arXiv subjects

César A. Uribe

Publications and source records attributed to César A. Uribe.

At least 37 records · Page 2Linked to original sources

arXiv2024

Adaptive Federated Learning with Auto-Tuned Clients

Federated learning (FL) is a distributed machine learning framework where the global model of a central server is trained via multiple collaborative steps by participating clients without sharing their data. While being a flexible framework, where the distribution of local data, participation rate, and computing power of each client can greatly vary, such flexibility gives rise to many new challenges, especially in the hyperparameter tuning on the client side. We propose $Δ$-SGD, a simple step size rule for SGD that enables each client to use its own step size by adapting to the local smoothness of the function each client is optimizing. We provide theoretical and empirical results where the benefit of the client adaptivity is shown in various FL scenarios.

cs.LG↗

arXiv2024

Decentralized and Equitable Optimal Transport

This paper considers the decentralized (discrete) optimal transport (D-OT) problem. In this setting, a network of agents seeks to design a transportation plan jointly, where the cost function is the sum of privately held costs for each agent. We reformulate the D-OT problem as a constraint-coupled optimization problem and propose a single-loop decentralized algorithm with an iteration complexity of O(1/ε) that matches existing centralized first-order approaches. Moreover, we propose the decentralized equitable optimal transport (DE-OT) problem. In DE-OT, in addition to cooperatively designing a transportation plan that minimizes transportation costs, agents seek to ensure equity in their individual costs. The iteration complexity of the proposed method to solve DE-OT is also O(1/ε). This rate improves existing centralized algorithms, where the best iteration complexity obtained is O(1/ε^2).

math.OC↗

arXiv2024

PIDformer: Transformer Meets Control Theory

In this work, we address two main shortcomings of transformer architectures: input corruption and rank collapse in their output representation. We unveil self-attention as an autonomous state-space model that inherently promotes smoothness in its solutions, leading to lower-rank outputs and diminished representation capacity. Moreover, the steady-state solution of the model is sensitive to input perturbations. We incorporate a Proportional-Integral-Derivative (PID) closed-loop feedback control system with a reference point into the model to improve robustness and representation capacity. This integration aims to preserve high-frequency details while bolstering model stability, rendering it more noise-resilient. The resulting controlled state-space model is theoretically proven robust and adept at addressing the rank collapse. Motivated by this control framework, we derive a novel class of transformers, PID-controlled Transformer (PIDformer), aimed at improving robustness and mitigating the rank-collapse issue inherent in softmax transformers. We empirically evaluate the model for advantages and robustness against baseline transformers across various practical tasks, including object classification, image segmentation, and language modeling.

cs.AI↗

arXiv2023

Improving Denoising Diffusion Probabilistic Models via Exploiting Shared Representations

In this work, we address the challenge of multi-task image generation with limited data for denoising diffusion probabilistic models (DDPM), a class of generative models that produce high-quality images by reversing a noisy diffusion process. We propose a novel method, SR-DDPM, that leverages representation-based techniques from few-shot learning to effectively learn from fewer samples across different tasks. Our method consists of a core meta architecture with shared parameters, i.e., task-specific layers with exclusive parameters. By exploiting the similarity between diverse data distributions, our method can scale to multiple tasks without compromising the image quality. We evaluate our method on standard image datasets and show that it outperforms both unconditional and conditional DDPM in terms of FID and SSIM metrics.

cs.LG↗

arXiv2023

PersA-FL: Personalized Asynchronous Federated Learning

We study the personalized federated learning problem under asynchronous updates. In this problem, each client seeks to obtain a personalized model that simultaneously outperforms local and global models. We consider two optimization-based frameworks for personalization: (i) Model-Agnostic Meta-Learning (MAML) and (ii) Moreau Envelope (ME). MAML involves learning a joint model adapted for each client through fine-tuning, whereas ME requires a bi-level optimization problem with implicit gradients to enforce personalization via regularized losses. We focus on improving the scalability of personalized federated learning by removing the synchronous communication assumption. Moreover, we extend the studied function class by removing boundedness assumptions on the gradient norm. Our main technical contribution is a unified proof for asynchronous federated learning with bounded staleness that we apply to MAML and ME personalization frameworks. For the smooth and non-convex functions class, we show the convergence of our method to a first-order stationary point. We illustrate the performance of our method and its tolerance to staleness through experiments for classification tasks over heterogeneous datasets.

