Hyungryul Baik · arXiv (Cornell University) · 2026
Han and Liu gave a geometric characterization of uniform perfectness for the Morse boundary of a proper geodesic metric space: the Morse boundary is uniformly perfect if and only if the space is Morse geodesically rich, equivalently center--exhaustive. In this paper we prove the analogous statement for the sublinearly Morse boundary $\partial_κX$. Here $κ$ is a fixed concave increasing sublinear function and $\partial_κX$ is the boundary introduced by Qing--Rafi for CAT(0) spaces and extended by Qing--Rafi--Tiozzo to proper geodesic spaces. Assuming that $\partial_κX$ has at least three points, we show that uniform perfectness of $\partial_κX$ (for any $κ$--visual metric based at a fixed basepoint) is equivalent to $κ$--Morse geodesic richness and to $κ$--center--exhaustiveness. The geometric input is a sublinear thin--triangle statement for $κ$--Morse geodesics, together with the renormalization map $ρ_κ(t)=\int_0^t \frac{ds}{κ(s)}$, which converts $κ$--scale errors at radius $R$ into bounded errors in the $ρ_κ$--scale. As applications we obtain quantitative lower bounds on the lower Assouad dimension (and, under doubling hypotheses, on the Hausdorff dimension) of $κ$--visual metrics on $\partial_κX$ in terms of the uniform perfectness constant. Finally, for $κ$--center--exhaustive spaces $X$ and $Y$ satisfying a mild additional growth condition on $κ$, we prove a rigidity statement in the sublinear category: every quasi-symmetric homeomorphism $\partial_κX\to\partial_κY$ is induced by a sublinear bilipschitz equivalence $X\to Y$.
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Igor E. Verbitsky · Mathematical Physics and Computer Simulation · 2017
We give a survey of recent results on positive solutions to sublinear elliptic equations of the type - + = , where is an elliptic operator in divergence form, 0 < < 1, 0 and is a function that may change sign, in a domain R , or in a weighted Riemannian manifold, with a positive Green's function . We discuss the existence, as well as global lower and upper pointwise estimates of classical and weak solutions , and conditions that ensure () or 1, ().
Cited by 4Open access
Dimitris Kastoris · Preprints.org · 2025
We numerically investigate improvements to the explicit Riemann–von Mangoldt formula for counting the zeros of the Riemann zeta function on the critical line. By introducing a nonlinear correction involving a shifted sine integral, we demonstrate significant reduction in maximum deviations from the true counting function. Numerical experiments across extensive ranges of zeros confirm robust improvements and hint at the presence of a sublinear saturation bound, potentially linked to spectral barriers in analytic number theory.
Cited by 0Open access
Hyungryul Baik · arXiv (Cornell University) · 2026
Han and Liu gave a geometric characterization of uniform perfectness for the Morse boundary of a proper geodesic metric space: the Morse boundary is uniformly perfect if and only if the space is Morse geodesically rich, equivalently center--exhaustive. In this paper we prove the analogous statement for the sublinearly Morse boundary $\partial_κX$. Here $κ$ is a fixed concave increasing sublinear function and $\partial_κX$ is the boundary introduced by Qing--Rafi for CAT(0) spaces and extended by Qing--Rafi--Tiozzo to proper geodesic spaces. Assuming that $\partial_κX$ has at least three points, we show that uniform perfectness of $\partial_κX$ (for any $κ$--visual metric based at a fixed basepoint) is equivalent to $κ$--Morse geodesic richness and to $κ$--center--exhaustiveness. The geometric input is a sublinear thin--triangle statement for $κ$--Morse geodesics, together with the renormalization map $ρ_κ(t)=\int_0^t \frac{ds}{κ(s)}$, which converts $κ$--scale errors at radius $R$ into bounded errors in the $ρ_κ$--scale. As applications we obtain quantitative lower bounds on the lower Assouad dimension (and, under doubling hypotheses, on the Hausdorff dimension) of $κ$--visual metrics on $\partial_κX$ in terms of the uniform perfectness constant. Finally, for $κ$--center--exhaustive spaces $X$ and $Y$ satisfying a mild additional growth condition on $κ$, we prove a rigidity statement in the sublinear category: every quasi-symmetric homeomorphism $\partial_κX\to\partial_κY$ is induced by a sublinear bilipschitz equivalence $X\to Y$.
