arXiv research

Sharp Gaussian bounds derived for Schrödinger kernel on Ricci solitons.

This paper introduces a set of algorithms for Monte-Carlo Bayesian reinforcement learning. Firstly, Monte-Carlo estimation of upper bounds on the Bayes-optimal value function is employed to construct an optimistic policy. Secondly, gradient-based algorithms for approximate upper and lower bounds are introduced. Finally…

SUSTAIN algorithm tackles stochastic bilevel optimization with near-optimal complexity.

New lower bounds for gradient methods in strongly convex finite-sum optimization.

We observe that stable integral simplicial volume of closed manifolds gives an upper bound for the rank gradient of the corresponding fundamental groups.

The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchhei…

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental soluti…

In online learning, the dynamic regret metric chooses the reference (optimal) solution that may change over time, while the typical (static) regret metric assumes the reference solution to be constant over the whole time horizon. The dynamic regret metric is particularly interesting for applications such as online reco…

A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension $m$ in $\mathbb{R}^n$ with the space of flat $m$-cochains, that is, the dual space of flat chains of dimension $m$ in $\mathbb{R}^n$. The main purpose of the present paper is to generalize Wolfe's theorem to the se…

Develops methods to calculate global index of real polynomials.

uHMC achieves fast mixing in high dimensions with gradient evaluations.

Paper improves stochastic bilevel optimization methods for highly-smooth problems.

Upper bound on CRN reaction rates derived using information geometry.

The paper studies heat kernels on modified manifolds and bounds their properties.

The paper sets limits on neural network sizes based on dataset shapes.

Estimates heat kernel gradients on fractal-like cable systems.

Sharp upper diameter limit found for Ricci solitons.

In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the $p$-Laplacian $Δ_{p}u=-λ|u|^{p-2}u$ with $p>1$ on a given complete Riemannian manifold. Consequently, we derive upper bound estimates of the first nontrivial eigenvalue of the $p$-Laplacian.

Estimates volume of convex Alexandrov spaces with boundary.

Study large deviations rates for SGD with strongly convex functions.

Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.

The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.

Novel method for bilevel optimization with convex lower-level problem.

This study tightens bounds on how GD and SGD generalize in smooth convex optimization problems.

Federated learning (FL) provides a communication-efficient approach to solve machine learning problems concerning distributed data, without sending raw data to a central server. However, existing works on FL only utilize first-order gradient descent (GD) and do not consider the preceding iterations to gradient update w…

New decay estimates for scalar curvature of steady gradient Ricci solitons.

In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the W…

Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.

Stochastic variational inference (SVI) plays a key role in Bayesian deep learning. Recently various divergences have been proposed to design the surrogate loss for variational inference. We present a simple upper bound of the evidence as the surrogate loss. This evidence upper bound (EUBO) equals to the log marginal li…

Study Markov chain gradient descent in Hilbert spaces for quadratic loss.

We derive and analyze learning algorithms for apprenticeship learning, policy evaluation, and policy gradient for average reward criteria. Existing algorithms explicitly require an upper bound on the mixing time. In contrast, we build on ideas from Markov chain theory and derive sampling algorithms that do not require …

The overall performance or expected excess risk of an iterative machine learning algorithm can be decomposed into training error and generalization error. While the former is controlled by its convergence analysis, the latter can be tightly handled by algorithmic stability. The machine learning community has a rich his…

In this paper, we first obtain an $L^q$ gradient estimate for $p$-harmonic maps, by assuming the target manifold supporting a certain function, whose gradient and Hessian satisfy some analysis conditions. From this $L^q$ gradient estimate, we get a corresponding Liouville type result for $p$-harmonic maps. Secondly, us…

Improved penalty-based methods for bilevel optimization with reduced complexity.

AdaGrad outperforms SGD in non-convex optimization problems by a factor of d.

We study the iteration complexity of stochastic gradient descent (SGD) for minimizing the gradient norm of smooth, possibly nonconvex functions. We provide several results, implying that the $\mathcal{O}(ε^{-4})$ upper bound of Ghadimi and Lan~\cite{ghadimi2013stochastic} (for making the average gradient norm less than…

In this paper we study the differentially private Empirical Risk Minimization (ERM) problem in different settings. For smooth (strongly) convex loss function with or without (non)-smooth regularization, we give algorithms that achieve either optimal or near optimal utility bounds with less gradient complexity compared …

The paper proves various inequalities on gradient shrinking Ricci solitons.

This paper bounds the Lipschitz constants of neural networks and their gradients.

Gradient estimate proved for Donaldson's equation on Kähler manifolds.

We show that sequences of compact gradient Ricci solitons converge to complete orbifold gradient solitons, assuming constraints on volume, the $L^{n/2}$-norm of curvature, and the auxiliary constant $C_1$. The strongest results are in dimension 4, where $L^2$ curvature bounds are equivalent to upper bounds on the Euler…

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…

A new differentiable UCB algorithm for linear bandits learns adaptive confidence bounds.

Study on gradient descent in Hilbert spaces with Markov chains, focusing on mixing coefficients.

We study both function theoretic and spectral properties of the weighted Laplacian $Δ_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$ with its Bakry-Émery curvature $Ric_f$ bounded from below by a constant. In particular, we establish a gradient estimate for positive $f-$harmonic functions and a sharp upper…

New method tackles bilevel optimization with polyhedral constraints.

In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR he…

We provide tight upper and lower bounds on the complexity of minimizing the average of $m$ convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs randomized optimization. For smooth functions, we show that accelerated gradient de…