cs.LG↗

arXiv2023

Near-optimal tensor methods for minimizing the gradient norm of convex functions and accelerated primal-dual tensor methods

Motivated, in particular, by the entropy-regularized optimal transport problem, we consider convex optimization problems with linear equality constraints, where the dual objective has Lipschitz $p$-th order derivatives, and develop two approaches for solving such problems. The first approach is based on the minimization of the norm of the gradient in the dual problem and then the reconstruction of an approximate primal solution. Recently, Grapiglia and Nesterov in their work showed lower complexity bounds for the problem of minimizing the gradient norm of the function with Lipschitz $p$-th order derivatives. Still, the question of optimal or near-optimal methods remained open as the algorithms presented in the paper achieve suboptimal bounds only. We close this gap by proposing two near-optimal (up to logarithmic factors) methods with complexity bounds $\tilde{O}(\varepsilon^{-2(p+1)/(3p+1)})$ and $\tilde{O}(\varepsilon^{-2/(3p+1)})$ with respect to the initial objective residual and the distance between the starting point and solution respectively. We then apply these results (having independent interest) to our primal-dual setting. As the second approach, we propose a direct accelerated primal-dual tensor method for convex problems with linear equality constraints, where the dual objective has Lipschitz $p$-th order derivatives. For this algorithm, we prove $\tilde O (\varepsilon^{-1 / (p + 1)})$ complexity in terms of the duality gap and the residual in the constraints. We illustrate the practical performance of the proposed algorithms in experiments on logistic regression, entropy-regularized optimal transport problem, and the minimal mutual information problem.

math.OC↗

arXiv2023

On First-Order Meta-Reinforcement Learning with Moreau Envelopes

Meta-Reinforcement Learning (MRL) is a promising framework for training agents that can quickly adapt to new environments and tasks. In this work, we study the MRL problem under the policy gradient formulation, where we propose a novel algorithm that uses Moreau envelope surrogate regularizers to jointly learn a meta-policy that is adjustable to the environment of each individual task. Our algorithm, called Moreau Envelope Meta-Reinforcement Learning (MEMRL), learns a meta-policy that can adapt to a distribution of tasks by efficiently updating the policy parameters using a combination of gradient-based optimization and Moreau Envelope regularization. Moreau Envelopes provide a smooth approximation of the policy optimization problem, which enables us to apply standard optimization techniques and converge to an appropriate stationary point. We provide a detailed analysis of the MEMRL algorithm, where we show a sublinear convergence rate to a first-order stationary point for non-convex policy gradient optimization. We finally show the effectiveness of MEMRL on a multi-task 2D-navigation problem.

cs.LG↗

arXiv2022

An energy management system model with power quality constraints for unbalanced multi-microgrids interacting in a local energy market

As multi-microgrids become readily available, some limited models have been proposed that study operational and power quality constraints with local energy markets independently. This paper proposes a convex optimization model of an energy management system with operational and power quality constraints and interactions in a Local Energy Market (LEM) for unbalanced microgrids (MGs). The LEM consists of a pre-dispatch step and an energy transactions step (ETS). The ETS combines the MGs' objectives while considering two strategies: minimize the cost of buyers or maximize the revenue of sellers. Our proposed model considers harmonic distortion and voltage limit power quality constraints in both steps. Moreover, we model operational constraints such as power flow, power balance, and distributed energy resources behaviors and capacities. We numerically evaluate the proposed model using three unbalanced MGs with residential, industrial, and commercial load profiles, where each microgrid manages its resources locally. Furthermore, we create two groups of cases to analyze the interactions in the local energy market. In the first group, the price of the DSO energy and the surplus from MGs to DSO are the same. The numerical results show that using the increasing revenue strategy promotes MGs to interact more while encouraging them to have high energy prices. When the reducing cost strategy is used, fewer energy interactions occur, and the price of MGs energy is encouraged to be lower.

eess.SY↗

arXiv2022

On the Performance of Gradient Tracking with Local Updates

We study the decentralized optimization problem where a network of $n$ agents seeks to minimize the average of a set of heterogeneous non-convex cost functions distributedly. State-of-the-art decentralized algorithms like Exact Diffusion~(ED) and Gradient Tracking~(GT) involve communicating every iteration. However, communication is expensive, resource intensive, and slow. In this work, we analyze a locally updated GT method (LU-GT), where agents perform local recursions before interacting with their neighbors. While local updates have been shown to reduce communication overhead in practice, their theoretical influence has not been fully characterized. We show LU-GT has the same communication complexity as the Federated Learning setting but allows arbitrary network topologies. In addition, we prove that the number of local updates does not degrade the quality of the solution achieved by LU-GT. Numerical examples reveal that local updates can lower communication costs in certain regimes (e.g., well-connected graphs).