Cited by 0Open access
A.M. Zimin, Rasmus Ibsen-Jensen, Krishnendu Chatterjee · arXiv (Cornell University) · 2014
We consider the problem of minimizing the regret in stochastic multi-armed bandit, when the measure of goodness of an arm is not the mean return, but some general function of the mean and the variance.We characterize the conditions under which learning is possible and present examples for which no natural algorithm can achieve sublinear regret.
Cited by 20Open access
Maciej Zworski · 2016
Abstract. We prove a local trace formula for Anosov flows. It relates Pollicott–Ruelle resonances to the periods of closed orbits. As an application, we show that the counting function for resonances in a sufficiently wide strip cannot have a sublinear growth. In particular, for any Anosov flow there exist strips with infinitely many resonances. 1.
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Shotaro Yagishita, Shummin Nakayama · JSIAM Letters · 2024
Recently, proximal Newton-type methods with metrics restricted to diagonal matrices have been proposed for solving composite optimization problems whose objective function is the sum of a smooth function and a possibly nonsmooth function. Although the effectiveness of one of them, the proximal diagonal Newton method (PDNM), has been reported theoretically and numerically, only 𝒪(1/k) sublinear convergence rate has been obtained at best for non-strongly convex problems. We propose an accelerated variant of the PDNM, which achieves the convergence rate of 𝒪(1/k2).
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George A. Anastassiou · Demonstratio Mathematica · 2018
Abstract Here we search quantitatively under convexity the approximation of multivariate function by general multivariate positive sublinear operators with applications to multivariate Max-product operators. These are of Bernstein type, of Favard-Szász-Mirakjan type, of Baskakov type, of sampling type, of Lagrange interpolation type and of Hermite-Fejér interpolation type. Our results are both: under the presence of smoothness and without any smoothness assumption on the function to be approximated which fulfills a convexity assumption.
Cited by 3Open access
Wen, Li, Wu, Yuan · ArXiv.org · 2025
We establish quantitative Green's function estimates for a class of quasi-periodic (QP) operators on $\mathbb{Z}^d$ with certain slowly decaying long-range hopping and analytic cosine type potentials. As applications, we prove the arithmetic spectral localization, and obtain upper bounds on quantum dynamics for all phase parameters. To deal with quantum dynamics estimates, we develop an approach employing separation property (rather than the sublinear bound) of resonant blocks in the regime of Green's function estimates.
Cited by 0Open access
Honghao Wei · Deep Blue (University of Michigan) · 2023
Safe reinforcement learning (RL) is an area of research focused on developing algorithms and methods that ensure the safety of RL agents during learning and decision-making processes. The goal is to enable RL agents to interact with their environments and learn optimal decisions while avoiding actions that can lead to harmful or undesirable outcomes. This dissertation provides a comprehensive study of {em model-free}, {em simulator-free} reinforcement learning algorithms for Constrained Markov Decision Processes (CMDPs) with sublinear regret and zero constraint violation, with the focus on three settings: $(1)$ episodic CMDPs; $(2)$ infinite-horizon average-reward CMDPs and $(3)$ non-stationary episodic CMDPs. The first part provides the first model-free, simulator-free safe-RL algorithm with sublinear regret and zero constraint violation. The algorithm is named Triple-Q because it includes three key components: a Q-function (also called action value function) for the cumulative reward, a Q-function for the cumulative utility of the constraint, and a virtual Queue that (over)-estimates the cumulative constraint violation. Under Triple-Q, at each step, an action is chosen based on the pseudo-Q-value that is a combination of the three Q values. The algorithm updates the reward and utility Q values with learning rates that depend on the visit counts to the corresponding (state, action) pairs and are periodically reset. In the episodic CMDP setting, Triple-Q achieves sublinear regret. Furthermore, Triple-Q guarantees zero constraint violation, both on expectation and with a high probability, when the number of episode is sufficiently large. Finally, the computational complexity of Triple-Q is similar to SARSA for unconstrained MDPs, and is computationally efficient. In Chapter III, the results are extended to infinite-horizon average-reward Constrained Markov Decision Processes (CMDPs). The proposed algorithm guarantees sublinear regret and zero constraint violation. Then in Chapter IV the dissertation studies safe-RL in a more challenging setting, non-stationary CMDPs, where the rewards/utilities and dynamics are time-varying and likely unknown a priori. In the nonstationary environment, reward, utility functions, and transition kernels