# Averaging Is Probably Not the Optimum Way of Aggregating Parameters in Federated Learning

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- All participating clients download the latest global model parameter vector ${w}^{,}$.
- (2)
- Each client improves downloaded model based on their local data using, e.g., stochastic gradient descent (SGD) [13] with a fixed learning rate $\eta $: ${w}_{q}={w}^{,}-\eta {g}_{q}$, where ${g}_{q}=\u25bd{F}_{q}\left({w}^{,}\right)$, and ${F}_{q}\left(w\right)$ is the local objective function for the qth device.
- (3)
- Each improved model parameter ${w}_{q}$ is sent from the client to the server.
- (4)
- The server aggregates all updated parameters to construct an enhanced global model ${w}^{*}$.

## 2. Related Work

## 3. Estimating the Mutual Information

#### 3.1. MI Derivative Formula under Multi-D Gaussian

#### 3.2. KNN Discretization Estimator

## 4. Simulation Results

#### 4.1. RNN on Signal Variation Prediction

#### 4.2. CNN on Digit Recognition

#### 4.3. Distance Metrics

**Proposition**

**1.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Poushter, J. Smartphone ownership and internet usage continues to climb in emerging economies. Pew Res. Center
**2016**, 22, 1–44. [Google Scholar] - Haghi, M.; Thurow, K.; Stoll, R. Wearable devices in medical internet of things: Scientific research and commercially available devices. Healthc. Inform. Res.
**2017**, 23, 4–15. [Google Scholar] [CrossRef] [PubMed] - Taylor, R.; Baron, D.; Schmidt, D. The world in 2025-predictions for the next ten years. In Proceedings of the 2015 10th International Microsystems, Packaging, Assembly and Circuits Technology Conference (IMPACT), Taipei, Taiwan, 21–23 October 2015; pp. 192–195. [Google Scholar]
- Bonomi, F.; Milito, R.; Zhu, J.; Addepalli, S. Fog computing and its role in the internet of things. In Proceedings of the First Edition of the MCC Workshop on Mobile Cloud Computing, Helsinki, Finland, 17 August 2012; pp. 13–16. [Google Scholar]
- Garcia Lopez, P.; Montresor, A.; Epema, D.; Datta, A.; Higashino, T.; Iamnitchi, A.; Barcellos, M.; Felber, P.; Riviere, E. Edge-centric computing: Vision and challenges. ACM SIGCOMM Comput. Commun. Rev.
**2015**, 45, 37–42. [Google Scholar] [CrossRef] - House, W. Consumer Data Privacy in a Networked World: A Framework for Protecting Privacy and Promoting Innovation in the Global Digital Economy; White House: Washington, DC, USA, 2012; pp. 1–62. [Google Scholar]
- McMahan, H.B.; Moore, E.; Ramage, D.; Hampson, S. Communication-efficient learning of deep networks from decentralized data. arXiv
**2016**, arXiv:1602.05629. [Google Scholar] - Dean, J.; Corrado, G.; Monga, R.; Chen, K.; Devin, M.; Mao, M.; Ranzato, M.; Senior, A.; Tucker, P.; Yang, K. Large scale distributed deep networks. In Proceedings of the Advances in Neural Information Processing Systems, Lake Tahoe, NV, USA, 3–6 December 2012; pp. 1223–1231. [Google Scholar]
- Hard, A.; Rao, K.; Mathews, R.; Ramaswamy, S.; Beaufays, F.; Augenstein, S.; Eichner, H.; Kiddon, C.; Ramage, D. Federated learning for mobile keyboard prediction. arXiv
**2018**, arXiv:1811.03604. [Google Scholar] - Ramaswamy, S.; Mathews, R.; Rao, K.; Beaufays, F. Federated learning for emoji prediction in a mobile keyboard. arXiv
**2019**, arXiv:1906.04329. [Google Scholar] - Huang, L.; Yin, Y.; Fu, Z.; Zhang, S.; Deng, H.; Liu, D. LoAdaBoost: Loss-Based AdaBoost Federated Machine Learning on medical Data. arXiv
**2018**, arXiv:1811.12629. [Google Scholar] - Samarakoon, S.; Bennis, M.; Saad, W.; Debbah, M. Federated learning for ultra-reliable low-latency V2V communications. In Proceedings of the 2018 IEEE Global Communications Conference (GLOBECOM), Abu Dhabi, UAE, 9–13 December 2018; pp. 1–7. [Google Scholar]
- Bottou, L. Large-scale machine learning with stochastic gradient descent. In Proceedings of the 19th International Conference on Computational Statistics, Paris, France, 22–27 August 2010; pp. 177–186. [Google Scholar]
- Sattler, F.; Wiedemann, S.; Müller, K.R.; Samek, W. Robust and communication-efficient federated learning from non-iid data. IEEE Trans. Neural Netw. Learn. Syst.
**2019**. [Google Scholar] [CrossRef] [PubMed][Green Version] - McMahan, H.B.; Ramage, D.; Talwar, K.; Zhang, L. Learning differentially private recurrent language models. In Proceedings of the International Conference on Learning Representations, Vancouver, BC, Canada, 30 April–3 May 2018. [Google Scholar]
- Agarwal, N.; Suresh, A.T.; Yu, F.X.X.; Kumar, S.; McMahan, B. cpSGD: Communication-efficient and differentially- private distributed SGD. In Proceedings of the Advances in Neural Information Processing Systems, Montreal, QC, Canada, 3–8 December 2018; pp. 7564–7575. [Google Scholar]
