Neural networks are a priori biased towards Boolean functions with low entropy
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 input neurons, one output neuron, and no threshold bias term -- we prove that upon random initialisation of weights, the a priori probability that it represents a Boolean function that classifies points in as has a remarkably simple form: . Since a perceptron can express far fewer Boolean functions with small or large values of (low *entropy*) than with intermediate values of (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 , 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.
This paper studies the a-priori bias of a feed-forward neural network when the weights are initialised uniformly at random and independent of the network architecture. The paper claims that this initialisation leads to biases towards low entropy functions when the input and output are binary values. The paper starts with a single layer perceptron without the bias term, and generalises the analysis to networks with multiple hidden layers and ReLU activations. The proposed approach seems rigorous, but I have a hard time to follow the paper as many of the important results are presented in appendix. In addition, the analysis is based on a feed-forward neural network with binary inputs and a single binary output, it is not clear whether these results can be generalised to architectures of practical importance such as convolutional/recurrent neural networks. Overall an interesting piece of work that contributes to the understanding of deep neural networks. Minor comments: Section 4.2: *as as predicted by Eq. 1* -> remove duplicated *as* Section 5.2: *some interesting recent work* -> *Some ...* *produce.At first sight* -> add a space before *At* *If there is not bias* -> *If there is no bias*
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Neural networks are a priori biased towards Boolean functions with low entropy
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 input neurons, one output neuron, and no threshold bias term -- we prove that upon random initialisation of weights, the a priori probability that it represents a Boolean function that classifies points in as has a remarkably simple form: . Since a perceptron can express far fewer Boolean functions with small or large values of (low *entropy*) than with intermediate values of (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 , 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.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



This paper studies the a-priori bias of a feed-forward neural network when the weights are initialised uniformly at random and independent of the network architecture. The paper claims that this initialisation leads to biases towards low entropy functions when the input and output are binary values. The paper starts with a single layer perceptron without the bias term, and generalises the analysis to networks with multiple hidden layers and ReLU activations. The proposed approach seems rigorous, but I have a hard time to follow the paper as many of the important results are presented in appendix. In addition, the analysis is based on a feed-forward neural network with binary inputs and a single binary output, it is not clear whether these results can be generalised to architectures of practical importance such as convolutional/recurrent neural networks. Overall an interesting piece of work that contributes to the understanding of deep neural networks. Minor comments: Section 4.2: *as as predicted by Eq. 1* -> remove duplicated *as* Section 5.2: *some interesting recent work* -> *Some ...* *produce.At first sight* -> add a space before *At* *If there is not bias* -> *If there is no bias*
assistant
This paper studies the a-priori bias of a feed-forward neural network when the weights are initialised uniformly at random and independent of the network architecture. The paper claims that this initialisation leads to biases towards low entropy functions when the input and output are binary values. The paper starts with a single layer perceptron without the bias term, and generalises the analysis to networks with multiple hidden layers and ReLU activations. The proposed approach seems rigorous, but I have a hard time to follow the paper as many of the important results are presented in appendix. In addition, the analysis is based on a feed-forward neural network with binary inputs and a single binary output, it is not clear whether these results can be generalised to architectures of practical importance such as convolutional/recurrent neural networks. Overall an interesting piece of work that contributes to the understanding of deep neural networks. Minor comments: Section 4.2: *as as predicted by Eq. 1* -> remove duplicated *as* Section 5.2: *some interesting recent work* -> *Some ...* *produce.At first sight* -> add a space before *At* *If there is not bias* -> *If there is no bias*