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Hongyu Cheng

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math.DS Machine Learning math-ph math.MP math.OC math.SP

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preprint2026arXiv

Sample Complexity of Stochastic Optimization with Integer Variables

We establish sample complexity results for stochastic optimization over the integers, especially with a view to understand the complexity with respect to the corresponding continuous optimization problem. We show that integer optimization can sometimes require strictly more samples and sometimes strictly smaller number of samples, depending on the structure of the objective and constraints. 1. For Lipschitz objectives over subsets of the $\ell_\infty$ ball, the statistical complexity of general stochastic mixed-integer, nonlinear, nonconvex optimization is exactly the same as stochastic linear optimization with just bound constraints. 2. For Lipschitz objectives over subsets of the $\ell_2$ ball, we show that integer optimization can require strictly *smaller* sample size compared to the continuous setting in a certain regime. To get to this result, we also establish tight sample complexity results for nonconvex continuous stochastic optimization which, to the best of our knowledge, do not appear in prior work. 3. For strongly convex, smooth objectives, integer optimization has high statistical complexity compared to the continuous setting. In particular, we show that integer optimization requires $Ω(1/ε^2)$ samples to report an $ε$ -approximate solution, compared to the well-known $O(1/ε)$ sample complexity from the continuous optimization literature.

preprint2022arXiv

Invariant tori for area-preserving maps with ultra-differentiable perturbation and Liouvillean frequency

We prove the existence of invariant tori to the area-preserving maps defined on $\mathbb{R}^2\times\mathbb{T}$ \begin{equation*} \bar{x}=F(x,θ), \qquad \barθ=θ+α\, \,(α\in \mathbb{R}\setminus\mathbb{Q}), \end{equation*} where $F$ is closed to a linear rotation, and the perturbation is ultra-differentiable in $θ\in \mathbb{T},$ which is very closed to $C^{\infty}$ regularity. Moreover, we assume that the frequency $α$ is any irrational number without other arithmetic conditions and the smallness of the perturbation does not depend on $α$ . Thus, both the difficulties from the ultra-differentiability of the perturbation and Liouvillean frequency will appear in this work. The proof of the main result is based on the Kolmogorov-Arnold-Moser (KAM) scheme about the area-preserving maps with some new techniques.

preprint2021arXiv

Global rigidity for ultra-differentiable quasiperiodic cocycles and its spectral applications

For quasiperiodic Schrödinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schrödinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy. From spectral theory side, the "Schrödinger conjecture" and the "Last's intersection spectrum conjecture" have been verified. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, see open problems by Fayad-Krikorian and Jitomirskaya-Mar. In this paper, we prove the above mentioned results for ultra-differentiable potentials.

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