Machine learning is a typical optimisation problem, and its learning process is an iterative optimisation process in the parameter space. It is a natural way of thinking to consider the iterative motion process of this algorithm as a kinetic process. The theory of dynamics has been developed over a long period of time and is very complete, with quantum dynamics, Newtonian dynamics, thermodynamics, electrodynamics and molecular dynamics, which theoretically describes the laws of motion by establishing a set of kinetic equations. The establishment of the dynamics theory of machine learning is expected to address the lack of theoretical models in this field, thus advancing the development of machine learning theory and applications. The theoretical modeling of optimization problems using a dynamics approach was pioneered by Metropolis in 1953, drawing from thermodynamics [1].

Recent years have witnessed a thriving growth of computing facilities connected at the network edge, cultivating edge computing networks as a fundamental infrastructure for supporting miscellaneous intelligent services. Meanwhile, Artificial Intelligence frontiers have extrapolated Machine Learning to the graph domain and promoted Graph Intelligence (GI), which unlocks unprecedented ability in learning from massive data in graph structures. Given the inherent relation between graphs and networks, the interdiscipline of graph representation learning and edge networks, i.e., Edge GI or EGI, has revealed a novel interplay between them -- GI models principally open a new door for modeling, understanding, and optimizing edge networks, and conversely, edge networks serve as physical support for training, deploying, and accelerating GI models. Driven by this delicate closed-loop, EGI can be widely recognized as a promising solution to fully unleash the potential of edge computing power and is garnering significant attention. Nevertheless, research on EGI yet remains nascent, and there is a soaring demand within both the communications and AI communities for a dedicated venue to share recent advancements. To this end, this paper promotes the concept of EGI, explores its scope and core principles, and conducts a comprehensive survey concerning recent research efforts on this emerging field and specifically, introduces and discusses: 1) fundamentals of edge computing and graph representation learning, 2) emerging techniques centering on the closed loop between graph intelligence and edge networks, and 3) open challenges and research opportunities of future EGI. By bridging the gap across communication, networking, and graph learning areas, we believe that this survey can garner increased attention, foster meaningful discussions, and inspire further research ideas in EGI.

Probabilistic state estimation is essential for robots navigating uncertain environments. Accurately and efficiently managing uncertainty in estimated states is key to robust robotic operation. However, nonlinearities in robotic platforms pose significant challenges that require advanced estimation techniques. Gaussian variational inference (GVI) offers an optimization perspective on the estimation problem, providing analytically tractable solutions and efficiencies derived from the geometry of Gaussian space. We propose a Sequential Gaussian Variational Inference (S-GVI) method to address nonlinearity and provide efficient sequential inference processes. Our approach integrates sequential Bayesian principles into the GVI framework, which are addressed using statistical approximations and gradient updates on the information geometry. Validations through simulations and real-world experiments demonstrate significant improvements in state estimation over the Maximum A Posteriori (MAP) estimation method.

Identifying causal effects is a key problem of interest across many disciplines. The two long-standing approaches to estimate causal effects are observational and experimental (randomized) studies. Observational studies can suffer from unmeasured confounding, which may render the causal effects unidentifiable. On the other hand, direct experiments on the target variable may be too costly or even infeasible to conduct. A middle ground between these two approaches is to estimate the causal effect of interest through proxy experiments, which are conducted on variables with a lower cost to intervene on compared to the main target. Akbari et al. [2022] studied this setting and demonstrated that the problem of designing the optimal (minimum-cost) experiment for causal effect identification is NP-complete and provided a naive algorithm that may require solving exponentially many NP-hard problems as a sub-routine in the worst case. In this work, we provide a few reformulations of the problem that allow for designing significantly more efficient algorithms to solve it as witnessed by our extensive simulations. Additionally, we study the closely-related problem of designing experiments that enable us to identify a given effect through valid adjustments sets.

The rise of advanced air mobility (AAM) is expected to become a multibillion-dollar industry in the near future. Market-based mechanisms are touted to be an integral part of AAM operations, which comprise heterogeneous operators with private valuations. In this work, we study the problem of designing a mechanism to coordinate the movement of electric vertical take-off and landing (eVTOL) aircraft, operated by multiple operators each having heterogeneous valuations associated with their fleet, between vertiports, while enforcing the arrival, departure, and parking constraints at vertiports. Particularly, we propose an incentive-compatible and individually rational vertiport reservation mechanism that maximizes a social welfare metric, which encapsulates the objective of maximizing the overall valuations of all operators while minimizing the congestion at vertiports. Additionally, we improve the computational tractability of designing the reservation mechanism by proposing a mixed binary linear programming approach that is based on constructing network flow graph corresponding to the underlying problem.

