On the Universal Approximability and Complexity Bounds of Quantized ReLU Neural Networks
Compression is a key step to deploy large neural networks on resource-constrained platforms. As a popular compression technique, quantization constrains the number of distinct weight values and thus reducing the number of bits required to represent and store each weight. In this paper, we study the representation power of quantized neural networks. First, we prove the universal approximability of quantized ReLU networks on a wide class of functions. Then we provide upper bounds on the number of weights and the memory size for a given approximation error bound and the bit-width of weights for function-independent and function-dependent structures. Our results reveal that, to attain an approximation error bound of , the number of weights needed by a quantized network is no more than times that of an unquantized network. This overhead is of much lower order than the lower bound of the number of weights needed for the error bound, supporting the empirical success of various quantization techniques. To the best of our knowledge, this is the first in-depth study on the complexity bounds of quantized neural networks.
 The authors propose in this paper a series of results on the approximation capabilities of neural networks based on ReLU using quantized weights. Results include upper bounds on the depth and on the number of weights needed to reach a certain approximation level given the number of distinct weights usable. The paper is clear and as far as I know the results are both new and significant. My only negative remark is about the appendix that could be clearer. In particular, I think that figure 2 obscures the proof of Proposition 1 rather than the contrary. I think it might be much clearer to give an explicit neural network approximation of x^2 for say r=2, for instance.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
On the Universal Approximability and Complexity Bounds of Quantized ReLU Neural Networks
Compression is a key step to deploy large neural networks on resource-constrained platforms. As a popular compression technique, quantization constrains the number of distinct weight values and thus reducing the number of bits required to represent and store each weight. In this paper, we study the representation power of quantized neural networks. First, we prove the universal approximability of quantized ReLU networks on a wide class of functions. Then we provide upper bounds on the number of weights and the memory size for a given approximation error bound and the bit-width of weights for function-independent and function-dependent structures. Our results reveal that, to attain an approximation error bound of , the number of weights needed by a quantized network is no more than times that of an unquantized network. This overhead is of much lower order than the lower bound of the number of weights needed for the error bound, supporting the empirical success of various quantization techniques. To the best of our knowledge, this is the first in-depth study on the complexity bounds of quantized neural networks.
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>



 The authors propose in this paper a series of results on the approximation capabilities of neural networks based on ReLU using quantized weights. Results include upper bounds on the depth and on the number of weights needed to reach a certain approximation level given the number of distinct weights usable. The paper is clear and as far as I know the results are both new and significant. My only negative remark is about the appendix that could be clearer. In particular, I think that figure 2 obscures the proof of Proposition 1 rather than the contrary. I think it might be much clearer to give an explicit neural network approximation of x^2 for say r=2, for instance.
assistant
 The authors propose in this paper a series of results on the approximation capabilities of neural networks based on ReLU using quantized weights. Results include upper bounds on the depth and on the number of weights needed to reach a certain approximation level given the number of distinct weights usable. The paper is clear and as far as I know the results are both new and significant. My only negative remark is about the appendix that could be clearer. In particular, I think that figure 2 obscures the proof of Proposition 1 rather than the contrary. I think it might be much clearer to give an explicit neural network approximation of x^2 for say r=2, for instance.