arXiv subjects

Huachen Ren

Publications and source records attributed to Huachen Ren.

3 recordsLinked to original sources

arXiv2026

Statistical Inference for Additive Monotone Models under the Fixed Lattice Design

We study statistical inference for least squares estimators (LSEs) in additive monotone models under a general fixed lattice design. We establish joint limiting distributions for the LSEs and show that the estimators of different additive components are asymptotically independent. The form of the limiting distribution of each component is determined by how fast the number of design points along the corresponding coordinate grows relative to the total sample size n. Apart from this growth rate, the limit depends only on the noise level and, in the non-Gaussian regimes, on the local derivative of the component. In particular, the limit does not depend on the dimension of the model. When the number of design points along a coordinate grows faster than n^(1/3), we construct tuning-free pointwise confidence intervals based on a pivotal limiting distribution, and we validate the theory in numerical simulations. We further show that the block-size normalization underlying these intervals fails to be pivotal at the critical growth rate n^(1/3). We also prove a switching lemma, of independent interest, that simplifies the derivation of limiting distributions in isotonic regression.

math.ST↗

arXiv2024

On Lai's Upper Confidence Bound in Multi-Armed Bandits

In this memorial paper, we honor Tze Leung Lai's seminal contributions to the topic of multi-armed bandits, with a specific focus on his pioneering work on the upper confidence bound. We establish sharp non-asymptotic regret bounds for an upper confidence bound index with a constant level of exploration for Gaussian rewards. Furthermore, we establish a non-asymptotic regret bound for the upper confidence bound index of Lai (1987) which employs an exploration function that decreases with the sample size of the corresponding arm. The regret bounds have leading constants that match the Lai-Robbins lower bound. Our results highlight an aspect of Lai's seminal works that deserves more attention in the machine learning literature.

stat.ML↗

arXiv2022

Gaussian random projections of convex cones: approximate kinematic formulae and applications

Understanding the stochastic behavior of random projections of geometric sets constitutes a fundamental problem in high dimension probability that finds wide applications in diverse fields. This paper provides a kinematic description for the behavior of Gaussian random projections of closed convex cones, in analogy to that of randomly rotated cones studied in [ALMT14]. Formally, let $K$ be a closed convex cone in $\mathbb{R}^n$, and $G\in \mathbb{R}^{m\times n}$ be a Gaussian matrix with i.i.d. $\mathcal{N}(0,1)$ entries. We show that $GK\equiv \{G\mu: \mu \in K\}$ behaves like a randomly rotated cone in $\mathbb{R}^m$ with statistical dimension $\min\{\delta(K),m\}$, in the following kinematic sense: for any fixed closed convex cone $L$ in $\mathbb{R}^m$, \begin{align*} &\delta(L)+\delta(K)\ll m\, \Rightarrow\, L\cap GK = \{0\} \hbox{ with high probability},\\ &\delta(L)+\delta(K)\gg m\, \Rightarrow\, L\cap GK \neq \{0\} \hbox{ with high probability}. \end{align*} A similar kinematic description is obtained for $G^{-1}L\equiv \{\mu \in \mathbb{R}^n: G\mu \in L\}$. The practical utility and broad applicability of the prescribed approximate kinematic formulae are demonstrated in a number of distinct problems arising from statistical learning, mathematical programming and asymptotic geometric analysis. In particular, we prove (i) new phase transitions of the existence of cone constrained maximum likelihood estimators in logistic regression, (ii) new phase transitions of the cost optimum of deterministic conic programs with random constraints, and (iii) a local version of the Gaussian Dvoretzky-Milman theorem that describes almost deterministic, low-dimensional behaviors of subspace sections of randomly projected convex sets.

math.PR↗