math.OC↗

arXiv2022

A State Feedback Controller for Mitigation of Continuous-Time Networked SIS Epidemics

The paper considers continuous-time networked susceptible-infected-susceptible (SIS) diseases spreading over a population. Each agent represents a sub-population and has its own healing rate and infection rate; the state of the agent at a time instant denotes what fraction of the said sub-population is infected with the disease at the said time instant. By taking account of the changes in behaviors of the agents in response to the infection rates in real-time, our goal is to devise a feedback strategy such that the infection level for each agent strictly stays below a pre-specified value. Furthermore, we are also interested in ensuring that the closed-loop system converges either to the disease-free equilibrium or, when it exists, to the endemic equilibrium. The upshot of devising such a strategy is that it allows health administration officials to ensure that there is sufficient capacity in the healthcare system to treat the most severe cases. We demonstrate the effectiveness of our controller via numerical examples.

eess.SY↗

arXiv2022

Hyperfast Second-Order Local Solvers for Efficient Statistically Preconditioned Distributed Optimization

Statistical preconditioning enables fast methods for distributed large-scale empirical risk minimization problems. In this approach, multiple worker nodes compute gradients in parallel, which are then used by the central node to update the parameter by solving an auxiliary (preconditioned) smaller-scale optimization problem. The recently proposed Statistically Preconditioned Accelerated Gradient (SPAG) method has complexity bounds superior to other such algorithms but requires an exact solution for computationally intensive auxiliary optimization problems at every iteration. In this paper, we propose an Inexact SPAG (InSPAG) and explicitly characterize the accuracy by which the corresponding auxiliary subproblem needs to be solved to guarantee the same convergence rate as the exact method. We build our results by first developing an inexact adaptive accelerated Bregman proximal gradient method for general optimization problems under relative smoothness and strong convexity assumptions, which may be of independent interest. Moreover, we explore the properties of the auxiliary problem in the InSPAG algorithm assuming Lipschitz third-order derivatives and strong convexity. For such problem class, we develop a linearly convergent Hyperfast second-order method and estimate the total complexity of the InSPAG method with hyperfast auxiliary problem solver. Finally, we illustrate the proposed method's practical efficiency by performing large-scale numerical experiments on logistic regression models. To the best of our knowledge, these are the first empirical results on implementing high-order methods on large-scale problems, we work with data where the dimension is of the order of~$3$ million, and the number of samples is~$700$ million.

math.OC↗

arXiv2022

Unbounded Gradients in Federated Learning with Buffered Asynchronous Aggregation

Synchronous updates may compromise the efficiency of cross-device federated learning once the number of active clients increases. The \textit{FedBuff} algorithm (Nguyen et al., 2022) alleviates this problem by allowing asynchronous updates (staleness), which enhances the scalability of training while preserving privacy via secure aggregation. We revisit the \textit{FedBuff} algorithm for asynchronous federated learning and extend the existing analysis by removing the boundedness assumptions from the gradient norm. This paper presents a theoretical analysis of the convergence rate of this algorithm when heterogeneity in data, batch size, and delay are considered.

cs.LG↗

arXiv2022

The Role of Local Steps in Local SGD

We consider the distributed stochastic optimization problem where $n$ agents want to minimize a global function given by the sum of agents' local functions, and focus on the heterogeneous setting when agents' local functions are defined over non-i.i.d. data sets. We study the Local SGD method, where agents perform a number of local stochastic gradient steps and occasionally communicate with a central node to improve their local optimization tasks. We analyze the effect of local steps on the convergence rate and the communication complexity of Local SGD. In particular, instead of assuming a fixed number of local steps across all communication rounds, we allow the number of local steps during the $i$-th communication round, $H_i$, to be different and arbitrary numbers. Our main contribution is to characterize the convergence rate of Local SGD as a function of $\{H_i\}_{i=1}^R$ under various settings of strongly convex, convex, and nonconvex local functions, where $R$ is the total number of communication rounds. Based on this characterization, we provide sufficient conditions on the sequence $\{H_i\}_{i=1}^R$ such that Local SGD can achieve linear speed-up with respect to the number of workers. Furthermore, we propose a new communication strategy with increasing local steps superior to existing communication strategies for strongly convex local functions. On the other hand, for convex and nonconvex local functions, we argue that fixed local steps are the best communication strategy for Local SGD and recover state-of-the-art convergence rate results. Finally, we justify our theoretical results through extensive numerical experiments.

