arXiv subjects

Mikael Johansson

Publications and source records attributed to Mikael Johansson.

At least 37 records · Page 2Linked to original sources

arXiv2024

Nonconvex Federated Learning on Compact Smooth Submanifolds With Heterogeneous Data

Many machine learning tasks, such as principal component analysis and low-rank matrix completion, give rise to manifold optimization problems. Although there is a large body of work studying the design and analysis of algorithms for manifold optimization in the centralized setting, there are currently very few works addressing the federated setting. In this paper, we consider nonconvex federated learning over a compact smooth submanifold in the setting of heterogeneous client data. We propose an algorithm that leverages stochastic Riemannian gradients and a manifold projection operator to improve computational efficiency, uses local updates to improve communication efficiency, and avoids client drift. Theoretically, we show that our proposed algorithm converges sub-linearly to a neighborhood of a first-order optimal solution by using a novel analysis that jointly exploits the manifold structure and properties of the loss functions. Numerical experiments demonstrate that our algorithm has significantly smaller computational and communication overhead than existing methods.

cs.LG↗

arXiv2024

Stable MPC with maximal terminal sets and quadratic terminal costs

This paper develops a technique for computing a quadratic terminal cost for linear model predictive controllers that is valid for all states in the maximal control invariant set. This maximizes the set of recursively feasible states for the controller, ensures asymptotic stability using standard proofs, and allows for easy tuning of the controller in linear operation.

math.OC↗

arXiv2024

Model-free fast charging of lithium-ion batteries by online gradient descent

A data-driven solution is provided for the fast-charging problem of lithium-ion batteries with multiple safety and aging constraints. The proposed method optimizes the charging current based on the observed history of measurable battery quantities, such as the input current, terminal voltage, and temperature. The proposed method does not need any detailed battery model or full-charging training episodes. The theoretical convergence is proven under mild conditions and is validated numerically on several linear and nonlinear battery models, including single-particle and equivalent-circuit models.

eess.SY↗

arXiv2024

Novel Limited Memory Quasi-Newton Methods Based On Optimal Matrix Approximation

Update formulas for the Hessian approximations in quasi-Newton methods such as BFGS can be derived as analytical solutions to certain nearest-matrix problems. In this article, we propose a similar idea for deriving new limited memory versions of quasi-Newton methods. Most limited memory quasi-Newton methods make use of Hessian approximations that can be written as a scaled identity matrix plus a symmetric matrix with limited rank. We derive a way of finding the nearest matrix of this type to an arbitrary symmetric matrix, in either the Frobenius norm, the induced $l^2$ norm, or a dissimilarity measure for positive definite matrices in terms of trace and determinant. In doing so, we lay down a framework for more general matrix optimization problems with unitarily invariant matrix norms and arbitrary constraints on the set of eigenvalues. We then propose a trust region method in which the Hessian approximation, after having been updated by a Broyden class formula and used to solve a trust-region problem, is replaced by one of its closest limited memory approximations. We propose to store the Hessian approximation in terms of its eigenvectors and eigenvalues in a way that completely defines its eigenvalue decomposition, as this simplifies both the solution of the trust region subproblem and the nearest limited memory matrix problem. Our method is compared to a reference trust region method with the usual limited memory BFGS updates, and is shown to require fewer iterations and the storage of fewer vectors for a variety of test problems.

math.OC↗

arXiv2024

Soft quasi-Newton: Guaranteed positive definiteness by relaxing the secant constraint

We propose a novel algorithm, termed soft quasi-Newton (soft QN), for optimization in the presence of bounded noise. Traditional quasi-Newton algorithms are vulnerable to such perturbations. To develop a more robust quasi-Newton method, we replace the secant condition in the matrix optimization problem for the Hessian update with a penalty term in its objective and derive a closed-form update formula. A key feature of our approach is its ability to maintain positive definiteness of the Hessian inverse approximation. Furthermore, we establish the following properties of soft QN: it recovers the BFGS method under specific limits, it treats positive and negative curvature equally, and it is scale invariant. Collectively, these features enhance the efficacy of soft QN in noisy environments. For strongly convex objective functions and Hessian approximations obtained using soft QN, we develop an algorithm that exhibits linear convergence toward a neighborhood of the optimal solution, even if gradient and function evaluations are subject to bounded perturbations. Through numerical experiments, we demonstrate superior performance of soft QN compared to state-of-the-art methods in various scenarios.

