Deep neural networks (DNNs) have significantly advanced machine learning, with model depth playing a central role in their successes. Deep residual networks (ResNets) fall into one of the most important classes of DNNs. In this paper, we study the compatible generalization behavior of deep residual networks with a broad class of activation functions in the deep-layer limit, from a dynamical system perspective. We first show a refined contraction inequality for Rademacher complexity of function classes composed by that type of activation functions. We then conduct an equivalence analysis of component hypothesis sets in Rademacher complexity and forward convergence analysis of network state variables, which enable us to establish compatible uniform generalization bounds that simultaneously achieve the optimal \(O(1/\sqrt{S})\) sample rate for both discrete- and continuous-time ResNets. The bounds also share a common layer-number independent leading term and non-positive refinement terms that capture the decomposition structure of the activation function, revealing a consistent generalization behavior in the deep-layer limit. Compared to previous works considering both discrete- and continuous-time ResNets simultaneously, our theoretical results not only have better sample rate, but also are established under weaker assumptions on the learnable parameters. We also support our theoretical findings with numerical experiments.
Neural operators are widely used as fast surrogates for numerical PDE solvers, mapping input functions to solution functions. However, their generalizability and robustness are not yet clearly defined in the operator-learning setting, which differs from traditional adversarial robustness definitions. This paper studies the generalizability and robustness of a learned neural operator from a solver-integrated perspective, addressing the challenge that the output of a learned operator and a numerical solver tends to change in tandem under input perturbation. First, we formalize the definition of generalization and robustness through a model-solver error operator, identifying fixed-input model-solver loss as generalization metric, and norm-bounded adversarial attack loss increase and Jacobian-error function norm as robustness metric. Second, we identify the solver-integrated adversarial attack as appropriate for PDE operator learning and show why model-only or fixed-ground-truth attacks can be insufficient when the solver output also changes with the input. Third, we develop solver-integrated adversarial training methods for neural operators. Experiments on representative PDE benchmarks show that this solver-integrated adversarial training clearly improves both generalizability and robustness. Deeper solver integration yields more effective attacks, more informative samples, and more efficient training than less integrated alternatives. These results provide a general framework for robust operator training and automatic sample selection without heavy manual intervention. More broadly, the formulation applies to adversarial regression whenever a ground-truth oracle can evaluate, and ideally differentiate, the true input-output map; PDE operator learning is one such case.
Heavy-tailed stochastic gradient noise, commonly observed in transformer models, can destabilize the optimization process. Recent works mainly focus on developing and understanding approaches to address heavy-tailed noise in the centralized or distributed, synchronous setting, while asynchronous optimization under such noise remains underexplored, especially in the practical settings with local updates. In this work, we first investigate two asynchronous communication schemes (server- and client-centric) that handle stragglers under heavy-tailed gradient noise. We further propose and theoretically analyze algorithmic modifications based on delay-aware learning rate scheduling and delay compensation to enhance the performance of asynchronous algorithms. Our convergence guarantees achieve the optimal rate under heavy-tailed noise and improve delay tolerance compared with existing asynchronous approaches. Empirically, our approaches outperform prior synchronous and asynchronous methods in terms of accuracy/runtime trade-offs and are more robust to hyperparameters in both image and language tasks.