Hancheng Min

Low-rank adaptation (LoRA) is the de facto method for parameter-efficient fine-tuning of large neural networks, achieving strong task adaptation with minimal trainable parameters and low optimization cost. Beyond this efficiency, practitioners have observed two striking properties: *robustness to catastrophic forgetting* and the ability to *merge independently trained adapters* into a single model that performs competitively across multiple tasks. These properties are central to continual and multi-task learning, yet remain poorly understood. In this work, we provide a theoretical explanation through a unified *margin-based perspective*. We analyze LoRA in multiclass linear classification under a near-orthogonal task regime with l_2-regularization, and characterize the optimal LoRA adapter across regularization regimes. Our analysis shows that, in an intermediate regime of regularization parameters, the optimal adapter aligns with the *max-margin solution* on the fine-tuning data. Building on this characterization, we derive two key consequences. First, we obtain closed-form expressions for the margins on pre-training and fine-tuning data, revealing a precise margin trade-off: the regularization parameter controls the balance between retention and adaptation. Second, we analyze adapter merging, proving that merged models achieve positive margin on each task and deriving optimal mixing coefficients that balance margins across tasks and maximize the margin over their union. These results lead to a simple, training-free merging rule, which we term *margin-based LoRA merging* (MaLoRA). Experiments on modern architectures and real datasets validate our theoretical predictions, showing that MaLoRA matches or outperforms several adapter-merging baselines across a range of vision and language classification tasks.

Transformer-based diffusion models have demonstrated remarkable performance at generating high-quality samples. However, our theoretical understanding of the reasons for this success remains limited. For instance, existing models are typically trained by minimizing a denoising objective, which is equivalent to fitting the score function of the training data. However, we do not know why transformer-based models can match the score function for denoising, or why gradient-based methods converge to the optimal denoising model despite the non-convex loss landscape. To the best of our knowledge, this paper provides the first convergence analysis for training transformer-based diffusion models. More specifically, we consider the population Denoising Diffusion Probabilistic Model (DDPM) objective for denoising data that follow a multi-token Gaussian mixture distribution. We theoretically quantify the required number of tokens per data point and training iterations for the global convergence towards the Bayes optimal risk of the denoising objective, thereby achieving a desired score matching error. A deeper investigation reveals that the self-attention module of the trained transformer implements a mean denoising mechanism that enables the trained model to approximate the oracle Minimum Mean Squared Error (MMSE) estimator of the injected noise in the diffusion steps. Numerical experiments validate these findings.

Deep neural networks (DNNs) trained via gradient descent (GD) with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. To theoretically understand this puzzling phenomenon, many works on convergence analysis for GD algorithms on NNs have been developed over the last half-decade. In this article, we review these research efforts and discuss how they address three specific questions related to this puzzle. The first question is why GD finds a global minimum efficiently, which the literature has addressed by studying what level of overparametrization (width, depth, etc.) and what type of initialization lead to a benign optimization landscape along the training trajectory, facilitating a linear convergence rate of GD. The next question is why the global minimum found by GD generalizes well, which has been addressed by showing that overparametrization induces an implicit simplicity bias along the GD trajectory. More recently, it has been observed that in practice, training DNNs with far larger learning rates than theoretically permissible results in faster convergence and better generalization. This leads to the third question of why faster convergence and better generalization can be achieved with a large learning rate, which has been recently addressed by identifying a self-stabilization mechanism and implicit bias toward flat minima.

Among many mysteries behind the success of deep networks lies the exceptional discriminative power of their learned representations as manifested by the intriguing Neural Collapse (NC) phenomenon, where simple feature structures emerge in the last layer of a trained neural network. Prior work on understanding NC theoretically has focused on analyzing the optimization landscape of matrix-factorization-like problems by considering the last-layer features as unconstrained free optimization variables and showing that their global minima exhibit NC. In this paper, we show that gradient flow on a two-layer ReLU network for classifying orthogonally separable data provably exhibits NC, thereby advancing prior results in two ways: First, we relax the assumption of unconstrained features, showing the effect of data structure and nonlinear activations on NC characterizations. Second, we reveal the role of the implicit bias of the training dynamics in facilitating the emergence of NC.

