In this paper, we study the classification problem by estimating the conditional probability function of the given data. Different from the traditional expected risk estimation theory on empirical data, we calculate the probability via Fredholm equation, this leads to estimate the distribution of the data. Based on the Fredholm equation, a new expected risk estimation theory by estimating the cumulative distribution function is presented. The main characteristics of the new expected risk estimation is to measure the risk on the distribution of the input space. The corresponding empirical risk estimation is also presented, and an $\varepsilon$-insensitive $L_{1}$ cumulative support vector machines ($\varepsilon$-$L_{1}VSVM$) is proposed by introducing an insensitive loss. It is worth mentioning that the classification models and the classification evaluation indicators based on the new mechanism are different from the traditional one. Experimental results show the effectiveness of the proposed $\varepsilon$-$L_{1}VSVM$ and the corresponding cumulative distribution function indicator on validity and interpretability of small data classification.

We study an algorithmic equivalence technique between non-convex gradient descent and convex mirror descent. We start by looking at a harder problem of regret minimization in online non-convex optimization. We show that under certain geometric and smoothness conditions, online gradient descent applied to non-convex functions is an approximation of online mirror descent applied to convex functions under reparameterization. In continuous time, the gradient flow with this reparameterization was shown to be exactly equivalent to continuous-time mirror descent by Amid and Warmuth 2020, but theory for the analogous discrete time algorithms is left as an open problem. We prove an $O(T^{\frac{2}{3}})$ regret bound for non-convex online gradient descent in this setting, answering this open problem. Our analysis is based on a new and simple algorithmic equivalence method.

Improving fairness between privileged and less-privileged sensitive attribute groups (e.g, {race, gender}) has attracted lots of attention. To enhance the model performs uniformly well in different sensitive attributes, we propose a principled \underline{R}obust \underline{A}dversarial \underline{A}ttribute \underline{N}eighbourhood (RAAN) loss to debias the classification head and promote a fairer representation distribution across different sensitive attribute groups. The key idea of RAAN is to mitigate the differences of biased representations between different sensitive attribute groups by assigning each sample an adversarial robust weight, which is defined on the representations of adversarial attribute neighbors, i.e, the samples from different protected groups. To provide efficient optimization algorithms, we cast the RAAN into a sum of coupled compositional functions and propose a stochastic adaptive (Adam-style) and non-adaptive (SGD-style) algorithm framework SCRAAN with provable theoretical guarantee. Extensive empirical studies on fairness-related benchmark datasets verify the effectiveness of the proposed method.

We present a novel kernel over the space of probability measures based on the dual formulation of optimal regularized transport. We propose an Hilbertian embedding of the space of probabilities using their Sinkhorn potentials, which are solutions of the dual entropic relaxed optimal transport between the probabilities and a reference measure $\mathcal{U}$. We prove that this construction enables to obtain a valid kernel, by using the Hilbert norms. We prove that the kernel enjoys theoretical properties such as universality and some invariances, while still being computationally feasible. Moreover we provide theoretical guarantees on the behaviour of a Gaussian process based on this kernel. The empirical performances are compared with other traditional choices of kernels for processes indexed on distributions.

Consider the problem setting of Interaction-Grounded Learning (IGL), in which a learner's goal is to optimally interact with the environment with no explicit reward to ground its policies. The agent observes a context vector, takes an action, and receives a feedback vector, using this information to effectively optimize a policy with respect to a latent reward function. Prior analyzed approaches fail when the feedback vector contains the action, which significantly limits IGL's success in many potential scenarios such as Brain-computer interface (BCI) or Human-computer interface (HCI) applications. We address this by creating an algorithm and analysis which allows IGL to work even when the feedback vector contains the action, encoded in any fashion. We provide theoretical guarantees and large-scale experiments based on supervised datasets to demonstrate the effectiveness of the new approach.

