Chihao Zhang

Houshuang Chen, Yuchen He, Chihao Zhang

Learning with expert advice and multi-armed bandit are two classic online decision problems which differ on how the information is observed in each round of the game. We study a family of problems interpolating the two. For a vector $\mathbf{m}=(m_1,\dots,m_K)\in \mathbb{N}^K$, an instance of $\mathbf{m}$-MAB indicates that the arms are partitioned into $K$ groups and the $i$-th group contains $m_i$ arms. Once an arm is pulled, the losses of all arms in the same group are observed. We prove tight minimax regret bounds for $\mathbf{m}$-MAB and design an optimal PAC algorithm for its pure exploration version, $\mathbf{m}$-BAI, where the goal is to identify the arm with minimum loss with as few rounds as possible. We show that the minimax regret of $\mathbf{m}$-MAB is $Θ\left(\sqrt{T\sum_{k=1}^K\log (m_k+1)}\right)$ and the minimum number of pulls for an $(ε,0.05)$-PAC algorithm of $\mathbf{m}$-BAI is $Θ\left(\frac{1}{ε^2}\cdot \sum_{k=1}^K\log (m_k+1)\right)$. Both our upper bounds and lower bounds for $\mathbf{m}$-MAB can be extended to a more general setting, namely the bandit with graph feedback, in terms of the clique cover and related graph parameters. As consequences, we obtained tight minimax regret bounds for several families of feedback graphs.

Yuchen He, Chihao Zhang

The problem of bandit with graph feedback generalizes both the multi-armed bandit (MAB) problem and the learning with expert advice problem by encoding in a directed graph how the loss vector can be observed in each round of the game. The mini-max regret is closely related to the structure of the feedback graph and their connection is far from being fully understood. We propose a new algorithmic framework for the problem based on a partition of the feedback graph. Our analysis reveals the interplay between various parts of the graph by decomposing the regret to the sum of the regret caused by small parts and the regret caused by their interaction. As a result, our algorithm can be viewed as an interpolation and generalization of the optimal algorithms for MAB and learning with expert advice. Our framework unifies previous algorithms for both strongly observable graphs and weakly observable graphs, resulting in improved and optimal regret bounds on a wide range of graph families including graphs of bounded degree and strongly observable graphs with a few corrupted arms.

Yuchen He, Chihao Zhang

We study the problem of sampling from a $d$-dimensional distribution with density $p(x)\propto e^{-f(x)}$, which does not necessarily satisfy good isoperimetric conditions. Specifically, we show that for any $L,M$ satisfying $LM\ge d\ge 5$, $ε\in \left(0,\frac{1}{32}\right)$, and any algorithm with query accesses to the value of $f(x)$ and $\nabla f(x)$, there exists an $L$-log-smooth distribution with second moment at most $M$ such that the algorithm requires $\left(\frac{LM}{dε}\right)^{Ω(d)}$ queries to compute a sample whose distribution is within $ε$ in total variation distance to the target distribution. We complement the lower bound with an algorithm requiring $\left(\frac{LM}{dε}\right)^{\mathcal{O}(d)}$ queries, thereby characterizing the tight (up to the constant in the exponent) query complexity for sampling from the family of non-log-concave distributions. Our results are in sharp contrast with the recent work of Huang et al. (COLT'24), where an algorithm with quasi-polynomial query complexity was proposed for sampling from a non-log-concave distribution when $M=\mathtt{poly}(d)$. Their algorithm works under the stronger condition that all distributions along the trajectory of the Ornstein-Uhlenbeck process, starting from the target distribution, are $\mathcal{O}(1)$-log-smooth. We investigate this condition and prove that it is strictly stronger than requiring the target distribution to be $\mathcal O(1)$-log-smooth. Additionally, we study this condition in the context of mixtures of Gaussians. Finally, we place our results within the broader theme of ``sampling versus optimization'', as studied in Ma et al. (PNAS'19). We show that for a wide range of parameters, sampling is strictly easier than optimization by a super-exponential factor in the dimension $d$.

Xikun Huang, Tianyu Ruan, Chihao Zhang et al.

Graph generation is a fundamental task with wide applications in modeling complex systems. Although existing methods align the spectrum or degree profile of the target graph, they often ignore the geometry induced by eigenvectors and the global structure of the graph. In this work, we propose Spectral Geodesic Flow Matching (SFMG), a novel framework that uses spectral eigenmaps to embed both input and target graphs into continuous Riemannian manifolds. We then define geodesic flows between embeddings and match distributions along these flows to generate output graphs. Our method yields several advantages: (i) captures geometric structure beyond eigenvalues, (ii) supports flexible generation of diverse graphs, and (iii) scales efficiently. Empirically, SFMG matches the performance of state-of-the-art approaches on graphlet, degree, and spectral metrics across diverse benchmarks. In particular, it achieves up to 30$\times$ speedup over diffusion-based models, offering a substantial advantage in scalability and training efficiency. We also demonstrate its ability to generalize to unseen graph scales. Overall, SFMG provides a new approach to graph synthesis by integrating spectral geometry with flow matching.

Yuchen He, Zhehan Lei, Jianan Shao et al.

