Hierarchical Federated Learning (HFL) has recently emerged as a promising solution for intelligent decision-making in vehicular networks, helping to address challenges such as limited communication resources, high vehicle mobility, and data heterogeneity. However, HFL remains vulnerable to adversarial and unreliable vehicles, whose misleading updates can significantly compromise the integrity and convergence of the global model. To address these challenges, we propose a novel defense framework that integrates dynamic vehicle selection with robust anomaly detection within a cluster-based HFL architecture, specifically designed to counter Gaussian noise and gradient ascent attacks. The framework performs a comprehensive reliability assessment for each vehicle by evaluating historical accuracy, contribution frequency, and anomaly records. Anomaly detection combines Z-score and cosine similarity analyses on model updates to identify both statistical outliers and directional deviations in model updates. To further refine detection, an adaptive thresholding mechanism is incorporated into the cosine similarity metric, dynamically adjusting the threshold based on the historical accuracy of each vehicle to enforce stricter standards for consistently high-performing vehicles. In addition, a weighted gradient averaging mechanism is implemented, which assigns higher weights to gradient updates from more trustworthy vehicles. To defend against coordinated attacks, a cross-cluster consistency check is applied to identify collaborative attacks in which multiple compromised clusters coordinate misleading updates. Together, these mechanisms form a multi-level defense strategy to filter out malicious contributions effectively. Simulation results show that the proposed algorithm significantly reduces convergence time compared to benchmark methods across both 1-hop and 3-hop topologies.

Robotic automation is a key technology that increases the efficiency and flexibility of manufacturing processes. However, one of the challenges in deploying robots in novel environments is finding the optimal base pose for the robot, which affects its reachability and deployment cost. Yet, the existing research for automatically optimizing the base pose of robots has not been compared. We address this problem by optimizing the base pose of industrial robots with Bayesian optimization, exhaustive search, genetic algorithms, and stochastic gradient descent and find that all algorithms can reduce the cycle time for various evaluated tasks in synthetic and real-world environments. Stochastic gradient descent shows superior performance with regard to success rate solving over 90% of our real-world tasks, while genetic algorithms show the lowest final costs. All benchmarks and implemented methods are available as baselines against which novel approaches can be compared.

Gradient descent based optimization methods are the methods of choice to train deep neural networks in machine learning. Beyond the standard gradient descent method, also suitable modified variants of standard gradient descent involving acceleration techniques such as the momentum method and/or adaptivity techniques such as the RMSprop method are frequently considered optimization methods. These days the most popular of such sophisticated optimization schemes is presumably the Adam optimizer that has been proposed in 2014 by Kingma and Ba. A highly relevant topic of research is to investigate the speed of convergence of such optimization methods. In particular, in 1964 Polyak showed that the standard gradient descent method converges in a neighborhood of a strict local minimizer with rate (x - 1)(x + 1)^{-1} while momentum achieves the (optimal) strictly faster convergence rate (\sqrt{x} - 1)(\sqrt{x} + 1)^{-1} where x \in (1,\infty) is the condition number (the ratio of the largest and the smallest eigenvalue) of the Hessian of the objective function at the local minimizer. It is the key contribution of this work to reveal that Adam also converges with the strictly faster convergence rate (\sqrt{x} - 1)(\sqrt{x} + 1)^{-1} while RMSprop only converges with the convergence rate (x - 1)(x + 1)^{-1}.

This paper introduces Tempered Fractional Gradient Descent (TFGD), a novel optimization framework that synergizes fractional calculus with exponential tempering to enhance gradient-based learning. Traditional gradient descent methods often suffer from oscillatory updates and slow convergence in high-dimensional, noisy landscapes. TFGD addresses these limitations by incorporating a tempered memory mechanism, where historical gradients are weighted by fractional coefficients $|w_j| = \binomα{j}$ and exponentially decayed via a tempering parameter $λ$. Theoretical analysis establishes TFGD's convergence guarantees: in convex settings, it achieves an $\mathcal{O}(1/K)$ rate with alignment coefficient $d_{α,λ} = (1 - e^{-λ})^{-α}$, while stochastic variants attain $\mathcal{O}(1/k^α)$ error decay. The algorithm maintains $\mathcal{O}(n)$ time complexity equivalent to SGD, with memory overhead scaling as $\mathcal{O}(d/λ)$ for parameter dimension $d$. Empirical validation on the Breast Cancer Wisconsin dataset demonstrates TFGD's superiority, achieving 98.25\% test accuracy (vs. 92.11\% for SGD) and 2$\times$ faster convergence. The tempered memory mechanism proves particularly effective in medical classification tasks, where feature correlations benefit from stable gradient averaging. These results position TFGD as a robust alternative to conventional optimizers in both theoretical and applied machine learning.

We study continual learning on multiple linear classification tasks by sequentially running gradient descent (GD) for a fixed budget of iterations per task. When all tasks are jointly linearly separable and are presented in a cyclic/random order, we show the directional convergence of the trained linear classifier to the joint (offline) max-margin solution. This is surprising because GD training on a single task is implicitly biased towards the individual max-margin solution for the task, and the direction of the joint max-margin solution can be largely different from these individual solutions. Additionally, when tasks are given in a cyclic order, we present a non-asymptotic analysis on cycle-averaged forgetting, revealing that (1) alignment between tasks is indeed closely tied to catastrophic forgetting and backward knowledge transfer and (2) the amount of forgetting vanishes to zero as the cycle repeats. Lastly, we analyze the case where the tasks are no longer jointly separable and show that the model trained in a cyclic order converges to the unique minimum of the joint loss function.

