Stefan Vlaski

Stefan Vlaski, Soummya Kar, Ali H. Sayed et al. · cmu

The article reviews significant advances in networked signal and information processing, which have enabled in the last 25 years extending decision making and inference, optimization, control, and learning to the increasingly ubiquitous environments of distributed agents. As these interacting agents cooperate, new collective behaviors emerge from local decisions and actions. Moreover, and significantly, theory and applications show that networked agents, through cooperation and sharing, are able to match the performance of cloud or federated solutions, while offering the potential for improved privacy, increasing resilience, and saving resources.

Roula Nassif, Stefan Vlaski, Marco Carpentiero et al.

In this paper, we consider decentralized optimization problems where agents have individual cost functions to minimize subject to subspace constraints that require the minimizers across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus or single-task optimization as special cases, and allows for more general task relatedness models such as multitask smoothness and coupled optimization. In order to cope with communication constraints, we propose and study an adaptive decentralized strategy where the agents employ differential randomized quantizers to compress their estimates before communicating with their neighbors. The analysis shows that, under some general conditions on the quantization noise, and for sufficiently small step-sizes $μ$, the strategy is stable both in terms of mean-square error and average bit rate: by reducing $μ$, it is possible to keep the estimation errors small (on the order of $μ$) without increasing indefinitely the bit rate as $μ\rightarrow 0$. Simulations illustrate the theoretical findings and the effectiveness of the proposed approach, revealing that decentralized learning is achievable at the expense of only a few bits.

Elsa Rizk, Stefan Vlaski, Ali H. Sayed

We study the privatization of distributed learning and optimization strategies. We focus on differential privacy schemes and study their effect on performance. We show that the popular additive random perturbation scheme degrades performance because it is not well-tuned to the graph structure. For this reason, we exploit two alternative graph-homomorphic constructions and show that they improve performance while guaranteeing privacy. Moreover, contrary to most earlier studies, the gradient of the risks is not assumed to be bounded (a condition that rarely holds in practice; e.g., quadratic risk). We avoid this condition and still devise a differentially private scheme with high probability. We examine optimization and learning scenarios and illustrate the theoretical findings through simulations.

Mert Kayaalp, Virginia Bordignon, Stefan Vlaski et al.

This work studies networked agents cooperating to track a dynamical state of nature under partial information. The proposed algorithm is a distributed Bayesian filtering algorithm for finite-state hidden Markov models (HMMs). It can be used for sequential state estimation tasks, as well as for modeling opinion formation over social networks under dynamic environments. We show that the disagreement with the optimal centralized solution is asymptotically bounded for the class of geometrically ergodic state transition models, which includes rapidly changing models. We also derive recursions for calculating the probability of error and establish convergence under Gaussian observation models. Simulations are provided to illustrate the theory and to compare against alternative approaches.

Ying Cao, Elsa Rizk, Stefan Vlaski et al.

This work focuses on adversarial learning over graphs. We propose a general adversarial training framework for multi-agent systems using diffusion learning. We analyze the convergence properties of the proposed scheme for convex optimization problems, and illustrate its enhanced robustness to adversarial attacks.

Elsa Rizk, Stefan Vlaski, Ali H. Sayed

Federated learning is a semi-distributed algorithm, where a server communicates with multiple dispersed clients to learn a global model. The federated architecture is not robust and is sensitive to communication and computational overloads due to its one-master multi-client structure. It can also be subject to privacy attacks targeting personal information on the communication links. In this work, we introduce graph federated learning (GFL), which consists of multiple federated units connected by a graph. We then show how graph homomorphic perturbations can be used to ensure the algorithm is differentially private. We conduct both convergence and privacy theoretical analyses and illustrate performance by means of computer simulations.

Ying Cao, Elsa Rizk, Stefan Vlaski et al.

The vulnerability of machine learning models to adversarial attacks has been attracting considerable attention in recent years. Most existing studies focus on the behavior of stand-alone single-agent learners. In comparison, this work studies adversarial training over graphs, where individual agents are subjected to perturbations of varied strength levels across space. It is expected that interactions by linked agents, and the heterogeneity of the attack models that are possible over the graph, can help enhance robustness in view of the coordination power of the group. Using a min-max formulation of distributed learning, we develop a decentralized adversarial training framework for multi-agent systems. Specifically, we devise two decentralized adversarial training algorithms by relying on two popular decentralized learning strategies--diffusion and consensus. We analyze the convergence properties of the proposed framework for strongly-convex, convex, and non-convex environments, and illustrate the enhanced robustness to adversarial attacks.

