Bin Shi

Mingwei Fu, Bin Shi

A numerical framework for constructing full-dimensional quasi-periodic solutions in nearly integrable systems was recently developed by Fu and Shi[2026]. Based on an alternating scheme, this approach effectively overcomes the secular drift in angle variables, a fundamental limitation of symplectic integrators. However, in many applications, such as the restricted three-body problem, lower-dimensional quasi-periodic solutions hold greater significance. The construction of these solutions is considerably more challenging due to the presence of normal frequencies, leading to intricate resonance phenomena. Beyond the subspace resonance, one must also account for the first and second Melnikov conditions to eliminate small divisors. In this study, we extend the proposed alternating numerical scheme to compute the elliptic lower-dimensional quasi-periodic solutions. Numerical experiments are presented for the Hénon-Heiles model and the Fermi--Pasta--Ulam (FPU) model, demonstrating the effectiveness of the proposed method. Furthermore, we emphasize that the perturbation is not merely a polynomial with real coefficients but is a real-valued function. As a result, the associated perturbation operator exhibits Gevrey decay without possessing a Hankel structure. Meanwhile, we further simplify the multi-scale analysis by exploiting the resolvent identity, showing that the global inverse can be expressed linearly in terms of local inverses via the gluing procedure. This representation reveals a regime-dependent interaction structure: weak interactions dominate at short range, while strong interactions emerge at long range. This balance ensures that the Gevrey decay of the inverse remains uniformly controlled. Moreover, within this linear representation, the inversion conditions provide a clearer characterization of the localization properties.

Rui Zhao, Bin Shi, Zhiming Liang et al.

Graph Neural Networks (GNNs) have been widely employed for semi-supervised node classification tasks on graphs. However, the performance of GNNs is significantly affected by label noise, that is, a small amount of incorrectly labeled nodes can substantially misguide model training. Mainstream solutions define node classification with label noise (NCLN) as a reliable labeling task, often introducing node similarity with quadratic computational complexity to more accurately assess label reliability. To this end, in this paper, we introduce the Label Ensemble Graph Neural Network (LEGNN), a lower complexity method for robust GNNs training against label noise. LEGNN reframes NCLN as a label ensemble task, gathering informative multiple labels instead of constructing a single reliable label, avoiding high-complexity computations for reliability assessment. Specifically, LEGNN conducts a two-step process: bootstrapping neighboring contexts and robust learning with gathered multiple labels. In the former step, we apply random neighbor masks for each node and gather the predicted labels as a high-probability label set. This mitigates the impact of inaccurately labeled neighbors and diversifies the label set. In the latter step, we utilize a partial label learning based strategy to aggregate the high-probability label information for model training. Additionally, we symmetrically gather a low-probability label set to counteract potential noise from the bootstrapped high-probability label set. Extensive experiments on six datasets demonstrate that LEGNN achieves outstanding performance while ensuring efficiency. Moreover, it exhibits good scalability on dataset with over one hundred thousand nodes and one million edges.

Bin Shi

Following the same routine as [SSJ20], we continue to present the theoretical analysis for stochastic gradient descent with momentum (SGD with momentum) in this paper. Differently, for SGD with momentum, we demonstrate it is the two hyperparameters together, the learning rate and the momentum coefficient, that play the significant role for the linear rate of convergence in non-convex optimization. Our analysis is based on the use of a hyperparameters-dependent stochastic differential equation (hp-dependent SDE) that serves as a continuous surrogate for SGD with momentum. Similarly, we establish the linear convergence for the continuous-time formulation of SGD with momentum and obtain an explicit expression for the optimal linear rate by analyzing the spectrum of the Kramers-Fokker-Planck operator. By comparison, we demonstrate how the optimal linear rate of convergence and the final gap for SGD only about the learning rate varies with the momentum coefficient increasing from zero to one when the momentum is introduced. Then, we propose a mathematical interpretation why the SGD with momentum converges faster and more robust about the learning rate than the standard SGD in practice. Finally, we show the Nesterov momentum under the existence of noise has no essential difference with the standard momentum.

Bin Shi, Weijie J. Su, Michael I. Jordan

The learning rate is perhaps the single most important parameter in the training of neural networks and, more broadly, in stochastic (nonconvex) optimization. Accordingly, there are numerous effective, but poorly understood, techniques for tuning the learning rate, including learning rate decay, which starts with a large initial learning rate that is gradually decreased. In this paper, we present a general theoretical analysis of the effect of the learning rate in stochastic gradient descent (SGD). Our analysis is based on the use of a learning-rate-dependent stochastic differential equation (lr-dependent SDE) that serves as a surrogate for SGD. For a broad class of objective functions, we establish a linear rate of convergence for this continuous-time formulation of SGD, highlighting the fundamental importance of the learning rate in SGD, and contrasting to gradient descent and stochastic gradient Langevin dynamics. Moreover, we obtain an explicit expression for the optimal linear rate by analyzing the spectrum of the Witten-Laplacian, a special case of the Schrödinger operator associated with the lr-dependent SDE. Strikingly, this expression clearly reveals the dependence of the linear convergence rate on the learning rate -- the linear rate decreases rapidly to zero as the learning rate tends to zero for a broad class of nonconvex functions, whereas it stays constant for strongly convex functions. Based on this sharp distinction between nonconvex and convex problems, we provide a mathematical interpretation of the benefits of using learning rate decay for nonconvex optimization.

Bin Shi, Simon S. Du, Weijie J. Su et al.

We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme. We show that the optimization algorithm generated by applying the symplectic scheme to a high-resolution ODE proposed by Shi et al. [2018] achieves an accelerated rate for minimizing smooth strongly convex functions. On the other hand, the resulting algorithm either fails to achieve acceleration or is impractical when the scheme is implicit, the ODE is low-resolution, or the scheme is explicit.

Bin Shi, Simon S. Du, Michael I. Jordan et al.

Gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distinguish between two fundamentally different algorithms---Nesterov's accelerated gradient method for strongly convex functions (NAG-SC) and Polyak's heavy-ball method---we study an alternative limiting process that yields high-resolution ODEs. We show that these ODEs permit a general Lyapunov function framework for the analysis of convergence in both continuous and discrete time. We also show that these ODEs are more accurate surrogates for the underlying algorithms; in particular, they not only distinguish between NAG-SC and Polyak's heavy-ball method, but they allow the identification of a term that we refer to as "gradient correction" that is present in NAG-SC but not in the heavy-ball method and is responsible for the qualitative difference in convergence of the two methods. We also use the high-resolution ODE framework to study Nesterov's accelerated gradient method for (non-strongly) convex functions, uncovering a hitherto unknown result---that NAG-C minimizes the squared gradient norm at an inverse cubic rate. Finally, by modifying the high-resolution ODE of NAG-C, we obtain a family of new optimization methods that are shown to maintain the accelerated convergence rates of NAG-C for smooth convex functions.

Bin Shi

We propose some algorithms to find local minima in nonconvex optimization and to obtain global minima in some degree from the Newton Second Law without friction. With the key observation of the velocity observable and controllable in the motion, the algorithms simulate the Newton Second Law without friction based on symplectic Euler scheme. From the intuitive analysis of analytical solution, we give a theoretical analysis for the high-speed convergence in the algorithm proposed. Finally, we propose the experiments for strongly convex function, non-strongly convex function and nonconvex function in high-dimension.