cs.LG↗

arXiv2022

Consensus ADMM-Based Distributed Simultaneous Imaging & Communication

This paper takes the first steps toward enabling wireless networks to perform both imaging and communication in a distributed manner. We propose Distributed Simultaneous Imaging and Symbol Detection (DSISD), a provably convergent distributed simultaneous imaging and communication scheme based on the alternating direction method of multipliers. We show that DSISD achieves similar imaging and communication performance as centralized schemes, with order-wise reduction in computational complexity. We evaluate the performance of DSISD via 2.4 GHz Wi-Fi simulations.

eess.SY↗

arXiv2022

Distributed Generalized Wirtinger Flow for Interferometric Imaging on Networks

We study the problem of decentralized interferometric imaging over networks, where agents have access to a subset of local radar measurements and can compute pair-wise correlations with their neighbors. We propose a primal-dual distributed algorithm named Distributed Generalized Wirtinger Flow (DGWF). We use the theory of low rank matrix recovery to show when the interferometric imaging problem satisfies the Regularity Condition, which implies the Polyak-Lojasiewicz inequality. Moreover, we show that DGWF converges geometrically for smooth functions. Numerical simulations for single-scattering radar interferometric imaging demonstrate that DGWF can achieve the same mean-squared error image reconstruction quality as its centralized counterpart for various network connectivity and size.

math.OC↗

arXiv2022

Local Stochastic Factored Gradient Descent for Distributed Quantum State Tomography

We propose a distributed Quantum State Tomography (QST) protocol, named Local Stochastic Factored Gradient Descent (Local SFGD), to learn the low-rank factor of a density matrix over a set of local machines. QST is the canonical procedure to characterize the state of a quantum system, which we formulate as a stochastic nonconvex smooth optimization problem. Physically, the estimation of a low-rank density matrix helps characterizing the amount of noise introduced by quantum computation. Theoretically, we prove the local convergence of Local SFGD for a general class of restricted strongly convex/smooth loss functions, i.e., Local SFGD converges locally to a small neighborhood of the global optimum at a linear rate with a constant step size, while it locally converges exactly at a sub-linear rate with diminishing step sizes. With a proper initialization, local convergence results imply global convergence. We validate our theoretical findings with numerical simulations of QST on the Greenberger-Horne-Zeilinger (GHZ) state.

quant-ph↗

arXiv2022

Approximate Wasserstein Attraction Flows for Dynamic Mass Transport over Networks

This paper presents a Wasserstein attraction approach for solving dynamic mass transport problems over networks. In the transport problem over networks, we start with a distribution over the set of nodes that needs to be "transported" to a target distribution accounting for the network topology. We exploit the specific structure of the problem, characterized by the computation of implicit gradient steps, and formulate an approach based on discretized flows. As a result, our proposed algorithm relies on the iterative computation of constrained Wasserstein barycenters. We show how the proposed method finds approximate solutions to the network transport problem, taking into account the topology of the network, the capacity of the communication channels, and the capacity of the individual nodes. Finally, we show the performance of this approach applied to large-scale water transportation networks.

math.OC↗

arXiv2022

On Arbitrary Compression for Decentralized Consensus and Stochastic Optimization over Directed Networks

We study the decentralized consensus and stochastic optimization problems with compressed communications over static directed graphs. We propose an iterative gradient-based algorithm that compresses messages according to a desired compression ratio. The proposed method provably reduces the communication overhead on the network at every communication round. Contrary to existing literature, we allow for arbitrary compression ratios in the communicated messages. We show a linear convergence rate for the proposed method on the consensus problem. Moreover, we provide explicit convergence rates for decentralized stochastic optimization problems on smooth functions that are either (i) strongly convex, (ii) convex, or (iii) non-convex. Finally, we provide numerical experiments to illustrate convergence under arbitrary compression ratios and the communication efficiency of our algorithm.

math.OC↗