@article {5466, title = {The Janus effects of SGD vs GD: high noise and low rank}, number = {144}, year = {2023}, month = {12/2024}, abstract = {

It was always obvious that \ SGD with small minibatch size yields for neural networks much higher asymptotic fluctuations in the updates of the weight matrices than GD. It has also been often reported that SGD in deep RELU networks shows empirically a low-rank bias in the weight matrices. A recent \ theoretical analysis derived a bound on the rank and linked it to the size of the SGD fluctuations [25]. In this paper, we provide an empirical and \ theoretical analysis of the convergence of SGD vs GD, first for deep RELU networks and then for the case of linear regression, where sharper estimates can be obtained and which is of independent interest. In the linear case, we prove that the component $W^\perp$ of the matrix $W$ corresponding to the null space of the data matrix $X$ converges to zero for both SGD and GD, provided the regularization term is non-zero. Because of the larger number of updates required to go through all the training data, the convergence rate {\it per epoch} of these components is much faster for SGD than for GD. In practice, SGD has a much stronger bias than GD towards solutions for weight matrices $W$ with high fluctuations -- even when the choice of mini batches is deterministic -- and low rank, provided the initialization is from a random matrix. Thus SGD \ with non-zero regularization, shows the coupled phenomenon of \ asymptotic noise and a low-rank bias-- unlike GD.

}, author = {Mengjia Xu and Tomer Galanti and Akshay Rangamani and Lorenzo Rosasco and Andrea Pinto and Tomaso Poggio} } @article {5242, title = {Norm-Based Generalization Bounds for Compositionally Sparse Neural Networks}, year = {2023}, abstract = {

In this paper, we investigate the Rademacher complexity of deep sparse neural networks, where each neuron receives a small number of inputs. We prove generalization bounds for multilayered sparse ReLU neural networks, including convolutional neural networks. These bounds differ from previous ones, as they consider the norms of the convolutional filters instead of the norms of the associated Toeplitz matrices, independently of weight sharing between neurons.

As we show theoretically, these bounds may be orders of magnitude better than standard norm-based generalization bounds and empirically, they are almost non-vacuous in estimating generalization in various simple classification problems. Taken together, these results suggest that compositional sparsity of the underlying target function is critical to the success of deep neural networks.

}, author = {Tomer Galanti and Mengjia Xu and Liane Galanti and Tomaso Poggio} } @conference {5496, title = {Norm-based Generalization Bounds for Sparse Neural Networks}, booktitle = {NeurIPS 2023}, year = {2023}, month = {12/2023}, address = {New Orleans}, abstract = {

In this paper, we derive norm-based generalization bounds for sparse ReLU neural networks, including convolutional neural networks. These bounds differ from previous ones because they consider the sparse structure of the neural network architecture and the norms of the convolutional filters, rather than the norms of the (Toeplitz) matrices associated with the convolutional layers. Theoretically, we demonstrate that these bounds are significantly tighter than standard norm-based generalization bounds. Empirically, they offer relatively tight estimations of generalization for various simple classification problems. Collectively, these findings suggest that the sparsity of the underlying target function and the model{\textquoteright}s architecture plays a crucial role in the success of deep learning.

}, url = {https://proceedings.neurips.cc/paper_files/paper/2023/file/8493e190ff1bbe3837eca821190b61ff-Paper-Conference.pdf}, author = {Tomer Galanti and Mengjia Xu and Liane Galanti and Tomaso Poggio} } @article {4625, title = {Implicit dynamic regularization in deep networks}, year = {2020}, month = {08/2020}, abstract = {

Square loss has been observed to perform well in classification tasks, at least as well as crossentropy. However, a theoretical justification is lacking. Here we develop a theoretical analysis for the square loss that \ complements the existing asymptotic analysis for the exponential loss.

}, author = {Tomaso Poggio and Qianli Liao and Mengjia Xu} }