More Like this

1. FL-NTK: A Neural Tangent Kernel-based Framework for Federated Learning Convergence Analysis

   Huang, Baihe ; Li, Xiaoxiao ; Song, Zhao ; Yang, Xin ( May 2021 , ArXivorg)

   null (Ed.)

   Federated Learning (FL) is an emerging learning scheme that allows different distributed clients to train deep neural networks together without data sharing. Neural networks have become popular due to their unprecedented success. To the best of our knowledge, the theoretical guarantees of FL concerning neural networks with explicit forms and multi-step updates are unexplored. Nevertheless, training analysis of neural networks in FL is non-trivial for two reasons: first, the objective loss function we are optimizing is non-smooth and non-convex, and second, we are even not updating in the gradient direction. Existing convergence results for gradient descent-based methods heavily rely on the fact that the gradient direction is used for updating. This paper presents a new class of convergence analysis for FL, Federated Learning Neural Tangent Kernel (FL-NTK), which corresponds to over-paramterized ReLU neural networks trained by gradient descent in FL and is inspired by the analysis in Neural Tangent Kernel (NTK). Theoretically, FL-NTK converges to a global-optimal solution at a linear rate with properly tuned learning parameters. Furthermore, with proper distributional assumptions, FL-NTK can also achieve good generalization. 

   more » « less
2. FL-NTK: A Neural Tangent Kernel-based Framework for Federated Learning Analysis

   Huang, Baihe ; Li, Xiaoxiao ; Song, Zhao ; Yang, Xin ( July 2021 , Proceedings of Machine Learning Research)

   Meila, Marina ; Zhang, Tong (Ed.)

   Federated Learning (FL) is an emerging learning scheme that allows different distributed clients to train deep neural networks together without data sharing. Neural networks have become popular due to their unprecedented success. To the best of our knowledge, the theoretical guarantees of FL concerning neural networks with explicit forms and multi-step updates are unexplored. Nevertheless, training analysis of neural networks in FL is non-trivial for two reasons: first, the objective loss function we are optimizing is non-smooth and non-convex, and second, we are even not updating in the gradient direction. Existing convergence results for gradient descent-based methods heavily rely on the fact that the gradient direction is used for updating. The current paper presents a new class of convergence analysis for FL, Federated Neural Tangent Kernel (FL-NTK), which corresponds to overparamterized ReLU neural networks trained by gradient descent in FL and is inspired by the analysis in Neural Tangent Kernel (NTK). Theoretically, FL-NTK converges to a global-optimal solution at a linear rate with properly tuned learning parameters. Furthermore, with proper distributional assumptions, FL-NTK can also achieve good generalization. The proposed theoretical analysis scheme can be generalized to more complex neural networks. 

   more » « less
3. Diving into the shallows: a computational perspective on large-scale shallow learning

   Ma, Siyuan ; Belkin, Mikhail ( October 2017 , Advances in neural information processing systems)

   Remarkable recent success of deep neural networks has not been easy to analyze theoretically. It has been particularly hard to disentangle relative significance of architecture and optimization in achieving accurate classification on large datasets. On the flip side, shallow methods (such as kernel methods) have encountered obstacles in scaling to large data, despite excellent performance on smaller datasets, and extensive theoretical analysis. Practical methods, such as variants of gradient descent used so successfully in deep learning, seem to perform below par when applied to kernel methods. This difficulty has sometimes been attributed to the limitations of shallow architecture. In this paper we identify a basic limitation in gradient descent-based optimization methods when used in conjunctions with smooth kernels. Our analysis demonstrates that only a vanishingly small fraction of the function space is reachable after a polynomial number of gradient descent iterations. That drastically limits the approximating power of gradient descent leading to over-regularization. The issue is purely algorithmic, persisting even in the limit of infinite data. To address this shortcoming in practice, we introduce EigenPro iteration, a simple and direct preconditioning scheme using a small number of approximately computed eigenvectors. It can also be viewed as learning a kernel optimized for gradient descent. Injecting this small, computationally inexpensive and SGD-compatible, amount of approximate second-order information leads to major improvements in convergence. For large data, this leads to a significant performance boost over the state-of-the-art kernel methods. In particular, we are able to match or improve the results reported in the literature at a small fraction of their computational budget. 

   more » « less
4. Superpolynomial Lower Bounds for Learning One-Layer Neural Networks using Gradient Descent

   Goel, Surbhi ; Gollakota, Aravind ; Jin, Zhihan ; Karmalkar, Sushrut ; Klivans, Adam ( July 2020 , International Conference on Machine Learning)

   We prove the first superpolynomial lower bounds for learning one-layer neural networks with respect to the Gaussian distribution using gradient descent. We show that any classifier trained using gradient descent with respect to square-loss will fail to achieve small test error in polynomial time given access to samples labeled by a one-layer neural network. For classification, we give a stronger result, namely that any statistical query (SQ) algorithm (including gradient descent) will fail to achieve small test error in polynomial time. Prior work held only for gradient descent run with small batch sizes, required sharp activations, and applied to specific classes of queries. Our lower bounds hold for broad classes of activations including ReLU and sigmoid. The core of our result relies on a novel construction of a simple family of neural networks that are exactly orthogonal with respect to all spherically symmetric distributions. 

   more » « less
5. Taming the Noisy Gradient: Train Deep Neural Networks with Small Batch Sizes

   https://doi.org/10.24963/ijcai.2019/604  

   Zhang, Yikai ; Qu, Hui ; Chen, Chao ; Metaxas, Dimitris ( August 2019 , The Twenty-Eighth International Joint Conference on Artificial Intelligence (IJCAI))

   Deep learning architectures are usually proposed with millions of parameters, resulting in a memory issue when training deep neural networks with stochastic gradient descent type methods using large batch sizes. However, training with small batch sizes tends to produce low quality solution due to the large variance of stochastic gradients. In this paper, we tackle this problem by proposing a new framework for training deep neural network with small batches/noisy gradient. During optimization, our method iteratively applies a proximal type regularizer to make loss function strongly convex. Such regularizer stablizes the gradient, leading to better training performance. We prove that our algorithm achieves comparable convergence rate as vanilla SGD even with small batch size. Our framework is simple to implement and can be potentially combined with many existing optimization algorithms. Empirical results show that our method outperforms SGD and Adam when batch size is small. Our implementation is available at https://github.com/huiqu18/TRAlgorithm.

    

   more » « less