Optimizing deep neural networks is largely thought to be an empirical process, requiring manual tuning of several parameters, such as learning rate, weight decay, and dropout rate. Arguably, the learning rate is the most important of these to tune, and this has gained more attention in recent works. In this paper, we propose a novel method to compute the learning rate for training deep neural networks. We derive a theoretical framework to compute learning rates dynamically, and then show experimental results on standard datasets and architectures to demonstrate the efficacy of our approach.
Our work is also motivated by recent works that theoretically show that stochastic gradient descent is sufficient to optimize over-parameterized neural networks, making minimal assumptions @cite_7 @cite_35 . Our aim is to mathematically identify an optimal learning rate, rejecting the notion that only small learning rates must be used, and then experimentally show the validity of our claims.
@cite_35 @cite_7
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Whereas optimizing deep neural networks using stochastic gradient descent has been showed great performances in practice, the rule for setting step size (i.e. learning rate) of gradient descent is not well studied. Although it appears that some intriguing learning rate rules such as ADAM (Kingma and Ba, 2014) have since been developed, they concentrated on improving convergence, not on improving generalization capabilities. Recently, the improved generalization property of the flat minima was revisited, and this research guides us towards promising solutions to many current optimization problems. In this paper, we analyze the flatness of loss surfaces in the lens of robustness to input perturbations and advocate that gradient descent should be guided to reach flatter region of loss surfaces to achieve generalization. Finally, we suggest a learning rate rule for escaping sharp regions of loss surfaces, and we demonstrate the capacity of our approach by performing numerous experiments. The performance of stochastic gradient descent (SGD) depends critically on how learning rates are tuned and decreased over time. We propose a method to automatically adjust multiple learning rates so as to minimize the expected error at any one time. The method relies on local gradient variations across samples. In our approach, learning rates can increase as well as decrease, making it suitable for non-stationary problems. Using a number of convex and non-convex learning tasks, we show that the resulting algorithm matches the performance of SGD or other adaptive approaches with their best settings obtained through systematic search, and effectively removes the need for learning rate tuning. The performance of stochastic gradient descent (SGD) depends critically on how learning rates are tuned and decreased over time. We propose a method to automatically adjust multiple learning rates so as to minimize the expected error at any one time. The method relies on local gradient variations across samples. In our approach, learning rates can increase as well as decrease, making it suitable for non-stationary problems. Using a number of convex and non-convex learning tasks, we show that the resulting algorithm matches the performance of SGD or other adaptive approaches with their best settings obtained through systematic search, and effectively removes the need for learning rate tuning. We study the Stochastic Gradient Langevin Dynamics (SGLD) algorithm for non-convex optimization. The algorithm performs stochastic gradient descent, where in each step it injects appropriately scaled Gaussian noise to the update. We analyze the algorithm's hitting time to an arbitrary subset of the parameter space. Two results follow from our general theory: First, we prove that for empirical risk minimization, if the empirical risk is point-wise close to the (smooth) population risk, then the algorithm achieves an approximate local minimum of the population risk in polynomial time, escaping suboptimal local minima that only exist in the empirical risk. Second, we show that SGLD improves on one of the best known learnability results for learning linear classifiers under the zero-one loss.