The recent success of Shampoo in the AlgoPerf contest has sparked renewed interest in Kronecker-factorization-based optimization algorithms for training neural networks. Despite its success, Shampoo relies heavily on several heuristics such as learning rate grafting and stale preconditioning to achieve performance at-scale. These heuristics increase algorithmic complexity, necessitate further hyperparameter tuning, and lack theoretical justification. This paper investigates these heuristics from the angle of Frobenius norm approximation to full-matrix Adam and decouples the preconditioner's eigenvalues and eigenbasis updates. We show that grafting from Adam mitigates the staleness and mis-scaling of the preconditioner's eigenvalues and how correcting the eigenvalues directly can eliminate the need for learning rate grafting. To manage the error induced by infrequent eigenbasis computations, we propose an adaptive criterion for determining the eigenbasis computation frequency motivated by terminating a warm-started QR algorithm. This criterion decouples the update frequency of different preconditioner matrices and enables us to investigate the impact of approximation error on convergence. These practical techniques offer a principled angle towards removing Shampoo's heuristics and developing improved Kronecker-factorization-based training algorithms.

We present a generic Branch-and-Bound procedure designed to solve L0-penalized optimization problems. Existing approaches primarily focus on quadratic losses and construct relaxations using "Big-M" constraints and/or L2-norm penalties. In contrast, our method accommodates a broader class of loss functions and allows greater flexibility in relaxation design through a general penalty term, encompassing existing techniques as special cases. We establish theoretical results ensuring that all key quantities required for the Branch-and-Bound implementation admit closed-form expressions under the general blanket assumptions considered in our work. Leveraging this framework, we introduce El0ps, an open-source Python solver with a plug-and-play workflow that enables user-defined losses and penalties in L0-penalized problems. Through extensive numerical experiments, we demonstrate that El0ps achieves state-of-the-art performance on classical instances and extends computational feasibility to previously intractable ones.

Stochastic Frank-Wolfe is a classical optimization method for solving constrained optimization problems. On the other hand, recent optimizers such as Lion and Muon have gained quite significant popularity in deep learning. In this work, we provide a unifying perspective by interpreting these seemingly disparate methods through the lens of Stochastic Frank-Wolfe. Specifically, we show that Lion and Muon with weight decay can be viewed as special instances of a Stochastic Frank-Wolfe, and we establish their convergence guarantees in terms of the Frank-Wolfe gap, a standard stationarity measure in non-convex optimization for Frank-Wolfe methods. We further find that convergence to this gap implies convergence to a KKT point of the original problem under a norm constraint for Lion and Muon. Moreover, motivated by recent empirical findings that stochastic gradients in modern machine learning tasks often exhibit heavy-tailed distributions, we extend Stochastic Frank-Wolfe to settings with heavy-tailed noise by developing two robust variants with strong theoretical guarantees, which in turn yields new variants of Lion and Muon.

Abstract-- In this work, we present a method for minimizing the time required for a redundant dual-arm robot to follow a desired relative Cartesian path at constant path speed by optimizing its joint trajectories, subject to position, ve locity, and acceleration limits. The problem is reformulated as a bi - level optimization whose lower level is a convex, closed-fo rm subproblem that maximizes path speed for a fixed trajectory, while the upper level updates the trajectory using a single-chain kinematic formulation and the subgradient of the lower-lev el value. Numerical results demonstrate the effectiveness of the proposed approach. In several industrial tasks performed by robots, the Cartesian path to be followed by the end-effector requires fewer degrees of freedom than what the manipulator possesses. In many applications [1]-[5], rotation around the attached tool center point (TCP) is considered a free axis, resulting in a kinematically redundant system [6], [7].

Fuzzy clustering, which allows an article to belong to multiple clusters with soft membership degrees, plays a vital role in analyzing publication data. This problem can be formulated as a constrained optimization model, where the goal is to minimize the discrepancy between the similarity observed from data and the similarity derived from a predicted distribution. While this approach benefits from leveraging state-of-the-art optimization algorithms, tailoring them to work with real, massive databases like OpenAlex or Web of Science - containing about 70 million articles and a billion citations - poses significant challenges. We analyze potentials and challenges of the approach from both mathematical and computational perspectives. Among other things, second-order optimality conditions are established, providing new theoretical insights, and practical solution methods are proposed by exploiting the structure of the problem. Specifically, we accelerate the gradient projection method using GPU-based parallel computing to efficiently handle large-scale data.