math.OC↗

arXiv2023

Generalized Polyak Step Size for First Order Optimization with Momentum

In machine learning applications, it is well known that carefully designed learning rate (step size) schedules can significantly improve the convergence of commonly used first-order optimization algorithms. Therefore how to set step size adaptively becomes an important research question. A popular and effective method is the Polyak step size, which sets step size adaptively for gradient descent or stochastic gradient descent without the need to estimate the smoothness parameter of the objective function. However, there has not been a principled way to generalize the Polyak step size for algorithms with momentum accelerations. This paper presents a general framework to set the learning rate adaptively for first-order optimization methods with momentum, motivated by the derivation of Polyak step size. It is shown that the resulting techniques are much less sensitive to the choice of momentum parameter and may avoid the oscillation of the heavy-ball method on ill-conditioned problems. These adaptive step sizes are further extended to the stochastic settings, which are attractive choices for stochastic gradient descent with momentum. Our methods are demonstrated to be more effective for stochastic gradient methods than prior adaptive step size algorithms in large-scale machine learning tasks.

math.OC↗

arXiv2023

Composite federated learning with heterogeneous data

We propose a novel algorithm for solving the composite Federated Learning (FL) problem. This algorithm manages non-smooth regularization by strategically decoupling the proximal operator and communication, and addresses client drift without any assumptions about data similarity. Moreover, each worker uses local updates to reduce the communication frequency with the server and transmits only a $d$-dimensional vector per communication round. We prove that our algorithm converges linearly to a neighborhood of the optimal solution and demonstrate the superiority of our algorithm over state-of-the-art methods in numerical experiments.

cs.LG↗

arXiv2023

Delay-agnostic Asynchronous Distributed Optimization

Existing asynchronous distributed optimization algorithms often use diminishing step-sizes that cause slow practical convergence, or fixed step-sizes that depend on an assumed upper bound of delays. Not only is such a delay bound hard to obtain in advance, but it is also large and therefore results in unnecessarily slow convergence. This paper develops asynchronous versions of two distributed algorithms, DGD and DGD-ATC, for solving consensus optimization problems over undirected networks. In contrast to alternatives, our algorithms can converge to the fixed-point set of their synchronous counterparts using step-sizes that are independent of the delays. We establish convergence guarantees under both partial and total asynchrony. The practical performance of our algorithms is demonstrated by numerical experiments.

math.OC↗

arXiv2023

Dynamic Privacy Allocation for Locally Differentially Private Federated Learning with Composite Objectives

This paper proposes a locally differentially private federated learning algorithm for strongly convex but possibly nonsmooth problems that protects the gradients of each worker against an honest but curious server. The proposed algorithm adds artificial noise to the shared information to ensure privacy and dynamically allocates the time-varying noise variance to minimize an upper bound of the optimization error subject to a predefined privacy budget constraint. This allows for an arbitrarily large but finite number of iterations to achieve both privacy protection and utility up to a neighborhood of the optimal solution, removing the need for tuning the number of iterations. Numerical results show the superiority of the proposed algorithm over state-of-the-art methods.

cs.LG↗

arXiv2023

Bringing regularized optimal transport to lightspeed: a splitting method adapted for GPUs

We present an efficient algorithm for regularized optimal transport. In contrast to previous methods, we use the Douglas-Rachford splitting technique to develop an efficient solver that can handle a broad class of regularizers. The algorithm has strong global convergence guarantees, low per-iteration cost, and can exploit GPU parallelization, making it considerably faster than the state-of-the-art for many problems. We illustrate its competitiveness in several applications, including domain adaptation and learning of generative models.

cs.LG↗

arXiv2023

Delay-agnostic Asynchronous Coordinate Update Algorithm

We propose a delay-agnostic asynchronous coordinate update algorithm (DEGAS) for computing operator fixed points, with applications to asynchronous optimization. DEGAS includes novel asynchronous variants of ADMM and block-coordinate descent as special cases. We prove that DEGAS converges under both bounded and unbounded delays under delay-free parameter conditions. We also validate by theory and experiments that DEGAS adapts well to the actual delays. The effectiveness of DEGAS is demonstrated by numerical experiments on classification problems.

math.OC↗

arXiv2023

Asynchronous Iterations in Optimization: New Sequence Results and Sharper Algorithmic Guarantees

We introduce novel convergence results for asynchronous iterations that appear in the analysis of parallel and distributed optimization algorithms. The results are simple to apply and give explicit estimates for how the degree of asynchrony impacts the convergence rates of the iterates. Our results shorten, streamline and strengthen existing convergence proofs for several asynchronous optimization methods and allow us to establish convergence guarantees for popular algorithms that were thus far lacking a complete theoretical understanding. Specifically, we use our results to derive better iteration complexity bounds for proximal incremental aggregated gradient methods, to obtain tighter guarantees depending on the average rather than maximum delay for the asynchronous stochastic gradient descent method, to provide less conservative analyses of the speedup conditions for asynchronous block-coordinate implementations of Krasnoselskii-Mann iterations, and to quantify the convergence rates for totally asynchronous iterations under various assumptions on communication delays and update rates.

