arXiv research

New rigidity found for 3D warped product domains.

In this paper, we prove the concavity of $p$-entropy power of probability densities solving the $p$-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of $L^p$-Euclidean Nash inequality and $L^p$-Euclidean Logarithmic Sobolev inequality, moreover, an improv…

Near-logarithmic regret per switch achieved for mixable/exp-concave losses.

This paper resolves a longstanding open question pertaining to the design of near-optimal first-order algorithms for smooth and strongly-convex-strongly-concave minimax problems. Current state-of-the-art first-order algorithms find an approximate Nash equilibrium using $\tilde{O}(κ_{\mathbf x}+κ_{\mathbf y})$ or $\tild…

We discuss a certain Riemannian metric, related to the toric Kahler-Einstein equation, that is associated in a linearly-invariant manner with a given log-concave measure in R^n. We use this metric in order to bound the second derivatives of the solution to the toric Kahler-Einstein equation, and in order to obtain spec…

The study improves fundamental gap estimates for surfaces with non-constant positive curvature.

The overarching goal of this paper is to derive excess risk bounds for learning from exp-concave loss functions in passive and sequential learning settings. Exp-concave loss functions encompass several fundamental problems in machine learning such as squared loss in linear regression, logistic loss in classification, a…

Polynomial mixing times for simulated tempering in mixture sampling problems.

ULA estimates covariance of log-concave distributions efficiently.

Paper analyzes complexity of solving nonconvex-strongly-concave problems.

Lower bounds found for nonconvex-strongly-concave min-max optimization problems.

For $φ$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $\int_X e^{-φ}.$ We prove that the logarithm of the volume is concave along continuous geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes f…

Let M be a compact, connected symplectic 2n-dimensional manifold on which an(n-2)-dimensional torus T acts effectively and Hamiltonianly. Under the assumption that there is an effective complementary 2-torus acting on M with symplectic orbits, we show that the Duistermaat-Heckman measure of the T-action is log-concave.…

MALA mixes optimally in κ√d steps for log-concave sampling.

We study the decades-old problem of online portfolio management and propose the first algorithm with logarithmic regret that is not based on Cover's Universal Portfolio algorithm and admits much faster implementation. Specifically Universal Portfolio enjoys optimal regret $\mathcal{O}(N\ln T)$ for $N$ financial instrum…

Paper approximates Kelly betting for wealth growth.

Study minimax risk of score estimation for log-concave distributions.

An online learning framework for survival analysis with real-time adaptation.

Investigates how rebalancing frequency and transaction costs affect log-optimal portfolios.

Sharp 2-Wasserstein bounds for DDPMs derived from Föllmer process.

The paper extends entropy formulas to super Ricci flows on metric measure spaces.

New algorithm for reinforcement learning reduces complexity and guarantees convergence.

Paper proposes efficient cost functions for automated market makers in DeFi.

Improved sampling for high-dimensional posteriors with underdamped Langevin.

Improved sampling algorithm with state-of-the-art complexity bounds.

New schemes improve error estimates for sampling from non-log-concave distributions.

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero poten…

Previous studies on stochastic primal-dual algorithms for solving min-max problems with faster convergence heavily rely on the bilinear structure of the problem, which restricts their applicability to a narrowed range of problems. The main contribution of this paper is the design and analysis of new stochastic primal-d…

We consider a contextual version of multi-armed bandit problem with global knapsack constraints. In each round, the outcome of pulling an arm is a scalar reward and a resource consumption vector, both dependent on the context, and the global knapsack constraints require the total consumption for each resource to be bel…

For $φ$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $\int_X e^{-φ}.$ We prove that the logarithm of the volume is concave along bounded geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from…

The unadjusted Langevin algorithm converges faster for some variables in high dimensions.

New algorithms minimize dynamic regret in non-stationary online learning.

We consider saddle point problems which objective functions are the average of $n$ strongly convex-concave individual components. Recently, researchers exploit variance reduction methods to solve such problems and achieve linear-convergence guarantees. However, these methods have a slow convergence when the condition n…

This paper tackles bandit optimization with a new pairwise comparison oracle for unknown strongly concave functions.

The paper establishes conditions for strict power concavity in convolutions.

In this paper, we give an easy proof of the main results of Andrews and Clutterbuck's paper [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916], which gives both a sharp lower bound for the spectral gap of a Schröinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We…

Minimal graph level sets are concave if boundary is concave.

Paper optimizes approximating high-dimensional diffusions by independent coordinates.

Proves log-concavity of cluster algebra coefficients for type $A_n$.

Study improves sampling from non-log-concave distributions using Fisher information.

Established concavity principle for curved spaces.

Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.

New saddle network architectures preserve convex-concave geometry in optimization problems.

Heat flow fails to preserve concavity in curved spaces.

We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set f…

Investigates concavity of spacetimes, showing conditions for local concavity.

We define a class of L-convex-concave subsets of $\Bbb{R}P^n$, where L is a projective subspace of dimension l in $\Bbb{R}P^n$. These are sets whose sections by any (l+1)-dimensional space L' containing L are convex and concavely depend on L'. We introduce an L-duality for these sets, and prove that the L-dual to an L-…

Geodesic concavity and hypersymplectic structures in $G2$-structures space.