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In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to its ability to capture intricate geometric properties of probability distributions while remaining computationally tractable, making it a valuable tool for various applications, including generative modeling and domain adaptation. Our study aims to provide a rigorous analysis of the critical points arising from the optimization of the SW objective. By computing explicit perturbations, we establish that stable critical points of SW cannot concentrate on segments. This stability analysis is crucial for understanding the behaviour of optimization algorithms for models trained using the SW objective. Furthermore, we investigate the properties of the SW objective, shedding light on the existence and convergence behavior of critical points. We illustrate our theoretical results through numerical experiments., Comment: 32p

Density ratio estimation in high dimensions can be reframed as integrating a certain quantity, the time score, over probability paths which interpolate between the two densities. In practice, the time score has to be estimated based on samples from the two densities. However, existing methods for this problem remain computationally expensive and can yield inaccurate estimates. Inspired by recent advances in generative modeling, we introduce a novel framework for time score estimation, based on a conditioning variable. Choosing the conditioning variable judiciously enables a closed-form objective function. We demonstrate that, compared to previous approaches, our approach results in faster learning of the time score and competitive or better estimation accuracies of the density ratio on challenging tasks. Furthermore, we establish theoretical guarantees on the error of the estimated density ratio.

The rapid growth of decentralized finance (DeFi) has led to the widespread use of automated agents, or bots, within blockchain ecosystems like Ethereum, Binance Smart Chain, and Solana. While these bots enhance market efficiency and liquidity, they also raise concerns due to exploitative behaviors that threaten network integrity and user trust. This paper presents a decentralized federated learning (DFL) approach for detecting financial bots within Ethereum Virtual Machine (EVM)-based blockchains. The proposed framework leverages federated learning, orchestrated through smart contracts, to detect malicious bot behavior while preserving data privacy and aligning with the decentralized nature of blockchain networks. Addressing the limitations of both centralized and rule-based approaches, our system enables each participating node to train local models on transaction history and smart contract interaction data, followed by on-chain aggregation of model updates through a permissioned consensus mechanism. This design allows the model to capture complex and evolving bot behaviors without requiring direct data sharing between nodes. Experimental results demonstrate that our DFL framework achieves high detection accuracy while maintaining scalability and robustness, providing an effective solution for bot detection across distributed blockchain networks., Comment: Paper accepted at the 2025 IEEE International Conference on Communications (ICC): Communication and Information System Security Symposium

The integration of Internet of Things (IoT) technology in various domains has led to operational advancements, but it has also introduced new vulnerabilities to cybersecurity threats, as evidenced by recent widespread cyberattacks on IoT devices. Intrusion detection systems are often reactive, triggered by specific patterns or anomalies observed within the network. To address this challenge, this work proposes a proactive approach to anticipate and preemptively mitigate malicious activities, aiming to prevent potential damage before it occurs. This paper proposes an innovative intrusion prediction framework empowered by Pre-trained Large Language Models (LLMs). The framework incorporates two LLMs: a fine-tuned Bidirectional and AutoRegressive Transformers (BART) model for predicting network traffic and a fine-tuned Bidirectional Encoder Representations from Transformers (BERT) model for evaluating the predicted traffic. By harnessing the bidirectional capabilities of BART the framework then identifies malicious packets among these predictions. Evaluated using the CICIoT2023 IoT attack dataset, our framework showcases a notable enhancement in predictive performance, attaining an impressive 98% overall accuracy, providing a powerful response to the cybersecurity challenges that confront IoT networks.

