This work introduces Residual TPP, a novel, unified, and lightweight approach for analyzing event stream data. It leverages the strengths of both simple statistical TPPs and expressive neural TPPs to achieve superior performance. Specifically, we propose the Residual Events Decomposition (RED) technique in temporal point processes, which defines a weight function to quantify how well the intensity function captures the event characteristics. The RED serves as a flexible, plug-and-play module that can be integrated with any TPP model in a wide range of tasks. It enables the identification of events for which the intensity function provides a poor fit, referred to as residual events. By combining RED with a Hawkes process, we capture the self-exciting nature of the data and identify residual events. Then an arbitrary neural TPP is employed to take care of residual events. Extensive experimental results demonstrate that Residual TPP consistently achieves state-of-the-art goodness-of-fit and prediction performance in multiple domains and offers significant computational advantages as well.

Temporal point process (TPP) is an important tool for modeling and predicting irregularly timed events across various domains. Recently, the recurrent neural network (RNN)-based TPPs have shown practical advantages over traditional parametric TPP models. However, in the current literature, it remains nascent in understanding neural TPPs from theoretical viewpoints. In this paper, we establish the excess risk bounds of RNN-TPPs under many well-known TPP settings. We especially show that an RNN-TPP with no more than four layers can achieve vanishing generalization errors. Our technical contributions include the characterization of the complexity of the multi-layer RNN class, the construction of $\tanh$ neural networks for approximating dynamic event intensity functions, and the truncation technique for alleviating the issue of unbounded event sequences. Our results bridge the gap between TPP's application and neural network theory.

We study an important variant of the stochastic multi-armed bandit (MAB) problem, which takes penalization into consideration. Instead of directly maximizing cumulative expected reward, we need to balance between the total reward and fairness level. In this paper, we present some new insights into MAB and formulate the problem in the penalization framework, where a rigorous penalized regret can be well defined and a more sophisticated regret analysis is possible. Under such a framework, we propose a hard-threshold UCB-like algorithm, which enjoys many merits including the asymptotic fairness, nearly optimal regret, good tradeoff between reward and fairness. Both gap-dependent and gap-independent regret bounds have been established. Multiple insightful comments are given to illustrate the soundness of our theoretical analysis. Numerous experimental results corroborate the theory and show the usefulness of our formulation of the problem and our method to solve it.

Graph representation learning (GRL) is critical for extracting insights from complex network structures, but it also raises security concerns due to potential privacy vulnerabilities in these representations. This paper investigates the structural vulnerabilities in graph neural models where sensitive topological information can be inferred through edge reconstruction attacks. Our research primarily addresses the theoretical underpinnings of similarity-based edge reconstruction attacks (SERA), furnishing a non-asymptotic analysis of their reconstruction capacities. Moreover, we present empirical corroboration indicating that such attacks can perfectly reconstruct sparse graphs as graph size increases. Conversely, we establish that sparsity is a critical factor for SERA's effectiveness, as demonstrated through analysis and experiments on (dense) stochastic block models. Finally, we explore the resilience of private graph representations produced via noisy aggregation (NAG) mechanism against SERA. Through theoretical analysis and empirical assessments, we affirm the mitigation of SERA using NAG . In parallel, we also empirically delineate instances wherein SERA demonstrates both efficacy and deficiency in its capacity to function as an instrument for elucidating the trade-off between privacy and utility.

Piecewise stationary stochastic multi-armed bandits have been extensively explored in the risk-neutral and sub-Gaussian setting. In this work, we consider a multi-armed bandit framework in which the reward distributions are heavy-tailed and non-stationary, and evaluate the performance of algorithms using general risk criteria. Specifically, we make the following contributions: (i) We first propose a non-parametric change detection algorithm that can detect general distributional changes in heavy-tailed distributions. (ii)We then propose a truncation-based UCB-type bandit algorithm integrating the above regime change detection algorithm to minimize the regret of the non-stationary learning problem. (iii) Finally, we establish the regret bounds for the proposed bandit algorithm by characterizing the statistical properties of the general change detection algorithm, along with a novel regret analysis.

