Better depth-width trade-offs for neural networks through the lens of dynamical systems

Vaggos Chatziafratis, Sai Ganesh Nagarajan, Ioannis Panageas
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:1469-1478, 2020.

Abstract

The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory. Recently, depth separation results for ReLU networks were obtained via a new connection with dynamical systems, using a generalized notion of fixed points of a continuous map $f$, called periodic points. In this work, we strengthen the connection with dynamical systems and we improve the existing width lower bounds along several aspects. Our first main result is period-specific width lower bounds that hold under the stronger notion of $L^1$-approximation error, instead of the weaker classification error. Our second contribution is that we provide sharper width lower bounds, still yielding meaningful exponential depth-width separations, in regimes where previous results wouldn’t apply. A byproduct of our results is that there exists a universal constant characterizing the depth-width trade-offs, as long as $f$ has odd periods. Technically, our results follow by unveiling a tighter connection between the following three quantities of a given function: its period, its Lipschitz constant and the growth rate of the number of oscillations arising under compositions of the function $f$ with itself.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-chatziafratis20a, title = {Better depth-width trade-offs for neural networks through the lens of dynamical systems}, author = {Chatziafratis, Vaggos and Nagarajan, Sai Ganesh and Panageas, Ioannis}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {1469--1478}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/chatziafratis20a/chatziafratis20a.pdf}, url = {https://proceedings.mlr.press/v119/chatziafratis20a.html}, abstract = {The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory. Recently, depth separation results for ReLU networks were obtained via a new connection with dynamical systems, using a generalized notion of fixed points of a continuous map $f$, called periodic points. In this work, we strengthen the connection with dynamical systems and we improve the existing width lower bounds along several aspects. Our first main result is period-specific width lower bounds that hold under the stronger notion of $L^1$-approximation error, instead of the weaker classification error. Our second contribution is that we provide sharper width lower bounds, still yielding meaningful exponential depth-width separations, in regimes where previous results wouldn’t apply. A byproduct of our results is that there exists a universal constant characterizing the depth-width trade-offs, as long as $f$ has odd periods. Technically, our results follow by unveiling a tighter connection between the following three quantities of a given function: its period, its Lipschitz constant and the growth rate of the number of oscillations arising under compositions of the function $f$ with itself.} }
Endnote
%0 Conference Paper %T Better depth-width trade-offs for neural networks through the lens of dynamical systems %A Vaggos Chatziafratis %A Sai Ganesh Nagarajan %A Ioannis Panageas %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-chatziafratis20a %I PMLR %P 1469--1478 %U https://proceedings.mlr.press/v119/chatziafratis20a.html %V 119 %X The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory. Recently, depth separation results for ReLU networks were obtained via a new connection with dynamical systems, using a generalized notion of fixed points of a continuous map $f$, called periodic points. In this work, we strengthen the connection with dynamical systems and we improve the existing width lower bounds along several aspects. Our first main result is period-specific width lower bounds that hold under the stronger notion of $L^1$-approximation error, instead of the weaker classification error. Our second contribution is that we provide sharper width lower bounds, still yielding meaningful exponential depth-width separations, in regimes where previous results wouldn’t apply. A byproduct of our results is that there exists a universal constant characterizing the depth-width trade-offs, as long as $f$ has odd periods. Technically, our results follow by unveiling a tighter connection between the following three quantities of a given function: its period, its Lipschitz constant and the growth rate of the number of oscillations arising under compositions of the function $f$ with itself.
APA
Chatziafratis, V., Nagarajan, S.G. & Panageas, I.. (2020). Better depth-width trade-offs for neural networks through the lens of dynamical systems. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:1469-1478 Available from https://proceedings.mlr.press/v119/chatziafratis20a.html.

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