The traditional Artificial Potential Field (APF) method exhibits limitations in its force distribution: excessive attraction when UAVs are far from the target may cause collisions with obstacles, while insufficient attraction near the goal often results in failure to reach the target. Furthermore, APF is highly susceptible to local minima, compromising motion reliability in complex environments. To address these challenges, this paper presents a novel hybrid obstacle avoidance algorithm-Deflected Simulated Annealing-Adaptive Artificial Potential Field (DSA-AAPF)-which combines an improved simulated annealing mechanism with an enhanced APF model. The proposed approach integrates a Leader-Follower distributed formation strategy with the APF framework, where the resultant force formulation is redefined to smooth UAV trajectories. An adaptive gravitational gain function is introduced to dynamically adjust UAV velocity based on environmental context, and a fast-converging controller ensures accurate and efficient convergence to the target. Moreover, a directional deflection mechanism is embedded within the simulated annealing process, enabling UAVs to escape local minima caused by semi-enclosed obstacles through continuous rotational motion. The simulation results, covering formation reconfiguration, complex obstacle avoidance, and entrapment escape, demonstrate the feasibility, robustness, and superiority of the proposed DSA-AAPF algorithm.

Artificial neural networks (ANNs) define parametrized families of f unctions (the realization functions) whose definition is inspired by biological neural networks. Running optimization algorithms on these parametrized families (the t raining of neural networks) has proven to be very efficient in various mach ine learning tasks, including image recognition, natural language processing, a utonomous systems, protein folding, climate modelling. The preferred method for the training of artificial neural networ ks (ANNs) are Stochastic Gradient Descent (SGD) algorithms. The vanilla SGD a lgorithm was first applied in Rumelhart et al. [ 1986 ]. Today, variants such as momentum-based methods [ Polyak, 1964 ], AMSProp [ Hinton, 2012 ] and the Adam optimizer [ Kingma and Ba, 2015 ] are more commonly used. Generally, the efficiency of optimization algorithms is significantly affec ted by the structure of the optimization landscape. The smoothing of u pdates in the momentum approaches seem to help with saddle points and adapt ive methods like RMSProp and Adam seem to adjust learning rates better to n avigate complex landscapes effectively. Mathematically rigorous approaches often assume that the SGD sc heme converges to a (local) minimum with a strongly convex neighborhood ( meaning that the Hessian of the landscape is strictly positive definite) or t hat a Polyak-null Lojasiewicz inequality (in the strong sense with exponent 2) applies.

We present a manifestly covariant formulation of the gradient descent method, ensuring consistency across arbitrary coordinate systems and general curved trainable spaces. The optimization dynamics is defined using a covariant force vector and a covariant metric tensor, both computed from the first and second statistical moments of the gradients. These moments are estimated through time-averaging with an exponential weight function, which preserves linear computational complexity. We show that commonly used optimization methods such as RMSProp, Adam and AdaBelief correspond to special limits of the covariant gradient descent (CGD) and demonstrate how these methods can be further generalized and improved.

Motivated by robust and quantile regression problems, we investigate the stochastic gradient descent (SGD) algorithm for minimizing an objective function $f$ that is locally strongly convex with a sub--quadratic tail. This setting covers many widely used online statistical methods. We introduce a novel piecewise Lyapunov function that enables us to handle functions $f$ with only first-order differentiability, which includes a wide range of popular loss functions such as Huber loss. Leveraging our proposed Lyapunov function, we derive finite-time moment bounds under general diminishing stepsizes, as well as constant stepsizes. We further establish the weak convergence, central limit theorem and bias characterization under constant stepsize, providing the first geometrical convergence result for sub--quadratic SGD. Our results have wide applications, especially in online statistical methods. In particular, we discuss two applications of our results. 1) Online robust regression: We consider a corrupted linear model with sub--exponential covariates and heavy--tailed noise. Our analysis provides convergence rates comparable to those for corrupted models with Gaussian covariates and noise. 2) Online quantile regression: Importantly, our results relax the common assumption in prior work that the conditional density is continuous and provide a more fine-grained analysis for the moment bounds.

Existing distribution compression methods, like Kernel Herding (KH), were originally developed for unlabelled data. However, no existing approach directly compresses the conditional distribution of labelled data. To address this gap, we first introduce the Average Maximum Conditional Mean Discrepancy (AMCMD), a natural metric for comparing conditional distributions. We then derive a consistent estimator for the AMCMD and establish its rate of convergence. Next, we make a key observation: in the context of distribution compression, the cost of constructing a compressed set targeting the AMCMD can be reduced from $\mathcal{O}(n^3)$ to $\mathcal{O}(n)$. Building on this, we extend the idea of KH to develop Average Conditional Kernel Herding (ACKH), a linear-time greedy algorithm that constructs a compressed set targeting the AMCMD. To better understand the advantages of directly compressing the conditional distribution rather than doing so via the joint distribution, we introduce Joint Kernel Herding (JKH), a straightforward adaptation of KH designed to compress the joint distribution of labelled data. While herding methods provide a simple and interpretable selection process, they rely on a greedy heuristic. To explore alternative optimisation strategies, we propose Joint Kernel Inducing Points (JKIP) and Average Conditional Kernel Inducing Points (ACKIP), which jointly optimise the compressed set while maintaining linear complexity. Experiments show that directly preserving conditional distributions with ACKIP outperforms both joint distribution compression (via JKH and JKIP) and the greedy selection used in ACKH. Moreover, we see that JKIP consistently outperforms JKH.

