Haishan Ye

Haishan Ye

Rank-based zeroth-order (ZO) optimization -- which relies only on the ordering of function evaluations -- offers strong robustness to noise and monotone transformations, and underlies many successful algorithms such as CMA-ES, natural evolution strategies, and rank-based genetic algorithms. Despite its widespread use, the theoretical understanding of rank-based ZO methods remains limited: existing analyses provide only asymptotic insights and do not yield explicit convergence rates for algorithms selecting the top-$k$ directions. This work closes this gap by analyzing a simple rank-based ZO algorithm and establishing the first \emph{explicit}, and \emph{non-asymptotic} query complexities. For a $d$-dimension problem, if the function is $L$-smooth and $μ$-strongly convex, the algorithm achieves $\widetilde{\mathcal O}\!\left(\frac{dL}μ\log\!\frac{dL}{μδ}\log\!\frac{1}{\varepsilon}\right)$ to find an $\varepsilon$-suboptimal solution, and for smooth nonconvex objectives it reaches $\mathcal O\!\left(\frac{dL}{\varepsilon}\log\!\frac{1}{\varepsilon}\right)$. Notation $\cO(\cdot)$ hides constant terms and $\widetilde{\mathcal O}(\cdot)$ hides extra $\log\log\frac{1}{\varepsilon}$ term. These query complexities hold with a probability at least $1-δ$ with $0<δ<1$. The analysis in this paper is novel and avoids classical drift and information-geometric techniques. Our analysis offers new insight into why rank-based heuristics lead to efficient ZO optimization.

Luo Luo, Cheng Chen, Guangzeng Xie et al.

We study the streaming model for approximate matrix multiplication (AMM). We are interested in the scenario that the algorithm can only take one pass over the data with limited memory. The state-of-the-art deterministic sketching algorithm for streaming AMM is the co-occurring directions (COD), which has much smaller approximation errors than randomized algorithms and outperforms other deterministic sketching methods empirically. In this paper, we provide a tighter error bound for COD whose leading term considers the potential approximate low-rank structure and the correlation of input matrices. We prove COD is space optimal with respect to our improved error bound. We also propose a variant of COD for sparse matrices with theoretical guarantees. The experiments on real-world sparse datasets show that the proposed algorithm is more efficient than baseline methods.