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Oliver Limoyo, Filip Marić, Matthew Giamou, Petra Alexson, Ivan Petrović, Jonathan Kelly

Quickly and reliably finding accurate inverse kinematics (IK) solutions remains a challenging problem for many robot manipulators. Existing numerical solvers are broadly applicable but typically only produce a single solution and rely on local search techniques to minimize nonconvex objective functions. More recent learning-based approaches that approximate the entire feasible set of solutions have shown promise as a means to generate multiple fast and accurate IK results in parallel. However, existing learning-based techniques have a significant drawback: each robot of interest requires a specialized model that must be trained from scratch. To address this key shortcoming, we propose a novel distance-geometric robot representation coupled with a graph structure that allows us to leverage the sample efficiency of Euclidean equivariant functions and the generalizability of graph neural networks (GNNs). Our approach is generative graphical inverse kinematics (GGIK), the first learned IK solver able to accurately and efficiently produce a large number of diverse solutions in parallel while also displaying the ability to generalize -- a single learned model can be used to produce IK solutions for a variety of different robots. When compared to several other learned IK methods, GGIK provides more accurate solutions with the same amount of data. GGIK can generalize reasonably well to robot manipulators unseen during training. Additionally, GGIK can learn a constrained distribution that encodes joint limits and scales efficiently to larger robots and a high number of sampled solutions. Finally, GGIK can be used to complement local IK solvers by providing reliable initializations for a local optimization process.

Mohit Kumar, Alexander Valentinitsch, Magdalena Fuchs, Mathias Brucker, Juliana Bowles, Adnan Husakovic, Ali Abbas, Bernhard A. Moser

This paper develops a novel mathematical framework for collaborative learning by means of geometrically inspired kernel machines which includes statements on the bounds of generalisation and approximation errors, and sample complexity. For classification problems, this approach allows us to learn bounded geometric structures around given data points and hence solve the global model learning problem in an efficient way by exploiting convexity properties of the related optimisation problem in a Reproducing Kernel Hilbert Space (RKHS). In this way, we can reduce classification problems to determining the closest bounded geometric structure from a given data point. Further advantages that come with our solution is that our approach does not require clients to perform multiple epochs of local optimisation using stochastic gradient descent, nor require rounds of communication between client/server for optimising the global model. We highlight that numerous experiments have shown that the proposed method is a competitive alternative to the state-of-the-art.

Jatin Chaudhary, Dipak Nidhi, Jukka Heikkonen, Harri Merisaari, Rajeev Kumar Kanth

This paper presents a theoretically grounded optimization framework for neural network training that integrates an Exponentially Decaying Learning Rate with Lyapunov-based stability analysis. We develop a dynamic learning rate algorithm and prove that it induces connected and stable descent paths through the loss landscape by maintaining the connectivity of super-level sets Sλ = {θ ∈ ℝn : ℒ(θ) ≥ λ}. Under the condition that the Lyapunov function V(θ) = ℒ(θ) satisfies Δ V(θ) ⋅ Δ ℒ(θ) ≥ 0, we establish that these super-level sets are not only connected but also equiconnected across epochs, providing uniform topological stability. We further derive convergence guarantees using a second-order Taylor expansion and demonstrate that our exponentially scheduled learning rate with gradient-based modulation leads to a monotonic decrease in loss. The proposed algorithm incorporates this schedule into a stability-aware update mechanism that adapts step sizes based on both curvature and energy-level geometry. This work formalizes the role of topological structure in convergence dynamics and introduces a provably stable optimization algorithm for high-dimensional, non-convex neural networks.

Yuval Shapira, Dana Drachsler-Cohen

Few-pixel attacks mislead a classifier by modifying a few pixels of an image. Their perturbation space is an ℓ₀-ball, which is not convex, unlike ℓₚ-balls for p ≥ 1. However, existing local robustness verifiers typically scale by relying on linear bound propagation, which captures convex perturbation spaces. We show that the convex hull of an ℓ₀-ball is the intersection of its bounding box and an asymmetrically scaled ℓ₁-like polytope. The volumes of the convex hull and this polytope are nearly equal as the input dimension increases. We then show a linear bound propagation that precisely computes bounds over the convex hull and is significantly tighter than bound propagations over the bounding box or our ℓ₁-like polytope. This bound propagation scales the state-of-the-art ℓ₀ verifier on its most challenging robustness benchmarks by 1.24x-7.07x, with a geometric mean of 3.16.

