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1. Correlation Aware Sparsified Mean Estimation Using Random Projection

   Jiang, Shuli ; Sharma, Pranay ; Joshi, Gauri ( December 2023 , Advances in neural information processing systems)

   We study the problem of communication-efficient distributed vector mean estimation, which is a commonly used subroutine in distributed optimization and Federated Learning (FL). Rand-k sparsification is a commonly used technique to reduce communication cost, where each client sends of its coordinates to the server. However, Rand-k is agnostic to any correlations, that might exist between clients in practical scenarios. The recently proposed Rand-k-Spatial estimator leverages the cross-client correlation information at the server to improve Rand-k's performance. Yet, the performance of Rand-k-Spatial is suboptimal, and improving mean estimation is key to faster convergence in distributed optimization. We propose the Rand-Proj-Spatial estimator with a more flexible encoding-decoding procedure, which generalizes the encoding of Rand- by projecting the client vectors to a random k-dimensional subspace. We utilize Subsampled Randomized Hadamard Transform (SRHT) as the projection matrix and show that Rand-Proj-Spatial with SRHT outperforms Rand-k-Spatial, using the correlation information more efficiently. Furthermore, we propose an approach to incorporate varying degrees of correlation and suggest a practical variant of Rand-Proj-Spatial when the correlation information is not available to the server. Finally, experiments on real-world distributed optimization tasks showcase the superior performance of Rand-Proj-Spatial compared to Rand-k-Spatial and other more sophisticated sparsification techniques. 

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2. Byzantine-Resilient High-Dimensional SGD with Local Iterations on Heterogeneous Data

   Data, Deepesh ; Diggavi, Suhas N. ( July 2021 , International Conference on Machine Learning (ICML))

   We study stochastic gradient descent (SGD) with local iterations in the presence of malicious/Byzantine clients, motivated by the federated learning. The clients, instead of communicating with the central server in every iteration, maintain their local models, which they update by taking several SGD iterations based on their own datasets and then communicate the net update with the server, thereby achieving communication-efficiency. Furthermore, only a subset of clients communicate with the server, and this subset may be different at different synchronization times. The Byzantine clients may collaborate and send arbitrary vectors to the server to disrupt the learning process. To combat the adversary, we employ an efficient high-dimensional robust mean estimation algorithm from Steinhardt et al.~i̧te[ITCS 2018]Resilience_SCV18 at the server to filter-out corrupt vectors; and to analyze the outlier-filtering procedure, we develop a novel matrix concentration result that may be of independent interest. We provide convergence analyses for strongly-convex and non-convex smooth objectives in the heterogeneous data setting, where different clients may have different local datasets, and we do not make any probabilistic assumptions on data generation. We believe that ours is the first Byzantine-resilient algorithm and analysis with local iterations. We derive our convergence results under minimal assumptions of bounded variance for SGD and bounded gradient dissimilarity (which captures heterogeneity among local datasets). We also extend our results to the case when clients compute full-batch gradients. 

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3. Byzantine-Resilient High-Dimensional SGD with Local Iterations on Heterogeneous Data

   Data, Deepesh ; Diggavi, Suhas N. ( January 2021 , International Conference on Machine Learning (ICML))

   We study stochastic gradient descent (SGD) with local iterations in the presence of malicious/Byzantine clients, motivated by the federated learning. The clients, instead of communicating with the central server in every iteration, maintain their local models, which they update by taking several SGD iterations based on their own datasets and then communicate the net update with the server, thereby achieving communication-efficiency. Furthermore, only a subset of clients communicate with the server, and this subset may be different at different synchronization times. The Byzantine clients may collaborate and send arbitrary vectors to the server to disrupt the learning process. To combat the adversary, we employ an efficient high-dimensional robust mean estimation algorithm from Steinhardt et al.at the server to filter-out corrupt vectors; and to analyze the outlier-filtering procedure, we develop a novel matrix concentration result that may be of independent interest. We provide convergence analyses for strongly-convex and non-convex smooth objectives in the heterogeneous data setting, where different clients may have different local datasets, and we do not make any probabilistic assumptions on data generation. We believe that ours is the first Byzantine-resilient algorithm and analysis with local iterations. We derive our convergence results under minimal assumptions of bounded variance for SGD and bounded gradient dissimilarity (which captures heterogeneity among local datasets). We also extend our results to the case when clients compute full-batch gradients. 

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4. Tracking the Adjacency Spectral Embedding for Streaming Graphs

   https://doi.org/10.1109/IEEECONF56349.2022.10051861  

   Larroca, Federico ; Bermolen, Paola ; Fiori, Marcelo ; Marenco, Bernardo ; Mateos, Gonzalo ( October 2022 , 2022 56th Asilomar Conference on Signals, Systems, and Computers)

   The popular Random Dot Product Graph (RDPG) generative model postulates that each node has an associated (latent) vector, and the probability of existence of an edge between two nodes is their inner-product (with variants to consider directed and weighted graphs). In any case, the latent vectors may be estimated through a spectral decomposition of the adjacency matrix, the so-called Adjacency Spectral Embedding (ASE). Assume we are monitoring a stream of graphs and the objective is to track the latent vectors. Examples include recommender systems or monitoring of a wireless network. It is clear that performing the ASE of each graph separately may result in a prohibitive computation load. Furthermore, the invariance to rotations of the inner product complicates comparing the latent vectors at different time-steps. By considering the minimization problem underlying ASE, we develop an iterative algorithm that updates the latent vectors' estimation as new graphs from the stream arrive. Differently to other proposals, our method does not accumulate errors and thus does not requires periodically re-computing the spectral decomposition. Furthermore, the pragmatic setting where nodes leave or join the graph (e.g. a new product in the recommender system) can be accommodated as well. Our code is available at https://github.com/marfiori/efficient-ASE 

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5. Learning the Structure of Large Networked Systems Obeying Conservation Laws

   Rayas, A. ; Anguluri, R. ; Dasarathy, G. ( January 2022 , Advances in Neural Information Processing Systems 35 (NeurIPS 2022))

   Many networked systems such as electric networks, the brain, and social networks of opinion dynamics are known to obey conservation laws. Examples of this phenomenon include the Kirchoff laws in electric networks and opinion consensus in social networks. Conservation laws in networked systems are modeled as balance equations of the form X=B*Y, where the sparsity pattern of B*∈R^{p×p} captures the connectivity of the network on p nodes, and Y, X ∈ R^p are vectors of ''potentials'' and ''injected flows'' at the nodes respectively. The node potentials Y cause flows across edges which aim to balance out the potential difference, and the flows X injected at the nodes are extraneous to the network dynamics. In several practical systems, the network structure is often unknown and needs to be estimated from data to facilitate modeling, management, and control. To this end, one has access to samples of the node potentials Y, but only the statistics of the node injections X. Motivated by this important problem, we study the estimation of the sparsity structure of the matrix B* from n samples of Y under the assumption that the node injections X follow a Gaussian distribution with a known covariance Σ_X. We propose a new ℓ1-regularized maximum likelihood estimator for tackling this problem in the high-dimensional regime where the size of the network may be vastly larger than the number of samples n. We show that this optimization problem is convex in the objective and admits a unique solution. Under a new mutual incoherence condition, we establish sufficient conditions on the triple (n,p,d) for which exact sparsity recovery of B* is possible with high probability; d is the degree of the underlying graph. We also establish guarantees for the recovery of B* in the element-wise maximum, Frobenius, and operator norms. Finally, we complement these theoretical results with experimental validation of the performance of the proposed estimator on synthetic and real-world data. 

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