Elsa Rizk, Stefan Vlaski, Ali H. Sayed

We study the generation of dependent random numbers in a distributed fashion in order to enable privatized distributed learning by networked agents. We propose a method that we refer to as local graph-homomorphic processing; it relies on the construction of particular noises over the edges to ensure a certain level of differential privacy. We show that the added noise does not affect the performance of the learned model. This is a significant improvement to previous works on differential privacy for distributed algorithms, where the noise was added in a less structured manner without respecting the graph topology and has often led to performance deterioration. We illustrate the theoretical results by considering a linear regression problem over a network of agents.

Stefan Vlaski, Christian Schroth, Michael Muma et al.

Distributed learning paradigms, such as federated and decentralized learning, allow for the coordination of models across a collection of agents, and without the need to exchange raw data. Instead, agents compute model updates locally based on their available data, and subsequently share the update model with a parameter server or their peers. This is followed by an aggregation step, which traditionally takes the form of a (weighted) average. Distributed learning schemes based on averaging are known to be susceptible to outliers. A single malicious agent is able to drive an averaging-based distributed learning algorithm to an arbitrarily poor model. This has motivated the development of robust aggregation schemes, which are based on variations of the median and trimmed mean. While such procedures ensure robustness to outliers and malicious behavior, they come at the cost of significantly reduced sample efficiency. This means that current robust aggregation schemes require significantly higher agent participation rates to achieve a given level of performance than their mean-based counterparts in non-contaminated settings. In this work we remedy this drawback by developing statistically efficient and robust aggregation schemes for distributed learning.

Roula Nassif, Virginia Bordignon, Stefan Vlaski et al.

Observations collected by agents in a network may be unreliable due to observation noise or interference. This paper proposes a distributed algorithm that allows each node to improve the reliability of its own observation by relying solely on local computations and interactions with immediate neighbors, assuming that the field (graph signal) monitored by the network lies in a low-dimensional subspace and that a low-rank noise is present in addition to the usual full-rank noise. While oblique projections can be used to project measurements onto a low-rank subspace along a direction that is oblique to the subspace, the resulting solution is not distributed. Starting from the centralized solution, we propose an algorithm that performs the oblique projection of the overall set of observations onto the signal subspace in an iterative and distributed manner. We then show how the oblique projection framework can be extended to handle distributed learning and adaptation problems over networks.

Christian A. Schroth, Stefan Vlaski, Abdelhak M. Zoubir

Distributed learning paradigms, such as federated or decentralized learning, allow a collection of agents to solve global learning and optimization problems through limited local interactions. Most such strategies rely on a mixture of local adaptation and aggregation steps, either among peers or at a central fusion center. Classically, aggregation in distributed learning is based on averaging, which is statistically efficient, but susceptible to attacks by even a small number of malicious agents. This observation has motivated a number of recent works, which develop robust aggregation schemes by employing robust variations of the mean. We present a new attack based on sensitivity curve maximization (SCM), and demonstrate that it is able to disrupt existing robust aggregation schemes by injecting small, but effective perturbations.

Shreya Wadehra, Roula Nassif, Stefan Vlaski

Classical paradigms for distributed learning, such as federated or decentralized gradient descent, employ consensus mechanisms to enforce homogeneity among agents. While these strategies have proven effective in i.i.d. scenarios, they can result in significant performance degradation when agents follow heterogeneous objectives or data. Distributed strategies for multitask learning, on the other hand, induce relationships between agents in a more nuanced manner, and encourage collaboration without enforcing consensus. We develop a generalization of the exact diffusion algorithm for subspace constrained multitask learning over networks, and derive an accurate expression for its mean-squared deviation when utilizing noisy gradient approximations. We verify numerically the accuracy of the predicted performance expressions, as well as the improved performance of the proposed approach over alternatives based on approximate projections.

Jie Peng, Weiyu Li, Stefan Vlaski et al.

