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For non-autonomous linear stochastic differential equations (SDEs), we establish that the top Lyapunov exponent is continuous if the coefficients "almost" uniformly converge. For autonomous SDEs, assuming the existence of invariant measures and the convergence of coefficients and their derivatives in pointwise sense, we get the continuity of all Lyapunov exponents. Furthermore, we demonstrate that for autonomous SDEs with strict monotonicity condition, all Lyapunov exponents are Lipschitz continuous with respect to the coefficients under the $L^{p,p}$ norm ($p>2$). Similarly, the H\"older continuity of Lyapunov exponents holds under weaker regularity conditions. It seems that the continuity of Lyapunov exponents has not been studied for SDEs so far, in spite that there are many results in this direction for discrete-time dynamical systems., Comment: 27 pages, no figures

We obtain the upper error bounds of robust estimators for mean vector, using the median-of-means (MOM) method. The method is designed to handle data with heavy tails and contamination, with only a finite second moment, which is weaker than many others, relying on the VC dimension rather than the Rademacher complexity to measure statistical complexity. This allows us to implement MOM in covariance estimation, without imposing conditions such as $L$-sub-Gaussian or $L_{4}-L_{2}$ norm equivalence. In particular, we derive a new robust estimator, the MOM version of the halfspace depth, along with error bounds for mean estimation in any norm.

Although the independent censoring assumption is commonly used in survival analysis, it can be violated when the censoring time is related to the survival time, which often happens in many practical applications. To address this issue, we propose a flexible semiparametric method for dependent censored data. Our approach involves fitting the survival time and the censoring time with a joint transformed linear model, where the transformed function is unspecified. This allows for a very general class of models that can account for possible covariate effects, while also accommodating administrative censoring. We assume that the transformed variables have a bivariate nonnormal distribution based on parametric copulas and parametric marginals, which further enhances the flexibility of our method. We demonstrate the identifiability of the proposed model and establish the consistency and asymptotic normality of the model parameters under appropriate regularity conditions and assumptions. Furthermore, we evaluate the performance of our method through extensive simulation studies, and provide a real data example for illustration.

Recommendation systems rely on historical clicks to learn user interests and provide appropriate items. However, current studies tend to treat clicks equally, which may ignore the assorted intensities of user interests in different clicks. In this paper, we aim to achieve multi-granularity Click confidence Learning via Self-Distillation in recommendation (CLSD). Due to the lack of supervised signals in click confidence, we first apply self-supervised learning to obtain click confidence scores via a global self-distillation method. After that, we define a local confidence function to adapt confidence scores at the user group level, since the confidence distributions can be varied among user groups. With the combination of multi-granularity confidence learning, we can distinguish the quality of clicks and model user interests more accurately without involving extra data and model structures. The significant improvements over different backbones on industrial offline and online experiments in a real-world recommender system prove the effectiveness of our model. Recently, CLSD has been deployed on a large-scale recommender system, affecting over 400 million users.

Personalized recommendation relies on user historical behaviors to provide user-interested items, and thus seriously struggles with the data sparsity issue. A powerful positive item augmentation is beneficial to address the sparsity issue, while few works could jointly consider both the accuracy and diversity of these augmented training labels. In this work, we propose a novel model-agnostic Diversified self-distillation guided positive augmentation (DivSPA) for accurate and diverse positive item augmentations. Specifically, DivSPA first conducts three types of retrieval strategies to collect high-quality and diverse positive item candidates according to users' overall interests, short-term intentions, and similar users. Next, a self-distillation module is conducted to double-check and rerank these candidates as the final positive augmentations. Extensive offline and online evaluations verify the effectiveness of our proposed DivSPA on both accuracy and diversity. DivSPA is simple and effective, which could be conveniently adapted to other base models and systems. Currently, DivSPA has been deployed on multiple widely-used real-world recommender systems.