math.OC↗

arXiv2023

Logarithmically Completely Monotonic Rational Functions

This paper studies the class of logarithmically completely monotonic (LCM) functions. These functions play an important role in characterising externally positive linear systems which find applications in important control problems such as non-overshooting reference tracking. Conditions are proposed to ensure a rational function is LCM, a result that enables the known space of linear continuous-time externally positive systems to be enlarged and an efficient and optimal pole-placement procedure for the monotonic tracking controller synthesis problem to be developed. The presented conditions are shown to be less conservative than existing approaches whilst being computationally tractable.

math.OC↗

arXiv2023

Over-the-Air Computation for Distributed Systems: Something Old and Something New

Facing the upcoming era of Internet-of-Things and connected intelligence, efficient information processing, computation, and communication design becomes a key challenge in large-scale intelligent systems. Recently, Over-the-Air (OtA) computation has been proposed for data aggregation and distributed computation of functions over a large set of network nodes. Theoretical foundations for this concept exist for a long time, but it was mainly investigated within the context of wireless sensor networks. There are still many open questions when applying OtA computation in different types of distributed systems where modern wireless communication technology is applied. In this article, we provide a comprehensive overview of the OtA computation principle and its applications in distributed learning, control, and inference systems, for both server-coordinated and fully decentralized architectures. Particularly, we highlight the importance of the statistical heterogeneity of data and wireless channels, the temporal evolution of model updates, and the choice of performance metrics, for the communication design in OtA federated learning (FL) systems. Several key challenges in privacy, security, and robustness aspects of OtA FL are also identified for further investigation.

cs.IT↗

arXiv2022

Inferring origin-destination distribution of agent transfer in a complex network using deep gated recurrent units

Predicting the origin-destination (OD) probability distribution of agent transfer is an important problem for managing complex systems. However, prediction accuracy of associated statistical estimators suffer from underdetermination. While specific techniques have been proposed to overcome this deficiency, there still lacks a general approach. Here, we propose a deep neural network framework with gated recurrent units (DNNGRU) to address this gap. Our DNNGRU is \emph{network-free}, as it is trained by supervised learning with time-series data on the volume of agents passing through edges. We use it to investigate how network topologies affect OD prediction accuracy, where performance enhancement is observed to depend on the degree of overlap between paths taken by different ODs. By comparing against methods that give exact results, we demonstrate the near-optimal performance of our DNNGRU, which we found to consistently outperform existing methods and alternative neural network architectures, under diverse data generation scenarios.

physics.soc-ph↗

arXiv2022

Distributed Safe Resource Allocation using Barrier Functions

Resource allocation plays a central role in many networked systems such as smart grids, communication networks and urban transportation systems. In these systems, many constraints have physical meaning and having feasible allocation is often vital to avoid system breakdown. Hence, algorithms with asymptotic feasibility guarantees are often insufficient since it is impractical to run algorithms for an infinite number of rounds. This paper proposes a distributed feasible method (DFM) for safe resource allocation based on barrier functions. In DFM, every iterate is feasible and thus safe to implement. We prove that under mild conditions, DFM converges to an arbitrarily small neighbourhood of the optimal solution. Numerical experiments demonstrate the competitive performance of DFM.

math.OC↗

arXiv2022

On Uniform Boundedness Properties of SGD and its Momentum Variants

A theoretical, and potentially also practical, problem with stochastic gradient descent is that trajectories may escape to infinity. In this note, we investigate uniform boundedness properties of iterates and function values along the trajectories of the stochastic gradient descent algorithm and its important momentum variant. Under smoothness and $R$-dissipativity of the loss function, we show that broad families of step-sizes, including the widely used step-decay and cosine with (or without) restart step-sizes, result in uniformly bounded iterates and function values. Several important applications that satisfy these assumptions, including phase retrieval problems, Gaussian mixture models, and some neural network classifiers, are discussed in detail. We further extend the uniform boundedness of SGD and its momentum variant under the generalized dissipativity for the functions whose tails grow slower than quadratic functions. This includes some interesting applications, for example, Bayesian logistic regression and logistic regression with $\ell_1$ regularization.

cs.LG↗

arXiv2022

Delay-adaptive step-sizes for asynchronous learning

In scalable machine learning systems, model training is often parallelized over multiple nodes that run without tight synchronization. Most analysis results for the related asynchronous algorithms use an upper bound on the information delays in the system to determine learning rates. Not only are such bounds hard to obtain in advance, but they also result in unnecessarily slow convergence. In this paper, we show that it is possible to use learning rates that depend on the actual time-varying delays in the system. We develop general convergence results for delay-adaptive asynchronous iterations and specialize these to proximal incremental gradient descent and block-coordinate descent algorithms. For each of these methods, we demonstrate how delays can be measured on-line, present delay-adaptive step-size policies, and illustrate their theoretical and practical advantages over the state-of-the-art.

cs.LG↗