Understanding the robustness difference between stochastic gradient descent and adaptive gradient methods

Stochastic gradient descent (SGD) and adaptive gradient methods, such as Adam and RMSProp, have been widely used in training deep neural networks. We empirically show that while the difference between the standard generalization performance of models trained using these methods is small, those trained using SGD exhibit far greater robustness under input perturbations. Notably, our investigation demonstrates the presence of irrelevant frequencies in natural datasets, where alterations do not affect models' generalization performance. However, models trained with adaptive methods show sensitivity to these changes, suggesting that their use of irrelevant frequencies can lead to solutions sensitive to perturbations. To better understand this difference, we study the learning dynamics of gradient descent (GD) and sign gradient descent (signGD) on a synthetic dataset that mirrors natural signals. With a three-dimensional input space, the models optimized with GD and signGD have standard risks close to zero but vary in their adversarial risks. Our result shows that linear models' robustness to $\ell_2$-norm bounded changes is inversely proportional to the model parameters' weight norm: a smaller weight norm implies better robustness. In the context of deep learning, our experiments show that SGD-trained neural networks show smaller Lipschitz constants, explaining the better robustness to input perturbations than those trained with adaptive gradient methods

Bellman Error Based Feature Generation using Random Projections on Sparse Spaces

We address the problem of automatic generation of features for value function approximation. Bellman Error Basis Functions (BEBFs) have been shown to improve the error of policy evaluation with function approximation, with a convergence rate similar to that of value iteration. We propose a simple, fast and robust algorithm based on random projections to generate BEBFs for sparse feature spaces. We provide a finite sample analysis of the proposed method, and prove that projections logarithmic in the dimension of the original space are enough to guarantee contraction in the error. Empirical results demonstrate the strength of this method

Efficient and Accurate Optimal Transport with Mirror Descent and Conjugate Gradients

We design a novel algorithm for optimal transport by drawing from the entropic optimal transport, mirror descent and conjugate gradients literatures. Our scalable and GPU parallelizable algorithm is able to compute the Wasserstein distance with extreme precision, reaching relative error rates of $10^{-8}$ without numerical stability issues. Empirically, the algorithm converges to high precision solutions more quickly in terms of wall-clock time than a variety of algorithms including log-domain stabilized Sinkhorn's Algorithm. We provide careful ablations with respect to algorithm and problem parameters, and present benchmarking over upsampled MNIST images, comparing to various recent algorithms over high-dimensional problems. The results suggest that our algorithm can be a useful addition to the practitioner's optimal transport toolkit