- He, L.; Bian, A.; Jaggi, M. Cola: Decentralized linear learning. In Proceedings of the Advances in Neural Information Processing Systems, Montreal, QC, Canada, 3–8 December 2018; pp. 4536–4546. [Google Scholar]
- Woodworth, B.E.; Wang, J.; Smith, A.; McMahan, B.; Srebro, N. Graph oracle models, lower bounds, and gaps for parallel stochastic optimization. In Proceedings of the Advances in Neural Information Processing Systems, Montreal, QC, Canada, 3–8 December 2018; pp. 8496–8506. [Google Scholar]
- Eichner, H.; Koren, T.; Mcmahan, B.; Srebro, N.; Talwar, K. Semi-Cyclic Stochastic Gradient Descent. In Proceedings of the International Conference on Machine Learning, Long Beach, CA, USA, 10–15 June 2019; pp. 1764–1773. [Google Scholar]
- Cover, T.M.; Thomas, J.A. Elements of Information Theory; John Wiley & Sons: New York, NY, USA, 2012. [Google Scholar]
- Kraskov, A.; Stögbauer, H.; Grassberger, P. Estimating mutual information. Phys. Rev. E
**2004**, 69, 066138. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ross, B.C. Mutual information between discrete and continuous data sets. PLoS ONE
**2014**, 9, e87357. [Google Scholar] [CrossRef] [PubMed] - Krizhevsky, A.; Sutskever, I.; Hinton, G.E. Imagenet classification with deep convolutional neural networks. In Proceedings of the Advances in Neural Information Processing Systems, Lake Tahoe, NV, USA, 3–6 December 2012; pp. 1097–1105. [Google Scholar]
- Mikolov, T.; Karafiát, M.; Burget, L.; Černockỳ, J.; Khudanpur, S. Recurrent neural network based language model. In Proceedings of the Eleventh Annual Conference of the International Speech Communication Association, Chiba, Japan, 26–30 September 2010. [Google Scholar]
- Danielsson, P.E. Euclidean distance mapping. Comput. Gr. Image Process.
**1980**, 14, 227–248. [Google Scholar] [CrossRef][Green Version] - Krause, E.F. Taxicab Geometry: An Adventure in Non-Euclidean Geometry; Courier Corporation: Chelmsford, MA, USA, 1986. [Google Scholar]
- Cantrell, C.D. Modern Mathematical Methods for Physicists and Engineers; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Tishby, N.; Zaslavsky, N. Deep learning and the information bottleneck principle. In Proceedings of the 2015 IEEE Information Theory Workshop (ITW), Jerusalem, Israel, 26 April–1 May 2015; pp. 1–5. [Google Scholar]
- Shwartz-Ziv, R.; Tishby, N. Opening the black box of deep neural networks via information. arXiv
**2017**, arXiv:1703.00810. [Google Scholar] - Saxe, A.M.; Bansal, Y.; Dapello, J.; Advani, M.; Kolchinsky, A.; Tracey, B.D.; Cox, D.D. On the information bottleneck theory of deep learning. In Proceedings of the International Conference on Learning Representations, Vancouver, BC, Canada, 30 April–3 May 2018. [Google Scholar]
- Noshad, M.; Zeng, Y.; Hero, A.O. Scalable mutual information estimation using dependence graphs. In Proceedings of the ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK, 12–17 May 2019; pp. 2962–2966. [Google Scholar]
- Kolchinsky, A.; Tracey, B.D.; Wolpert, D.H. Nonlinear information bottleneck. Entropy
**2019**, 21, 1181. [Google Scholar] [CrossRef][Green Version] - Wickstrøm, K.; Løkse, S.; Kampffmeyer, M.; Yu, S.; Principe, J.; Jenssen, R. Information Plane Analysis of Deep Neural Networks via Matrix-Based Renyi’s Entropy and Tensor Kernels. arXiv
**2019**, arXiv:1909.11396. [Google Scholar] - Adilova, L.; Rosenzweig, J.; Kamp, M. Information-Theoretic Perspective of Federated Learning. arXiv
**2019**, arXiv:1911.07652. [Google Scholar] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef][Green Version] - Deng, L. The MNIST database of handwritten digit images for machine learning research [best of the web]. IEEE Signal Process. Mag.
**2012**, 29, 141–142. [Google Scholar] [CrossRef] - Kingma, D.P.; Ba, J. Adam: A method for stochastic optimization. arXiv
**2014**, arXiv:1412.6980. [Google Scholar] - Gal, Y.; Ghahramani, Z. Dropout as a bayesian approximation: Representing model uncertainty in deep learning. Proceedings of The International Conference on Machine Learning, New York, NY, USA, 19–24 June 2016; pp. 1050–1059. [Google Scholar]
- LeCun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. Gradient-based learning applied to document recognition. Proc. IEEE
**1998**, 86, 2278–2324. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**(