Robustness to malicious attacks is of paramount importance for distributed learning. Existing works usually consider the classical Byzantine attacks model, which assumes that some workers can send arbitrarily malicious messages to the server and disturb the aggregation steps of the distributed learning process. To defend against such worst-case Byzantine attacks, various robust aggregators have been proposed. They are proven to be effective and much superior to the often-used mean aggregator. In this paper, however, we demonstrate that the robust aggregators are too conservative for a class of weak but practical malicious attacks, as known as label poisoning attacks, where the sample labels of some workers are poisoned. Surprisingly, we are able to show that the mean aggregator is more robust than the state-of-the-art robust aggregators in theory, given that the distributed data are sufficiently heterogeneous. In fact, the learning error of the mean aggregator is proven to be order-optimal in this case. Experimental results corroborate our theoretical findings, showing the superiority of the mean aggregator under label poisoning attacks.

Christian A. Schroth, Stefan Vlaski, Abdelhak M. Zoubir

In distributed learning agents aim at collaboratively solving a global learning problem. It becomes more and more likely that individual agents are malicious or faulty with an increasing size of the network. This leads to a degeneration or complete breakdown of the learning process. Classical aggregation schemes are prone to breakdown at small contamination rates, therefore robust aggregation schemes are sought for. While robust aggregation schemes can generally tolerate larger contamination rates, many have been shown to be susceptible to carefully crafted malicious attacks. In this work, we show how the sensitivity curve (SC), a classical tool from robust statistics, can be used to systematically derive optimal attack patterns against arbitrary robust aggregators, in most cases rendering them ineffective. We show the effectiveness of the proposed attack in multiple simulations.

Zirui Wan, Stefan Vlaski

Classical consensus-based strategies for federated and decentralized learning are statistically suboptimal in the presence of heterogeneous local data or task distributions. As a result, in recent years, there has been growing interest in multitask or personalized strategies, which allow individual agents to benefit from one another in pursuing locally optimal models without enforcing consensus. Existing strategies require either precise prior knowledge of the underlying task relationships or are fully non-parametric and instead rely on meta-learning or proximal constructions. In this work, we introduce an algorithmic framework that strikes a balance between these extremes. By modeling task relationships through a Gaussian Markov Random Field with an unknown precision matrix, we develop a strategy that jointly learns both the task relationships and the local models, allowing agents to self-organize in a way consistent with their individual data distributions. Our theoretical analysis quantifies the quality of the learned relationship, and our numerical experiments demonstrate its practical effectiveness.

Xue Xian Zheng, Weihang Liu, Xin Lou et al.

This paper introduces an innovative error feedback framework designed to mitigate quantization noise in distributed graph filtering, where communications are constrained to quantized messages. It comes from error spectrum shaping techniques from state-space digital filters, and therefore establishes connections between quantized filtering processes over different domains. In contrast to existing error compensation methods, our framework quantitatively feeds back the quantization noise for exact compensation. We examine the framework under three key scenarios: (i) deterministic graph filtering, (ii) graph filtering over random graphs, and (iii) graph filtering with random node-asynchronous updates. Rigorous theoretical analysis demonstrates that the proposed framework significantly reduces the effect of quantization noise, and we provide closed-form solutions for the optimal error feedback coefficients. Moreover, this quantitative error feedback mechanism can be seamlessly integrated into communication-efficient decentralized optimization frameworks, enabling lower error floors. Numerical experiments validate the theoretical results, consistently showing that our method outperforms conventional quantization strategies in terms of both accuracy and robustness.

Sangwoo Park, Stefan Vlaski, Lajos Hanzo

In multi-objective optimization, minimizing the worst objective can be preferable to minimizing the average objective, as this ensures improved fairness across objectives. Due to the non-smooth nature of the resultant min-max optimization problem, classical subgradient-based approaches typically exhibit slow convergence. Motivated by primal-dual consensus techniques in multi-agent optimization and learning, we formulate a smooth variant of the min-max problem based on the augmented Lagrangian. The resultant Exact Pareto Optimization via Augmented Lagrangian (EPO-AL) algorithm scales better with the number of objectives than subgradient-based strategies, while exhibiting lower per-iteration complexity than recent smoothing-based counterparts. We establish that every fixed-point of the proposed algorithm is both Pareto and min-max optimal under mild assumptions and demonstrate its effectiveness in numerical simulations.