Convex programming plays a fundamental role in machine learning, data science, and engineering. Testing convexity structure in nonlinear programs relies on verifying the convexity of objectives and constraints. \citet{grant2006disciplined} introduced a framework, Disciplined Convex Programming (DCP), for automating this verification task for a wide range of convex functions that can be decomposed into basic convex functions (atoms) using convexity-preserving compositions and transformations (rules). However, the restriction to Euclidean convexity concepts can limit the applicability of the framework. For instance, many notable instances of statistical estimators and matrix-valued (sub)routines in machine learning applications are Euclidean non-convex, but exhibit geodesic convexity through a more general Riemannian lens. In this work, we extend disciplined programming to this setting by introducing Disciplined Geodesically Convex Programming (DGCP). We determine convexity-preserving compositions and transformations for geodesically convex functions on general Cartan-Hadamard manifolds, as well as for the special case of symmetric positive definite matrices, a common setting in matrix-valued optimization. For the latter, we also define a basic set of atoms. Our paper is accompanied by a Julia package SymbolicAnalysis.jl, which provides functionality for testing and certifying DGCP-compliant expressions. Our library interfaces with manifold optimization software, which allows for directly solving verified geodesically convex programs.

In this paper, we consider a multi-stage dynamic assortment optimization problem with multi-nomial choice modeling (MNL) under resource knapsack constraints. Given the current resource inventory levels, the retailer makes an assortment decision at each period, and the goal of the retailer is to maximize the total profit from purchases. With the exact optimal dynamic assortment solution being computationally intractable, a practical strategy is to adopt the re-solving technique that periodically re-optimizes deterministic linear programs (LP) arising from fluid approximation. However, the fractional structure of MNL makes the fluid approximation in assortment optimization highly non-linear, which brings new technical challenges. To address this challenge, we propose a new epoch-based re-solving algorithm that effectively transforms the denominator of the objective into the constraint. Theoretically, we prove that the regret (i.e., the gap between the resolving policy and the optimal objective of the fluid approximation) scales logarithmically with the length of time horizon and resource capacities.

DNA sequences encode vital genetic and biological information, yet these unfixed-length sequences cannot serve as the input of common data mining algorithms. Hence, various representation schemes have been developed to transform DNA sequences into fixed-length numerical representations. However, these schemes face difficulties in learning high-quality representations due to the complexity and sparsity of DNA data. Additionally, DNA sequences are inherently noisy because of mutations. While several schemes have been proposed for their effectiveness, they often lack semantic structure, making it difficult for biologists to validate and leverage the results. To address these challenges, we propose \textbf{Dy-mer}, an explainable and robust DNA representation scheme based on sparse recovery. Leveraging the underlying semantic structure of DNA, we modify the traditional sparse recovery to capture recurring patterns indicative of biological functions by representing frequent K-mers as basis vectors and reconstructing each DNA sequence through simple concatenation. Experimental results demonstrate that \textbf{Dy-mer} achieves state-of-the-art performance in DNA promoter classification, yielding a remarkable \textbf{13\%} increase in accuracy. Moreover, its inherent explainability facilitates DNA clustering and motif detection, enhancing its utility in biological research.

Data-driven methods such as reinforcement and imitation learning have achieved remarkable success in robot autonomy. However, their data-centric nature still hinders them from generalizing well to ever-changing environments. Moreover, collecting large datasets for robotic tasks is often impractical and expensive. To overcome these challenges, we introduce a new self-supervised neural-symbolic (NeSy) computational framework, imperative learning (IL), for robot autonomy, leveraging the generalization abilities of symbolic reasoning. The framework of IL consists of three primary components: a neural module, a reasoning engine, and a memory system. We formulate IL as a special bilevel optimization (BLO), which enables reciprocal learning over the three modules. This overcomes the label-intensive obstacles associated with data-driven approaches and takes advantage of symbolic reasoning concerning logical reasoning, physical principles, geometric analysis, etc. We discuss several optimization techniques for IL and verify their effectiveness in five distinct robot autonomy tasks including path planning, rule induction, optimal control, visual odometry, and multi-robot routing. Through various experiments, we show that IL can significantly enhance robot autonomy capabilities and we anticipate that it will catalyze further research across diverse domains.

Bayesian causal discovery offers the power to quantify epistemic uncertainties among a broad range of structurally diverse causal theories potentially explaining the data, represented in forms of directed acyclic graphs (DAGs). However, existing methods struggle with efficient DAG sampling due to the complex acyclicity constraint. In this study, we propose a scalable Bayesian approach to effectively learn the posterior distribution over causal graphs given observational data thanks to the ability to generate DAGs without explicitly enforcing acyclicity. Specifically, we introduce a novel differentiable DAG sampling method that can generate a valid acyclic causal graph by mapping an unconstrained distribution of implicit topological orders to a distribution over DAGs. Given this efficient DAG sampling scheme, we are able to model the posterior distribution over causal graphs using a simple variational distribution over a continuous domain, which can be learned via the variational inference framework. Extensive empirical experiments on both simulated and real datasets demonstrate the superior performance of the proposed model compared to several state-of-the-art baselines.