Xin Li

Aritra Dutta, El Houcine Bergou, Soumia Boucherouite et al.

Stochastic gradient descent (SGD) and its variants are the main workhorses for solving large-scale optimization problems with nonconvex objective functions. Although the convergence of SGDs in the (strongly) convex case is well-understood, their convergence for nonconvex functions stands on weak mathematical foundations. Most existing studies on the nonconvex convergence of SGD show the complexity results based on either the minimum of the expected gradient norm or the functional sub-optimality gap (for functions with extra structural property) by searching the entire range of iterates. Hence the last iterations of SGDs do not necessarily maintain the same complexity guarantee. This paper shows that an $ε$-stationary point exists in the final iterates of SGDs, given a large enough total iteration budget, $T$, not just anywhere in the entire range of iterates -- a much stronger result than the existing one. Additionally, our analyses allow us to measure the density of the $ε$-stationary points in the final iterates of SGD, and we recover the classical $O(\frac{1}{\sqrt{T}})$ asymptotic rate under various existing assumptions on the objective function and the bounds on the stochastic gradient. As a result of our analyses, we addressed certain myths and legends related to the nonconvex convergence of SGD and posed some thought-provoking questions that could set new directions for research.

Chengbo He, Bochao Zou, Xin Li et al.

Agents have demonstrated their potential in scientific reasoning tasks through large language models. However, they often face challenges such as insufficient accuracy and degeneration of thought when handling complex reasoning tasks, which impede their performance. To overcome these issues, we propose the Reactive and Reflection agents with Multi-Path Reasoning (RR-MP) Framework, aimed at enhancing the reasoning capabilities of LLMs. Our approach improves scientific reasoning accuracy by employing a multi-path reasoning mechanism where each path consists of a reactive agent and a reflection agent that collaborate to prevent degeneration of thought inherent in single-agent reliance. Additionally, the RR-MP framework does not require additional training; it utilizes multiple dialogue instances for each reasoning path and a separate summarizer to consolidate insights from all paths. This design integrates diverse perspectives and strengthens reasoning across each path. We conducted zero-shot and few-shot evaluations on tasks involving moral scenarios, college-level physics, and mathematics. Experimental results demonstrate that our method outperforms baseline approaches, highlighting the effectiveness and advantages of the RR-MP framework in managing complex scientific reasoning tasks.

El Houcine Bergou, Soumia Boucherouite, Aritra Dutta et al.

The randomized Kaczmarz (RK) algorithm is one of the most computationally and memory-efficient iterative algorithms for solving large-scale linear systems. However, practical applications often involve noisy and potentially inconsistent systems. While the convergence of RK is well understood for consistent systems, the study of RK on noisy, inconsistent linear systems is limited. This paper investigates the asymptotic behavior of RK iterates in expectation when solving noisy and inconsistent systems, addressing the locations of their limit points. We explore the roles of singular vectors of the (noisy) coefficient matrix and derive bounds on the convergence horizon, which depend on the noise levels and system characteristics. Finally, we provide extensive numerical experiments that validate our theoretical findings, offering practical insights into the algorithm's performance under realistic conditions. These results establish a deeper understanding of the RK algorithm's limitations and robustness in noisy environments, paving the way for optimized applications in real-world scientific and engineering problems.

Xu He, Bo An, Yanghua Li et al.

Online recommendation services recommend multiple commodities to users. Nowadays, a considerable proportion of users visit e-commerce platforms by mobile devices. Due to the limited screen size of mobile devices, positions of items have a significant influence on clicks: 1) Higher positions lead to more clicks for one commodity. 2) The 'pseudo-exposure' issue: Only a few recommended items are shown at first glance and users need to slide the screen to browse other items. Therefore, some recommended items ranked behind are not viewed by users and it is not proper to treat this kind of items as negative samples. While many works model the online recommendation as contextual bandit problems, they rarely take the influence of positions into consideration and thus the estimation of the reward function may be biased. In this paper, we aim at addressing these two issues to improve the performance of online mobile recommendation. Our contributions are four-fold. First, since we concern the reward of a set of recommended items, we model the online recommendation as a contextual combinatorial bandit problem and define the reward of a recommended set. Second, we propose a novel contextual combinatorial bandit method called UBM-LinUCB to address two issues related to positions by adopting the User Browsing Model (UBM), a click model for web search. Third, we provide a formal regret analysis and prove that our algorithm achieves sublinear regret independent of the number of items. Finally, we evaluate our algorithm on two real-world datasets by a novel unbiased estimator. An online experiment is also implemented in Taobao, one of the most popular e-commerce platforms in the world. Results on two CTR metrics show that our algorithm outperforms the other contextual bandit algorithms.