Explainable artificial intelligence (XAI) twin systems will be a fundamental enabler of zero-touch network and service management (ZSM) for sixth-generation (6G) wireless networks. A reliable XAI twin system for ZSM requires two composites: an extreme analytical ability for discretizing the physical behavior of the Internet of Everything (IoE) and rigorous methods for characterizing the reasoning of such behavior. In this paper, a novel neuro-symbolic explainable artificial intelligence twin framework is proposed to enable trustworthy ZSM for a wireless IoE. The physical space of the XAI twin executes a neural-network-driven multivariate regression to capture the time-dependent wireless IoE environment while determining unconscious decisions of IoE service aggregation. Subsequently, the virtual space of the XAI twin constructs a directed acyclic graph (DAG)-based Bayesian network that can infer a symbolic reasoning score over unconscious decisions through a first-order probabilistic language model. Furthermore, a Bayesian multi-arm bandits-based learning problem is proposed for reducing the gap between the expected explained score and the current obtained score of the proposed neuro-symbolic XAI twin. To address the challenges of extensible, modular, and stateless management functions in ZSM, the proposed neuro-symbolic XAI twin framework consists of two learning systems: 1) an implicit learner that acts as an unconscious learner in physical space, and 2) an explicit leaner that can exploit symbolic reasoning based on implicit learner decisions and prior evidence. Experimental results show that the proposed neuro-symbolic XAI twin can achieve around 96.26% accuracy while guaranteeing from 18% to 44% more trust score in terms of reasoning and closed-loop automation.

Clustering is an important exploratory data analysis technique to group objects based on their similarity. The widely used $K$-means clustering method relies on some notion of distance to partition data into a fewer number of groups. In the Euclidean space, centroid-based and distance-based formulations of the $K$-means are equivalent. In modern machine learning applications, data often arise as probability distributions and a natural generalization to handle measure-valued data is to use the optimal transport metric. Due to non-negative Alexandrov curvature of the Wasserstein space, barycenters suffer from regularity and non-robustness issues. The peculiar behaviors of Wasserstein barycenters may make the centroid-based formulation fail to represent the within-cluster data points, while the more direct distance-based $K$-means approach and its semidefinite program (SDP) relaxation are capable of recovering the true cluster labels. In the special case of clustering Gaussian distributions, we show that the SDP relaxed Wasserstein $K$-means can achieve exact recovery given the clusters are well-separated under the $2$-Wasserstein metric. Our simulation and real data examples also demonstrate that distance-based $K$-means can achieve better classification performance over the standard centroid-based $K$-means for clustering probability distributions and images.

Survival analysis is a set of statistical approaches used to determine the time it takes for an event of interest to occur. We use survival analysis to study the time until some event of interest occurs. Time is usually measured in years, months, weeks, days, and other time measuring units. The event of interest could be anything of interest. It could be an actual death, a birth, a retirement, along with others.

In this work, we consider a distributed multi-agent stochastic optimization problem, where each agent holds a local objective function that is smooth and convex, and that is subject to a stochastic process. The goal is for all agents to collaborate to find a common solution that optimizes the sum of these local functions. With the practical assumption that agents can only obtain noisy numerical function queries at exactly one point at a time, we extend the distributed stochastic gradient-tracking method to the bandit setting where we don't have an estimate of the gradient, and we introduce a zero-order (ZO) one-point estimate (1P-DSGT). We analyze the convergence of this novel technique for smooth and convex objectives using stochastic approximation tools, and we prove that it converges almost surely to the optimum. We then study the convergence rate for when the objectives are additionally strongly convex. We obtain a rate of $O(\frac{1}{\sqrt{k}})$ after a sufficient number of iterations $k > K_2$ which is usually optimal for techniques utilizing one-point estimators. We also provide a regret bound of $O(\sqrt{k})$, which is exceptionally good compared to the aforementioned techniques. We further illustrate the usefulness of the proposed technique using numerical experiments.

As one of the most popular machine learning models today, graph neural networks (GNNs) have attracted intense interest recently, and so does their explainability. Users are increasingly interested in a better understanding of GNN models and their outcomes. Unfortunately, today's evaluation frameworks for GNN explainability often rely on few inadequate synthetic datasets, leading to conclusions of limited scope due to a lack of complexity in the problem instances. As GNN models are deployed to more mission-critical applications, we are in dire need for a common evaluation protocol of explainability methods of GNNs. In this paper, we propose, to our best knowledge, the first systematic evaluation framework for GNN explainability, considering explainability on three different "user needs". We propose a unique metric that combines the fidelity measures and classifies explanations based on their quality of being sufficient or necessary. We scope ourselves to node classification tasks and compare the most representative techniques in the field of input-level explainability for GNNs. For the inadequate but widely used synthetic benchmarks, surprisingly shallow techniques such as personalized PageRank have the best performance for a minimum computation time. But when the graph structure is more complex and nodes have meaningful features, gradient-based methods are the best according to our evaluation criteria. However, none dominates the others on all evaluation dimensions and there is always a trade-off. We further apply our evaluation protocol in a case study for frauds explanation on eBay transaction graphs to reflect the production environment.