Why Transformers Need Adam: A Hessian Perspective

SGD performs worse than Adam by a significant margin on Transformers, but the reason remains unclear. In this work, we provide an explanation of SGD's failure on Transformers through the lens of Hessian: (i) Transformers are ``heterogeneous'': the Hessian spectrum across parameter blocks vary dramatically, a phenomenon we call ``block heterogeneity"; (ii) Heterogeneity hampers SGD: SGD performs badly on problems with block heterogeneity. To validate that heterogeneity hampers SGD, we check various Transformers, CNNs, MLPs, and quadratic problems, and find that SGD works well on problems without block heterogeneity but performs badly when the heterogeneity exists. Our initial theoretical analysis indicates that SGD fails because it applies one single learning rate for all blocks, which cannot handle the heterogeneity among blocks. The failure could be rescued if we could assign different learning rates across blocks, as designed in Adam.

Federated Learning with Lossy Distributed Source Coding: Analysis and Optimization

Recently, federated learning (FL), which replaces data sharing with model sharing, has emerged as an efficient and privacy-friendly paradigm for machine learning (ML). A main challenge of FL is its huge uplink communication cost. In this paper, we tackle this challenge from an information-theoretic perspective. Specifically, we put forth a distributed source coding (DSC) framework for FL uplink, which unifies the encoding, transmission, and aggregation of the local updates as a lossy DSC problem, thus providing a systematic way to exploit the correlation between local updates to improve the uplink efficiency. Under this DSC-FL framework, we propose an FL uplink scheme based on the modified Berger-Tung coding (MBTC), which supports separate encoding and joint decoding by modifying the achievability scheme of the Berger-Tung inner bound. The achievable region of the MBTC-based uplink scheme is also derived. To unleash the potential of the MBTC-based FL scheme, we carry out a convergence analysis and then formulate a convergence rate maximization problem to optimize the parameters of MBTC. To solve this problem, we develop two algorithms, respectively for small- and large-scale FL systems, based on the majorization-minimization (MM) technique. Numerical results demonstrate the superiority of the MBTC-based FL scheme in terms of aggregation distortion, convergence performance, and communication cost, revealing the great potential of the DSC-FL framework.

The Global Landscape of Neural Networks: An Overview

One of the major concerns for neural network training is that the non-convexity of the associated loss functions may cause bad landscape. The recent success of neural networks suggests that their loss landscape is not too bad, but what specific results do we know about the landscape? In this article, we review recent findings and results on the global landscape of neural networks. First, we point out that wide neural nets may have sub-optimal local minima under certain assumptions. Second, we discuss a few rigorous results on the geometric properties of wide networks such as "no bad basin", and some modifications that eliminate sub-optimal local minima and/or decreasing paths to infinity. Third, we discuss visualization and empirical explorations of the landscape for practical neural nets. Finally, we briefly discuss some convergence results and their relation to landscape results.

Sub-Optimal Local Minima Exist for Almost All Over-parameterized Neural Networks

Does over-parameterization eliminate sub-optimal local minima for neural network problems? On one hand, existing positive results do not prove the claim, but often weaker claims. On the other hand, existing negative results have strong assumptions on the activation functions and/or data samples, causing a large gap with positive results. It was unclear before whether there is a clean answer of "yes" or "no". In this paper, we answer this question with a strong negative result. In particular, we prove that for deep and over-parameterized networks, sub-optimal local minima exist for generic input data samples and generic nonlinear activation. This is the setting widely studied in the global landscape of over-parameterized networks, thus our result corrects a possible misconception that "over-parameterization eliminates sub-optimal local-min". Our construction is based on fundamental optimization analysis, and thus rather principled.

Over-Parameterized Deep Neural Networks Have No Strict Local Minima For Any Continuous Activations

In this paper, we study the loss surface of the over-parameterized fully connected deep neural networks. We prove that for any continuous activation functions, the loss function has no bad strict local minimum, both in the regular sense and in the sense of sets. This result holds for any convex and continuous loss function, and the data samples are only required to be distinct in at least one dimension. Furthermore, we show that bad local minima do exist for a class of activation functions.