[Paper Review] Size and Depth Separation in Approximating Benign Functions with Neural Networks

Gal Vardi, Daniel Reichman+2 more|arXiv (Cornell University)|Jan 30, 2021

Machine Learning and Algorithms43 references4 citations

TL;DR

This paper investigates the expressive power of ReLU neural networks in approximating 'benign' functions—functions with polynomially bounded Lipschitz constants and efficient computability. It establishes complexity-theoretic barriers to proving depth and size separations for such functions, showing that proving depth separation beyond depth 4 or size separation beyond O(d log²d) would resolve longstanding open problems in computational complexity. It also constructs an explicit benign function that requires Ω(d/log d) size, demonstrating a strict separation between O(d) and o(d/log d) size networks.

ABSTRACT

When studying the expressive power of neural networks, a main challenge is to understand how the size and depth of the network affect its ability to approximate real functions. However, not all functions are interesting from a practical viewpoint: functions of interest usually have a polynomially-bounded Lipschitz constant, and can be computed efficiently. We call functions that satisfy these conditions "benign", and explore the benefits of size and depth for approximation of benign functions with ReLU networks. As we show, this problem is more challenging than the corresponding problem for non-benign functions. We give barriers to showing depth-lower-bounds: Proving existence of a benign function that cannot be approximated by polynomial-size networks of depth $4$ would settle longstanding open problems in computational complexity. It implies that beyond depth $4$ there is a barrier to showing depth-separation for benign functions, even between networks of constant depth and networks of nonconstant depth. We also study size-separation, namely, whether there are benign functions that can be approximated with networks of size $O(s(d))$, but not with networks of size $O(s'(d))$. We show a complexity-theoretic barrier to proving such results beyond size $O(d\log^2(d))$, but also show an explicit benign function, that can be approximated with networks of size $O(d)$ and not with networks of size $o(d/\log d)$. For approximation in $L_\infty$ we achieve such separation already between size $O(d)$ and size $o(d)$. Moreover, we show superpolynomial size lower bounds and barriers to such lower bounds, depending on the assumptions on the function. Our size-separation results rely on an analysis of size lower bounds for Boolean functions, which is of independent interest: We show linear size lower bounds for computing explicit Boolean functions with neural networks and threshold circuits.

Motivation & Objective

• To understand how network size and depth affect the approximation of 'benign' functions—functions with polynomially bounded Lipschitz constants and efficient computability.
• To investigate whether depth and size provide meaningful advantages in approximating such functions, especially in the context of practical, real-world learning tasks.
• To identify fundamental complexity-theoretic barriers to proving depth and size separations for benign functions.
• To construct explicit examples of benign functions that demonstrate strict separations in required network size or depth.

Proposed method

• Defining 'benign' functions as those that are poly(d)-Lipschitz and computable in exponential or polynomial time.
• Reducing the problem of proving depth separation to known complexity-theoretic assumptions, such as EXP ⊈ TC⁰_{k-2}.
• Using probabilistic averaging over shifted inputs to show that a single network can approximate a function over a shifted domain, enabling size reduction.
• Constructing a neural network of size O(d) to compute an explicit Boolean function (e.g., set disjointness) with poly(d)-Lipschitz properties.
• Applying a thresholding transformation to approximate networks to recover Boolean outputs, linking network size to Boolean circuit lower bounds.
• Leveraging known linear size lower bounds for threshold circuits to derive size lower bounds for ReLU networks.

Experimental results

Research questions

• RQ1 Can depth provide a significant advantage in approximating benign functions, particularly beyond depth 4?
• RQ2 Is there a fundamental complexity-theoretic barrier to proving depth separation for benign functions in networks of constant depth?
• RQ3 Can size separation be proven for benign functions, and what are the limits of such proofs?
• RQ4 What is the minimal network size required to approximate explicit benign functions, and how does it compare to known lower bounds?
• RQ5 How do size and depth separations for benign functions relate to complexity classes like TC⁰ and P?

Key findings

• Proving that a benign function cannot be approximated by polynomial-sized networks of depth 4 or more would imply EXP ⊈ TC⁰_{k-2}, a major open problem in complexity theory.
• There is a complexity-theoretic barrier to proving size separation beyond O(d log²d), as such results would resolve open problems in circuit complexity.
• An explicit benign function exists that can be approximated by networks of size O(d) but not by networks of size o(d/log d), establishing a super-constant size separation.
• For L∞ approximation, a strict separation exists between size O(d) and size o(d), demonstrating that O(d) is asymptotically optimal for certain benign functions.
• Linear size lower bounds for threshold circuits imply linear size lower bounds for ReLU networks computing the same Boolean functions, providing a new pathway to proving network size lower bounds.
• The paper constructs a polynomial-time benign function computable by a constant-depth, O(d)-sized ReLU network, but requiring Ω(d) size for any network approximating it within L∞ error 1/3.