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تسجيل مستخدم جديدThe Low-Rank Simplicity Bias in Deep Networks
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Modern deep neural networks are highly over-parameterized compared to the data on which they are trained, yet they often generalize remarkably well. A flurry of recent work has asked: why do deep networks not overfit to their training data? We investigate the hypothesis that deeper nets are implicitly biased to find lower rank solutions and that these are the solutions that generalize well. We prove for the asymptotic case that the percent volume of low effective-rank solutions increases monotonically as linear neural networks are made deeper. We then show empirically that our claim holds true on finite width models. We further empirically find that a similar result holds for non-linear networks: deeper non-linear networks learn a feature space whose kernel has a lower rank. We further demonstrate how linear over-parameterization of deep non-linear models can be used to induce low-rank bias, improving generalization performance without changing the effective model capacity. We evaluate on various model architectures and demonstrate that linearly over-parameterized models outperform existing baselines on image classification tasks, including ImageNet.
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Several works have proposed Simplicity Bias (SB)---the tendency of standard training procedures such as Stochastic Gradient Descent (SGD) to find simple models---to justify why neural networks generalize well [Arpit et al. 2017, Nakkiran et al. 2019,
Soudry et al. 2018]. However, the precise notion of simplicity remains vague. Furthermore, previous settings that use SB to theoretically justify why neural networks generalize well do not simultaneously capture the non-robustness of neural networks---a widely observed phenomenon in practice [Goodfellow et al. 2014, Jo and Bengio 2017]. We attempt to reconcile SB and the superior standard generalization of neural networks with the non-robustness observed in practice by designing datasets that (a) incorporate a precise notion of simplicity, (b) comprise multiple predictive features with varying levels of simplicity, and (c) capture the non-robustness of neural networks trained on real data. Through theory and empirics on these datasets, we make four observations: (i) SB of SGD and variants can be extreme: neural networks can exclusively rely on the simplest feature and remain invariant to all predictive complex features. (ii) The extreme aspect of SB could explain why seemingly benign distribution shifts and small adversarial perturbations significantly degrade model performance. (iii) Contrary to conventional wisdom, SB can also hurt generalization on the same data distribution, as SB persists even when the simplest feature has less predictive power than the more complex features. (iv) Common approaches to improve generalization and robustness---ensembles and adversarial training---can fail in mitigating SB and its pitfalls. Given the role of SB in training neural networks, we hope that the proposed datasets and methods serve as an effective testbed to evaluate novel algorithmic approaches aimed at avoiding the pitfalls of SB.
To enable DNNs on edge devices like mobile phones, low-rank approximation has been widely adopted because of its solid theoretical rationale and efficient implementations. Several previous works attempted to directly approximate a pretrained model by
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