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1. Deep Classifiers trained with the Square Loss

   Xu, Mengjia; Rangamani, Akshay; Banburski, Andrzej; Liao, Qianli; Galanti, Tomer; Poggio, Tomaso (October 2022, Center for Brains, Minds and Machines (CBMM))

   We overview several properties -- old and new -- of training overparametrized deep networks under the square loss. We first consider a model of the dynamics of gradient flow under the square loss in deep homogeneous ReLU networks. We study the convergence to a solution with the absolute minimum $$\rho$$, which is the product of the Frobenius norms of each layer weight matrix, when normalization by Lagrange multipliers (LM) is used together with Weight Decay (WD) under different forms of gradient descent. A main property of the minimizers that bounds their expected error {\it for a specific network architecture} is $$\rho$$. In particular, we derive novel norm-based bounds for convolutional layers that are orders of magnitude better than classical bounds for dense networks. Next we prove that quasi-interpolating solutions obtained by Stochastic Gradient Descent (SGD) in the presence of WD have a bias towards low rank weight matrices -- that, as we also explain, should improve generalization. The same analysis predicts the existence of an inherent SGD noise for deep networks. In both cases, we verify our predictions experimentally. We then predict Neural Collapse and its properties without any specific assumption -- unlike other published proofs. Our analysis supports the idea that the advantage of deep networks relative to other classifiers is greater for the problems that are appropriate for sparse deep architectures such as CNNs. The deep reason compositionally sparse target functions can be approximated well by ``sparse'' deep networks without incurring in the curse of dimensionality. 

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2. SGD Noise and Implicit Low-Rank Bias in Deep Neural Networks

   Galanti, Tomer; Poggio, Tomaso (March 2022, Center for Brains, Minds and Machines (CBMM))

   We analyze deep ReLU neural networks trained with mini-batch stochastic gradient decent and weight decay. We prove that the source of the SGD noise is an implicit low rank constraint across all of the weight matrices within the network. Furthermore, we show, both theoretically and empirically, that when training a neural network using Stochastic Gradient Descent (SGD) with a small batch size, the resulting weight matrices are expected to be of small rank. Our analysis relies on a minimal set of assumptions and the neural networks may include convolutional layers, residual connections, as well as batch normalization layers. 

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3. SGD and Weight Decay Provably Induce a Low-Rank Bias in Deep Neural Networks

   Galanti, Tomer; Siegel, Zachary; Gupte, Aparna; Poggio, Tomaso (February 2023, Center for Brains, Minds and Machines (CBMM))

   In this paper, we study the bias of Stochastic Gradient Descent (SGD) to learn low-rank weight matrices when training deep ReLU neural networks. Our results show that training neural networks with mini-batch SGD and weight decay causes a bias towards rank minimization over the weight matrices. Specifically, we show, both theoretically and empirically, that this bias is more pronounced when using smaller batch sizes, higher learning rates, or increased weight decay. Additionally, we predict and observe empirically that weight decay is necessary to achieve this bias. Finally, we empirically investigate the connection between this bias and generalization, finding that it has a marginal effect on generalization. Our analysis is based on a minimal set of assumptions and applies to neural networks of any width or depth, including those with residual connections and convolutional layers. 

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4. Convergence of a Relaxed Variable Splitting Method for Learning Sparse Neural Networks via L1, L0, and transformed-L1 Penalties

   Dinh, Thu; Xin, Jack (September 2020, Intelligent Systems Conference (IntelliSys))

   Sparsification of neural networks is one of the effective complexity reduction methods to improve efficiency and generalizability. We consider the problem of learning a one hidden layer convolutional neural network with ReLU activation function via gradient descent under sparsity promoting penalties. It is known that when the input data is Gaussian distributed, no-overlap networks (without penalties) in regression problems with ground truth can be learned in polynomial time at high probability. We propose a relaxed variable splitting method integrating thresholding and gradient descent to overcome the non-smoothness in the loss function. The sparsity in network weight is realized during the optimization (training) process. We prove that under L1, L0, and transformed-L1 penalties, no-overlap networks can be learned with high probability, and the iterative weights converge to a global limit which is a transformation of the true weight under a novel thresholding operation. Numerical experiments confirm theoretical findings, and compare the accuracy and sparsity trade-off among the penalties. 

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5. Superpolynomial Lower Bounds for Learning One-Layer Neural Networks using Gradient Descent

   Goel, Surbhi; Gollakota, Aravind; Jin, Zhihan; Karmalkar, Sushrut; Klivans, Adam (July 2020, International Conference on Machine Learning)

   We prove the first superpolynomial lower bounds for learning one-layer neural networks with respect to the Gaussian distribution using gradient descent. We show that any classifier trained using gradient descent with respect to square-loss will fail to achieve small test error in polynomial time given access to samples labeled by a one-layer neural network. For classification, we give a stronger result, namely that any statistical query (SQ) algorithm (including gradient descent) will fail to achieve small test error in polynomial time. Prior work held only for gradient descent run with small batch sizes, required sharp activations, and applied to specific classes of queries. Our lower bounds hold for broad classes of activations including ReLU and sigmoid. The core of our result relies on a novel construction of a simple family of neural networks that are exactly orthogonal with respect to all spherically symmetric distributions. 

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