Shigeki Kusaka, Keita Saito, Mikoto Kudo, Takumi Tanabe, Akifumi Wachi, Youhei Akimoto

Large language models (LLMs) are increasingly deployed in real-world systems, making it critical to understand their vulnerabilities. While data poisoning attacks during RLHF/DPO alignment have been studied empirically, their theoretical foundations remain unclear. We investigate the minimum-cost poisoning attack required to steer an LLM’s policy toward an attacker’s target by flipping preference labels during RLHF/DPO, without altering the compared outputs. We formulate this as a convex optimization problem with linear constraints, deriving lower and upper bounds on the minimum attack cost. As a byproduct of this theoretical analysis, we show that any existing label-flipping attack can be post-processed via our proposed method to reduce the number of label flips required while preserving the intended poisoning effect. Empirical results demonstrate that this cost-minimization post-processing can significantly reduce poisoning costs over baselines, particularly when the reward model’s feature dimension is small relative to the dataset size. These findings highlight fundamental vulnerabilities in RLHF/DPO pipelines and provide tools to evaluate their robustness against low-cost poisoning attacks.

Mohammad Mahdi Omati, Yasin Salajeghe, Mahshad Moradi, Arash Amini

Semi-supervised learning (SSL) on graphs is critical in applications where labeled data are scarce and costly, yet existing graph-based methods often degrade under extreme label sparsity or class imbalance, yielding trivial or unstable solutions. We introduce \textbf{CombCut}, the first exact combinatorial optimization framework for multi-class graph-based semi-supervised learning that operates directly on binary one-hot assignments, without any convex relaxation or heuristic volume constraints. By employing a minorization–maximization (MM) scheme, CombCut transforms each step into a structured linear assignment problem solved efficiently via network-flow algorithms. Total unimodularity guarantees integral iterates, and our theoretical analysis establishes both monotonic ascent of the true discrete objective and convergence of every limit point to a Karush–Kuhn–Tucker (KKT) stationary solution of the original combinatorial problem. Our approach requires no hyperparameter tuning and scales near-linearly in the number of vertices. Empirical evaluation on MNIST, Fashion-MNIST, and CIFAR-10 with as few as 1–5 labels per class shows that CombCut excels in worst-case labeling scenarios, significantly outperforming state-of-the-art graph-SSL baselines and yielding more stable and accurate label propagation under severe supervision constraints.

Yedidya Kfir, Elad Sarafian, Yoram Louzoun, Sarit Kraus

Black-box algorithms aim to optimize functions without access to their analytical structure or gradient information, making them essential when gradients are unavailable or computationally expensive to obtain. Traditional methods for black-box optimization (BBO) primarily utilize non-parametric models, but these approaches often struggle to scale effectively in large input spaces. Conversely, parametric approaches, which rely on neural estimators and gradient signals via backpropagation, frequently encounter substantial gradient estimation errors, limiting their reliability. Explicit Gradient Learning (EGL), a recent advancement, directly learns gradients using a first-order Taylor approximation and has demonstrated superior performance compared to both parametric and non-parametric methods. However, EGL inherently remains local and myopic, often faltering on highly non-convex optimization landscapes. In this work, we address this limitation by integrating global statistical insights from the evolutionary algorithm CMA-ES into the gradient learning framework, effectively biasing gradient estimates towards regions with higher optimization potential. Moreover, we enhance the gradient learning process by estimating the Hessian matrix, allowing us to correct the second-order residual of the Taylor series approximation. Our proposed algorithm, EvoGrad2 (Evolutionary Gradient Learning with second-order approximation), achieves state-of-the-art results on the synthetic COCO test suite, exhibiting significant advantages in high-dimensional optimization problems. We further demonstrate EvoGrad2's effectiveness on challenging real-world machine learning tasks, including adversarial training and code generation, highlighting its ability to produce more robust, high-quality solutions. Our results underscore EvoGrad2's potential as a powerful tool for researchers and practitioners facing complex, high-dimensional, and non-linear optimization problems.

