The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods.

Decentralized learning provides a scalable alternative to parameter-server-based training, yet its performance is often hindered by limited peer-to-peer communication. In this paper, we study how communication should be scheduled over time, including determining when and how frequently devices synchronize. Counterintuitive empirical results show that concentrating communication budgets in the later stages of decentralized training remarkably improves global test performance. Surprisingly, we uncover that fully connected communication at the final step, implemented by a single global merging, can significantly improve the performance of decentralized learning under high data heterogeneity. Our theoretical contributions, which explain these phenomena, are the first to establish that the globally merged model of decentralized SGD can match the convergence rate of parallel SGD. Technically, we reinterpret part of the discrepancy among local models, which were previously considered as detrimental noise, as constructive components essential for matching this rate. This work provides evidence that decentralized learning is able to generalize under high data heterogeneity and limited communication, while offering broad new avenues for model merging research.

Different federated optimization algorithms typically employ distinct client-selection strategies: some methods communicate only with a randomly sampled subset of clients at each round, while others need to periodically communicate with all clients or use a hybrid scheme that combines both strategies. However, existing metrics for comparing optimization methods typically do not distinguish between these strategies, which often incur different communication costs in practice. To address this disparity, we introduce a simple and natural model of federated optimization that quantifies communication and local computation complexities. This new model allows for several commonly used client-selection strategies and explicitly associates each with a distinct cost. Within this setting, we propose a new algorithm that achieves the best-known communication and local complexities among existing federated optimization methods for non-convex optimization. This algorithm is based on the inexact composite gradient method with a carefully constructed gradient estimator and a special procedure for solving the auxiliary subproblem at each iteration. The gradient estimator is based on SAGA, a popular variance-reduced gradient estimator. We first derive a new variance bound for it, showing that SAGA can exploit functional similarity. We then introduce the Recursive-Gradient technique as a general way to potentially improve the error bound of a given conditionally unbiased gradient estimator, including both SAGA and SVRG. By applying this technique to SAGA, we obtain a new estimator, RG-SAGA, which has an improved error bound compared to the original one.

This paper focuses on Geodesic Principal Component Analysis (GPCA) on a collection of probability distributions using the Otto-Wasserstein geometry. The goal is to identify geodesic curves in the space of probability measures that best capture the modes of variation of the underlying dataset. We first address the case of a collection of Gaussian distributions, and show how to lift the computations in the space of invertible linear maps. For the more general setting of absolutely continuous probability measures, we leverage a novel approach to parameterizing geodesics in Wasserstein space with neural networks. Finally, we compare to classical tangent PCA through various examples and provide illustrations on real-world datasets.

Scaling laws describe how learning performance improves with data, compute, or training time, and have become a central theme in modern deep learning. We study this phenomenon in a canonical nonlinear model: phase retrieval with anisotropic Gaussian inputs whose covariance spectrum follows a power law. Unlike the isotropic case, where dynamics collapse to a two-dimensional system, anisotropy yields a qualitatively new regime in which an infinite hierarchy of coupled equations governs the evolution of the summary statistics. We develop a tractable reduction that reveals a three-phase trajectory: (i) fast escape from low alignment, (ii) slow convergence of the summary statistics, and (iii) spectral-tail learning in low-variance directions. From this decomposition, we derive explicit scaling laws for the mean-squared error, showing how spectral decay dictates convergence times and error curves. Experiments confirm the predicted phases and exponents. These results provide the first rigorous characterization of scaling laws in nonlinear regression with anisotropic data, highlighting how anisotropy reshapes learning dynamics.

Neural networks (NNs) often have critical behavioural trade-offs that are set at design time with hyperparameters—such as reward weights in reinforcement learning or quantile targets in regression. Post-deployment, however, user preferences can evolve, making initial settings undesirable, necessitating potentially expensive retraining. To circumvent this, we introduce the task of Hyperparameter Trajectory Inference (HTI): to learn, from observed data, how a NN's conditional output distribution changes with its hyperparameters, and construct a surrogate model that approximates the NN at unobserved hyperparameter settings. HTI requires extending existing trajectory inference approaches to incorporate conditions, exacerbating the challenge of ensuring inferred paths are feasible. We propose an approach based on conditional Lagrangian optimal transport, jointly learning the Lagrangian function governing hyperparameter-induced dynamics along with the associated optimal transport maps and geodesics between observed marginals, which form the surrogate model. We incorporate inductive biases based on the manifold hypothesis and least-action principles into the learned Lagrangian, improving surrogate model feasibility. We empirically demonstrate that our approach reconstructs NN outputs across various hyperparameter spectra better than other alternatives.

Machine unlearning seeks to efficiently remove the influence of selected data while preserving generalization. Significant progress has been made in low dimensions, where the dimension of the parameter $p$ is much smaller than the sample size $n$, but high dimensions, including proportional regimes $p \sim n$, pose serious theoretical challenges as standard optimization assumptions of $\Omega(1)$ strong convexity and $O(1)$ smoothness of the per-example loss $f$ rarely hold simultaneously in proportional regimes $p\sim n$. In this work, we introduce $\varepsilon$-Gaussian certifiability, a canonical and robust notion well-suited to high-dimensional regimes, that optimally captures a broad class of noise adding mechanisms. Then we theoretically analyze the performance of a widely used unlearning algorithm based on one step of the Newton method in the high-dimensional setting described above. Our analysis shows that a single Newton step, followed by a well-calibrated Gaussian noise, is sufficient to achieve both privacy and accuracy in this setting. This result stands in sharp contrast to the only prior work that analyzes machine unlearning in high dimensions \citet{zou2025certified}, which relaxes some of the standard optimization assumptions for high-dimensional applicability, but operates under the notion of $\varepsilon$-certifiability. That work concludes %that a single Newton step is insufficient even for removing a single data point, and that at least two steps are required to ensure both privacy and accuracy. Our result leads us to conclude that the discrepancy in the number of steps arises because of the sub optimality of the notion of $\varepsilon$-certifiability and its incompatibility with noise adding mechanisms, which $\varepsilon$-Gaussian certifiability is able to overcome optimally.