This paper targets on the regularization effect of momentum-based methods in regression settings and analyzes the popular diagonal linear networks to precisely characterize the implicit bias of continuous versions of heavy-ball (HB) and Nesterov's method of accelerated gradients (NAG). We show that, HB and NAG exhibit different implicit bias compared to GD for diagonal linear networks, which is different from the one for classic linear regression problem where momentum-based methods share the same implicit bias with GD. Specifically, the role of momentum in the implicit bias of GD is twofold: (a) HB and NAG induce extra initialization mitigation effects similar to SGD that are beneficial for generalization of sparse regression; (b) the implicit regularization effects of HB and NAG also depend on the initialization of gradients explicitly, which may not be benign for generalization. As a result, whether HB and NAG have better generalization properties than GD jointly depends on the aforementioned twofold effects determined by various parameters such as learning rate, momentum factor, and integral of gradients. Our findings highlight the potential beneficial role of momentum and can help understand its advantages in practice such as when it will lead to better generalization performance.

Understanding the dynamics of causal structures is crucial for uncovering the underlying processes in time series data. Previous approaches rely on static assumptions, where contemporaneous and time-lagged dependencies are assumed to have invariant topological structures. However, these models fail to capture the evolving causal relationship between variables when the underlying process exhibits such dynamics. To address this limitation, we propose DyCAST, a novel framework designed to learn dynamic causal structures in time series using Neural Ordinary Differential Equations (Neural ODEs). The key innovation lies in modeling the temporal dynamics of the contemporaneous structure, drawing inspiration from recent advances in Neural ODEs on constrained manifolds. We reformulate the task of learning causal structures at each time step as solving the solution trajectory of a Neural ODE on the directed acyclic graph (DAG) manifold. To accommodate high-dimensional causal structures, we extend DyCAST by learning the temporal dynamics of the hidden state for contemporaneous causal structure. Experiments on both synthetic and real-world datasets demonstrate that DyCAST achieves superior or comparable performance compared to existing causal discovery models.

Transformers and many other deep learning models are empirically shown to predictably enhance their performance as a power law in training time, model size, or the number of training data points, which is termed as the neural scaling law. This paper studies this intriguing phenomenon particularly for the transformer architecture in theoretical setups. Specifically, we propose a framework for linear self-attention, the underpinning block of transformer without softmax, to learn in an in-context manner, where the corresponding learning dynamics is modeled as a non-linear ordinary differential equation (ODE) system. Furthermore, we establish a procedure to derive a tractable approximate solution for this ODE system by reformulating it as a *Riccati equation*, which allows us to precisely characterize neural scaling laws for linear self-attention with training time, model size, data size, and the optimal compute. In addition, we reveal that the linear self-attention shares similar neural scaling laws with several other architectures when the context sequence length of the in-context learning is fixed, otherwise it would exhibit a different scaling law of training time.

This paper establishes a continuous time approximation, a piece-wise continuous differential equation, for the discrete Heavy-Ball (HB) momentum method with explicit discretization error. Investigating continuous differential equations has been a promising approach for studying the discrete optimization methods. Despite the crucial role of momentum in gradient-based optimization methods, the gap between the original dynamics and the continuous time approximations due to the discretization error has not been comprehensively bridged yet. In this work, we study the HB momentum method in continuous time while putting more focus on the discretization error to provide additional theoretical tools to this area. In particular, we design a first-order piece-wise continuous differential equation, where we add a number of counter terms to account for the discretization error explicitly. As a result, we provide a continuous time model for the HB momentum method that allows the control of discretization error to arbitrary order of the learning rate. As an application, we leverage it to find a new implicit regularization of the directional smoothness and investigate the implicit bias of HB for diagonal linear networks, indicating how our results can be used in deep learning. Our theoretical findings are further supported by numerical experiments.

Adversarial training has been empirically demonstrated as an effective strategy to improve the robustness of deep neural networks (DNNs) against adversarial examples. However, the underlying reason of its effectiveness is still non-transparent. In this paper we conduct both extensive theoretical and empirical analysis on the implicit bias induced by adversarial training from a generalized margin perspective. Our results focus on adversarial training for homogeneous DNNs. In particular, (i) For deep linear networks with $\ell _{p}$ℓp-norm perturbation, we show that weight matrices of adjacent layers get aligned and the converged parameters maximize the margin of adversarial examples, which can be further viewed as a generalized margin of the original dataset that can be achieved by an interpolation solution between $\ell _{2}$ℓ2-SVM and $\ell _{q}$ℓq-SVM where $1/p + 1/q=1$1/p+1/q=1. (ii) For general homogeneous DNNs, including both linear and nonlinear ones, we investigate adversarial training with a variety of adversarial perturbations in a unified manner. Specifically, we show that the direction of the limit point of parameters converges to a KKT point of a constrained optimization problem that aims to maximize the margin for adversarial examples. Additionally, as an application of this general result for two special linear homogeneous DNNs, diagonal linear networks and linear convolutional networks, we show that adversarial training with $\ell _{p}$ℓp-norm perturbation equivalently minimizes an interpolation norm that depends on the depth, the architecture, and the value of $p$p in the predictor space. Extensive experiments are conducted to verify theoretical claims. Our results theoretically provide the basis for the longstanding folklore Madry et al. 2018 that adversarial training modifies the decision boundary by utilizing adversarial examples to improve robustness, and potentially provide insights for designing new robust training strategies.

Studying the implicit bias of gradient descent (GD) and stochastic gradient descent (SGD) is critical to unveil the underlying mechanism of deep learning. Unfortunately, even for standard linear networks in regression setting, a comprehensive characterization of the implicit bias is still an open problem. This paper proposes to investigate a new proxy model of standard linear network, rank-1 linear network, where each weight matrix is parameterized as a rank-1 form. For over-parameterized regression problem, we precisely analyze the implicit bias of GD and SGD---by identifying a “potential” function such that GD converges to its minimizer constrained by zero training error (i.e., interpolation solution), and further characterizing the role of the noise introduced by SGD in perturbing the form of this potential. Our results explicitly connect the depth of the network and the initialization with the implicit bias of GD and SGD. Furthermore, we emphasize a new implicit bias of SGD jointly induced by stochasticity and over-parameterization, which can reduce the dependence of the SGD's solution on the initialization. Our findings regarding the implicit bias are different from that of a recently popular model, the diagonal linear network. We highlight that the induced bias of our rank-1 model is more consistent with standard linear network while the diagonal one is not. This suggests that the proposed rank-1 linear network might be a plausible proxy for standard linear net.