Reconstructing 3D scenes from sparse viewpoints is a long-standing challenge with wide applications. Recent advances in feed-forward 3D Gaussian sparse-view reconstruction methods provide an efficient solution for real-time novel view synthesis by leveraging geometric priors learned from large-scale multi-view datasets and computing 3D Gaussian centers via back-projection. Despite offering strong geometric cues, both feed-forward multi-view depth estimation and flow-depth joint estimation face key limitations: the former suffers from mislocation and artifact issues in low-texture or repetitive regions, while the latter is prone to local noise and global inconsistency due to unreliable matches when ground-truth flow supervision is unavailable. To overcome this, we propose JointSplat, a unified framework that leverages the complementarity between optical flow and depth via a novel probabilistic optimization mechanism. Specifically, this pixel-level mechanism scales the information fusion between depth and flow based on the matching probability of optical flow during training. Building upon the above mechanism, we further propose a novel multi-view depth-consistency loss to leverage the reliability of supervision while suppressing misleading gradients in uncertain areas. Evaluated on RealEstate10K and ACID, JointSplat consistently outperforms state-of-the-art (SOTA) methods, demonstrating the effectiveness and robustness of our proposed probabilistic joint flow-depth optimization approach for high-fidelity sparse-view 3D reconstruction.

As the number of mobile devices continues to grow, interference has become a major bottleneck in improving data rates in wireless networks. Efficient joint channel and power allocation (JCPA) is crucial for managing interference. In this paper, we first propose an enhanced WMMSE (eWMMSE) algorithm to solve the JCPA problem in multi-channel wireless networks. To reduce the computational complexity of iterative optimization, we further introduce JCPGNN-M, a graph neural network-based solution that enables simultaneous multi-channel allocation for each user. We reformulate the problem as a Lagrangian function, which allows us to enforce the total power constraints systematically. Our solution involves combining this Lagrangian framework with GNNs and iteratively updating the Lagrange multipliers and resource allocation scheme. Unlike existing GNN-based methods that limit each user to a single channel, JCPGNN-M supports efficient spectrum reuse and scales well in dense network scenarios. Simulation results show that JCPGNN-M achieves better data rate compared to eWMMSE. Meanwhile, the inference time of JCPGNN-M is much lower than eWMMS, and it can generalize well to larger networks.

Proximal Policy Optimization (PPO) has become a widely adopted algorithm for reinforcement learning, offering a practical policy gradient method with strong empirical performance. Despite its popularity, PPO lacks formal theoretical guarantees for policy improvement and convergence. PPO is motivated by Trust Region Policy Optimization (TRPO) that utilizes a surrogate loss with a KL divergence penalty, which arises from linearizing the value function within a flat geometric space. In this paper, we derive a tighter surrogate in the Fisher-Rao (FR) geometry, yielding a novel variant, Fisher-Rao PPO (FR-PPO). Our proposed scheme provides strong theoretical guarantees, including monotonic policy improvement. Furthermore, in the tabular setting, we demonstrate that FR-PPO achieves sub-linear convergence without any dependence on the dimensionality of the action or state spaces, marking a significant step toward establishing formal convergence results for PPO-based algorithms.

The challenge of identifying the best feasible arm within a fixed budget has attracted considerable interest in recent years. However, a notable gap remains in the literature: the exact exponential rate at which the error probability approaches zero has yet to be established, even in the relatively simple setting of $K$-armed bandits with Gaussian noise. In this paper, we address this gap by examining the problem within the context of linear bandits. We introduce a novel algorithm for best feasible arm identification that guarantees an exponential decay in the error probability. Remarkably, the decay rate -- characterized by the exponent -- matches the theoretical lower bound derived using information-theoretic principles. Our approach leverages a posterior sampling framework embedded within a game-based sampling rule involving a min-learner and a max-learner. This strategy shares its foundations with Thompson sampling, but is specifically tailored to optimize the identification process under fixed-budget constraints. Furthermore, we validate the effectiveness of our algorithm through comprehensive empirical evaluations across various problem instances with different levels of complexity. The results corroborate our theoretical findings and demonstrate that our method outperforms several benchmark algorithms in terms of both accuracy and efficiency.

Edge computing addresses the growing data demands of connected-device networks by placing computational resources closer to end users through decentralized infrastructures. This decentralization challenges traditional, fully centralized orchestration, which suffers from latency and resource bottlenecks. We present \textbf{FAuNO} -- \emph{Federated Asynchronous Network Orchestrator} -- a buffered, asynchronous \emph{federated reinforcement-learning} (FRL) framework for decentralized task offloading in edge systems. FAuNO adopts an actor-critic architecture in which local actors learn node-specific dynamics and peer interactions, while a federated critic aggregates experience across agents to encourage efficient cooperation and improve overall system performance. Experiments in the \emph{PeersimGym} environment show that FAuNO consistently matches or exceeds heuristic and federated multi-agent RL baselines in reducing task loss and latency, underscoring its adaptability to dynamic edge-computing scenarios.