Deep learning-based classifiers are known to be vulnerable to adversarial attacks. Existing methods for defending against such attacks require adding a defense mechanism or modifying the learning procedure (e.g., by adding adversarial examples). This paper shows that for certain data distributions one can learn a provably robust classifier using standard learning methods and without adding a defense mechanism. More specifically, this paper addresses the problem of finding a robust classifier for a binary classification problem in which the data comes from an isotropic mixture of Gaussians with orthonormal cluster centers. First, we characterize the largest \ell_2-attack any classifier can defend against while maintaining high accuracy, and show the existence of optimal robust classifiers achieving this maximum \ell_2-robustness. Next, we show that given data sampled from the orthonormal Gaussian mixture model, gradient flow on a two-layer network with a polynomial ReLU activation and without adversarial examples provably finds an optimal robust classifier.

Deep neural networks (DNNs) trained via gradient descent (GD) with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. To theoretically understand this puzzling phenomenon, many works on convergence analysis for GD algorithms on NNs have been developed over the last half-decade. In this article, we review these research efforts and discuss how they address three specific questions related to this puzzle. The first question is why GD finds a global minimum efficiently, which the literature has addressed by studying what level of overparametrization (width, depth, etc.) and what type of initialization lead to a benign optimization landscape along the training trajectory, facilitating a linear convergence rate of GD. The next question is why the global minimum found by GD generalizes well, which has been addressed by showing that overparametrization induces an implicit simplicity bias along the GD trajectory. More recently, it has been observed that in practice, training DNNs with far larger learning rates than theoretically permissible results in faster convergence and better generalization. This leads to the third question of why faster convergence and better generalization can be achieved with a large learning rate, which has been recently addressed by identifying a self-stabilization mechanism and implicit bias toward flat minima.

Among many mysteries behind the success of deep networks lies the exceptional discriminative power of their learned representations as manifested by the intriguing Neural Collapse (NC) phenomenon, where simple feature structures emerge in the last layer of a trained neural network. Prior work on understanding NC theoretically has focused on analyzing the optimization landscape of matrix-factorization-like problems by considering the last-layer features as unconstrained free optimization variables and showing that their global minima exhibit NC. In this paper, we show that gradient flow on a two-layer ReLU network for classifying orthogonally separable data provably exhibits NC, thereby advancing prior results in two ways: First, we relax the assumption of unconstrained features, showing the effect of data structure and nonlinear activations on NC characterizations. Second, we reveal the role of the implicit bias of the training dynamics in facilitating the emergence of NC.

Deep learning-based classifiers are known to be vulnerable to adversarial attacks. Existing methods for defending against such attacks require adding a defense mechanism or modifying the learning procedure (e.g., by adding adversarial examples). This paper shows that for certain data distributions one can learn a provably robust classifier using standard learning methods and without adding a defense mechanism. More specifically, this paper addresses the problem of finding a robust classifier for a binary classification problem in which the data comes from an isotropic mixture of Gaussians with orthonormal cluster centers. First, we characterize the largest \ell_2-attack any classifier can defend against while maintaining high accuracy, and show the existence of optimal robust classifiers achieving this maximum \ell_2-robustness. Next, we show that given data sampled from the orthonormal Gaussian mixture model, gradient flow on a two-layer network with a polynomial ReLU activation and without adversarial examples provably finds an optimal robust classifier.

This paper studies the problem of training a two-layer ReLU network for binary classification using gradient flow with small initialization. We consider a training dataset with well-separated input vectors: Any pair of input data with the same label are positively correlated, and any pair with different labels are negatively correlated. Our analysis shows that, during the early phase of training, neurons in the first layer try to align with either the positive data or the negative data, depending on its corresponding weight on the second layer. A careful analysis of the neurons’ directional dynamics allows us to provide an \mathcalO(\frac\log n\sqrtμ) upper bound on the time it takes for all neurons to achieve good alignment with the input data, where n is the number of data points and μmeasures how well the data are separated. After the early alignment phase, the loss converges to zero at a \mathcalO(\frac1t) rate, and the weight matrix on the first layer is approximately low-rank. Numerical experiments on the MNIST dataset illustrate our theoretical findings.

In this paper, we analyze the convergence of gradient flow on a multi-layer linear model with a loss function of the form f(W_1W_2⋯W_L). We show that when f satisfies the gradient dominance property, proper weight initialization leads to exponential convergence of the gradient flow to a global minimum of the loss. Moreover, the convergence rate depends on two trajectory-specific quantities that are controlled by the weight initialization: the imbalance matrices, which measure the difference between the weights of adjacent layers, and the least singular value of the weight product W=W_1W_2⋯W_L. Our analysis exploits the fact that the gradient of the overparameterized loss can be written as the composition of the non-overparametrized gradient with a time-varying (weight-dependent) linear operator whose smallest eigenvalue controls the convergence rate. The key challenge we address is to derive a uniform lower bound for this time-varying eigenvalue that lead to improved rates for several multi-layer network models studied in the literature.