Mathematical Programs with Complementarity Constraints (MPCC) are critical in various real-world applications but notoriously challenging due to non-smoothness and degeneracy from complementarity constraints. The $\ell_1$-Exact Penalty-Barrier enhanced \texttt{IPOPT} improves performance and robustness by introducing additional inequality constraints and decision variables. However, this comes at the cost of increased computational complexity due to the higher dimensionality and additional constraints introduced in the centralized formulation. To mitigate this, we propose a distributed structure-splitting reformulation that decomposes these inequality constraints and auxiliary variables into independent sub-problems. Furthermore, we introduce Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN)-$\beta$, a novel approach that integrates the $\ell_1$-Exact Penalty-Barrier method with ALADIN to efficiently solve the distributed reformulation. Numerical experiments demonstrate that even without a globalization strategy, the proposed distributed approach achieves fast convergence while maintaining high precision.
This paper addresses the problem of collaboratively satisfying long-term spatial constraints in multi-agent systems. Each agent is subject to spatial constraints, expressed as inequalities, which may depend on the positions of other agents with whom they may or may not have direct communication. These constraints need to be satisfied asymptotically or after an unknown finite time. The agents' objective is to collectively achieve a formation that fulfills all constraints. The problem is initially framed as a centralized unconstrained optimization, where the solution yields the optimal configuration by maximizing an objective function that reflects the degree of constraint satisfaction. This function encourages collaboration, ensuring agents help each other meet their constraints while fulfilling their own. When the constraints are infeasible, agents converge to a least-violating solution. A distributed consensus-based optimization scheme is then introduced, which approximates the centralized solution, leading to the development of distributed controllers for single-integrator agents. Finally, simulations validate the effectiveness of the proposed approach.
The Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method is a cutting-edge distributed optimization algorithm known for its superior numerical performance. It relies on each agent transmitting information to a central coordinator for data exchange. However, in practical network optimization and federated learning, unreliable information transmission often leads to packet loss, posing challenges for the convergence analysis of ALADIN. To address this issue, this paper proposes Flexible ALADIN, a random polling variant of ALADIN, and presents a rigorous convergence analysis, including global convergence for convex problems and local convergence for non-convex problems.
By letting local clients perform multiple local updates before communicating with a parameter server, modern federated learning algorithms such as FedAvg tackle the communication bottleneck problem in distributed learning and have found many successful applications. However, this asynchrony between local updates and communication also leads to a ''client-drift'' problem when the data is heterogeneous (not independent and identically distributed), resulting in errors in the final learning result. In this paper, we propose a federated learning algorithm, which is called FedCET, to ensure accurate convergence even under heterogeneous distributions of data across clients. Inspired by the distributed optimization algorithm NIDS, we use learning rates to weight information received from local clients to eliminate the ''client-drift''. We prove that under appropriate learning rates, FedCET can ensure linear convergence to the exact solution. Different from existing algorithms which have to share both gradients and a drift-correction term to ensure accurate convergence under heterogeneous data distributions, FedCET only shares one variable, which significantly reduces communication overhead. Numerical comparison with existing counterpart algorithms confirms the effectiveness of FedCET.
Byzantine attacks present a critical challenge to Federated Learning (FL), where malicious participants can disrupt the training process, degrade model accuracy, and compromise system reliability. Traditional FL frameworks typically rely on aggregation-based protocols for model updates, leaving them vulnerable to sophisticated adversarial strategies. In this paper, we demonstrate that distributed optimization offers a principled and robust alternative to aggregation-centric methods. Specifically, we show that the Primal-Dual Method of Multipliers (PDMM) inherently mitigates Byzantine impacts by leveraging its fault-tolerant consensus mechanism. Through extensive experiments on three datasets (MNIST, FashionMNIST, and Olivetti), under various attack scenarios including bit-flipping and Gaussian noise injection, we validate the superior resilience of distributed optimization protocols. Compared to traditional aggregation-centric approaches, PDMM achieves higher model utility, faster convergence, and improved stability. Our results highlight the effectiveness of distributed optimization in defending against Byzantine threats, paving the way for more secure and resilient federated learning systems.
Modern machine learning tasks often involve massive datasets and models, necessitating distributed optimization algorithms with reduced communication overhead. Communication compression, where clients transmit compressed updates to a central server, has emerged as a key technique to mitigate communication bottlenecks. However, the theoretical understanding of stochastic distributed optimization with contractive compression remains limited, particularly in conjunction with Nesterov acceleration -- a cornerstone for achieving faster convergence in optimization. In this paper, we propose a novel algorithm, ADEF (Accelerated Distributed Error Feedback), which integrates Nesterov acceleration, contractive compression, error feedback, and gradient difference compression. We prove that ADEF achieves the first accelerated convergence rate for stochastic distributed optimization with contractive compression in the general convex regime. Numerical experiments validate our theoretical findings and demonstrate the practical efficacy of ADEF in reducing communication costs while maintaining fast convergence.
