We study finite sample expressivity, i.e., memorization power of ReLU networks.

From a computational standpoint, we show that under certain assumptions, adversarial training can detect the presence of backdoors in a training set.

From a computational standpoint, we show that under certain assumptions, adversarial training can detect the presence of backdoors in a training set.

The paper investigates the problem of expressiveness in neural networks w.r.t. The authors also show an upper bound for classification, a corollary of which is that a three hidden layer network with hidden layers of sized 2k-2k-4k can perfectly classify ImageNet. Moreover, they show that if the overall sum of hidden nodes in a ResNet is of order N/d_x, where d_x is the input dimension then again the network can perfectly realize the data. Lastly, an analysis is given showing batch SGD that is initialized close to a global minimum will come close to a point with value significantly smaller than the loss in the initialization (though a convergence guarantee could not be given). The paper is clear and easy to follow for the most part, and conveys a feeling that the authors did their best to make the analysis as thorough and exhausting as possible, providing results for various settings.

The topic is timely, and the results would be of interest to a wide audience. The reviewers found the paper well written and were also satisfied with the authors response. However, please do take the time to address their comments and revise what is necessary in the final version.

Recent research has explored the memorization capacity of multi-head attention, but these findings are constrained by unrealistic limitations on the context size. We present a novel proof for language-based Transformers that extends the current hypothesis to any context size. Our approach improves upon the state-of-the-art by achieving more effective exact memorization with an attention layer, while also introducing the concept of approximate memorization of distributions. Through experimental validation, we demonstrate that our proposed bounds more accurately reflect the true memorization capacity of language models, and provide a precise comparison with prior work.

We study finite sample expressivity, i.e., memorization power of ReLU networks. Recent results require N hidden nodes to memorize/interpolate arbitrary N data points. In contrast, by exploiting depth, we show that 3-layer ReLU networks with \Omega(\sqrt{N}) hidden nodes can perfectly memorize most datasets with N points. We also prove that width \Theta(\sqrt{N}) is necessary and sufficient for memorizing N data points, proving tight bounds on memorization capacity. The sufficiency result can be extended to deeper networks; we show that an L -layer network with W parameters in the hidden layers can memorize N data points if W \Omega(N) .

It is well known that modern deep neural networks are powerful enough to memorize datasets even when the labels have been randomized. Recently, Vershynin(2020) settled a long standing question by Baum(1988), proving that deep threshold networks can memorize n points in d dimensions using \widetilde{\mathcal{O}}(e {1/\delta 2} \sqrt{n}) neurons and \widetilde{\mathcal{O}}(e {1/\delta 2}(d \sqrt{n}) n) weights, where \delta is the minimum distance between the points. Our construction uses Gaussian random weights only in the first layer, while all the subsequent layers use binary or integer weights. We also prove new lower bounds by connecting memorization in neural networks to the purely geometric problem of separating n points on a sphere using hyperplanes.

In recent years, the Transformer architecture (Vaswani et al., 2017) has played a pivotal role in the field of machine learning, becoming indispensable for a variety of models in the community. In addition to the original breakthroughs in natural language processing, such as the GPT series (Brown et al., 2020; Radford et al., 2018, 2019), it has been observed that in numerous applications, higher accuracy can be achieved by replacing existing models with Transformers. Specifically, models such as the Vision Transformer (Dosovitskiy et al., 2021) in image processing and the Diffusion Transformer (Peebles & Xie, 2023) in generative tasks have demonstrated exceptional performances in a wide variety of tasks. These examples demonstrate how effective and versatile Transformers are for a diverse range of purposes. Although the high performance of Transformers has led to their widespread use in practice, there are ongoing attempts to theoretically analyze what exactly contributes to their superior performance.