Neural networks are a priori biased towards Boolean functions with low entropy
This work provides foundational insights into why neural networks generalize, which is critical for researchers in machine learning theory, though it is incremental as it builds on prior studies of inductive bias.
The paper tackles the problem of understanding neural networks' inductive bias by proving that a single-layer perceptron with random initialization has a strong a priori bias towards Boolean functions with low entropy, specifically showing that the probability of representing a function classifying t points as 1 is uniform at 2^{-n}, and this bias strengthens with added ReLU layers or increased bias variance.
Understanding the inductive bias of neural networks is critical to explaining their ability to generalise. Here, for one of the simplest neural networks -- a single-layer perceptron with n input neurons, one output neuron, and no threshold bias term -- we prove that upon random initialisation of weights, the a priori probability $P(t)$ that it represents a Boolean function that classifies t points in ${0,1}^n$ as 1 has a remarkably simple form: $P(t) = 2^{-n}$ for $0\leq t < 2^n$. Since a perceptron can express far fewer Boolean functions with small or large values of t (low entropy) than with intermediate values of t (high entropy) there is, on average, a strong intrinsic a-priori bias towards individual functions with low entropy. Furthermore, within a class of functions with fixed t, we often observe a further intrinsic bias towards functions of lower complexity. Finally, we prove that, regardless of the distribution of inputs, the bias towards low entropy becomes monotonically stronger upon adding ReLU layers, and empirically show that increasing the variance of the bias term has a similar effect.