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Adaptive gradient methods, such as AdaGrad, are among the most successful optimization algorithms for neural network training. While these methods are known to achieve better dimensional dependence than stochastic gradient descent (SGD) for stochastic convex optimization under favorable geometry, the theoretical justification for their success in stochastic non-convex optimization remains elusive. In fact, under standard assumptions of Lipschitz gradients and bounded noise variance, it is known that SGD is worst-case optimal in terms of finding a near-stationary point with respect to the l2-norm, making further improvements impossible. Motivated by this limitation, we introduce refined assumptions on the smoothness structure of the objective and the gradient noise variance, which better suit the coordinate-wise nature of adaptive gradient methods. Moreover, we adopt the l1-norm of the gradient as the stationarity measure, as opposed to the standard l2-norm, to align with the coordinate-wise analysis and obtain tighter convergence guarantees for AdaGrad. Under these new assumptions and the l1-norm stationarity measure, we establish an upper bound on the convergence rate of AdaGrad and a corresponding lower bound for SGD. In particular, we identify non-convex settings in which the iteration complexity of AdaGrad is favorable over SGD and show that, for certain configurations of problem parameters, it outperforms SGD by a factor of d, where d is the problem dimension. To the best of our knowledge, this is the first result to demonstrate a provable gain of adaptive gradient methods over SGD in a non-convex setting. We also present supporting lower bounds, including one specific to AdaGrad and one applicable to general deterministic first-order methods, showing that our upper bound for AdaGrad is tight and unimprovable up to a logarithmic factor under certain conditions.
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Revisiting Outer Optimization in Adversarial Training
Despite the fundamental distinction between adversarial and natural training (AT and NT), AT methods generally adopt momentum SGD (MSGD) for the outer optimization. This paper aims to analyze this choice by investigating the overlooked role of outer optimization in AT. Our exploratory evaluations reveal that AT induces higher gradient norm and variance compared to NT. This phenomenon hinders the outer optimization in AT since the convergence rate of MSGD is highly dependent on the variance of the gradients. To this end, we propose an optimization method called ENGM which regularizes the contribution of each input example to the average mini-batch gradients. We prove that the convergence rate of ENGM is independent of the variance of the gradients, and thus, it is suitable for AT. We introduce a trick to reduce the computational cost of ENGM using empirical observations on the correlation between the norm of gradients w.r.t. the network parameters and input examples. Our extensive evaluations and ablation studies on CIFAR-10, CIFAR-100, and TinyImageNet demonstrate that ENGM and its variants consistently improve the performance of a wide range of AT methods. Furthermore, ENGM alleviates major shortcomings of AT including robust overfitting and high sensitivity to hyperparameter settings.
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- Award ID(s):
- 1650474
- PAR ID:
- 10401298
- Date Published:
- Journal Name:
- 16th European Conference on Computer Vision (ECCV 2022)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-031-20065-6_15