Distributed optimization aims to leverage the local computation and communication capabilities of each agent to achieve a desired global objective. This paper addresses the distributed pose graph optimization (PGO) problem under non-convex constraints, with the goal of approximating the rotation and translation of each pose given relevant noisy measurements. To achieve this goal, the splitting method based on the concepts of the alternating direction method of multipliers (ADMM) and Bregman iteration are applied to solve the rotation subproblems. The proposed approach enables the iterative resolution of constrained problems, achieved through solving unconstrained problems and orthogonality-constrained quadratic problems that have analytical solutions. The performance of the proposed algorithm is compared against two practical methods in pose graph optimization: the Distributed Gauss-Seidel (DGS) algorithm and the centralized pose graph optimizer with an optimality certificate (SE-Sync). The efficiency of the proposed method is verified through its application to several simulated and real-world pose graph datasets. Unlike the DGS method, our approach attempts to solve distributed PGO problems without relaxing the non-convex constraints.
Federated Learning (FL) enables collaborative learning without directly sharing individual's raw data. FL can be implemented in either a centralized (server-based) or decentralized (peer-to-peer) manner. In this survey, we present a novel perspective: the fundamental difference between centralized FL (CFL) and decentralized FL (DFL) is not merely the network topology, but the underlying training protocol: separate aggregation vs. joint optimization. We argue that this distinction in protocol leads to significant differences in model utility, privacy preservation, and robustness to attacks. We systematically review and categorize existing works in both CFL and DFL according to the type of protocol they employ. This taxonomy provides deeper insights into prior research and clarifies how various approaches relate or differ. Through our analysis, we identify key gaps in the literature. In particular, we observe a surprising lack of exploration of DFL approaches based on distributed optimization methods, despite their potential advantages. We highlight this under-explored direction and call for more research on leveraging distributed optimization for federated learning. Overall, this work offers a comprehensive overview from centralized to decentralized FL, sheds new light on the core distinctions between approaches, and outlines open challenges and future directions for the field.
This paper addresses the challenge of packet-based information routing in large-scale wireless communication networks. The problem is framed as a constrained statistical learning task, where each network node operates using only local information. Opportunistic routing exploits the broadcast nature of wireless communication to dynamically select optimal forwarding nodes, enabling the information to reach the destination through multiple relay nodes simultaneously. To solve this, we propose a State-Augmentation (SA) based distributed optimization approach aimed at maximizing the total information handled by the source nodes in the network. The problem formulation leverages Graph Neural Networks (GNNs), which perform graph convolutions based on the topological connections between network nodes. Using an unsupervised learning paradigm, we extract routing policies from the GNN architecture, enabling optimal decisions for source nodes across various flows. Numerical experiments demonstrate that the proposed method achieves superior performance when training a GNN-parameterized model, particularly when compared to baseline algorithms. Additionally, applying the method to real-world network topologies and wireless ad-hoc network test beds validates its effectiveness, highlighting the robustness and transferability of GNNs.
Distributed methods are essential for handling machine learning pipelines comprising large-scale models and datasets. However, their benefits often come at the cost of increased communication overhead between the central server and agents, which can become the main bottleneck, making training costly or even unfeasible in such systems. Compression methods such as quantization and sparsification can alleviate this issue. Still, their robustness to large and heavy-tailed gradient noise, a phenomenon sometimes observed in language modeling, remains poorly understood. This work addresses this gap by analyzing Distributed Compressed SGD (DCSGD) and Distributed SignSGD (DSignSGD) using stochastic differential equations (SDEs). Our results show that DCSGD with unbiased compression is more vulnerable to noise in stochastic gradients, while DSignSGD remains robust, even under large and heavy-tailed noise. Additionally, we propose new scaling rules for hyperparameter tuning to mitigate performance degradation due to compression. These findings are empirically validated across multiple deep learning architectures and datasets, providing practical recommendations for distributed optimization.
A substantial body of work in machine learning (ML) and randomized numerical linear algebra (RandNLA) has exploited various sorts of random sketching methodologies, including random sampling and random projection, with much of the analysis using Johnson--Lindenstrauss and subspace embedding techniques. Recent studies have identified the issue of inversion bias -- the phenomenon that inverses of random sketches are not unbiased, despite the unbiasedness of the sketches themselves. This bias presents challenges for the use of random sketches in various ML pipelines, such as fast stochastic optimization, scalable statistical estimators, and distributed optimization. In the context of random projection, the inversion bias can be easily corrected for dense Gaussian projections (which are, however, too expensive for many applications). Recent work has shown how the inversion bias can be corrected for sparse sub-gaussian projections. In this paper, we show how the inversion bias can be corrected for random sampling methods, both uniform and non-uniform leverage-based, as well as for structured random projections, including those based on the Hadamard transform. Using these results, we establish problem-independent local convergence rates for sub-sampled Newton methods.