We study the complexity of online stochastic gradient descent (SGD) for learning a two-layer neural network with $P$ neurons on isotropic Gaussian data: $f_*(\boldsymbol{x}) = \sum_{p=1}^P a_p\cdot \sigma(\langle\boldsymbol{x},\boldsymbol{v}_p^*\rangle)$, $\boldsymbol{x} \sim \mathcal{N}(0,\boldsymbol{I}_d)$, where the activation $\sigma:\mathbb{R}\to\mathbb{R}$ is an even function with information exponent $k_*>2$ (defined as the lowest degree in the Hermite expansion), $\{\boldsymbol{v}^*_p\}_{p\in[P]}\subset \mathbb{R}^d$ are orthonormal signal directions, and the non-negative second-layer coefficients satisfy $\sum_{p} a_p^2=1$. We focus on the challenging ``extensive-width'' regime $P\gg 1$ and permit diverging condition number in the second-layer, covering as a special case the power-law scaling $a_p\asymp p^{-\beta}$ where $\beta\in\mathbb{R}_{\ge 0}$. We provide a precise analysis of SGD dynamics for the training of a student two-layer network to minimize the mean squared error (MSE) objective, and explicitly identify sharp transition times to recover each signal direction. In the power-law setting, we characterize scaling law exponents for the MSE loss with respect to the number of training samples and SGD steps, as well as the number of parameters in the student neural network. Our analysis entails that while the learning of individual teacher neurons exhibits abrupt transitions, the juxtaposition of $P\gg 1$ emergent learning curves at different timescales leads to a smooth scaling law in the cumulative objective.

This paper proposes and analyzes a gradient-type algorithm based on Burer-Monteiro factorization, called the Asymmetric Projected Gradient Descent (APGD), for reconstructing the point set configuration from partial Euclidean distance measurements, known as the Euclidean Distance Matrix Completion (EDMC) problem. By paralleling the incoherence matrix completion framework, we show for the first time that global convergence guarantee with exact recovery of this routine can be established given $\mathcal{O}(\mu^2 r^3 \kappa^2 n \log n)$ Bernoulli random observations without any sample splitting. Unlike leveraging the tangent space Restricted Isometry Property (RIP) and local curvature of the low-rank embedding manifold in some very recent works, our proof provides new upper bounds to replace the random graph lemma under EDMC setting. The APGD works surprisingly well and numerical experiments demonstrate exact linear convergence behavior in rich-sample regions yet deteriorates fast when compared with the performance obtained by optimizing the s-stress function, i.e., the standard but unexplained non-convex approach for EDMC, if the sample size is limited. While virtually matching our theoretical prediction, this unusual phenomenon might indicate that: (i) the power of implicit regularization is weakened when specified in the APGD case; (ii) the stabilization of such new gradient direction requires substantially more samples than the information-theoretic limit would suggest.

Imitation learning holds the promise of equipping robots with versatile skills by learning from expert demonstrations. However, policies trained on finite datasets often struggle to generalize beyond the training distribution. In this work, we present a unified perspective on the generalization capability of imitation learning, grounded in both information theorey and data distribution property. We first show that the generalization gap can be upper bounded by (i) the conditional information bottleneck on intermediate representations and (ii) the mutual information between the model parameters and the training dataset. This characterization provides theoretical guidance for designing effective training strategies in imitation learning, particularly in determining whether to freeze, fine-tune, or train large pretrained encoders (e.g., vision-language models or vision foundation models) from scratch to achieve better generalization. Furthermore, we demonstrate that high conditional entropy from input to output induces a flatter likelihood landscape, thereby reducing the upper bound on the generalization gap. In addition, it shortens the stochastic gradient descent (SGD) escape time from sharp local minima, which may increase the likelihood of reaching global optima under fixed optimization budgets. These insights explain why imitation learning often exhibits limited generalization and underscore the importance of not only scaling the diversity of input data but also enriching the variability of output labels conditioned on the same input.

W e generalize the connection between activation function and spline regression/smoothing and characterize how this choice may influence spectral bias within a 1D shallow network. W e then demonstrate how gradient descent (GD) can be reinterpreted as a shrinkage operator that masks the singular values of a neural network's Jacobian. Viewed this way, GD implicitly selects the number of frequency components to retain, thereby controlling the spectral bias. An explicit relationship is proposed between the choice of GD hyperparameters (learning rate & number of iterations) and bandwidth (the number of active components). GD regularization is shown to be effective only with monotonic activation functions. Finally, we highlight the utility of non-monotonic activation functions (sinc, Gaussian) as iteration-efficient surrogates for spectral bias.

Two-time-scale stochastic approximation is an algorithm with coupled iterations which has found broad applications in reinforcement learning, optimization and game control. While several prior works have obtained a mean square error bound of $O(1/k)$ for linear two-time-scale iterations, the best known bound in the non-linear contractive setting has been $O(1/k^{2/3})$. In this work, we obtain an improved bound of $O(1/k)$ for non-linear two-time-scale stochastic approximation. Our result applies to algorithms such as gradient descent-ascent and two-time-scale Lagrangian optimization. The key step in our analysis involves rewriting the original iteration in terms of an averaged noise sequence which decays sufficiently fast. Additionally, we use an induction-based approach to show that the iterates are bounded in expectation.