We revisit a classical assumption for analyzing stochastic gradient algorithms where the squared norm of the stochastic subgradient (or the variance for smooth problems) is allowed to grow as fast as the squared norm of the optimization variable. We contextualize this assumption in view of its inception in the 1960s, its seemingly independent appearance in the recent literature, its relationship to weakest-known variance assumptions for analyzing stochastic gradient algorithms, and its relevance in deterministic problems for non-Lipschitz nonsmooth convex optimization. We build on and extend a connection recently made between this assumption and the Halpern iteration. For convex nonsmooth, and potentially stochastic, optimization, we analyze horizon-free, anytime algorithms with last-iterate rates. For problems beyond simple constrained optimization, such as convex problems with functional constraints or regularized convex-concave min-max problems, we obtain rates for optimality measures that do not require boundedness of the feasible set.

Machine unlearning is a promising approach to improve LLM safety by removing unwanted knowledge from the model. However, prevailing gradient-based unlearning methods suffer from issues such as high computational costs, hyperparameter instability, poor sequential unlearning capability, vulnerability to relearning attacks, low data efficiency, and lack of interpretability. While Sparse Autoencoders are well-suited to improve these aspects by enabling targeted activation-based unlearning, prior approaches underperform gradient-based methods. This work demonstrates that, contrary to these earlier findings, SAEs can significantly improve unlearning when employed dynamically. We introduce $\textbf{Dynamic DAE Guardrails}$ (DSG), a novel method for precision unlearning that leverages principled feature selection and a dynamic classifier. Our experiments show DSG substantially outperforms leading unlearning methods, achieving superior forget-utility trade-offs. DSG addresses key drawbacks of gradient-based approaches for unlearning -- offering enhanced computational efficiency and stability, robust performance in sequential unlearning, stronger resistance to relearning attacks, better data efficiency including zero-shot settings, and more interpretable unlearning.

In practical instances of nonconvex matrix factorization, the rank of the true solution $r^{\star}$ is often unknown, so the rank $r$ of the model can be overspecified as $r>r^{\star}$. This over-parameterized regime of matrix factorization significantly slows down the convergence of local search algorithms, from a linear rate with $r=r^{\star}$ to a sublinear rate when $r>r^{\star}$. We propose an inexpensive preconditioner for the matrix sensing variant of nonconvex matrix factorization that restores the convergence rate of gradient descent back to linear, even in the over-parameterized case, while also making it agnostic to possible ill-conditioning in the ground truth. Classical gradient descent in a neighborhood of the solution slows down due to the need for the model matrix factor to become singular. Our key result is that this singularity can be corrected by $\ell_{2}$ regularization with a specific range of values for the damping parameter. In fact, a good damping parameter can be inexpensively estimated from the current iterate. The resulting algorithm, which we call preconditioned gradient descent or PrecGD, is stable under noise, and converges linearly to an information theoretically optimal error bound. Our numerical experiments find that PrecGD works equally well in restoring the linear convergence of other variants of nonconvex matrix factorization in the over-parameterized regime.

Modern neural networks are usually highly over-parameterized. Behind the wide usage of over-parameterized networks is the belief that, if the data are simple, then the trained network will be automatically equivalent to a simple predictor. Following this intuition, many existing works have studied different notions of "ranks" of neural networks and their relation to the rank of data. In this work, we study the rank of convolutional neural networks (CNNs) trained by gradient descent, with a specific focus on the robustness of the rank to image background noises. Specifically, we point out that, when adding background noises to images, the rank of the CNN trained with gradient descent is affected far less compared with the rank of the data. We support our claim with a theoretical case study, where we consider a particular data model to characterize low-rank clean images with added background noises. We prove that CNNs trained by gradient descent can learn the intrinsic dimension of clean images, despite the presence of relatively large background noises. We also conduct experiments on synthetic and real datasets to further validate our claim.

We propose smoothed primal-dual algorithms for solving stochastic and smooth nonconvex optimization problems with linear inequality constraints. Our algorithms are single-loop and only require a single stochastic gradient based on one sample at each iteration. A distinguishing feature of our algorithm is that it is based on an inexact gradient descent framework for the Moreau envelope, where the gradient of the Moreau envelope is estimated using one step of a stochastic primal-dual augmented Lagrangian method. To handle inequality constraints and stochasticity, we combine the recently established global error bounds in constrained optimization with a Moreau envelope-based analysis of stochastic proximal algorithms. For obtaining $\varepsilon$-stationary points, we establish the optimal $O(\varepsilon^{-4})$ sample complexity guarantee for our algorithms and provide extensions to stochastic linear constraints. We also show how to improve this complexity to $O(\varepsilon^{-3})$ by using variance reduction and the expected smoothness assumption. Unlike existing methods, the iterations of our algorithms are free of subproblems, large batch sizes or increasing penalty parameters and use dual variable updates to ensure feasibility.