In this article, we provide a stochastic sampling algorithm with polynomial complexity in fixed dimension that leverages the recent advances on diffusion models where it is shown that under mild conditions, sampling can be achieved via an accurate estimation of intermediate scores across the marginals $(p_t)_{t\ge 0}$ of the standard Ornstein-Uhlenbeck process started at $\mu$, the density we wish to sample from. The heart of our method consists into approaching these scores via a computationally cheap estimator and relating the variance of this estimator to the smoothness properties of the forward process. Under the assumption that the density to sample from is $L$-log-smooth and that the forward process is semi-log-concave: $-\nabla^2 \log(p_t) \succeq -\beta I_d$ for some $\beta \geq 0$, we prove that our algorithm achieves an expected $\epsilon$ error in $\text{KL}$ divergence in $O(d^7(L+\beta)^2L^{d+2}\epsilon^{-2(d+3)}(d+m_2(\mu))^{2(d+1)})$ time with $m_2(\mu)$ the second order moment of $\mu$. In particular, our result allows to fully transfer the problem of sampling from a log-smooth distribution into a regularity estimate problem. As an application, we derive an exponential complexity improvement for the problem of sampling from an $L$-log-smooth distribution that is $\alpha$-strongly log-concave outside some ball of radius $R$: after proving that such distributions verify the semi-log-concavity assumption, a result which might be of independent interest, we recover a $poly(R, L, \alpha^{-1}, \epsilon^{-1})$ complexity in fixed dimension which exponentially improves upon the previously known $poly(e^{LR^2}, L,\alpha^{-1}, \log(\epsilon^{-1}))$ complexity in the low precision regime.

This work considers the problem of sampling from a probability distribution known up to a normalization constant while satisfying a set of statistical constraints specified by the expected values of general nonlinear functions. This problem finds applications in, e.g., Bayesian inference, where it can constrain moments to evaluate counterfactual scenarios or enforce desiderata such as prediction fairness. Methods developed to handle support constraints, such as those based on mirror maps, barriers, and penalties, are not suited for this task. This work therefore relies on gradient descent-ascent dynamics in Wasserstein space to put forward a discrete-time primal-dual Langevin Monte Carlo algorithm (PD-LMC) that simultaneously constrains the target distribution and samples from it. We analyze the convergence of PD-LMC under standard assumptions on the target distribution and constraints, namely (strong) convexity and log-Sobolev inequalities. To do so, we bring classical optimization arguments for saddle-point algorithms to the geometry of Wasserstein space. We illustrate the relevance and effectiveness of PD-LMC in several applications., Comment: 39 pages, 14 figures. Published at NeurIPS 2024

Geometric tempering is a popular approach to sampling from challenging multi-modal probability distributions by instead sampling from a sequence of distributions which interpolate, using the geometric mean, between an easier proposal distribution and the target distribution. In this paper, we theoretically investigate the soundness of this approach when the sampling algorithm is Langevin dynamics, proving both upper and lower bounds. Our upper bounds are the first analysis in the literature under functional inequalities. They assert the convergence of tempered Langevin in continuous and discrete-time, and their minimization leads to closed-form optimal tempering schedules for some pairs of proposal and target distributions. Our lower bounds demonstrate a simple case where the geometric tempering takes exponential time, and further reveal that the geometric tempering can suffer from poor functional inequalities and slow convergence, even when the target distribution is well-conditioned. Overall, our results indicate that geometric tempering may not help, and can even be harmful for convergence.

We introduce a (de)-regularization of the Maximum Mean Discrepancy (DrMMD) and its Wasserstein gradient flow. Existing gradient flows that transport samples from source distribution to target distribution with only target samples, either lack tractable numerical implementation ($f$-divergence flows) or require strong assumptions, and modifications such as noise injection, to ensure convergence (Maximum Mean Discrepancy flows). In contrast, DrMMD flow can simultaneously (i) guarantee near-global convergence for a broad class of targets in both continuous and discrete time, and (ii) be implemented in closed form using only samples. The former is achieved by leveraging the connection between the DrMMD and the $\chi^2$-divergence, while the latter comes by treating DrMMD as MMD with a de-regularized kernel. Our numerical scheme uses an adaptive de-regularization schedule throughout the flow to optimally trade off between discretization errors and deviations from the $\chi^2$ regime. The potential application of the DrMMD flow is demonstrated across several numerical experiments, including a large-scale setting of training student/teacher networks.

