Yuliang Wang

Lei Li, Jian-Guo Liu, Yuliang Wang

We consider the geometric ergodicity of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm under nonconvexity settings. Via the technique of reflection coupling, we prove the Wasserstein contraction of SGLD when the target distribution is log-concave only outside some compact set. The time discretization and the minibatch in SGLD introduce several difficulties when applying the reflection coupling, which are addressed by a series of careful estimates of conditional expectations. As a direct corollary, the SGLD with constant step size has an invariant distribution and we are able to obtain its geometric ergodicity in terms of $W_1$ distance. The generalization to non-gradient drifts is also included.

Arjun Sridharkumar, Sara Al Hajj Ibrahim, Jiayuan Zhou et al.

Timely resolution and disclosure of vulnerabilities are essential for maintaining the security of open-source software. However, many vulnerabilities remain unreported, unpatched, or undisclosed for extended periods, exposing users to prolonged security threats. While various vulnerability detection tools exist, they primarily focus on predicting or identifying known vulnerabilities, often failing to capture vulnerabilities that experience significant delays in resolution. In this study, we examine the vulnerability lifecycle by analyzing protracted vulnerabilities (PCVEs), which remain unresolved or undisclosed over long periods. We construct a dataset of PCVEs and conduct a qualitative analysis to uncover underlying causes of delay. To assess current automated solutions, we evaluate four state-of-the-art (SOTA) vulnerability detectors on our dataset. These tools detect only 1,059 out of 2,402 PCVEs, achieving approximately 44% coverage. To address this limitation, we propose DeeptraVul, an enhanced detection approach designed specifically for protracted cases. DeeptraVul integrates multiple development artifacts and code signals, supported by a Large Language Model (LLM)-based summarization component. For comparison, we also evaluate a standalone LLM. Our results show that DeeptraVul improves detection performance, achieving a 14% increase in coverage across all PCVEs and reaching 90% coverage on the DeeptraVul PCVE subset, outperforming existing SOTA detectors and standalone LLM based inference.

Hongyu Liu, Yuliang Wang, Can Yang

In this paper, we present a conceptual design of a novel gesture-based instruction/input device using wave detection. The device recogonizes/detects gestures from a person and based on which to give the specific orders/inputs to the computing machine that is connected to it. The gestures are modelled as the shapes of some impenetrable or penetrable scatterers from a certain admissible class, called a dictionary. The device generates time-harmonic point signals for the gesture recognition/detection. It then collects the scattered wave in a relatively small backscattering aperture on a bounded surface containing the point sources. The recognition algorithm consists of two steps and requires only two incident waves of different wavenumbers. The approximate location of the scatterer is first determined by using the measured data at a small wavenumber and the shape of the scatterer is then identified using the computed location of the scatterer and the measured data at a regular wavenumber. We provide the mathematical principle with rigorous justifications underlying the design. Numerical experiments show that the proposed device works effectively and efficiently in some practical scenarios.

Lei Li, Yuliang Wang

We establish a sharp uniform-in-time error estimate for the Stochastic Gradient Langevin Dynamics (SGLD), which is a widely-used sampling algorithm. Under mild assumptions, we obtain a uniform-in-time $O(η^2)$ bound for the KL-divergence between the SGLD iteration and the Langevin diffusion, where $η$ is the step size (or learning rate). Our analysis is also valid for varying step sizes. Consequently, we are able to derive an $O(η)$ bound for the distance between the invariant measures of the SGLD iteration and the Langevin diffusion, in terms of Wasserstein or total variation distances. Our result can be viewed as a significant improvement compared with existing analysis for SGLD in related literature.

Lei Li, Yuliang Wang

The diffusion approximation of stochastic gradient descent (SGD) in current literature is only valid on a finite time interval. In this paper, we establish the uniform-in-time diffusion approximation of SGD, by only assuming that the expected loss is strongly convex and some other mild conditions, without assuming the convexity of each random loss function. The main technique is to establish the exponential decay rates of the derivatives of the solution to the backward Kolmogorov equation. The uniform-in-time approximation allows us to study asymptotic behaviors of SGD via the continuous stochastic differential equation (SDE) even when the random objective function $f(\cdot;ξ)$ is not strongly convex.

Yuchen Guo, Nicholas Hanoian, Zhexiao Lin et al.

We propose a novel model for a topic-aware chatbot by combining the traditional Recurrent Neural Network (RNN) encoder-decoder model with a topic attention layer based on Nonnegative Matrix Factorization (NMF). After learning topic vectors from an auxiliary text corpus via NMF, the decoder is trained so that it is more likely to sample response words from the most correlated topic vectors. One of the main advantages in our architecture is that the user can easily switch the NMF-learned topic vectors so that the chatbot obtains desired topic-awareness. We demonstrate our model by training on a single conversational data set which is then augmented with topic matrices learned from different auxiliary data sets. We show that our topic-aware chatbot not only outperforms the non-topic counterpart, but also that each topic-aware model qualitatively and contextually gives the most relevant answer depending on the topic of question.

Eemeli Blåsten, Xiaofei Li, Hongyu Liu et al.

This paper is concerned with the intrinsic geometric structure of interior transmission eigenfunctions arising in wave scattering theory. We numerically show that the aforementioned geometric structure can be much delicate and intriguing. The major findings can be roughly summarized as follows. If there is a cusp on the support of the underlying potential function, then the interior transmission eigenfunction vanishes near the cusp if its interior angle is less than $π$, whereas the interior transmission eigenfunction localizes near the cusp if its interior angle is bigger than $π$. Furthermore, we show that the vanishing and blowup orders are inversely proportional to the interior angle of the cusp: the sharper the angle, the higher the convergence order. Our results are first of its type in the spectral theory for transmission eigenvalue problems, and the existing studies in the literature concentrate more on the intrinsic properties of the transmission eigenvalues instead of the transmission eigenfunctions. Due to the limitedness of the computing resources, our study is by no means exclusive and complete. We consider our study only in a certain geometric setup including corner, curved corner and edge singularities. Nevertheless, we believe that similar results hold for more general cusp singularities and rigorous theoretical justifications are much desirable. Our study enriches the spectral theory for transmission eigenvalue problems. We also discuss its implication to inverse scattering theory.