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1. Byzantine-Resilient SGD in High Dimensions on Heterogeneous Data

   https://doi.org/10.1109/ISIT45174.2021.9518248  

   Data, Deepesh ; Diggavi, Suhas ( July 2021 , IEEE International Symposium on Information Theory (ISIT))

   We study distributed stochastic gradient descent (SGD) in the master-worker architecture under Byzantine attacks. We consider the heterogeneous data model, where different workers may have different local datasets, and we do not make any probabilistic assumptions on data generation. At the core of our algorithm, we use the polynomial-time outlier-filtering procedure for robust mean estimation proposed by Steinhardt et al. (ITCS 2018) to filter-out corrupt gradients. In order to be able to apply their filtering procedure in our heterogeneous data setting where workers compute stochastic gradients, we derive a new matrix concentration result, which may be of independent interest. We provide convergence analyses for smooth strongly-convex and non-convex objectives and show that our convergence rates match that of vanilla SGD in the Byzantine-free setting. In order to bound the heterogeneity, we assume that the gradients at different workers have bounded deviation from each other, and we also provide concrete bounds on this deviation in the statistical heterogeneous data model. 

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2. On Byzantine-Resilient High-Dimensional Stochastic Gradient Descent

   https://doi.org/10.1109/ISIT44484.2020.9174363  

   Data, Deepesh ; Diggavi, Suhas ( June 2020 , IEEE International Symposium on Information Theory (ISIT))

   We study stochastic gradient descent (SGD) in the master-worker architecture under Byzantine attacks. Building upon the recent advances in algorithmic high-dimensional robust statistics, in each SGD iteration, master employs a non-trivial decoding to estimate the true gradient from the unbiased stochastic gradients received from workers, some of which may be corrupt. We provide convergence analyses for both strongly-convex and non-convex smooth objectives under standard SGD assumptions. We can control the approximation error of our solution in both these settings by the mini-batch size of stochastic gradients; and we can make the approximation error as small as we want, provided that workers use a sufficiently large mini-batch size. Our algorithm can tolerate less than 1/3 fraction of Byzantine workers. It can approximately find the optimal parameters in the strongly-convex setting exponentially fast, and reaches to an approximate stationary point in the non-convex setting with linear speed, i.e., with a rate of 1/T, thus, matching the convergence rates of vanilla SGD in the Byzantine-free setting. 

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3. 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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4. 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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5. A High Probability Analysis of Adaptive SGD with Momentum

   Li, Xiaoyu ; Orabona, Francesco ( January 2020 , Workshop on Beyond First Order Methods in ML Systems at ICML'20)

   null (Ed.)

   Stochastic Gradient Descent (SGD) and its variants are the most used algorithms in machine learning applications. In particular, SGD with adaptive learning rates and momentum is the industry standard to train deep networks. Despite the enormous success of these methods, our theoretical understanding of these variants in the non-convex setting is not complete, with most of the results only proving convergence in expectation and with strong assumptions on the stochastic gradients. In this paper, we present a high probability analysis for adaptive and momentum algorithms, under weak assumptions on the function, stochastic gradients, and learning rates. We use it to prove for the first time the convergence of the gradients to zero in high probability in the smooth nonconvex setting for Delayed AdaGrad with momentum. 

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