In this paper, we study the statistical and geometrical properties of the Kullback-Leibler divergence with kernel covariance operators (KKL) introduced by Bach [2022]. Unlike the classical Kullback-Leibler (KL) divergence that involves density ratios, the KKL compares probability distributions through covariance operators (embeddings) in a reproducible kernel Hilbert space (RKHS), and compute the Kullback-Leibler quantum divergence. This novel divergence hence shares parallel but different aspects with both the standard Kullback-Leibler between probability distributions and kernel embeddings metrics such as the maximum mean discrepancy. A limitation faced with the original KKL divergence is its inability to be defined for distributions with disjoint supports. To solve this problem, we propose in this paper a regularised variant that guarantees that the divergence is well defined for all distributions. We derive bounds that quantify the deviation of the regularised KKL to the original one, as well as finite-sample bounds. In addition, we provide a closed-form expression for the regularised KKL, specifically applicable when the distributions consist of finite sets of points, which makes it implementable. Furthermore, we derive a Wasserstein gradient descent scheme of the KKL divergence in the case of discrete distributions, and study empirically its properties to transport a set of points to a target distribution.

In recent years, numerous large-scale cyberattacks have exploited Internet of Things (IoT) devices, a phenomenon that is expected to escalate with the continuing proliferation of IoT technology. Despite considerable efforts in attack detection, intrusion detection systems remain mostly reactive, responding to specific patterns or observed anomalies. This work proposes a proactive approach to anticipate and mitigate malicious activities before they cause damage. This paper proposes a novel network intrusion prediction framework that combines Large Language Models (LLMs) with Long Short Term Memory (LSTM) networks. The framework incorporates two LLMs in a feedback loop: a fine-tuned Generative Pre-trained Transformer (GPT) model for predicting network traffic and a fine-tuned Bidirectional Encoder Representations from Transformers (BERT) for evaluating the predicted traffic. The LSTM classifier model then identifies malicious packets among these predictions. Our framework, evaluated on the CICIoT2023 IoT attack dataset, demonstrates a significant improvement in predictive capabilities, achieving an overall accuracy of 98%, offering a robust solution to IoT cybersecurity challenges.

The introduction of 6G technology into the Internet of Vehicles (IoV) promises to revolutionize connectivity with ultra-high data rates and seamless network coverage. However, this technological leap also brings significant challenges, particularly for the dynamic and diverse IoV landscape, which must meet the rigorous reliability and security requirements of 6G networks. Furthermore, integrating 6G will likely increase the IoV's susceptibility to a spectrum of emerging cyber threats. Therefore, it is crucial for security mechanisms to dynamically adapt and learn new attack patterns, keeping pace with the rapid evolution and diversification of these threats - a capability currently lacking in existing systems. This paper presents a novel intrusion detection system leveraging the paradigm of life-long (or continual) learning. Our methodology combines class-incremental learning with federated learning, an approach ideally suited to the distributed nature of the IoV. This strategy effectively harnesses the collective intelligence of Connected and Automated Vehicles (CAVs) and edge computing capabilities to train the detection system. To the best of our knowledge, this study is the first to synergize class-incremental learning with federated learning specifically for cyber attack detection. Through comprehensive experiments on a recent network traffic dataset, our system has exhibited a robust adaptability in learning new cyber attack patterns, while effectively retaining knowledge of previously encountered ones. Additionally, it has proven to maintain high accuracy and a low false positive rate.