The assumption that response and predictor belong to the same statistical unit may be violated in practice. Unbiased estimation and recovery of true label ordering based on unlabeled data are challenging tasks and have attracted increasing attentions in the recent literature. In this paper, we present a relatively complete analysis of label permutation problem for the generalized linear model with multivariate responses. The theory is established under different scenarios, with knowledge of true parameters, with partial knowledge of underlying label permutation matrix and without any knowledge. Our results remove the stringent conditions required by the current literature and are further extended to the missing observation setting which has never been considered in the field of label permutation problem. On computational side, we propose two methods, "maximum likelihood estimation" algorithm and "two-step estimation" algorithm, to accommodate for different settings. When the proportion of permuted labels is moderate, both methods work effectively. Multiple numerical experiments are provided and corroborate our theoretical findings.

Instance-wise feature selection and ranking methods can achieve a good selection of task-friendly features for each sample in the context of neural networks. However, existing approaches that assume feature subsets to be independent are imperfect when considering the dependency between features. To address this limitation, we propose to incorporate the Gaussian copula, a powerful mathematical technique for capturing correlations between variables, into the current feature selection framework with no additional changes needed. Experimental results on both synthetic and real datasets, in terms of performance comparison and interpretability, demonstrate that our method is capable of capturing meaningful correlations.

We present a new finite-sample analysis of Catoni’s M-estimator under adversarial contamination, where an adversary is allowed to corrupt a fraction of the samples arbitrarily. We make minimal assumptions on the distribution of the uncontaminated random variables, namely, we only assume the existence of a known upper bound $\upsilon_{\varepsilon} > 0$ on the $(1+\varepsilon)^{th}$ central moment of the random variables, namely, for $\varepsilon \in (0,1]$ \[ \mathbb{E}_{X_1 \sim \mathcal{D}} \Big| X_1 - \mu \Big|^{1+\varepsilon} \leq \upsilon_{\varepsilon}. \]{We} provide a lower bound on the minimax error rate for the mean estimation problem under adversarial corruption under this weak assumption, and establish that the proposed M-estimator achieves this lower bound (up to multiplicative constants). When the variance is infinite, the tolerance to contamination of any estimator reduces as $\varepsilon \downarrow 0$. We establish a tight upper bound that characterizes this bargain. To illustrate the usefulness of the derived robust M-estimator in an online setting, we present a bandit algorithm for the partially identifiable best arm identification problem that improves upon the sample complexity of the state of the art algorithms.

In this paper, we extend the remarkable M-estimator of Catoni \citep{Cat12} to situations where the variance is infinite. In particular, given a sequence of i.i.d random variables $\{X_i\}_{i=1}^n$ from distribution $\mathcal{D}$ over $\mathbb{R}$ with mean $\mu$, we only assume the existence of a known upper bound $\upsilon_{\varepsilon} > 0$ on the $(1+\varepsilon)^{th}$ central moment of the random variables, namely, for $\varepsilon \in (0,1]$ \[ \mathbb{E}_{X_1 \sim \mathcal{D}} \Big| X_1 - \mu \Big|^{1+\varepsilon} \leq \upsilon_{\varepsilon}. \]{The} extension is non-trivial owing to the difficulty in characterizing the roots of certain polynomials of degree smaller than $2$. The proposed estimator has the same order of magnitude and the same asymptotic constant as in \citet{Cat12}, but for the case of bounded moments. We further propose a version of the estimator that does not require even the knowledge of $\upsilon_{\varepsilon}$, but adapts the moment bound in a data-driven manner. Finally, to illustrate the usefulness of the derived non-asymptotic confidence bounds, we consider an application in multi-armed bandits and propose best arm identification algorithms, in the fixed confidence setting, that outperform the state of the art.

This paper considers the sensitivity of the optimization problem minf(x) with the linear constraint Ax = b, where f is a general differentiable function quantifying sparsity, Ax=b is the under-determined linear system of equations, A ∈ℝn×p. Given small noises to A and b, we are able to show the difference between the perturbed solution and optimal solution. The new perturbation bound reveals the factors that affect the sensitivity of the optimal solution of the linear system. Different objective functions f’s lead to distinct perturbation bounds, whose magnitudes determine the robustness of the optimization problem. Our results shed a fresh insight in understanding the robustness of under-determined linear system.