Federated learning enables training machine learning models while preserving the privacy of participants. Surprisingly, there is no differentially private distributed method for smooth, non-convex optimization problems. The reason is that standard privacy techniques require bounding the participants' contributions, usually enforced via $\textit{clipping}$ of the updates. Existing literature typically ignores the effect of clipping by assuming the boundedness of gradient norms or analyzes distributed algorithms with clipping but ignores DP constraints. In this work, we study an alternative approach via $\textit{smoothed normalization}$ of the updates motivated by its favorable performance in the single-node setting. By integrating smoothed normalization with an error-feedback mechanism, we design a new distributed algorithm $\alpha$-$\sf NormEC$. We prove that our method achieves a superior convergence rate over prior works. By extending $\alpha$-$\sf NormEC$ to the DP setting, we obtain the first differentially private distributed optimization algorithm with provable convergence guarantees. Finally, our empirical results from neural network training indicate robust convergence of $\alpha$-$\sf NormEC$ across different parameter settings.
In recent years, non-convex optimization problems are more often described by generalized $(L_0, L_1)$-smoothness assumption rather than standard one. Meanwhile, severely corrupted data used in these problems has increased the demand for methods capable of handling heavy-tailed noises, i.e., noises with bounded $\kappa$-th moment. Motivated by these real-world trends and challenges, we explore sign-based methods in this setup and demonstrate their effectiveness in comparison with other popular solutions like clipping or normalization. In theory, we prove the first-known high probability convergence bounds under $(L_0, L_1)$-smoothness and heavy-tailed noises with mild parameter dependencies. In the case of standard smoothness, these bounds are novel for sign-based methods as well. In particular, SignSGD with batching achieves sample complexity $\tilde{O}\left(\left(\frac{\Delta L_0d}{\varepsilon^2} + \frac{\Delta L_1d^\frac{3}{2}}{\varepsilon}\right)\left[1 + \left(\frac{\sigma}{\varepsilon}\right)^\frac{\kappa}{\kappa-1}\right]\right), \kappa \in (1,2]$. Under the assumption of symmetric noises, SignSGD with Majority Voting can robustly work on the whole range of $\kappa \in (0,2]$ with complexity $\tilde{O}\left(\left(\frac{\Delta L_0d}{\varepsilon^2} + \frac{\Delta L_1d^\frac{3}{2}}{\varepsilon}\right)\left[\frac{1}{\kappa^2} + \frac{\sigma^2}{\varepsilon^2}\right]\right)$. We also obtain results for parameter-agnostic setups, Polyak-Lojasiewicz functions and momentum-based methods (in expectation). Our theoretical findings are supported by the superior performance of sign-based methods in training Large Language Models compared to clipping and normalization.
As data sets grow in size and complexity, it is becoming more difficult to pull useful features from them using hand-crafted feature extractors. For this reason, deep learning (DL) frameworks are now widely popular. The Holy Grail of DL and one of the most mysterious challenges in all of modern ML is to develop a fundamental understanding of DL optimization and generalization. While numerous optimization techniques have been introduced in the literature to navigate the exploration of the highly non-convex DL optimization landscape, many survey papers reviewing them primarily focus on summarizing these methodologies, often overlooking the critical theoretical analyses of these methods. In this paper, we provide an extensive summary of the theoretical foundations of optimization methods in DL, including presenting various methodologies, their convergence analyses, and generalization abilities. This paper not only includes theoretical analysis of popular generic gradient-based first-order and second-order methods, but it also covers the analysis of the optimization techniques adapting to the properties of the DL loss landscape and explicitly encouraging the discovery of well-generalizing optimal points. Additionally, we extend our discussion to distributed optimization methods that facilitate parallel computations, including both centralized and decentralized approaches. We provide both convex and non-convex analysis for the optimization algorithms considered in this survey paper. Finally, this paper aims to serve as a comprehensive theoretical handbook on optimization methods for DL, offering insights and understanding to both novice and seasoned researchers in the field.