Sean Xiao, Sangwoo Park, Stefan Vlaski

Stochastic first-order methods for empirical risk minimization employ gradient approximations based on sampled data in lieu of exact gradients. Such constructions introduce noise into the learning dynamics, which can be corrected through variance-reduction techniques. There is increasing evidence in the literature that in many modern learning applications noise can have a beneficial effect on optimization and generalization. To this end, the recently proposed variance-reduction technique, alpha-SVRG [Yin et al., 2023] allows for fine-grained control of the level of residual noise in the learning dynamics, and has been reported to empirically outperform both SGD and SVRG in modern deep learning scenarios. By focusing on strongly convex environments, we first provide a unified convergence rate expression for alpha-SVRG under fixed learning rate, which reduces to that of either SGD or SVRG by setting alpha=0 or alpha=1, respectively. We show that alpha-SVRG has faster convergence rate compared to SGD and SVRG under suitable choice of alpha. Simulation results on linear regression validate our theory.

Muyun Li, Aaron Fainman, Stefan Vlaski

Decentralized learning strategies allow a collection of agents to learn efficiently from local data sets without the need for central aggregation or orchestration. Current decentralized learning paradigms typically rely on an averaging mechanism to encourage agreement in the parameter space. We argue that in the context of deep neural networks, which are often over-parameterized, encouraging consensus of the neural network outputs, as opposed to their parameters can be more appropriate. This motivates the development of a new decentralized learning algorithm, termed DRT diffusion, based on deep relative trust (DRT), a recently introduced similarity measure for neural networks. We provide convergence analysis for the proposed strategy, and numerically establish its benefit to generalization, especially with sparse topologies, in an image classification task.

Roula Nassif, Stefan Vlaski, Marco Carpentiero et al.

Communication-constrained algorithms for decentralized learning and optimization rely on local updates coupled with the exchange of compressed signals. In this context, differential quantization is an effective technique to mitigate the negative impact of compression by leveraging correlations between successive iterates. In addition, the use of error feedback, which consists of incorporating the compression error into subsequent steps, is a powerful mechanism to compensate for the bias caused by the compression. Under error feedback, performance guarantees in the literature have so far focused on algorithms employing a fusion center or a special class of contractive compressors that cannot be implemented with a finite number of bits. In this work, we propose a new decentralized communication-efficient learning approach that blends differential quantization with error feedback. The approach is specifically tailored for decentralized learning problems where agents have individual risk functions to minimize subject to subspace constraints that require the minimizers across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus or single-task optimization as special cases, and allows for more general task relatedness models such as multitask smoothness and coupled optimization. We show that, under some general conditions on the compression noise, and for sufficiently small step-sizes $μ$, the resulting communication-efficient strategy is stable both in terms of mean-square error and average bit rate: by reducing $μ$, it is possible to keep the estimation errors small (on the order of $μ$) without increasing indefinitely the bit rate as $μ\rightarrow 0$. The results establish that, in the small step-size regime and with a finite number of bits, it is possible to attain the performance achievable in the absence of compression.

Virginia Bordignon, Stefan Vlaski, Vincenzo Matta et al.

This work proposes a decentralized architecture, where individual agents aim at solving a classification problem while observing streaming features of different dimensions and arising from possibly different distributions. In the context of social learning, several useful strategies have been developed, which solve decision making problems through local cooperation across distributed agents and allow them to learn from streaming data. However, traditional social learning strategies rely on the fundamental assumption that each agent has significant prior knowledge of the underlying distribution of the observations. In this work we overcome this issue by introducing a machine learning framework that exploits social interactions over a graph, leading to a fully data-driven solution to the distributed classification problem. In the proposed social machine learning (SML) strategy, two phases are present: in the training phase, classifiers are independently trained to generate a belief over a set of hypotheses using a finite number of training samples; in the prediction phase, classifiers evaluate streaming unlabeled observations and share their instantaneous beliefs with neighboring classifiers. We show that the SML strategy enables the agents to learn consistently under this highly-heterogeneous setting and allows the network to continue learning even during the prediction phase when it is deciding on unlabeled samples. The prediction decisions are used to continually improve performance thereafter in a manner that is markedly different from most existing static classification schemes where, following training, the decisions on unlabeled data are not re-used to improve future performance.