can vary arbitrarily over time as long as the cumulative variations do not exceed certain variation budgets. We propose the first model-free, simulator-free RL algorithms with sublinear regret and zero constraint violation for non-stationary CMDPs in both tabular and linear function approximation settings with provable performance guarantees. Our results on regret bound and constraint violation for the tabular case match the corresponding best results for stationary CMDPs when the total budget is known. Additionally, we present a general framework for addressing the well-known challenges associated with analyzing non-stationary CMDPs, without requiring prior knowledge of the variation budget. We apply the approach to both tabular and linear approximation settings.
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Zuyue Fu, Zhuoran Yang, Zhaoran Wang · arXiv (Cornell University) · 2020
We study the global convergence and global optimality of actor-critic, one of the most popular families of reinforcement learning algorithms. While most existing works on actor-critic employ bi-level or two-timescale updates, we focus on the more practical single-timescale setting, where the actor and critic are updated simultaneously. Specifically, in each iteration, the critic update is obtained by applying the Bellman evaluation operator only once while the actor is updated in the policy gradient direction computed using the critic. Moreover, we consider two function approximation settings where both the actor and critic are represented by linear or deep neural networks. For both cases, we prove that the actor sequence converges to a globally optimal policy at a sublinear $O(K^{-1/2})$ rate, where $K$ is the number of iterations. To the best of our knowledge, we establish the rate of convergence and global optimality of single-timescale actor-critic with linear function approximation for the first time. Moreover, under the broader scope of policy optimization with nonlinear function approximation, we prove that actor-critic with deep neural network finds the globally optimal policy at a sublinear rate for the first time.
Cited by 16Open access
Libo Liu · 2008
Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a dimension n. For α ∈ (0, ∞) denote by Hαp(X ), Hdp(X ), and H*,p(X ) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calder′on reproducing formula, it is shown that all these Hardy spaces coincide with Lp(X ) when p ∈ (1, ∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H*,p(X ) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1), 1], it is proved that the space H*,p(X ), the Hardy space Hp(X ) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman and Weiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from Hp(X ) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.
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Vjekoslav Kovač, Kristina Ana Škreb · arXiv (Cornell University) · 2021
We prove a bi-sublinear embedding for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form and for certain mutually dual pairs of Orlicz-space norms. This generalizes a result by Carbonaro and Dragičević from power functions to more general Young functions that still behave like powers. To achieve this, we generalize a Bellman function constructed by Nazarov and Treil.
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Xuanyu Cao, Junshan Zhang, H. Vincent Poor · IEEE Transactions on Automatic Control · 2020
In this article, we study constrained online convex optimization (OCO) in the presence of feedback delays, where a decision maker chooses sequential actions without knowing the loss functions and constraint functions <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> . The loss/constraint functions vary with time and their feedback information is revealed to the decision maker with delays, which arise in many applications. We first consider the scenario of delayed function feedback, in which the complete information of the loss/constraint functions is revealed to the decision maker with delays. We develop a modified online saddle point algorithm suitable for constrained OCO with feedback delays. Sublinear regret and sublinear constraint violation bounds are established for the algorithm in terms of the delays. In practice, the complete information (functional forms) of the loss/constraint functions may not be revealed to the decision maker. Thus, we further examine the scenario of delayed bandit feedback, where only the values of the loss/constraint functions at two random points close to the chosen action are revealed to the decision maker with delays. A delayed version of the bandit online saddle point algorithm is proposed by utilizing stochastic gradient estimates of the loss/constraint functions based on delayed bandit feedback. We also establish sublinear regret and sublinear constraint violation bounds for this bandit optimization algorithm in terms of the delays. Finally, numerical results for online quadratically constrained quadratic programs are presented to corroborate the efficacy of the proposed algorithms.