This paper addresses the problem of optimal linear filtering in a network of local estimators, commonly referred to as distributed Kalman filtering (DKF). The DKF problem is formulated within a distributed optimization framework, where coupling constraints require the exchange of local state and covariance updates between neighboring nodes to achieve consensus. To address these constraints, the problem is transformed into an unconstrained optimization form using the augmented Lagrangian method. The distributed alternating direction method of multipliers (ADMM) is then applied to derive update steps that achieve the desired performance while exchanging only the primal variables. Notably, the proposed method enhances communication efficiency by eliminating the need for dual variable exchange. We show that the design parameters depend on the maximum eigenvalue of the network's Laplacian matrix, yielding a significantly tighter bound compared to existing results. A rigorous convergence analysis is provided, proving that the state estimates converge to the true state and that the covariance matrices across all local estimators converge to a globally optimal solution. Numerical results are presented to validate the efficacy of the proposed approach.
This paper introduces a new approach that leverages Multi-agent Bayesian Optimization (MABO) to design Distributed Model Predictive Control (DMPC) schemes for multi-agent systems. The primary objective is to learn optimal DMPC schemes even when local model predictive controllers rely on imperfect local models. The proposed method invokes a dual decomposition-based distributed optimization framework, incorporating an Alternating Direction Method of Multipliers (ADMM)-based MABO algorithm to enable coordinated learning of parameterized DMPC schemes. This enhances the closed-loop performance of local controllers, despite discrepancies between their models and the actual multi-agent system dynamics. In addition to the newly proposed algorithms, this work also provides rigorous proofs establishing the optimality and convergence of the underlying learning method. Finally, numerical examples are given to demonstrate the efficacy of the proposed MABO-based learning approach.
This paper aims at distributed multi-agent convex optimization where the communications network among the agents are presented by a random sequence of possibly state-dependent weighted graphs. This is the first work to consider both random arbitrary communication networks and state-dependent interactions among agents. The state-dependent weighted random operator of the graph is shown to be quasi-nonexpansive; this property neglects a priori distribution assumption of random communication topologies to be imposed on the operator. Therefore, it contains more general class of random networks with or without asynchronous protocols. A more general mathematical optimization problem than that addressed in the literature is presented, namely minimization of a convex function over the fixed-value point set of a quasi-nonexpansive random operator. A discrete-time algorithm is provided that is able to converge both almost surely and in mean square to the global solution of the optimization problem. Hence, as a special case, it reduces to a totally asynchronous algorithm for the distributed optimization problem. The algorithm is able to converge even if the weighted matrix of the graph is periodic and irreducible under synchronous protocol. Finally, a case study on a network of robots in an automated warehouse is given where there is distribution dependency among random communication graphs.
Federated learning (FL) is a promising technology for data privacy and distributed optimization, but it suffers from data imbalance and heterogeneity among clients. Existing FL methods try to solve the problems by aligning client with server model or by correcting client model with control variables. These methods excel on IID and general Non-IID data but perform mediocrely in Simpson's Paradox scenarios. Simpson's Paradox refers to the phenomenon that the trend observed on the global dataset disappears or reverses on a subset, which may lead to the fact that global model obtained through aggregation in FL does not accurately reflect the distribution of global data. Thus, we propose FedCFA, a novel FL framework employing counterfactual learning to generate counterfactual samples by replacing local data critical factors with global average data, aligning local data distributions with the global and mitigating Simpson's Paradox effects. In addition, to improve the quality of counterfactual samples, we introduce factor decorrelation (FDC) loss to reduce the correlation among features and thus improve the independence of extracted factors. We conduct extensive experiments on six datasets and verify that our method outperforms other FL methods in terms of efficiency and global model accuracy under limited communication rounds.
In recent years, as data and problem sizes have increased, distributed learning has become an essential tool for training high-performance models. However, the communication bottleneck, especially for high-dimensional data, is a challenge. Several techniques have been developed to overcome this problem. These include communication compression and implementation of local steps, which work particularly well when there is similarity of local data samples. In this paper, we study the synergy of these approaches for efficient distributed optimization. We propose the first theoretically grounded accelerated algorithms utilizing unbiased and biased compression under data similarity, leveraging variance reduction and error feedback frameworks. Our results are of record and confirmed by experiments on different average losses and datasets.
Cooperative control is crucial for the effective operation of dynamical multi-agent systems. Especially for distributed control schemes, it is essential to exchange data between the agents. This becomes a privacy threat if the data is sensitive. Encrypted control has shown the potential to address this risk and ensure confidentiality. However, existing approaches mainly focus on cloud-based control and distributed schemes are restrictive. In this paper, we present a novel privacy-preserving cooperative control scheme based on encrypted distributed optimization. More precisely, we focus on a secure distributed solution of a general consensus problem, which has manifold applications in cooperative control, by means of the alternating direction method of multipliers (ADMM). As a unique feature of our approach, we explicitly take into account the common situation that local decision variables contain copies of quantities associated with neighboring agents and ensure the neighbor's privacy. We show the effectiveness of our method based on a numerical case study dealing with the formation of mobile robots.