Stefan Vlaski, Ali H. Sayed

Adaptive networks have the capability to pursue solutions of global stochastic optimization problems by relying only on local interactions within neighborhoods. The diffusion of information through repeated interactions allows for globally optimal behavior, without the need for central coordination. Most existing strategies are developed for cooperative learning settings, where the objective of the network is common to all agents. We consider in this work a team setting, where a subset of the agents form a team with a common goal while competing with the remainder of the network. We develop an algorithm for decentralized competition among teams of adaptive agents, analyze its dynamics and present an application in the decentralized training of generative adversarial neural networks.

Elsa Rizk, Stefan Vlaski, Ali H. Sayed

Federated learning encapsulates distributed learning strategies that are managed by a central unit. Since it relies on using a selected number of agents at each iteration, and since each agent, in turn, taps into its local data, it is only natural to study optimal sampling policies for selecting agents and their data in federated learning implementations. Usually, only uniform sampling schemes are used. However, in this work, we examine the effect of importance sampling and devise schemes for sampling agents and data non-uniformly guided by a performance measure. We find that in schemes involving sampling without replacement, the performance of the resulting architecture is controlled by two factors related to data variability at each agent, and model variability across agents. We illustrate the theoretical findings with experiments on simulated and real data and show the improvement in performance that results from the proposed strategies.

Stefan Vlaski, Elsa Rizk, Ali H. Sayed

Federated learning is a useful framework for centralized learning from distributed data under practical considerations of heterogeneity, asynchrony, and privacy. Federated architectures are frequently deployed in deep learning settings, which generally give rise to non-convex optimization problems. Nevertheless, most existing analysis are either limited to convex loss functions, or only establish first-order stationarity, despite the fact that saddle-points, which are first-order stationary, are known to pose bottlenecks in deep learning. We draw on recent results on the second-order optimality of stochastic gradient algorithms in centralized and decentralized settings, and establish second-order guarantees for a class of federated learning algorithms.

Elsa Rizk, Stefan Vlaski, Ali H. Sayed

Federated learning involves a mixture of centralized and decentralized processing tasks, where a server regularly selects a sample of the agents and these in turn sample their local data to compute stochastic gradients for their learning updates. This process runs continually. The sampling of both agents and data is generally uniform; however, in this work we consider non-uniform sampling. We derive optimal importance sampling strategies for both agent and data selection and show that non-uniform sampling without replacement improves the performance of the original FedAvg algorithm. We run experiments on a regression and classification problem to illustrate the theoretical results.

Virginia Bordignon, Stefan Vlaski, Vincenzo Matta et al.

This work proposes a new way of combining independently trained classifiers over space and time. Combination over space means that the outputs of spatially distributed classifiers are aggregated. Combination over time means that the classifiers respond to streaming data during testing and continue to improve their performance even during this phase. By doing so, the proposed architecture is able to improve prediction performance over time with unlabeled data. Inspired by social learning algorithms, which require prior knowledge of the observations distribution, we propose a Social Machine Learning (SML) paradigm that is able to exploit the imperfect models generated during the learning phase. We show that this strategy results in consistent learning with high probability, and it yields a robust structure against poorly trained classifiers. Simulations with an ensemble of feedforward neural networks are provided to illustrate the theoretical results.

Stefan Vlaski, Ali H. Sayed

Decentralized algorithms for stochastic optimization and learning rely on the diffusion of information as a result of repeated local exchanges of intermediate estimates. Such structures are particularly appealing in situations where agents may be hesitant to share raw data due to privacy concerns. Nevertheless, in the absence of additional privacy-preserving mechanisms, the exchange of local estimates, which are generated based on private data can allow for the inference of the data itself. The most common mechanism for guaranteeing privacy is the addition of perturbations to local estimates before broadcasting. These perturbations are generally chosen independently at every agent, resulting in a significant performance loss. We propose an alternative scheme, which constructs perturbations according to a particular nullspace condition, allowing them to be invisible (to first order in the step-size) to the network centroid, while preserving privacy guarantees. The analysis allows for general nonconvex loss functions, and is hence applicable to a large number of machine learning and signal processing problems, including deep learning.

