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Numerous theories of learning propose to prevent the gradient from exponential growth with depth or time, to stabilize and improve training. Typically, these analyses are conducted on feed-forward fully-connected neural networks or simple…
In this paper, we consider a non-linear fourth-order evolution equation of Cahn-Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order…
We study a stabilization problem of linear uncertain systems with parametric uncertainties via feedback control over data-rate-constrained channels. The objective is to find the limitation on the amount of information that must be conveyed…
Stochastic Gradient Descent (SGD) is widely used in machine learning problems to efficiently perform empirical risk minimization, yet, in practice, SGD is known to stall before reaching the actual minimizer of the empirical risk. SGD…
Multiple analytical and empirical stability criteria have been derived in the literature for two planet systems. But, the dependence of the stability limit on the initial mutual inclination between the inner and outer orbits is not well…
We revisited the central limit theorem (CLT) for stochastic gradient descent (SGD) type methods, including the vanilla SGD, momentum SGD and Nesterov accelerated SGD methods with constant or vanishing damping parameters. By taking advantage…
We analyze the fluctuations of incomplete $U$-statistics over a triangular array of independent random variables. We give criteria for a Central Limit Theorem (CLT, for short) to hold in the sense that we prove that an appropriately scaled…
In this work, we study online convex optimization with a fixed constraint function $g : \mathbb{R}^d \rightarrow \mathbb{R}$. Prior work on this problem has shown $O(\sqrt{T})$ regret and cumulative constraint satisfaction $\sum_{t=1}^{T}…
Using a new strategy, we extend the classical Nekhoroshev's estimates to the case of H\"older regular steep near-integrable hamiltonian systems, the stability times being polynomially long in the inverse of the size of the perturbation. We…
Regularization is a core component of modern inverse problems, as it helps establish the well-posedness of the solution of interest. Popular regularization approaches include variational regularization and iterative regularization. The…
Higher-order time integration methods that unconditionally preserve the positivity and linear invariants of the underlying differential equation system cannot belong to the class of general linear methods. This poses a major challenge for…
Non-normal transient growth of disturbances is considered as an essential prerequisite for subcritical transition in shear flows, i.e. transition to turbulence despite linear stability of the laminar flow. In this work we present numerical…
We consider the fluctuation of linear eigenvalue statistics of random band $n\times n$ matrices whose entries have the form $\mathcal{M}_{ij}=b^{-1/2}u^{1/2}(|i-j|)\tilde w_{ij}$ with i.i.d. $w_{ij}$ possessing the $(4+\varepsilon)$th…
The Allen-Cahn equation is a fundamental model for phase transitions, offering critical insights into the dynamics of interface evolution in various physical systems. This paper investigates the stability and robustness of frequently…
In this paper, the stability of $\theta$-methods for delay differential equations is studied based on the test equation $y'(t)=-A y(t) + B y(t-\tau)$, where $\tau$ is a constant delay and $A$ is a positive definite matrix. It is mainly…
Often in the analysis of first-order methods for both smooth and nonsmooth optimization, assuming the existence of a growth/error bound or KL condition facilitates much stronger convergence analysis. Hence separate analysis is typically…
We investigate the stability of katabatic slope flows over an infinitely wide and uniformly cooled planar surface subject to an additional forcing due to a uniform downslope wind field aloft. We adopt an extension of Prandtl's original…
Softmax feedback systems are a common mathematical core of entropy-regularized reinforcement learning, logit game dynamics, population choice, and mean-field variational updates. Their central stability question is simple: when does a…
We present a new approach, inspired by Stein's method, to prove a central limit theorem (CLT) for linear statistics of $\beta$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the…
Classical discrete-time adaptive controllers provide asymptotic stabilization and tracking; neither exponential stabilization nor a bounded noise gain is typically proven. In recent work it has been shown, in both the pole placement…