In the rapidly evolving landscape of cyber threats targeting the Internet of Things (IoT) ecosystem, and in light of the surge in botnet-driven Distributed Denial of Service (DDoS) and brute force attacks, this study focuses on the early detection of IoT bots. It specifically addresses the detection of stealth bot communication that precedes and orchestrates attacks. This study proposes a comprehensive methodology for analyzing IoT network traffic, including considerations for both unidirectional and bidirectional flow, as well as packet formats. It explores a wide spectrum of network features critical for representing network traffic and characterizing benign IoT traffic patterns effectively. Moreover, it delves into the modeling of traffic using various semi-supervised learning techniques. Through extensive experimentation with the IoT-23 dataset - a comprehensive collection featuring diverse botnet types and traffic scenarios - we have demonstrated the feasibility of detecting botnet traffic corresponding to different operations and types of bots, specifically focusing on stealth command and control (C2) communications. The results obtained have demonstrated the feasibility of identifying C2 communication with a 100% success rate through packet-based methods and 94% via flow based approaches, with a false positive rate of 1.53%.

Intrusion Detection Systems (IDS) play a crucial role in ensuring the security of computer networks. Machine learning has emerged as a popular approach for intrusion detection due to its ability to analyze and detect patterns in large volumes of data. However, current ML-based IDS solutions often struggle to keep pace with the ever-changing nature of attack patterns and the emergence of new attack types. Additionally, these solutions face challenges related to class imbalance, where the number of instances belonging to different classes (normal and intrusions) is significantly imbalanced, which hinders their ability to effectively detect minor classes. In this paper, we propose a novel multi-agent reinforcement learning (RL) architecture, enabling automatic, efficient, and robust network intrusion detection. To enhance the capabilities of the proposed model, we have improved the DQN algorithm by implementing the weighted mean square loss function and employing cost-sensitive learning techniques. Our solution introduces a resilient architecture designed to accommodate the addition of new attacks and effectively adapt to changes in existing attack patterns. Experimental results realized using CIC-IDS-2017 dataset, demonstrate that our approach can effectively handle the class imbalance problem and provide a fine grained classification of attacks with a very low false positive rate. In comparison to the current state-of-the-art works, our solution demonstrates a significant superiority in both detection rate and false positive rate.

The Advanced Metering Infrastructure (AMI) is one of the key components of the smart grid. It provides interactive services for managing billing and electricity consumption, but it also introduces new vectors for cyberattacks. Although, the devastating and severe impact of power overloading cyberattacks on smart grid AMI, few researches in the literature have addressed them. In the present paper, we propose a two-level anomaly detection framework based on regression decision trees. The introduced detection approach leverages the regularity and predictability of energy consumption to build reference consumption patterns for the whole neighborhood and each household within it. Using a reference consumption pattern enables detecting power overloading cyberattacks regardless of the attacker's strategy as they cause a drastic change in the consumption pattern. The continuous two-level monitoring of energy consumption load allows efficient and early detection of cyberattacks. We carried out an extensive experiment on a real-world publicly available energy consumption dataset of 500 customers in Ireland. We extracted, from the raw data, the relevant attributes for training the energy consumption patterns. The evaluation shows that our approach achieves a high detection rate, a low false alarm rate, and superior performances compared to existing solutions.

Deploying Connected and Automated Vehicles (CAVs) on top of 5G and Beyond networks (5GB) makes them vulnerable to increasing vectors of security and privacy attacks. In this context, a wide range of advanced machine/deep learning based solutions have been designed to accurately detect security attacks. Specifically, supervised learning techniques have been widely applied to train attack detection models. However, the main limitation of such solutions is their inability to detect attacks different from those seen during the training phase, or new attacks, also called zero-day attacks. Moreover, training the detection model requires significant data collection and labeling, which increases the communication overhead, and raises privacy concerns. To address the aforementioned limits, we propose in this paper a novel detection mechanism that leverages the ability of the deep auto-encoder method to detect attacks relying only on the benign network traffic pattern. Using federated learning, the proposed intrusion detection system can be trained with large and diverse benign network traffic, while preserving the CAVs privacy, and minimizing the communication overhead. The in-depth experiment on a recent network traffic dataset shows that the proposed system achieved a high detection rate while minimizing the false positive rate, and the detection delay.