Fares Fourati, Mohamed-Slim Alouini, Vaneet Aggarwal

We propose ECPv2, a scalable and theoretically grounded algorithm for global optimization of Lipschitz continuous functions with unknown Lipschitz constants. Building on the Every Call is Precious (ECP) framework, which ensures that each accepted function evaluation is potentially informative, ECPv2 addresses key limitations of ECP, including high computational cost and overly conservative early behavior. ECPv2 introduces three innovations: (i) an adaptive lower bound that prevents vacuous acceptance regions, (ii) a memory mechanism that restricts comparisons to a fixed-size subset of past evaluations, and (iii) a fixed random projection that accelerates distance computations in high dimensions. We theoretically show that ECPv2 retains ECP’s regret guarantees and expands the acceptance region with high probability. Extensive experiments and ablation studies empirically validate these findings. Using principled hyperparameter settings, we evaluate ECPv2 across a wide range of nonconvex optimization problems and find that it consistently matches or outperforms leading optimizers while significantly reducing wall clock time.

Maxim Divilkovskiy, Alexander Gasnikov

We study non-smooth stochastic decentralized optimization problems over time-varying networks, where objective functions are distributed across nodes and network connections may intermittently appear or break. Specifically, we consider two settings: (i) stochastic non-smooth (strongly) convex optimization, and (ii) stochastic non-smooth (strongly) convex–(strongly) concave saddle point optimization. Convex problems of this type commonly arise in deep neural network training, while saddle point problems are central to machine learning tasks such as the training of generative adversarial networks (GANs). Prior works have primarily focused on the smooth setting, or time-invariant network scenarios. We extend the existing theory to the more general non-smooth and stochastic setting over time-varying networks and saddle point problems. Our analysis establishes upper bounds on both the number of stochastic oracle calls and communication rounds, matching lower bounds for both convex and saddle point optimization problems.

Xin Chen, Chunfeng Cui, Deren Han, Liqun Qi

Pose graph optimization (PGO) is fundamental to robot perception and navigation systems, serving as the mathematical backbone for solving simultaneous localization and mapping (SLAM). Existing solvers suffer from polynomial growth in computational complexity with graph size, hindering real-time deployment in large-scale scenarios. In this paper, by duplicating variables and introducing equality constraints, we reformulate the problem and propose a Parallelizable Riemannian Alternating Direction Method of Multipliers (PRADMM) to solve it efficiently. Compared with the state-of-the-art methods that usually exhibit polynomial time complexity growth with graph size, PRADMM enables efficient parallel computation across vertices regardless of graph size. Crucially, all subproblems admit closed-form solutions, ensuring PRADMM maintains exceptionally stable performance. Furthermore, by carefully exploiting the structures of the coefficient matrices in the constraints, we establish the global convergence of PRADMM under mild conditions, enabling larger relaxation step sizes within the interval (0,2). Extensive empirical validation on two synthetic datasets and multiple real-world 3D SLAM benchmarks confirms the superior computational performance of PRADMM.

Haoyang Zheng, Hengrong Du, Ruqi Zhang, Guang Lin

Gradient-based Discrete Samplers (GDSs) are effective for sampling discrete energy landscapes. However, they often stagnate in complex, non-convex settings. To improve exploration, we introduce the Discrete Replica EXchangE Langevin (DREXEL) sampler and its variant with Adjusted Metropolis (DREAM). These samplers use two GDSs at different temperatures and step sizes: one focuses on local exploitation, while the other explores broader energy landscapes. When energy differences are significant, sample swaps occur, governed by a mechanism tailored for discrete sampling to ensure detailed balance. Theoretically, we prove that the proposed samplers satisfy detailed balance and converge to the target distribution under mild conditions. Experiments across 2d synthetic simulations, sampling from Ising models and restricted Boltzmann machines, and training deep energy-based models further confirm their efficiency in exploring non-convex discrete energy landscapes.

Bogdan Butyrin, Artemy Rubtsov, Alexey Naumov, Vladimir V. Ulyanov, Sergey Samsonov

In this paper, we establish non-asymptotic bounds for accuracy of normal approximation for linear two-timescale stochastic approximation (TTSA) algorithms driven by martingale difference or Markov noise. Focusing on both the last iterate and Polyak–Ruppert averaging regimes, we derive bounds for normal approximation in terms of the convex distance between probability distributions. Our analysis reveals a non-trivial interaction between the fast and slow timescales: the normal approximation rate for the last iterate improves as the timescale separation increases, while it decreases in the Polyak–Ruppert averaged setting. We also provide the high-order moment bounds for the error of linear TTSA algorithm, which may be of independent interest. Finally, we demonstrate that our theoretical results are directly applicable to reinforcement learning algorithms such as GTD and TDC.