We study the problem of sampling from a distribution $μ$ with density $\propto e^{-V}$ for some potential function $V:\mathbb R^d\to \mathbb R$ with query access to $V$ and $\nabla V$. We start with the following standard assumptions: (1) The potential function $V$ is $L$-smooth. (2) The second moment $\mathbf{E}_{X\sim μ}[\|X\|^2]\leq M$. Recently, He and Zhang (COLT'25) showed that the query complexity of sampling from such distributions is at least $\left(\frac{LM}{dε}\right)^{Ω(d)}$ where $ε$ is the desired accuracy in total variation distance, and the Poincaré constant can be arbitrarily large. Meanwhile, another common assumption in the study of diffusion based samplers (see e.g., the work of Chen, Chewi, Li, Li, Salim and Zhang (ICLR'23)) strengthens the smoothness condition (1) to the following: (1*) The potential function of *every* distribution along the Ornstein-Uhlenbeck process starting from $μ$ is $L$-smooth. We show that under the assumptions (1*) and (2), the query complexity of sampling from $μ$ can be $\mathrm{poly}(L,d)\cdot \left(\frac{Ld+M}{ε^2}\right)^{\mathcal{O}(L+1)}$, which is polynomial in $d$ and $\frac{1}ε$ when $L=\mathcal{O}(1)$ and $M=\mathrm{poly}(d)$. This improves the algorithm with quasi-polynomial query complexity developed by Huang et al. (COLT'24). Our results imply that the seemly moderate strengthening of the smoothness condition (1) to (1*) can lead to an exponential gap in the query complexity of sampling algorithms. Moreover, we show that together with the assumption (1*) and the stronger moment assumption that $\|X\|$ is $λ$-sub-Gaussian for $X\simμ$, the Poincaré constant of $μ$ is at most $\mathcal{O}(λ)^{2(L+1)}$. As an application of our technique, we obtain improved estimate of the Poincaré constant for mixture of Gaussians with the same covariance.

Houshuang Chen, Yuchen He, Chihao Zhang

This paper presents a comprehensive study on the problem of Best Arm Retention (BAR), which has recently found applications in streaming algorithms for multi-armed bandits. In the BAR problem, the goal is to retain $m$ arms with the best arm included from $n$ after some trials, in stochastic multi-armed bandit settings. We first investigate pure exploration for the BAR problem under different criteria, and then minimize the regret with specific constraints, in the context of further exploration in streaming algorithms. - We begin by revisiting the lower bound for the $(\varepsilon,δ)$-PAC algorithm for Best Arm Identification (BAI) and adapt the classical KL-divergence argument to derive optimal bounds for $(\varepsilon,δ)$-PAC algorithms for BAR. - We further study another variant of the problem, called $r$-BAR, which requires the expected gap between the best arm and the optimal arm retained is less than $r$. We prove tight sample complexity for the problem. - We explore the regret minimization problem for $r$-BAR and develop algorithm beyond pure exploration. We conclude with a conjecture on the optimal regret in this setting.

Houshuang Chen, Yaonan Jin, Pinyan Lu et al.

We examine fixed-price mechanisms in bilateral trade through the lens of regret minimization. Our main results are twofold. (i) For independent values, a near-optimal $\widetildeΘ(T^{2/3})$ tight bound for $\textsf{Global Budget Balance}$ fixed-price mechanisms with two-bit/one-bit feedback. (ii) For correlated/adversarial values, a near-optimal $Ω(T^{3/4})$ lower bound for $\textsf{Global Budget Balance}$ fixed-price mechanisms with two-bit/one-bit feedback, which improves the best known $Ω(T^{5/7})$ lower bound obtained in the work [BCCF24] and, up to polylogarithmic factors, matches the $\widetilde{\mathcal{O}}(T^{3 / 4})$ upper bound obtained in the same work. Our work in combination with the previous works [CCCFL24mor, CCCFL24jmlr, AFF24, BCCF24] (essentially) gives a thorough understanding of regret minimization for fixed-price bilateral trade. En route, we have developed two technical ingredients that might be of independent interest: (i) A novel algorithmic paradigm, called $\textit{fractal elimination}$, to address one-bit feedback and independent values. (ii) A new $\textit{lower-bound construction}$ with novel proof techniques, to address the $\textsf{Global Budget Balance}$ constraint and correlated values.

Houshuang Chen, Zengfeng Huang, Shuai Li et al.

The bandit problem with graph feedback, proposed in [Mannor and Shamir, NeurIPS 2011], is modeled by a directed graph $G=(V,E)$ where $V$ is the collection of bandit arms, and once an arm is triggered, all its incident arms are observed. A fundamental question is how the structure of the graph affects the min-max regret. We propose the notions of the fractional weak domination number $δ^*$ and the $k$-packing independence number capturing upper bound and lower bound for the regret respectively. We show that the two notions are inherently connected via aligning them with the linear program of the weakly dominating set and its dual -- the fractional vertex packing set respectively. Based on this connection, we utilize the strong duality theorem to prove a general regret upper bound $O\left(\left( δ^*\log |V|\right)^{\frac{1}{3}}T^{\frac{2}{3}}\right)$ and a lower bound $Ω\left(\left(δ^*/α\right)^{\frac{1}{3}}T^{\frac{2}{3}}\right)$ where $α$ is the integrality gap of the dual linear program. Therefore, our bounds are tight up to a $\left(\log |V|\right)^{\frac{1}{3}}$ factor on graphs with bounded integrality gap for the vertex packing problem including trees and graphs with bounded degree. Moreover, we show that for several special families of graphs, we can get rid of the $\left(\log |V|\right)^{\frac{1}{3}}$ factor and establish optimal regret.