There are roughly three stages in real industrial recommendation systems, candidates generation (retrieval), ranking and reranking. Individual-level diversity and system-level diversity are both important for industrial recommender systems. The former focus on each single user's experience, while the latter focus on the difference among users. Graph-based retrieval strategies are inevitably hijacked by heavy users and popular items, leading to the convergence of candidates for users and the lack of system-level diversity. Meanwhile, in the reranking phase, Determinantal Point Process (DPP) is deployed to increase individual-level diverisity. Heavily relying on the semantic information of items, DPP suffers from clickbait and inaccurate attributes. Besides, most studies only focus on one of the two levels of diversity, and ignore the mutual influence among different stages in real recommender systems. We argue that individual-level diversity and system-level diversity should be viewed as an integrated problem, and we provide an efficient and deployable solution for web-scale recommenders. Generally, we propose to employ the retrieval graph information in diversity-based reranking, by which to weaken the hidden similarity of items exposed to users, and consequently gain more graph explorations to improve the system-level diveristy. Besides, we argue that users' propensity for diversity changes over time in content feed recommendation. Therefore, with the explored graph, we also propose to capture the user's real-time personalized propensity to the diversity. We implement and deploy the combined system in WeChat App's Top Stories used by hundreds of millions of users. Offline simulations and online A/B tests show our solution can effectively improve both user engagement and system revenue.

In this paper, we study the existence of $n$-dimensional linear stochastic differential equations (SDEs) such that the sign of Lyapunov exponents is changed under an exponentially decaying perturbation. First, we show that the equation with all positive Lyapunov exponents will have $n-1$ linearly independent solutions with negative Lyapunov exponents under the perturbation. Meanwhile, we prove that the equation with all negative Lyapunov exponents will also have solutions with positive Lyapunov exponents under another similar perturbation. Finally, we also show that other three kinds of perturbations which appear at different positions of the equation will change the sign of Lyapunov exponents., Comment: 18 pages, no figure

Sequential recommendation models are primarily optimized to distinguish positive samples from negative ones during training in which negative sampling serves as an essential component in learning the evolving user preferences through historical records. Except for randomly sampling negative samples from a uniformly distributed subset, many delicate methods have been proposed to mine negative samples with high quality. However, due to the inherent randomness of negative sampling, false negative samples are inevitably collected in model training. Current strategies mainly focus on removing such false negative samples, which leads to overlooking potential user interests, lack of recommendation diversity, less model robustness, and suffering from exposure bias. To this end, we propose a novel method that can Utilize False Negative samples for sequential Recommendation (UFNRec) to improve model performance. We first devise a simple strategy to extract false negative samples and then transfer these samples to positive samples in the following training process. Furthermore, we construct a teacher model to provide soft labels for false negative samples and design a consistency loss to regularize the predictions of these samples from the student model and the teacher model. To the best of our knowledge, this is the first work to utilize false negative samples instead of simply removing them for the sequential recommendation. Experiments on three benchmark public datasets are conducted using three widely applied SOTA models. The experiment results demonstrate that our proposed UFNRec can effectively draw information from false negative samples and further improve the performance of SOTA models. The code is available at https://github.com/UFNRec-code/UFNRec.

As a crucial component of most modern deep recommender systems, feature embedding maps high-dimensional sparse user/item features into low-dimensional dense embeddings. However, these embeddings are usually assigned a unified dimension, which suffers from the following issues: (1) high memory usage and computation cost. (2) sub-optimal performance due to inferior dimension assignments. In order to alleviate the above issues, some works focus on automated embedding dimension search by formulating it as hyper-parameter optimization or embedding pruning problems. However, they either require well-designed search space for hyperparameters or need time-consuming optimization procedures. In this paper, we propose a Single-Shot Embedding Dimension Search method, called SSEDS, which can efficiently assign dimensions for each feature field via a single-shot embedding pruning operation while maintaining the recommendation accuracy of the model. Specifically, it introduces a criterion for identifying the importance of each embedding dimension for each feature field. As a result, SSEDS could automatically obtain mixed-dimensional embeddings by explicitly reducing redundant embedding dimensions based on the corresponding dimension importance ranking and the predefined parameter budget. Furthermore, the proposed SSEDS is model-agnostic, meaning that it could be integrated into different base recommendation models. The extensive offline experiments are conducted on two widely used public datasets for CTR prediction tasks, and the results demonstrate that SSEDS can still achieve strong recommendation performance even if it has reduced 90\% parameters. Moreover, SSEDS has also been deployed on the WeChat Subscription platform for practical recommendation services. The 7-day online A/B test results show that SSEDS can significantly improve the performance of the online recommendation model.