The Internet of Vehicles (IoV) is a crucial technology for Intelligent Transportation Systems (ITS) that integrates vehicles with the Internet and other entities. The emergence of 5G and the forthcoming 6G networks presents an enormous potential to transform the IoV by enabling ultra-reliable, low-latency, and high-bandwidth communications. Nevertheless, as connectivity expands, cybersecurity threats have become a significant concern. The issue has been further exacerbated by the rising number of zero-day (0-day) attacks, which can exploit unknown vulnerabilities and bypass existing Intrusion Detection Systems (IDSs). In this paper, we propose Zero-X, an innovative security framework that effectively detects both 0-day and N-day attacks. The framework achieves this by combining deep neural networks with Open-Set Recognition (OSR). Our approach introduces a novel scheme that uses blockchain technology to facilitate trusted and decentralized federated learning (FL) of the ZeroX framework. This scheme also prioritizes privacy preservation, enabling both CAVs and Security Operation Centers (SOCs) to contribute their unique knowledge while protecting the privacy of their sensitive data. To the best of our knowledge, this is the first work to leverage OSR in combination with privacy-preserving FL to identify both 0-day and N-day attacks in the realm of IoV. The in-depth experiments on two recent network traffic datasets show that the proposed framework achieved a high detection rate while minimizing the false positive rate. Comparison with related work showed that the Zero-X framework outperforms existing solutions.

Connected cars offer safety and efficiency for both individuals and fleets of private vehicles and public transportation companies. However, equipping vehicles with information and communication technologies raises privacy and security concerns, which significantly threaten the user's data and life. Using bot malware, a hacker may compromise a vehicle and control it remotely, for instance, he can disable breaks or start the engine remotely. In this paper, besides in-vehicle attacks existing in the literature, we consider new zeroday bot malware attacks specific to the vehicular context, WSMP-Flood, and Geo-WSMP Flood. Then, we propose AntibotV, a multilevel behaviour-based framework for vehicular botnets detection in vehicular networks. The proposed framework combines two main modules for attack detection, the first one monitors the vehicle's activity at the network level, whereas the second one monitors the in-vehicle activity. The two intrusion detection modules have been trained on a historical network and in-vehicle communication using decision tree algorithms. The experimental results showed that the proposed framework outperforms existing solutions, it achieves a detection rate higher than 97% and a false positive rate lower than 0.14%.

Diffusion models are state-of-the-art methods in generative modeling when samples from a target probability distribution are available, and can be efficiently sampled, using score matching to estimate score vectors guiding a Langevin process. However, in the setting where samples from the target are not available, e.g. when this target's density is known up to a normalization constant, the score estimation task is challenging. Previous approaches rely on Monte Carlo estimators that are either computationally heavy to implement or sample-inefficient. In this work, we propose a computationally attractive alternative, relying on the so-called dilation path, that yields score vectors that are available in closed-form. This path interpolates between a Dirac and the target distribution using a convolution. We propose a simple implementation of Langevin dynamics guided by the dilation path, using adaptive step-sizes. We illustrate the results of our sampling method on a range of tasks, and shows it performs better than classical alternatives.

As the problem of minimizing functionals on the Wasserstein space encompasses many applications in machine learning, different optimization algorithms on $\mathbb{R}^d$ have received their counterpart analog on the Wasserstein space. We focus here on lifting two explicit algorithms: mirror descent and preconditioned gradient descent. These algorithms have been introduced to better capture the geometry of the function to minimize and are provably convergent under appropriate (namely relative) smoothness and convexity conditions. Adapting these notions to the Wasserstein space, we prove guarantees of convergence of some Wasserstein-gradient-based discrete-time schemes for new pairings of objective functionals and regularizers. The difficulty here is to carefully select along which curves the functionals should be smooth and convex. We illustrate the advantages of adapting the geometry induced by the regularizer on ill-conditioned optimization tasks, and showcase the improvement of choosing different discrepancies and geometries in a computational biology task of aligning single-cells., Comment: Accepted as Spotlight at Conference on Neural Information Processing Systems (NeurIPS 2024)

