Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets

Pulkit Gopalani; Samyak Jha; Anirbit Mukherjee

Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets

Pulkit Gopalani, Samyak Jha, Anirbit Mukherjee

Machine Learning and Robotics

Indian Institute of Technology - Mumbai

Research output: Contribution to journal › Article › peer-review

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Abstract

In this note, we demonstrate a first-of-its-kind provable convergence of SGD to the global minima of appropriately regularized logistic empirical risk of depth 2 nets -- for arbitrary data and with any number of gates with adequately smooth and bounded activations like sigmoid and tanh. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence of Frobenius norm regularized logistic loss functions on constant-sized neural nets which are "Villani functions" and thus be able to build on recent progress with analyzing SGD on such objectives.

Original language	Undefined
Number of pages	21
Journal	Transactions on Machine Learning Research
Publication status	Published - 25 Feb 2024

Keywords

stochastic differential equations
deep-learning
stochastic gradient descent
stochastic optimization

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Institute for Data Science and AI

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TMLR-1589Final published version, 709 KB

https://openreview.net/pdf?id=9TqAUYB6tC

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MCAIF: Centre for AI Fundamentals
Kaski, S. (PI), Alvarez, M. (Researcher), Pan, W. (Researcher), Mu, T. (Researcher), Rivasplata, O. (PI), Sun, M. (Researcher), Mukherjee, A. (Researcher), Caprio, M. (PI), Sonee, A. (Researcher), Leroy, A. (Researcher), Wang, J. (Researcher), Lee, J. (Researcher), Parakkal Unni, M. (Researcher), Sloman, S. (Researcher), Menary, S. (Researcher), Quilter, T. (Researcher), Hosseinzadeh, A. (PGR student), Mousa, A. (PGR student), Glover, E. (PGR student), Das, A. (PGR student), DURSUN, F. (PGR student), Zhu, H. (PGR student), Abdi, H. (PGR student), Dandago, K. (PGR student), Piriyajitakonkij, M. (PGR student), Rachman, R. (PGR student), Shi, X. (PGR student), Keany, T. (PGR student), Liu, X. (PGR student), Jiang, Y. (PGR student), Wan, Z. (PGR student), Evans, I. (Support team), Harrison, M. (Support team) & Machado, M. (PI)
1/10/21 → 30/09/26
Project: Research

2 Article
1 Paper
1 Preprint

Depth-2 neural networks under a data-poisoning attack
Karmakar, S., Mukherjee, A. & Papamarkou, T., 21 Feb 2023, (E-pub ahead of print) In: Neurocomputing.
Research output: Contribution to journal › Article › peer-review

Open Access
Global Convergence of SGD On Two Layer Neural Nets
Mukherjee, A. & Gopalani, P., 2022, 23 p.
Research output: Preprint/Working paper › Preprint
Provable training of a ReLU gate with an iterative non-gradient algorithm
Mukherjee, A. & Karmakar, S., 4 Apr 2022, In: Neural Networks. 151, p. 264-275 12 p.
Research output: Contribution to journal › Article › peer-review

Open Access

Cite this

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title = "Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets",

abstract = "In this note, we demonstrate a first-of-its-kind provable convergence of SGD to the global minima of appropriately regularized logistic empirical risk of depth 2 nets -- for arbitrary data and with any number of gates with adequately smooth and bounded activations like sigmoid and tanh. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence of Frobenius norm regularized logistic loss functions on constant-sized neural nets which are {"}Villani functions{"} and thus be able to build on recent progress with analyzing SGD on such objectives.",

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N2 - In this note, we demonstrate a first-of-its-kind provable convergence of SGD to the global minima of appropriately regularized logistic empirical risk of depth 2 nets -- for arbitrary data and with any number of gates with adequately smooth and bounded activations like sigmoid and tanh. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence of Frobenius norm regularized logistic loss functions on constant-sized neural nets which are "Villani functions" and thus be able to build on recent progress with analyzing SGD on such objectives.

AB - In this note, we demonstrate a first-of-its-kind provable convergence of SGD to the global minima of appropriately regularized logistic empirical risk of depth 2 nets -- for arbitrary data and with any number of gates with adequately smooth and bounded activations like sigmoid and tanh. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence of Frobenius norm regularized logistic loss functions on constant-sized neural nets which are "Villani functions" and thus be able to build on recent progress with analyzing SGD on such objectives.

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ER -

Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets

Abstract

Keywords

Research Beacons, Institutes and Platforms

Access to Document

Projects

MCAIF: Centre for AI Fundamentals

Research output

Depth-2 neural networks under a data-poisoning attack

Global Convergence of SGD On Two Layer Neural Nets

Provable training of a ReLU gate with an iterative non-gradient algorithm

Cite this