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Register a new userTowards Understanding the Dynamics of the First-Order Adversaries
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An acknowledged weakness of neural networks is their vulnerability to adversarial perturbations to the inputs. To improve the robustness of these models, one of the most popular defense mechanisms is to alternatively maximize the loss over the constrained perturbations (or called adversaries) on the inputs using projected gradient ascent and minimize over weights. In this paper, we analyze the dynamics of the maximization step towards understanding the experimentally observed effectiveness of this defense mechanism. Specifically, we investigate the non-concave landscape of the adversaries for a two-layer neural network with a quadratic loss. Our main result proves that projected gradient ascent finds a local maximum of this non-concave problem in a polynomial number of iterations with high probability. To our knowledge, this is the first work that provides a convergence analysis of the first-order adversaries. Moreover, our analysis demonstrates that, in the initial phase of adversarial training, the scale of the inputs matters in the sense that a smaller input scale leads to faster convergence of adversarial training and a more regular landscape. Finally, we show that these theoretical findings are in excellent agreement with a series of experiments.
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An important research direction in machine learning has centered around developing meta-learning algorithms to tackle few-shot learning. An especially successful algorithm has been Model Agnostic Meta-Learning (MAML), a method that consists of two optimization loops, with the outer loop finding a meta-initialization, from which the inner loop can efficiently learn new tasks. Despite MAMLs popularity, a fundamental open question remains -- is the effectiveness of MAML due to the meta-initialization being primed for rapid learning (large, efficient changes in the representations) or due to feature reuse, with the meta initialization already containing high quality features? We investigate this question, via ablation studies and analysis of the latent representations, finding that feature reuse is the dominant factor. This leads to the ANIL (Almost No Inner Loop) algorithm, a simplification of MAML where we remove the inner loop for all but the (task-specific) head of a MAML-trained network. ANIL matches MAMLs performance on benchmark few-shot image classification and RL and offers computational improvements over MAML. We further study the precise contributions of the head and body of the network, showing that performance on the test tasks is entirely determined by the quality of the learned features, and we can remove even the head of the network (the NIL algorithm). We conclude with a discussion of the rapid learning vs feature reuse question for meta-learning algorithms more broadly.
Despite existing work on ensuring generalization of neural networks in terms of scale sensitive complexity measures, such as norms, margin and sharpness, these complexity measures do not offer an explanation of why neural networks generalize better with over-parametrization. In this work we suggest a novel complexity measure based on unit-wise capacities resulting in a tighter generalization bound for two layer ReLU networks. Our capacity bound correlates with the behavior of test error with increasing network sizes, and could potentially explain the improvement in generalization with over-parametrization. We further present a matching lower bound for the Rademacher complexity that improves over previous capacity lower bounds for neural networks.
Deep neural networks can empirically perform efficient hierarchical learning, in which the layers learn useful representations of the data. However, how they make use of the intermediate representations are not explained by recent theories that relate them to shallow learners such as kernels. In this work, we demonstrate that intermediate neural representations add more flexibility to neural networks and can be advantageous over raw inputs. We consider a fixed, randomly initialized neural network as a representation function fed into another trainable network. When the trainable network is the quadratic Taylor model of a wide two-layer network, we show that neural representation can achieve improved sample complexities compared with the raw input: For learning a low-rank degree-$p$ polynomial ($p geq 4$) in $d$ dimension, neural representation requires only $tilde{O}(d^{lceil p/2 rceil})$ samples, while the best-known sample complexity upper bound for the raw input is $tilde{O}(d^{p-1})$. We contrast our result with a lower bound showing that neural representations do not improve over the raw input (in the infinite width limit), when the trainable network is instead a neural tangent kernel. Our results characterize when neural representations are beneficial, and may provide a new perspective on why depth is important in deep learning.
We investigate the problem of machine learning with mislabeled training data. We try to make the effects of mislabeled training better understood through analysis of the basic model and equations that characterize the problem. This includes results about the ability of the noisy model to make the same decisions as the clean model and the effects of noise on model performance. In addition to providing better insights we also are able to show that the Maximum Likelihood (ML) estimate of the parameters of the noisy model determine those of the clean model. This property is obtained through the use of the ML invariance property and leads to an approach to developing a classifier when training has been mislabeled: namely train the classifier on noisy data and adjust the decision threshold based on the noise levels and/or class priors. We show how our approach to mislabeled training works with multi-layered perceptrons (MLPs).
Deep Convolutional Sparse Coding (D-CSC) is a framework reminiscent of deep convolutional neural networks (DCNNs), but by omitting the learning of the dictionaries one can more transparently analyse the role of the activation function and its ability to recover activation paths through the layers. Papyan, Romano, and Elad conducted an analysis of such an architecture, demonstrated the relationship with DCNNs and proved conditions under which the D-CSC is guaranteed to recover specific activation paths. A technical innovation of their work highlights that one can view the efficacy of the ReLU nonlinear activation function of a DCNN through a new variant of the tensors sparsity, referred to as stripe-sparsity. Using this they proved that representations with an activation density proportional to the ambient dimension of the data are recoverable. We extend their uniform guarantees to a modified model and prove that with high probability the true activation is typically possible to recover for a greater density of activations per layer. Our extension follows from incorporating the prior work on one step thresholding by Schnass and Vandergheynst.
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