Modeling multi-agent systems on networks is a fundamental challenge in a wide variety of disciplines. We jointly infer the weight matrix of the network and the interaction kernel, which determine respectively which agents interact with which others and the rules of such interactions from data consisting of multiple trajectories. The estimator we propose leads naturally to a non-convex optimization problem, and we investigate two approaches for its solution: one is based on the alternating least squares (ALS) algorithm; another is based on a new algorithm named operator regression with alternating least squares (ORALS). Both algorithms are scalable to large ensembles of data trajectories. We establish coercivity conditions guaranteeing identifiability and well-posedness. The ALS algorithm appears statistically efficient and robust even in the small data regime but lacks performance and convergence guarantees. The ORALS estimator is consistent and asymptotically normal under a coercivity condition. We conduct several numerical experiments ranging from Kuramoto particle systems on networks to opinion dynamics in leader-follower models.
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Statistical tools which satisfy rigorous privacy guarantees are necessary for modern data analysis. It is well-known that robustness against contamination is linked to differential privacy. Despite this fact, using multivariate medians for differentially private and robust multivariate location estimation has not been systematically studied. We develop novel finite-sample performance guarantees for differentially private multivariate depth-based medians, which are essentially sharp. Our results cover commonly used depth functions, such as the halfspace (or Tukey) depth, spatial depth, and the integrated dual depth. We show that under Cauchy marginals, the cost of heavy-tailed location estimation outweighs the cost of privacy. We demonstrate our results numerically using a Gaussian contamination model in dimensions up to d = 100, and compare them to a state-of-the-art private mean estimation algorithm. As a by-product of our investigation, we prove concentration inequalities for the output of the exponential mechanism about the maximizer of the population objective function. This bound applies to objective functions that satisfy a mild regularity condition.
As ChatGPT possesses powerful capabilities in natural language processing and code analysis, it has received widespread attention since its launch. Developers have applied its powerful capabilities to various domains through software projects which are hosted on the largest open-source platform (GitHub) worldwide. Simultaneously, these projects have triggered extensive discussions. In order to comprehend the research content of these projects and understand the potential requirements discussed, we collected ChatGPT-related projects from the GitHub platform and utilized the LDA topic model to identify the discussion topics. Specifically, we selected 200 projects, categorizing them into three primary categories through analyzing their descriptions: ChatGPT implementation & training, ChatGPT application, ChatGPT improvement & extension. Subsequently, we employed the LDA topic model to identify 10 topics from issue texts, and compared the distribution and evolution trend of the discovered topics within the three primary project categories. Our observations include (1) The number of projects growing in a single month for the three primary project categories are closely associated with the development of ChatGPT. (2) There exist significant variations in the popularity of each topic for the three primary project categories. (3) The monthly changes in the absolute impact of each topic for the three primary project categories are diverse, which is often closely associated with the variation in the number of projects owned by that category. (4) With the passage of time, the relative impact of each topic exhibits different development trends in the three primary project categories. Based on these findings, we discuss implications for developers and users.
Navigating multi-robot systems in complex terrains has always been a challenging task. This is due to the inherent limitations of traditional robots in collision avoidance, adaptation to unknown environments, and sustained energy efficiency. In order to overcome these limitations, this research proposes a solution by integrating living insects with miniature electronic controllers to enable robotic-like programmable control, and proposing a novel control algorithm for swarming. Although these creatures, called cyborg insects, have the ability to instinctively avoid collisions with neighbors and obstacles while adapting to complex terrains, there is a lack of literature on the control of multi-cyborg systems. This research gap is due to the difficulty in coordinating the movements of a cyborg system under the presence of insects' inherent individual variability in their reactions to control input. In response to this issue, we propose a novel swarm navigation algorithm addressing these challenges. The effectiveness of the algorithm is demonstrated through an experimental validation in which a cyborg swarm was successfully navigated through an unknown sandy field with obstacles and hills. This research contributes to the domain of swarm robotics and showcases the potential of integrating biological organisms with robotics and control theory to create more intelligent autonomous systems with real-world applications.
