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The Implicit Bias for Adaptive Optimization Algorithms on Homogeneous Neural Networks
Bohan Wang · Qi Meng · Wei Chen · Tie-Yan Liu

Wed Jul 21 05:00 PM -- 05:20 PM (PDT) @ None

Despite their overwhelming capacity to overfit, deep neural networks trained by specific optimization algorithms tend to generalize relatively well to unseen data. Recently, researchers explained it by investigating the implicit bias of optimization algorithms. A remarkable progress is the work (Lyu & Li, 2019), which proves gradient descent (GD) maximizes the margin of homogeneous deep neural networks. Except the first-order optimization algorithms like GD, adaptive algorithms such as AdaGrad, RMSProp and Adam are popular owing to their rapid training process. Mean-while, numerous works have provided empirical evidence that adaptive methods may suffer from poor generalization performance. However, theoretical explanation for the generalization of adaptive optimization algorithms is still lacking. In this paper, we study the implicit bias of adaptive optimization algorithms on homogeneous neural networks. In particular, we study the convergent direction of parameters when they are optimizing the logistic loss. We prove that the convergent direction of Adam and RMSProp is the same as GD, while for AdaGrad, the convergent direction depends on the adaptive conditioner. Technically, we provide a unified framework to analyze convergent direction of adaptive optimization algorithms by constructing novel and nontrivial adaptive gradient flow and surrogate margin. The theoretical findings explain the superiority on generalization of exponential moving average strategy that is adopted by RMSProp and Adam. To the best of knowledge, it is the first work to study the convergent direction of adaptive optimizations on non-linear deep neural networks

Author Information

Bohan Wang (Microsoft Research Asia)
Qi Meng (Microsoft)
Wei Chen (Microsoft Research)
Tie-Yan Liu (Microsoft Research Asia)

Tie-Yan Liu is a principal researcher of Microsoft Research Asia, leading the research on artificial intelligence and machine learning. He is very well known for his pioneer work on learning to rank and computational advertising, and his recent research interests include deep learning, reinforcement learning, and distributed machine learning. Many of his technologies have been transferred to Microsoft’s products and online services (such as Bing, Microsoft Advertising, and Azure), and open-sourced through Microsoft Cognitive Toolkit (CNTK), Microsoft Distributed Machine Learning Toolkit (DMTK), and Microsoft Graph Engine. On the other hand, he has been actively contributing to academic communities. He is an adjunct/honorary professor at Carnegie Mellon University (CMU), University of Nottingham, and several other universities in China. His papers have been cited for tens of thousands of times in refereed conferences and journals. He has won quite a few awards, including the best student paper award at SIGIR (2008), the most cited paper award at Journal of Visual Communications and Image Representation (2004-2006), the research break-through award (2012) and research-team-of-the-year award (2017) at Microsoft Research, and Top-10 Springer Computer Science books by Chinese authors (2015), and the most cited Chinese researcher by Elsevier (2017). He has been invited to serve as general chair, program committee chair, local chair, or area chair for a dozen of top conferences including SIGIR, WWW, KDD, ICML, NIPS, IJCAI, AAAI, ACL, ICTIR, as well as associate editor of ACM Transactions on Information Systems, ACM Transactions on the Web, and Neurocomputing. Tie-Yan Liu is a fellow of the IEEE, a distinguished member of the ACM, and a vice chair of the CIPS information retrieval technical committee.

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