**a**) The MI in the signal variation prediction task calculated by the first method (Section 3.1). (

**b**) The MI in the signal variation prediction task calculated by the second method (Section 3.2). (

**c**) The MSE variation with the training iterations.

**Figure 2.**(

**a**) The MI in the digit recognition task calculated by the first method (Section 3.1). (

**b**) The MI in the digit recognition task calculated by the second method (Section 3.2). (

**c**) The accuracy variation with the training iterations.

**Figure 3.**The distance metrics in the signal variation prediction task (

**a**) Euclidean distance. (

**b**) Manhattan distance. (

**c**) Chebyshev distance.

**Figure 4.**The distance metrics in the digit recognition task (

**a**) Euclidean distance. (

**b**) Manhattan distance. (

**c**) Chebyshev distance.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xiao, P.; Cheng, S.; Stankovic, V.; Vukobratovic, D.
Averaging Is Probably Not the Optimum Way of Aggregating Parameters in Federated Learning. *Entropy* **2020**, *22*, 314.
https://doi.org/10.3390/e22030314

**AMA Style**

Xiao P, Cheng S, Stankovic V, Vukobratovic D.
Averaging Is Probably Not the Optimum Way of Aggregating Parameters in Federated Learning. *Entropy*. 2020; 22(3):314.
https://doi.org/10.3390/e22030314

**Chicago/Turabian Style**

Xiao, Peng, Samuel Cheng, Vladimir Stankovic, and Dejan Vukobratovic.
2020. "Averaging Is Probably Not the Optimum Way of Aggregating Parameters in Federated Learning" *Entropy* 22, no. 3: 314.
https://doi.org/10.3390/e22030314