arXiv research

Study shows Stochastic Mirror Descent optimizes convex problems with infinite noise variance.

We give improved constants for data dependent and variance sensitive confidence bounds, called empirical Bernstein bounds, and extend these inequalities to hold uniformly over classes of functionswhose growth function is polynomial in the sample size n. The bounds lead us to consider sample variance penalization, a nov…

New Q-learning algorithm reduces sample complexity for large discount factors.

AdaGrad-Norm achieves optimal convergence rates for non-convex objectives without tuning.

The paper provides concentration inequalities for Markov chain variance estimators.

This paper introduces the first asymptotically optimal strategy for a multi armed bandit (MAB) model under side constraints. The side constraints model situations in which bandit activations are limited by the availability of certain resources that are replenished at a constant rate. The main result involves the deriva…

We consider the problem of streaming kernel regression, when the observations arrive sequentially and the goal is to recover the underlying mean function, assumed to belong to an RKHS. The variance of the noise is not assumed to be known. In this context, we tackle the problem of tuning the regularization parameter ada…

This paper develops the first method for the exact simulation of reflected Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The running time of generating exact samples of non-stationary RBM at any time $t$ is uniformly bounded by $\mathcal{O}(1/\barγ^2)$ where $\barγ$ is the average drift of…

Two algorithms tackle heavy-tailed rewards in reinforcement learning with linear function approximation.

Uniformly positive scalar curvature implies a lower bound on injectivity radius.

New concentration inequality for U-statistics of Markov chains.

In this paper, we prove the compactness theorem for gradient Ricci solitons. Let $(M_α, g_α)$ be a sequence of compact gradient Ricci solitons of dimension $n\geq 4$, whose curvatures have uniformly bounded $L^{\frac{n}{2}}$ norms, whose Ricci curvatures are uniformly bounded from below with uniformly lower bounded vol…

Develops Lefschetz theory for noncompact manifolds.

Jiang et al. (2020) found no uniformly tight generalization bounds for neural networks in the overparameterized setting.

Adam converges with high probability under unconstrained non-convex smooth stochastic optimizations.

New estimator reduces bias and variance in tensor and matrix denoising.

Method improves treatment effect estimation in randomized experiments.

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.

Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can be extended beyond $T$. We prove that if the Ricci curvature is uniformly bounded…

We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times $t$ \in[0, T), then the curvature tensor has to be uniformly bounded as well.

Paper analyzes convergence of two time-scale stochastic approximation using martingale approach.

Let $(X, P)$ be a toric variety. In this note, we show that the $C^0$-norm of the Calabi flow $\varphi(t)$ on $X$ is uniformly bounded in $[0, T)$ if the Sobolev constant of $\varphi(t)$ is uniformly bounded in $[0, T)$. We also show that if $(X, P)$ is uniform $K$-stable, then the modified Calabi flow converges expone…

Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical $\ell^\infty$-semi-norm. We show that, for uniformly discrete spaces of bounded geometry, this semi-norm on uniformly finite homology in …

Uniformly branching trees are equivalent to certain metric spaces.

We establish the equivalence between the family of closed uniformly regular Riemannian manifolds and the class of complete manifolds with bounded geometry.

The paper sets sample complexity bounds for learning high-dimensional simplices in noisy data.

We consider least squares estimation in a general nonparametric regression model. The rate of convergence of the least squares estimator (LSE) for the unknown regression function is well studied when the errors are sub-Gaussian. We find upper bounds on the rates of convergence of the LSE when the errors have uniformly …

Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.

Optimizes learning policies in average-reward MDPs with improved sample complexity.

We show the existence of uniformly bounded sequences of increasing numbers of orthonormal sections of powers $L^k$ of a positive holomorphic line bundle $L$ on a compact Kähler manifold $M$. In particular, we construct for each positive integer $k$, orthonormal sections $s^k_1,\dots,s^k_{n_k}$ in $H^0(M,L^k)$, $n_k\geβ…

Estimates spectral projections restricted to uniformly embedded submanifolds.

We show that a space with a finite asymptotic dimension is embeddable in a non-positively curved manifold. Then we prove that if a uniformly contractible manifold X is uniformly embeddable in $\R^n$ or non-positively curved n-dimensional simply connected manifold then $X\times\R^n$ is integrally hyperspherical. If a un…

It is known that Garside groups are strongly translation discrete. In this paper, we show that the translation numbers in a Garside group are rational with uniformly bounded denominators and can be computed in finite time. As an application, we give solutions to some group-theoretic problems.

A fundamental tool in the analysis of Ricci flow is a compactness result of Hamilton in the spirit of the work of Cheeger, Gromov and others. Roughly speaking it allows one to take a sequence of Ricci flows with uniformly bounded curvature and uniformly controlled injectivity radius, and extract a subsequence that conv…

If a normalized Kähler-Ricci flow $g(t),t\in[0,\infty),$ on a compact Kähler $n$-manifold, $n\geq 3$, of positive first Chern class satisfies $g(t)\in 2πc_{1}(M)$ and has $L^{n}$ curvature operator uniformly bounded, then the curvature operator will also uniformly bounded along the flow. Consequently the flow will conv…

Consider a sequence of minimal varieties M_i in a Riemannian manifold N such that the boundary measures are uniformly bounded on compact sets. Let Z be the set of points at which the areas of the M_i blow up. We prove that Z behaves in some ways like a minimal variety without boundary: in particular, it satisfies the s…

We prove that for a solution $(M^n,g(t))$, $t\in[0,T)$, where $T<\infty$, to the Ricci flow with bounded curvature on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant $C$ on $M^n\times [0,T)$, the curvature tensor stays uniformly bounded on $M^n\times [0,T)$.…

We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kahler surfaces, projective manifolds o…

Calabi flow works well with bounded curvature on compact manifolds.

For a domain $Ω\subset\mathbb R^n$, we introduce the concept of a uniformly $C^m$ defining function. We characterize uniformly $C^m$ defining functions in terms of the signed distance function for the boundary and provide a large class of examples of unbounded domains with uniformly $C^m$ defining functions. Some of ou…

VRER selectively reuses past observations to reduce variance in policy optimization.

Given a closed Riemannian manifold $(N^{n+1},g)$, $n+1 \geq 3$ we prove the compactness of the space of singular, minimal hypersurfaces in $N$ whose volumes are uniformly bounded from above and the $p$-th Jacobi eigenvalue $λ_p$'s are uniformly bounded from below. This generalizes the results of Sharp and Ambrozio-Carl…

This paper examines limits of Riemannian 2-manifolds with bounded curvature.

We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the compact case. The proof uses Ricci flow with surgery as well as arguments involvin…

A new gradient estimator reduces variance near boundaries for binary latent variables.

Study asymptotic behavior of Weingarten surfaces at infinity.

Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.

Aspherical manifolds with bounded curvature have non-trivial abelian subgroups in their fundamental groups.