Better Depth-Width Trade-offs for Neural Networks through the lens of Dynamical Systems
This work addresses theoretical foundations for neural network design, offering incremental improvements in depth-width separation results for researchers in deep learning theory.
The authors tackled the problem of understanding depth-width trade-offs in neural networks by strengthening the connection with dynamical systems, improving width lower bounds under stronger approximation error metrics and providing sharper bounds in previously inapplicable regimes, with results including a universal constant for trade-offs when functions have odd periods.
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.