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Shoham Sabach, Shimrit Shtern · arXiv (Cornell University) · 2017
In this paper we study convex bi-level optimization problems for which the inner level consists of minimization of the sum of smooth and nonsmooth functions. The outer level aims at minimizing a smooth and strongly convex function over the optimal solutions set of the inner problem. We analyze a first order method which is based on an existing fixed-point algorithm. Global sublinear rate of convergence of the method is established in terms of the inner objective function values.
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David Criens, Lars Niemann · Statistics & Probability Letters · 2024
In this note we consider a family of nonlinear (conditional) expectations that can be understood as a multidimensional diffusion with uncertain drift and certain volatility. Here, the drift is prescribed by a set-valued function that depends on time and path in a Markovian way. We establish the Feller property for the associated sublinear Markovian semigroup and we observe a smoothing effect as our framework carries enough randomness. Furthermore, we link the corresponding value function to a semilinear Kolmogorov equation.
Cited by 1Open access
Jérome Goddard, EunKyoung Lee, R. Shivaji · Boundary Value Problems · 2010
We study the positive solutions to boundary value problems of the form ; , ; , where is a bounded domain in with , is the Laplace operator, is a positive parameter, is a continuous function which is sublinear at , is the outward normal derivative, and is a smooth function nondecreasing in . In particular, we discuss the existence of at least two positive radial solutions for when is an annulus in . Further, we discuss the existence of a double S-shaped bifurcation curve when , , and with .
Cited by 8Open access
Pedro Terán, José M. Zapataa · arXiv (Cornell University) · 2024
We prove by counterexample that a large deviation principle established by Chen and Feng [{\em Comm. Statist. Theory Methods} {\bf 45} (2016), 400--412] in the framework of sublinear expectations is incorrect. That implies that the rate function cannot, in general, be obtained by computing the Fenchel transform of the cumulant generating function, as is the case for ordinary probabilities. We derive a corrected version of that result and show that the original presentation holds under a stronger independence assumption.
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Xinlei Yi, Xiuxian Li, Lihua Xie et al. · 2019
This paper considers distributed online optimization with time-varying coupled inequality constraints. The global objective function is composed of local convex cost and regularization functions and the coupled constraint function is the sum of local convex constraint functions. A distributed online primal-dual mirror descent algorithm is proposed to solve this problem, where the local cost, regularization, and constraint functions are held privately and revealed only after each time slot. We first derive regret and constraint violation bounds for the algorithm and show how they depend on the stepsize sequences, the accumulated variation of the comparator sequence, the number of agents, and the network connectivity. As a result, we prove that the algorithm achieves sublinear dynamic regret and constraint violation if the accumulated variation of the optimal sequence also grows sublinearly. We also prove that the algorithm achieves sublinear static regret and constraint violation under mild conditions. In addition, smaller bounds on the static regret are achieved when the objective functions are strongly convex. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.
Cited by 4
Makoto Yamashita, Naoki Hayashi, Shigemasa Takai · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences · 2021
This paper considers a distributed subgradient method for online optimization with event-triggered communication over multi-agent networks. At each step, each agent obtains a time-varying private convex cost function. To cooperatively minimize the global cost function, these agents need to communicate each other. The communication with neighbor agents is conducted by the event-triggered method that can reduce the number of communications. We demonstrate that the proposed online algorithm achieves a sublinear regret bound in a dynamic environment with slow dynamics.
Cited by 11