Learning in a Markov Decision Process (MDP) framework is a fundamental challenge for sequential decision making under uncertainty. In this paper, we present a new perspective on model-based learning in MDPs using ideas from renewal theory. In particular, we reformulate the problem of controlling a Markov chain to one of controlling a renewal reward process with bandit feedback. For this reformulated problem, we provide a regret decomposition that informs novel algorithm design for MDPs. We further provide a naive algorithm based¨ on this reformulation along with the regret analysis. A simple greedy variant of the proposed algorithm is shown to empirically outperform popular value-based methods for finite MDPs.

In this work, we propose a non-parametric and robust change detection algorithm to detect multiple change points in time series data under non-adversarial contamination. The algorithm is designed for the offline setting, where the objective is to detect changes when all data are received. We only make weak moment assumptions on the inliers (uncorrupted data) to handle a large class of distributions. The robust scan statistic in the change detection algorithm is fashioned using mean estimators based on influence functions. We establish the consistency of the estimated change point indexes as the number of samples increases, and provide empirical evidence to support the consistency results.

Latent variable models have been playing a central role in statistics, econometrics, machine learning with applications to repeated observation study, panel data inference, user behavior analysis, etc. In many modern applications, the inference based on latent variable models involves one or several of the following features: the presence of complex latent structure, the observed and latent variables being continuous or discrete, constraints on parameters, and data size being large. Therefore, solving an estimation problem for general latent variable models is highly non-trivial. In this paper, we consider a gradient based method via using variance reduction technique to accelerate estimation procedure. Theoretically, we show the convergence results for the proposed method under general and mild model assumptions. The algorithm has better computational complexity compared with the classical gradient methods and maintains nice statistical properties. Various numerical results corroborate our theory.

Biclustering structures exist ubiquitously in data matrices and the biclustering problem was first formalized by John Hartigan (1972) to cluster rows and columns simultaneously. In this paper, we develop a theory for the estimation of general biclustering models, where the data is assumed to follow certain statistical distribution with underlying biclustering structure. Due to the existence of latent variables, directly computing the maximal likelihood estimator is prohibitively difficult in practice and we instead consider the variational inference (VI) approach to solve the parameter estimation problem. Although variational inference method generally has good empirical performance, there are very few theoretical results around VI. In this paper, we obtain the precise estimation bound of variational estimator and show that it matches the minimax rate in terms of estimation error under mild assumptions in biclustering setting. Furthermore, we study the convergence property of the coordinate ascent variational inference algorithm, where both local and global convergence results have been provided. Numerical results validate our new theories.

The sparse generalized eigenvalue problem (SGEP) aims to find the leading eigenvector with sparsity structure. SGEP plays an important role in statistical learning and has wide applications including, but not limited to, sparse principal component analysis, sparse canonical correlation analysis and sparse Fisher discriminant analysis, etc. Due to the sparsity constraint, the solution of SGEP entails interesting properties from both numerical and statistical perspectives. In this paper, we provide a detailed sensitivity analysis for SGEP and establish the rate-optimal perturbation bound under the sparse setting. Specifically, we show that the bound is related to the perturbation/noise level and the recovery of the true support of the leading eigenvector as well. We also investigate the estimator of SGEP via imposing a non-convex regularization. Such estimator can achieve the optimal error rate and can recover the sparsity structure as well. Extensive numerical experiments corroborate our theoretical findings via using alternating direction method of multipliers (ADMM)-based computational method.

Greedy coordinate descent (GCD) is an efficient optimization algorithm for a wide range of machine learning and data mining applications. GCD could be significantly faster than randomized coordinate descent (RCD) if they have similar per iteration cost. Nevertheless, in some cases, the greedy rule used in GCD cannot be efficiently implemented, leading to huge per iteration cost and making GCD slower than RCD. To alleviate the cost per iteration, the existing solutions rely on maximum inner product search (MIPS) as an approximate greedy rule. But it has been empirically shown that GCD with approximate greedy rule could suffer from slow convergence even with the state-of-the-art MIPS algorithms. We propose a hybrid coordinate descent algorithm with a simple variable partition strategy to tackle the cases when greedy rule cannot be implemented efficiently. The convergence rate and theoretical properties of the new algorithm are presented. The proposed method is shown to be especially useful when the data matrix has a group structure. Numerical experiments with both synthetic and real-world data demonstrate that our new algorithm is competitive against RCD, GCD, approximate GCD with MIPS and their accelerated variants.