Mert Kayaalp, Stefan Vlaski, Ali H. Sayed

The objective of meta-learning is to exploit the knowledge obtained from observed tasks to improve adaptation to unseen tasks. As such, meta-learners are able to generalize better when they are trained with a larger number of observed tasks and with a larger amount of data per task. Given the amount of resources that are needed, it is generally difficult to expect the tasks, their respective data, and the necessary computational capacity to be available at a single central location. It is more natural to encounter situations where these resources are spread across several agents connected by some graph topology. The formalism of meta-learning is actually well-suited to this decentralized setting, where the learner would be able to benefit from information and computational power spread across the agents. Motivated by this observation, in this work, we propose a cooperative fully-decentralized multi-agent meta-learning algorithm, referred to as Diffusion-based MAML or Dif-MAML. Decentralized optimization algorithms are superior to centralized implementations in terms of scalability, avoidance of communication bottlenecks, and privacy guarantees. The work provides a detailed theoretical analysis to show that the proposed strategy allows a collection of agents to attain agreement at a linear rate and to converge to a stationary point of the aggregate MAML objective even in non-convex environments. Simulation results illustrate the theoretical findings and the superior performance relative to the traditional non-cooperative setting.

Stefan Vlaski, Elsa Rizk, Ali H. Sayed

The utilization of online stochastic algorithms is popular in large-scale learning settings due to their ability to compute updates on the fly, without the need to store and process data in large batches. When a constant step-size is used, these algorithms also have the ability to adapt to drifts in problem parameters, such as data or model properties, and track the optimal solution with reasonable accuracy. Building on analogies with the study of adaptive filters, we establish a link between steady-state performance derived under stationarity assumptions and the tracking performance of online learners under random walk models. The link allows us to infer the tracking performance from steady-state expressions directly and almost by inspection.

Stefan Vlaski, Ali H. Sayed

Rapid advances in data collection and processing capabilities have allowed for the use of increasingly complex models that give rise to nonconvex optimization problems. These formulations, however, can be arbitrarily difficult to solve in general, in the sense that even simply verifying that a given point is a local minimum can be NP-hard [1]. Still, some relatively simple algorithms have been shown to lead to surprisingly good empirical results in many contexts of interest. Perhaps the most prominent example is the success of the backpropagation algorithm for training neural networks. Several recent works have pursued rigorous analytical justification for this phenomenon by studying the structure of the nonconvex optimization problems and establishing that simple algorithms, such as gradient descent and its variations, perform well in converging towards local minima and avoiding saddle-points. A key insight in these analyses is that gradient perturbations play a critical role in allowing local descent algorithms to efficiently distinguish desirable from undesirable stationary points and escape from the latter. In this article, we cover recent results on second-order guarantees for stochastic first-order optimization algorithms in centralized, federated, and decentralized architectures.

Elsa Rizk, Stefan Vlaski, Ali H. Sayed

Federated learning has emerged as an umbrella term for centralized coordination strategies in multi-agent environments. While many federated learning architectures process data in an online manner, and are hence adaptive by nature, most performance analyses assume static optimization problems and offer no guarantees in the presence of drifts in the problem solution or data characteristics. We consider a federated learning model where at every iteration, a random subset of available agents perform local updates based on their data. Under a non-stationary random walk model on the true minimizer for the aggregate optimization problem, we establish that the performance of the architecture is determined by three factors, namely, the data variability at each agent, the model variability across all agents, and a tracking term that is inversely proportional to the learning rate of the algorithm. The results clarify the trade-off between convergence and tracking performance.

Roula Nassif, Stefan Vlaski, Cedric Richard et al.

The problem of learning simultaneously several related tasks has received considerable attention in several domains, especially in machine learning with the so-called multitask learning problem or learning to learn problem [1], [2]. Multitask learning is an approach to inductive transfer learning (using what is learned for one problem to assist in another problem) and helps improve generalization performance relative to learning each task separately by using the domain information contained in the training signals of related tasks as an inductive bias. Several strategies have been derived within this community under the assumption that all data are available beforehand at a fusion center. However, recent years have witnessed an increasing ability to collect data in a distributed and streaming manner. This requires the design of new strategies for learning jointly multiple tasks from streaming data over distributed (or networked) systems. This article provides an overview of multitask strategies for learning and adaptation over networks. The working hypothesis for these strategies is that agents are allowed to cooperate with each other in order to learn distinct, though related tasks. The article shows how cooperation steers the network limiting point and how different cooperation rules allow to promote different task relatedness models. It also explains how and when cooperation over multitask networks outperforms non-cooperative strategies.