Variational inference (VI) is a popular approach in Bayesian inference, that looks for the best approximation of the posterior distribution within a parametric family, minimizing a loss that is typically the (reverse) Kullback-Leibler (KL) divergence. Despite its empirical success, the theoretical properties of VI have only received attention recently, and mostly when the parametric family is the one of Gaussians. This work aims to contribute to the theoretical study of VI in the non-Gaussian case by investigating the setting of Mixture of Gaussians with fixed covariance and constant weights. In this view, VI over this specific family can be casted as the minimization of a Mollified relative entropy, i.e. the KL between the convolution (with respect to a Gaussian kernel) of an atomic measure supported on Diracs, and the target distribution. The support of the atomic measure corresponds to the localization of the Gaussian components. Hence, solving variational inference becomes equivalent to optimizing the positions of the Diracs (the particles), which can be done through gradient descent and takes the form of an interacting particle system. We study two sources of error of variational inference in this context when optimizing the mollified relative entropy. The first one is an optimization result, that is a descent lemma establishing that the algorithm decreases the objective at each iteration. The second one is an approximation error, that upper bounds the objective between an optimal finite mixture and the target distribution.

Off-policy learning (OPL) often involves minimizing a risk estimator based on importance weighting to correct bias from the logging policy used to collect data. However, this method can produce an estimator with a high variance. A common solution is to regularize the importance weights and learn the policy by minimizing an estimator with penalties derived from generalization bounds specific to the estimator. This approach, known as pessimism, has gained recent attention but lacks a unified framework for analysis. To address this gap, we introduce a comprehensive PAC-Bayesian framework to examine pessimism with regularized importance weighting. We derive a tractable PAC-Bayesian generalization bound that universally applies to common importance weight regularizations, enabling their comparison within a single framework. Our empirical results challenge common understanding, demonstrating the effectiveness of standard IW regularization techniques., Comment: Accepted at UAI 2024

In interactive systems, actions are often correlated, presenting an opportunity for more sample-efficient off-policy evaluation (OPE) and learning (OPL) in large action spaces. We introduce a unified Bayesian framework to capture these correlations through structured and informative priors. In this framework, we propose sDM, a generic Bayesian approach designed for OPE and OPL, grounded in both algorithmic and theoretical foundations. Notably, sDM leverages action correlations without compromising computational efficiency. Moreover, inspired by online Bayesian bandits, we introduce Bayesian metrics that assess the average performance of algorithms across multiple problem instances, deviating from the conventional worst-case assessments. We analyze sDM in OPE and OPL, highlighting the benefits of leveraging action correlations. Empirical evidence showcases the strong performance of sDM., Comment: 23 pages, 5 figures

We present a new algorithm to optimize distributions defined implicitly by parameterized stochastic diffusions. Doing so allows us to modify the outcome distribution of sampling processes by optimizing over their parameters. We introduce a general framework for first-order optimization of these processes, that performs jointly, in a single loop, optimization and sampling steps. This approach is inspired by recent advances in bilevel optimization and automatic implicit differentiation, leveraging the point of view of sampling as optimization over the space of probability distributions. We provide theoretical guarantees on the performance of our method, as well as experimental results demonstrating its effectiveness. We apply it to training energy-based models and finetuning denoising diffusions., Comment: 38 pages, 16 figures. Updated with additional experiments

This paper explores the connections between tempering (for Sequential Monte Carlo; SMC) and entropic mirror descent to sample from a target probability distribution whose unnormalized density is known. We establish that tempering SMC corresponds to entropic mirror descent applied to the reverse Kullback-Leibler (KL) divergence and obtain convergence rates for the tempering iterates. Our result motivates the tempering iterates from an optimization point of view, showing that tempering can be seen as a descent scheme of the KL divergence with respect to the Fisher-Rao geometry, in contrast to Langevin dynamics that perform descent of the KL with respect to the Wasserstein-2 geometry. We exploit the connection between tempering and mirror descent iterates to justify common practices in SMC and derive adaptive tempering rules that improve over other alternative benchmarks in the literature.

