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Better depth-width trade-offs for neural networks through the lens of dynamical systems
Evangelos Chatziafratis · Sai Ganesh Nagarajan · Ioannis Panageas

Thu Jul 16 07:00 AM -- 07:45 AM & Thu Jul 16 08:00 PM -- 08:45 PM (PDT) @ Virtual #None
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.

Author Information

Vaggos Chatziafratis (Stanford University)

I am a 3rd year PhD student at Stanford University working with Prof. Tim Roughgarden, Moses Charikar and Jan Vondrak. My research interests are in Algorithms and Learning Theory.

Sai Ganesh Nagarajan (SUTD)
Ioannis Panageas (Singapore University of Technology and Design)

I am an Assistant Professor in Information Systems at SUTD. Before that I was a MIT Postdoctoral Fellow working with Costis Daskalakis. I obtained my PhD in Algorithms, Combinatorics, and Optimization (ACO) at Georgia Tech, advised by Prasad Tetali. At Georgia Tech, I also obtained a MSc in Mathematics. I did my undergrad studies in National Technical University of Athens. I am interested in theory of computation and its interface with optimization, dynamical systems, probability and statistics, machine learning and their applications to game theory.

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