Jingtao Tang, Hang Ma

We study GCS-TSP, a variant of the Traveling Salesman Problem (TSP) defined over a Graph of Convex Sets (GCS) - a powerful representation for trajectory planning that decomposes the configuration space into convex regions connected by a sparse graph. In GCS-TSP, edge costs are not fixed but depend on the specific trajectory passing through each convex region, making classical TSP methods inapplicable. We introduce GHOST, a hierarchical framework that optimally solves GCS-TSP by combining combinatorial tour search with convex trajectory optimization. GHOST systematically explores tours on a complete graph induced by the GCS, using a novel abstract-path-unfolding algorithm to compute admissible lower bounds that guide best-first search at both the high level (over tours) and the low level (over feasible GCS paths realizing the tour). These bounds provide strong pruning power, reducing unnecessary optimization calls. We prove that GHOST guarantees optimality and present a bounded-suboptimal variant for time-critical settings. Experiments show that GHOST is orders-of-magnitude faster than unified mixed-integer convex programming baseline while uniquely handling complex problems involving high-order continuity constraints and incomplete GCSs.

Yannik Schnitzer, Alessandro Abate, David Parker

Robust Markov decision processes (r-MDPs) extend MDPs by explicitly modelling epistemic uncertainty about transition dynamics. Learning r-MDPs from interactions with an unknown environment enables the synthesis of robust policies with provable (PAC) guarantees on performance, but this can require a large number of sample interactions. We propose novel methods for solving and learning r-MDPs based on factored state-space representations that leverage the independence between model uncertainty across system components. Although policy synthesis for factored r-MDPs leads to hard, non-convex optimisation problems, we show how to reformulate these into tractable linear programs. Building on these, we also propose methods to learn factored model representations directly. Our experimental results show that exploiting factored structure can yield dimensional gains in sample efficiency, producing more effective robust policies with tighter performance guarantees than state-of-the-art methods.

Xiaowei Wang, Jie Yu, Yulong Wang

Tensor Compressive Sensing (TCS) has gained significant attention recently due to its strong ability to preserve the multidimensional structure of data. However, existing TCS methods face three critical challenges: 1) Biased approximation of tensor rank imposed by the convex surrogate Tensor Nuclear Norm (TNN) may interfere with the original low-rank structure of tensor data. 2) Vulnerability to non-Gaussian noise and outliers makes TCS methods highly susceptible to complex noise environments ubiquitous in real-world applications. 3) Most of them are confined to third-order tensors and cannot handle high-order tensor data effectively. Being aware of these, we propose Robust Tensor Compressive Sensing (RTCS) based on M-estimators with three key innovations: 1) We design a novel M-estimator-based low-rank regularizer for high-order tensors, which provides a superior approximation of tensor rank and better preserves the original data structure. 2) RTCS incorporates a robust Welsch estimator that adaptively mitigates the influence of complex noises and outliers in tensor recovery. 3) RTCS is developed to handle high-order tensors, thereby allowing for broader applicability beyond conventional third-order tensors. We further design an efficient algorithm based on the Alternating Direction Method of Multipliers (ADMM) to handle the complex optimization problem. Experiments show that RTCS consistently outperforms existing approaches across various noises.

Maoran Wang, Xingju Cai, Yongxin Chen

This paper investigates problems of large-scale distributed composite convex optimization, with motivations from a broad range of applications, including multi-agent systems, federated learning, smart grids, wireless sensor networks, compressed sensing, and so on. Stochastic gradient descent (SGD) and its variants are commonly employed to solve such problems. However, existing algorithms often rely on vanishing step sizes, strong convexity assumptions, or entail substantial computational overhead to ensure convergence or obtain favorable complexity. To bridge the gap between theory and practice, we integrate consensus optimization and operator splitting techniques (see Problem Reformulation) to develop a novel stochastic splitting algorithm, termed the stochastic distributed regularized splitting method (S-D-RSM). In practice, S-D-RSM performs parallel updates of proximal mappings and gradient information for only a randomly selected subset of agents at each iteration. By introducing regularization terms, it effectively mitigates consensus discrepancies among distributed nodes. In contrast to conventional stochastic methods, our theoretical analysis establishes that S-D-RSM achieves global convergence without requiring diminishing step sizes or strong convexity assumptions. Furthermore, it achieves an iteration complexity of 1/epsilon with respect to both the objective function value and the consensus error. Numerical experiments show that S-D-RSM achieves up to two to three times speedup compared with state-of-the-art baselines, while maintaining comparable or better accuracy. These results not only validate the algorithm's theoretical guarantees but also demonstrate its effectiveness in practical tasks such as compressed sensing and empirical risk minimization.