Off-policy learning (OPL) aims at finding improved policies from logged bandit data, often by minimizing the inverse propensity scoring (IPS) estimator of the risk. In this work, we investigate a smooth regularization for IPS, for which we derive a two-sided PAC-Bayes generalization bound. The bound is tractable, scalable, interpretable and provides learning certificates. In particular, it is also valid for standard IPS without making the assumption that the importance weights are bounded. We demonstrate the relevance of our approach and its favorable performance through a set of learning tasks. Since our bound holds for standard IPS, we are able to provide insight into when regularizing IPS is useful. Namely, we identify cases where regularization might not be needed. This goes against the belief that, in practice, clipped IPS often enjoys favorable performance than standard IPS in OPL., Comment: ICML 2023 (Oral and Poster)

Sampling from a target measure whose density is only known up to a normalization constant is a fundamental problem in computational statistics and machine learning. In this paper, we present a new optimization-based method for sampling called mollified interaction energy descent (MIED). MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs). These energies rely on mollifier functions -- smooth approximations of the Dirac delta originated from PDE theory. We show that as the mollifier approaches the Dirac delta, the MIE converges to the chi-square divergence with respect to the target measure and the gradient flow of the MIE agrees with that of the chi-square divergence. Optimizing this energy with proper discretization yields a practical first-order particle-based algorithm for sampling in both unconstrained and constrained domains. We show experimentally that for unconstrained sampling problems our algorithm performs on par with existing particle-based algorithms like SVGD, while for constrained sampling problems our method readily incorporates constrained optimization techniques to handle more flexible constraints with strong performance compared to alternatives.

This paper studies the Variational Inference (VI) used for training Bayesian Neural Networks (BNN) in the overparameterized regime, i.e., when the number of neurons tends to infinity. More specifically, we consider overparameterized two-layer BNN and point out a critical issue in the mean-field VI training. This problem arises from the decomposition of the lower bound on the evidence (ELBO) into two terms: one corresponding to the likelihood function of the model and the second to the Kullback-Leibler (KL) divergence between the prior distribution and the variational posterior. In particular, we show both theoretically and empirically that there is a trade-off between these two terms in the overparameterized regime only when the KL is appropriately re-scaled with respect to the ratio between the the number of observations and neurons. We also illustrate our theoretical results with numerical experiments that highlight the critical choice of this ratio.

Many problems in machine learning can be formulated as optimizing a convex functional over a vector space of measures. This paper studies the convergence of the mirror descent algorithm in this infinite-dimensional setting. Defining Bregman divergences through directional derivatives, we derive the convergence of the scheme for relatively smooth and convex pairs of functionals. Such assumptions allow to handle non-smooth functionals such as the Kullback--Leibler (KL) divergence. Applying our result to joint distributions and KL, we show that Sinkhorn's primal iterations for entropic optimal transport in the continuous setting correspond to a mirror descent, and we obtain a new proof of its (sub)linear convergence. We also show that Expectation Maximization (EM) can always formally be written as a mirror descent. When optimizing only on the latent distribution while fixing the mixtures parameters -- which corresponds to the Richardson--Lucy deconvolution scheme in signal processing -- we derive sublinear rates of convergence.

Adaptive importance sampling is a widely spread Monte Carlo technique that uses a re-weighting strategy to iteratively estimate the so-called target distribution. A major drawback of adaptive importance sampling is the large variance of the weights which is known to badly impact the accuracy of the estimates. This paper investigates a regularization strategy whose basic principle is to raise the importance weights at a certain power. This regularization parameter, that might evolve between zero and one during the algorithm, is shown (i) to balance between the bias and the variance and (ii) to be connected to the mirror descent framework. Using a kernel density estimate to build the sampling policy, the uniform convergence is established under mild conditions. Finally, several practical ways to choose the regularization parameter are discussed and the benefits of the proposed approach are illustrated empirically., Comment: 35 pages, 5 figures