Many classical inferential approaches fail to hold when interference exists among the population units. This amounts to the treatment status of one unit affecting the potential outcome of other units in the population. Testing for such spillover effects in this setting makes the null hypothesis non-sharp. An interesting approach to tackling the non-sharp nature of the null hypothesis in this setup is constructing conditional randomization tests such that the null is sharp on the restricted population. In randomized experiments, conditional randomized tests hold finite sample validity. Such approaches can pose computational challenges as finding these appropriate sub-populations based on experimental design can involve solving an NP-hard problem. In this paper, we view the network amongst the population as a random variable instead of being fixed. We propose a new approach that builds a conditional quasi-randomization test. Our main idea is to build the (non-sharp) null distribution of no spillover effects using random graph null models. We show that our method is exactly valid in finite-samples under mild assumptions. Our method displays enhanced power over other methods, with substantial improvement in complex experimental designs. We highlight that the method reduces to a simple permutation test, making it easy to implement in practice. We conduct a simulation study to verify the finite-sample validity of our approach and illustrate our methodology to test for interference in a weather insurance adoption experiment run in rural China.
Regularization of inverse problems is of paramount importance in computational imaging. The ability of neural networks to learn efficient image representations has been recently exploited to design powerful data-driven regularizers. While state-of-the-art plug-and-play methods rely on an implicit regularization provided by neural denoisers, alternative Bayesian approaches consider Maximum A Posteriori (MAP) estimation in the latent space of a generative model, thus with an explicit regularization. However, state-of-the-art deep generative models require a huge amount of training data compared to denoisers. Besides, their complexity hampers the optimization involved in latent MAP derivation. In this work, we first propose to use compressive autoencoders instead. These networks, which can be seen as variational autoencoders with a flexible latent prior, are smaller and easier to train than state-of-the-art generative models. As a second contribution, we introduce the Variational Bayes Latent Estimation (VBLE) algorithm, which performs latent estimation within the framework of variational inference. Thanks to a simple yet efficient parameterization of the variational posterior, VBLE allows for fast and easy (approximate) posterior sampling. Experimental results on image datasets BSD and FFHQ demonstrate that VBLE reaches similar performance than state-of-the-art plug-and-play methods, while being able to quantify uncertainties faster than other existing posterior sampling techniques.
We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.
Deep generative models aim to learn the underlying distribution of data and generate new ones. Despite the diversity of generative models and their high-quality generation performance in practice, most of them lack rigorous theoretical convergence proofs. In this work, we aim to establish some convergence results for OT-Flow, one of the deep generative models. First, by reformulating the framework of OT-Flow model, we establish the $\Gamma$-convergence of the formulation of OT-flow to the corresponding optimal transport (OT) problem as the regularization term parameter $\alpha$ goes to infinity. Second, since the loss function will be approximated by Monte Carlo method in training, we established the convergence between the discrete loss function and the continuous one when the sample number $N$ goes to infinity as well. Meanwhile, the approximation capability of the neural network provides an upper bound for the discrete loss function of the minimizers. The proofs in both aspects provide convincing assurances for OT-Flow.
Differential privacy is a mathematical concept that provides an information-theoretic security guarantee. While differential privacy has emerged as a de facto standard for guaranteeing privacy in data sharing, the known mechanisms to achieve it come with some serious limitations. Utility guarantees are usually provided only for a fixed, a priori specified set of queries. Moreover, there are no utility guarantees for more complex - but very common - machine learning tasks such as clustering or classification. In this paper we overcome some of these limitations. Working with metric privacy, a powerful generalization of differential privacy, we develop a polynomial-time algorithm that creates a private measure from a data set. This private measure allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools. Moreover, we prove an asymptotically sharp min-max result for private measures and synthetic data for general compact metric spaces. A key ingredient in our construction is a new superregular random walk, whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmicaly slowly.
We propose a method for obtaining parsimonious decompositions of networks into higher order interactions which can take the form of arbitrary motifs.The method is based on a class of analytically solvable generative models, where vertices are connected via explicit copies of motifs, which in combination with non-parametric priors allow us to infer higher order interactions from dyadic graph data without any prior knowledge on the types or frequencies of such interactions. Crucially, we also consider 'degree--corrected' models that correctly reflect the degree distribution of the network and consequently prove to be a better fit for many real world--networks compared to non-degree corrected models. We test the presented approach on simulated data for which we recover the set of underlying higher order interactions to a high degree of accuracy. For empirical networks the method identifies concise sets of atomic subgraphs from within thousands of candidates that cover a large fraction of edges and include higher order interactions of known structural and functional significance. The method not only produces an explicit higher order representation of the network but also a fit of the network to analytically tractable models opening new avenues for the systematic study of higher order network structures.
The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.
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