integrating Blockwise LR into AdamW, we consistently achieve lower terminal loss and nearly $2\times$ speedup compared to vanilla AdamW. This improvement is demonstrated across GPT-2 and LLaMA models, with model sizes ranging from 0.12B to 1.1B and datasets including OpenWebText and MiniPile. 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 savings. These results underscore the potential of leveraging the sharpness disparity principle to improve LLM training.

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.

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 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, SE-Merging achieves dynamic model merging without additional training. Extensive experiments demonstrate that SE-Merging achieves significant performance improvements while remaining compatible with existing model merging techniques.

We prove that when we do the Taylor series expansion of the loss function, 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. We believe that proving the blocking of certain loss terms provides an analytic perspective for potential detects of a deep model with BN operations, although the blocking problem is not fully equivalent to significant damages in all tasks on benchmark datasets. Experiments show that the BN operation significantly affects feature representations in specific tasks.

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. We have also drawn several interesting empirical findings, including: 1) the larger the network, the more redundant features it learns; 2) in particular, we show how to prune the networks based on our finding using direct equivalent feature merging, without fine-tuning which is often needed in peer network pruning methods; 3) same structured networks with higher feature complexity achieve better performance; 4) through the layers of a neural network, the feature complexity first increase then decrease; 5) for the image classification task, a group of functionally equivalent features may correspond to a specific semantic meaning. Source code will be made publicly available.

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.

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.

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 permutation approaches. The study of LLFC transcends and advances our understanding of LMC by adopting a feature-learning perspective.

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 robustness-boosting methods in a principle way. Besides, our findings also make a revision of previous inaccurate understanding of the shape bias of adversarially learned features. Our code is available online at https://github.com/Jie-Ren/A-Unified-Game-Theoretic-Interpretation-of-Adversarial-Robustness.