Optimal Transport (OT) theory seeks to determine the map $T:X \to Y$ that transports a source measure $P$ to a target measure $Q$, minimizing the cost $c(\mathbf{x}, T(\mathbf{x}))$ between $\mathbf{x}$ and its image $T(\mathbf{x})$. Building upon the Input Convex Neural Network OT solver and incorporating the concept of displacement-sparse maps, we introduce a sparsity penalty into the minimax Wasserstein formulation, promote sparsity in displacement vectors $\Delta(\mathbf{x}) := T(\mathbf{x}) - \mathbf{x}$, and enhance the interpretability of the resulting map. However, increasing sparsity often reduces feasibility, causing $T_{\#}(P)$ to deviate more significantly from the target measure. In low-dimensional settings, we propose a heuristic framework to balance the trade-off between sparsity and feasibility by dynamically adjusting the sparsity intensity parameter during training. For high-dimensional settings, we directly constrain the dimensionality of displacement vectors by enforcing $\dim(\Delta(\mathbf{x})) \leq l$, where $l < d$ for $X \subseteq \mathbb{R}^d$. Among maps satisfying this constraint, we aim to identify the most feasible one. This goal can be effectively achieved by adapting our low-dimensional heuristic framework without resorting to dimensionality reduction. We validate our method on both synthesized sc-RNA and real 4i cell perturbation datasets, demonstrating improvements over existing methods.

Adaptive optimization algorithms -- such as Adagrad, Adam, and their variants -- have found widespread use in machine learning, signal processing and many other settings. Several methods in this family are not rotationally equivariant, meaning that simple reparameterizations (i.e. change of basis) can drastically affect their convergence. However, their sensitivity to the choice of parameterization has not been systematically studied; it is not clear how to identify a "favorable" change of basis in which these methods perform best. In this paper we propose a reparameterization method and demonstrate both theoretically and empirically its potential to improve their convergence behavior. Our method is an orthonormal transformation based on the expected gradient outer product (EGOP) matrix, which can be approximated using either full-batch or stochastic gradient oracles. We show that for a broad class of functions, the sensitivity of adaptive algorithms to choice-of-basis is influenced by the decay of the EGOP matrix spectrum. We illustrate the potential impact of EGOP reparameterization by presenting empirical evidence and theoretical arguments that common machine learning tasks with "natural" data exhibit EGOP spectral decay.

Federated multi-task learning (FMTL) aims to simultaneously learn multiple related tasks across clients without sharing sensitive raw data. However, in the decentralized setting, existing FMTL frameworks are limited in their ability to capture complex task relationships and handle feature and sample heterogeneity across clients. To address these challenges, we introduce a novel sheaf-theoretic-based approach for FMTL. By representing client relationships using cellular sheaves, our framework can flexibly model interactions between heterogeneous client models. We formulate the sheaf-based FMTL optimization problem using sheaf Laplacian regularization and propose the Sheaf-FMTL algorithm to solve it. We show that the proposed framework provides a unified view encompassing many existing federated learning (FL) and FMTL approaches. Furthermore, we prove that our proposed algorithm, Sheaf-FMTL, achieves a sublinear convergence rate in line with state-of-the-art decentralized FMTL algorithms. Extensive experiments demonstrate that Sheaf-FMTL exhibits communication savings by sending significantly fewer bits compared to decentralized FMTL baselines.

Bilevel optimization, addressing challenges in hierarchical learning tasks, has gained significant interest in machine learning. The practical implementation of the gradient descent method to bilevel optimization encounters computational hurdles, notably the computation of the exact lower-level solution and the inverse Hessian of the lower-level objective. Although these two aspects are inherently connected, existing methods typically handle them separately by solving the lowerlevel problem and a linear system for the inverse Hessian-vector product. In this paper, we introduce a general framework to address these computational challenges in a coordinated manner. Specifically, we leverage quasi-Newton algorithms to accelerate the resolution of the lower-level problem while efficiently approximating the inverse Hessian-vector product. Furthermore, by exploiting the superlinear convergence properties of BFGS, we establish the non-asymptotic convergence analysis of the BFGS adaptation within our framework. Numerical experiments demonstrate the comparable or superior performance of the proposed algorithms in real-world learning tasks, including hyperparameter optimization, data hypercleaning, and few-shot meta-learning. Bilevel optimization (BLO), which addresses challenges in hierarchical decision process, has gained significant interest in many real-world applications.

Vision sensors are extensively used for localizing a robot's pose, particularly in environments where global localization tools such as GPS or motion capture systems are unavailable. In many visual navigation systems, localization is achieved by detecting and tracking visual features or landmarks, which provide information about the sensor's relative pose. For reliable feature tracking and accurate pose estimation, it is crucial to maintain visibility of a sufficient number of features. This requirement can sometimes conflict with the robot's overall task objective. In this paper, we approach it as a constrained control problem. By leveraging the invariance properties of visibility constraints within the robot's kinematic model, we propose a real-time safety filter based on quadratic programming. This filter takes a reference velocity command as input and produces a modified velocity that minimally deviates from the reference while ensuring the information score from the currently visible features remains above a user-specified threshold. Numerical simulations demonstrate that the proposed safety filter preserves the invariance condition and ensures the visibility of more features than the required minimum. We also validated its real-world performance by integrating it into a visual simultaneous localization and mapping (SLAM) algorithm, where it maintained high estimation quality in challenging environments, outperforming a simple tracking controller.

