We show that suitably regular functions can be approximated in the $\mathcal{C}^1$-norm both with rational functions and rational neural networks, including approximation rates with respect to width and depth of the network, and degree of the rational functions. As consequence of our results, we further obtain $\mathcal{C}^1$-approximation results for rational neural networks with the $\text{EQL}^÷$ and ParFam architecture, both of which are important in particular in the context of symbolic regression for physical law learning.

We consider imitation learning with access only to expert demonstrations, whose real-world application is often limited by covariate shift due to compounding errors during execution. We investigate the effectiveness of the Continuity-based Corrective Labels for Imitation Learning (CCIL) framework in mitigating this issue for real-world fine manipulation tasks. CCIL generates corrective labels by learning a locally continuous dynamics model from demonstrations to guide the agent back toward expert states. Through extensive experiments on peg insertion and fine grasping, we provide the first empirical validation that CCIL can significantly improve imitation learning performance despite discontinuities present in contact-rich manipulation. We find that: (1) real-world manipulation exhibits sufficient local smoothness to apply CCIL, (2) generated corrective labels are most beneficial in low-data regimes, and (3) label filtering based on estimated dynamics model error enables performance gains. To effectively apply CCIL to robotic domains, we offer a practical instantiation of the framework and insights into design choices and hyperparameter selection. Our work demonstrates CCIL's practicality for alleviating compounding errors in imitation learning on physical robots.

In this paper we consider a class of structured monotone inclusion (MI) problems that consist of finding a zero in the sum of two monotone operators, in which one is maximal monotone while another is locally Lipschitz continuous. In particular, we first propose a primal-dual extrapolation (PDE) method for solving a structured strongly MI problem by modifying the classical forward-backward splitting method by using a point and operator extrapolation technique, in which the parameters are adaptively updated by a backtracking line search scheme. The proposed PDE method is almost parameter-free, equipped with a verifiable termination criterion, and enjoys an operation complexity of ${\cal O}(\log \epsilon^{-1})$, measured by the amount of fundamental operations consisting only of evaluations of one operator and resolvent of another operator, for finding an $\epsilon$-residual solution of the structured strongly MI problem. We then propose another PDE method for solving a structured non-strongly MI problem by applying the above PDE method to approximately solve a sequence of structured strongly MI problems. The resulting PDE method is parameter-free, equipped with a verifiable termination criterion, and enjoys an operation complexity of ${\cal O}(\epsilon^{-1}\log \epsilon^{-1})$ for finding an $\epsilon$-residual solution of the structured non-strongly MI problem. As a consequence, we apply the latter PDE method to convex conic optimization, conic constrained saddle point, and variational inequality problems, and obtain complexity results for finding an $\epsilon$-KKT or $\epsilon$-residual solution of them under local Lipschitz continuity. To the best of our knowledge, no prior studies were conducted to investigate methods with complexity guarantees for solving the aforementioned problems under local Lipschitz continuity. All the complexity results obtained in this paper are entirely new.