Stefan Vlaski, Ali H. Sayed

Under appropriate cooperation protocols and parameter choices, fully decentralized solutions for stochastic optimization have been shown to match the performance of centralized solutions and result in linear speedup (in the number of agents) relative to non-cooperative approaches in the strongly-convex setting. More recently, these results have been extended to the pursuit of first-order stationary points in non-convex environments. In this work, we examine in detail the dependence of second-order convergence guarantees on the spectral properties of the combination policy for non-convex multi agent optimization. We establish linear speedup in saddle-point escape time in the number of agents for symmetric combination policies and study the potential for further improvement by employing asymmetric combination weights. The results imply that a linear speedup can be expected in the pursuit of second-order stationary points, which exclude local maxima as well as strict saddle-points and correspond to local or even global minima in many important learning settings.

Stefan Vlaski, Lieven Vandenberghe, Ali H. Sayed

The purpose of this work is to develop and study a distributed strategy for Pareto optimization of an aggregate cost consisting of regularized risks. Each risk is modeled as the expectation of some loss function with unknown probability distribution while the regularizers are assumed deterministic, but are not required to be differentiable or even continuous. The individual, regularized, cost functions are distributed across a strongly-connected network of agents and the Pareto optimal solution is sought by appealing to a multi-agent diffusion strategy. To this end, the regularizers are smoothed by means of infimal convolution and it is shown that the Pareto solution of the approximate, smooth problem can be made arbitrarily close to the solution of the original, non-smooth problem. Performance bounds are established under conditions that are weaker than assumed before in the literature, and hence applicable to a broader class of adaptation and learning problems.

Stefan Vlaski, Ali H. Sayed

Recent years have seen increased interest in performance guarantees of gradient descent algorithms for non-convex optimization. A number of works have uncovered that gradient noise plays a critical role in the ability of gradient descent recursions to efficiently escape saddle-points and reach second-order stationary points. Most available works limit the gradient noise component to be bounded with probability one or sub-Gaussian and leverage concentration inequalities to arrive at high-probability results. We present an alternate approach, relying primarily on mean-square arguments and show that a more relaxed relative bound on the gradient noise variance is sufficient to ensure efficient escape from saddle-points without the need to inject additional noise, employ alternating step-sizes or rely on a global dispersive noise assumption, as long as a gradient noise component is present in a descent direction for every saddle-point.

Stefan Vlaski, Ali H. Sayed

The diffusion strategy for distributed learning from streaming data employs local stochastic gradient updates along with exchange of iterates over neighborhoods. In Part I [2] of this work we established that agents cluster around a network centroid and proceeded to study the dynamics of this point. We established expected descent in non-convex environments in the large-gradient regime and introduced a short-term model to examine the dynamics over finite-time horizons. Using this model, we establish in this work that the diffusion strategy is able to escape from strict saddle-points in O(1/$μ$) iterations; it is also able to return approximately second-order stationary points in a polynomial number of iterations. Relative to prior works on the polynomial escape from saddle-points, most of which focus on centralized perturbed or stochastic gradient descent, our approach requires less restrictive conditions on the gradient noise process.

Bicheng Ying, Kun Yuan, Stefan Vlaski et al.

In empirical risk optimization, it has been observed that stochastic gradient implementations that rely on random reshuffling of the data achieve better performance than implementations that rely on sampling the data uniformly. Recent works have pursued justifications for this behavior by examining the convergence rate of the learning process under diminishing step-sizes. This work focuses on the constant step-size case and strongly convex loss function. In this case, convergence is guaranteed to a small neighborhood of the optimizer albeit at a linear rate. The analysis establishes analytically that random reshuffling outperforms uniform sampling by showing explicitly that iterates approach a smaller neighborhood of size $O(μ^2)$ around the minimizer rather than $O(μ)$. Furthermore, we derive an analytical expression for the steady-state mean-square-error performance of the algorithm, which helps clarify in greater detail the differences between sampling with and without replacement. We also explain the periodic behavior that is observed in random reshuffling implementations.