In this paper, we study the machine learning elements which we are interested in together as a machine learning system, consisting of a collection of machine learning elements and a collection of relations between the elements. The relations we concern are algebraic operations, binary relations, and binary relations with composition that can be reasoned categorically. A machine learning system transformation between two systems is a map between the systems, which preserves the relations we concern. The system transformations given by quotient or clustering, representable functor, and Yoneda embedding are highlighted and discussed by machine learning examples. An adjunction between machine learning systems, a special machine learning system transformation loop, provides the optimal way of solving problems. Machine learning system transformations are linked and compared by their maps at 2-cell, natural transformations. New insights and structures can be obtained from universal properties and algebraic structures given by monads, which are generated from adjunctions.

Non-smooth and non-convex global optimization poses significant challenges across various applications, where standard gradient-based methods often struggle. We propose the Ball-Proximal Point Method, Broximal Point Method, or Ball Point Method (BPM) for short - a novel algorithmic framework inspired by the classical Proximal Point Method (PPM) (Rockafellar, 1976), which, as we show, sheds new light on several foundational optimization paradigms and phenomena, including non-convex and non-smooth optimization, acceleration, smoothing, adaptive stepsize selection, and trust-region methods. At the core of BPM lies the ball-proximal ("broximal") operator, which arises from the classical proximal operator by replacing the quadratic distance penalty by a ball constraint. Surprisingly, and in sharp contrast with the sublinear rate of PPM in the nonsmooth convex regime, we prove that BPM converges linearly and in a finite number of steps in the same regime. Furthermore, by introducing the concept of ball-convexity, we prove that BPM retains the same global convergence guarantees under weaker assumptions, making it a powerful tool for a broader class of potentially non-convex optimization problems. Just like PPM plays the role of a conceptual method inspiring the development of practically efficient algorithms and algorithmic elements, e.g., gradient descent, adaptive step sizes, acceleration (Ahn & Sra, 2020), and "W" in AdamW (Zhuang et al., 2022), we believe that BPM should be understood in the same manner: as a blueprint and inspiration for further development.

Variational Quantum Eigensolvers (VQEs) are a powerful class of hybrid quantum-classical algorithms designed to approximate the ground state of a quantum system described by its Hamiltonian. VQEs hold promise for various applications, including lattice field theory. However, the inherent noise of Noisy Intermediate-Scale Quantum (NISQ) devices poses a significant challenge for running VQEs as these algorithms are particularly susceptible to noise, e.g., measurement shot noise and hardware noise. In a recent work, it was proposed to enhance the classical optimization of VQEs with Gaussian Processes (GPs) and Bayesian Optimization, as these machine-learning techniques are well-suited for handling noisy data. In these proceedings, we provide additional insights into this new algorithm and present further numerical experiments. In particular, we examine the impact of hardware noise and error mitigation on the algorithm's performance. We validate the algorithm using classical simulations of quantum hardware, including hardware noise benchmarks, which have not been considered in previous works. Our numerical experiments demonstrate that GP-enhanced algorithms can outperform state-of-the-art baselines, laying the foundation for future research on deploying these techniques to real quantum hardware and lattice field theory setups.

Hyperparameter optimization (HPO) is a crucial step in achieving strong predictive performance. However, the impact of individual hyperparameters on model generalization is highly context-dependent, prohibiting a one-size-fits-all solution and requiring opaque automated machine learning (AutoML) systems to find optimal configurations. The black-box nature of most AutoML systems undermines user trust and discourages adoption. To address this, we propose a game-theoretic explainability framework for HPO that is based on Shapley values and interactions. Our approach provides an additive decomposition of a performance measure across hyperparameters, enabling local and global explanations of hyperparameter importance and interactions. The framework, named HyperSHAP, offers insights into ablations, the tunability of learning algorithms, and optimizer behavior across different hyperparameter spaces. We evaluate HyperSHAP on various HPO benchmarks by analyzing the interaction structure of the HPO problem. Our results show that while higher-order interactions exist, most performance improvements can be explained by focusing on lower-order representations.

Deep graph learning and network science both analyze graphs but approach similar problems from different perspectives. Whereas network science focuses on models and measures that reveal the organizational principles of complex systems with explicit assumptions, deep graph learning focuses on flexible and generalizable models that learn patterns in graph data in an automated fashion. Despite these differences, both fields share the same goal: to better model and understand patterns in graph-structured data. Early efforts to integrate methods, models, and measures from network science and deep graph learning indicate significant untapped potential. In this position, we explore opportunities at their intersection. We discuss open challenges in deep graph learning, including data augmentation, improved evaluation practices, higher-order models, and pooling methods. Likewise, we highlight challenges in network science, including scaling to massive graphs, integrating continuous gradient-based optimization, and developing standardized benchmarks.