Kaizheng Wang, Fabio Cuzzolin, David Moens, Hans Hallez

Deep ensembles (DE) have emerged as a powerful approach for quantifying predictive uncertainty and distinguishing its aleatoric and epistemic components, thereby enhancing model robustness and reliability. However, their high computational and memory costs during inference pose significant challenges for wide practical deployment. To overcome this issue, we propose credal ensemble distillation (CED), a novel framework that compresses a DE into a single model, CREDIT, for classification tasks. Instead of a single softmax probability distribution, CREDIT predicts class-wise probability intervals that define a credal set, a convex set of probability distributions, for uncertainty quantification. Empirical results on out-of-distribution detection benchmarks demonstrate that CED achieves superior or comparable uncertainty estimation compared to several existing baselines, while substantially reducing inference overhead compared to DE.

Vladimir Solodkin, Andrey Veprikov, Alexander Chernyavskiy, Aleksandr Beznosikov

This paper examines a variety of classical optimization problems, including well-known minimization tasks and more general variational inequalities. We consider a stochastic formulation of these problems and, unlike most previous work, we take into account the complex Markov nature of the noise. We also consider the geometry of the problem in an arbitrary non-Euclidean setting and propose four methods based on the Mirror Descent iteration technique. The theoretical analysis is provided for smooth and convex minimization problems and variational inequalities with Lipschitz and monotone operators. The convergence guarantees obtained are optimal for first-order stochastic methods, as evidenced by the lower bound estimates provided in this paper. In order to validate the theoretical results, we present the relevant numerical experiments on various reinforcement learning tasks.

Chang-Wei Shi, Shi-Shang Wang, Wu-Jun Li

Momentum SGD (MSGD) serves as a foundational optimizer in training deep models due to momentum's key role in accelerating convergence and enhancing generalization. Meanwhile, asynchronous distributed learning is crucial for training large-scale deep models, especially when the computing capabilities of the workers in the cluster are heterogeneous. To reduce communication frequency, local updates are widely adopted in distributed learning. However, how to implement asynchronous distributed MSGD with local updates remains unexplored. To solve this problem, we propose a novel method, called ordered local momentum (OrLoMo), for asynchronous distributed learning. In OrLoMo, each worker runs MSGD locally. Then the local momentum from each worker will be aggregated by the server in order based on its global iteration index. To the best of our knowledge, OrLoMo is the first method to implement asynchronous distributed MSGD with local updates. We prove the convergence of OrLoMo for non-convex problems under arbitrary delays. Experiments validate that OrLoMo can outperform its synchronous counterpart and other asynchronous methods.

Hao Qiu, Mengxiao Zhang, Juliette Achddou

Decentralized online convex optimization (D-OCO), where multiple agents within a network collaboratively learn optimal decisions in real-time, arises naturally in applications such as federated learning, sensor networks, and multi-agent control. In this paper, we study D-OCO under unknown, time- and agent-varying feedback delays. While recent work has addressed this problem~\citep{nguyen2024handling}, existing algorithms assume prior knowledge of the total delay over agents and still suffer from suboptimal dependence on both the delay and network parameters. To overcome these limitations, we propose a novel algorithm that achieves an improved regret bound of Õ(N √d_tot + N √( T / √(1 − σ₂) )), where d_tot denotes the average total delay across agents, N is the number of agents, and 1 − σ₂ is the spectral gap of the network. We also prove a lower bound showing that our upper bound is tight up to logarithmic factors. Our approach builds upon recent advances in D-OCO~\citep{wan2024nearly}, but crucially incorporates an adaptive learning rate mechanism via a decentralized communication protocol. This enables each agent to estimate delays locally using a gossip-based strategy without the prior knowledge of the total delay. We further extend our framework to the strongly convex setting and derive a sharper regret bound. Experimental results validate the effectiveness of our approach, showing improvements over existing benchmark algorithms.