Adam: A Method for Stochastic Optimization

We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.

[1]  Shun-ichi Amari,et al.  Natural Gradient Works Efficiently in Learning , 1998, Neural Computation.

[2]  Geoffrey Zweig,et al.  Recent advances in deep learning for speech research at Microsoft , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[3]  Alex Graves,et al.  Generating Sequences With Recurrent Neural Networks , 2013, ArXiv.

[4]  Boris Polyak,et al.  Acceleration of stochastic approximation by averaging , 1992 .

[5]  Christopher Potts,et al.  Learning Word Vectors for Sentiment Analysis , 2011, ACL.

[6]  Martin Zinkevich,et al.  Online Convex Programming and Generalized Infinitesimal Gradient Ascent , 2003, ICML.

[7]  Max Welling,et al.  Auto-Encoding Variational Bayes , 2014, ICLR.

[8]  Sebastian Ruder,et al.  An overview of gradient descent optimization algorithms , 2016, ArXiv.

[9]  Tom Schaul,et al.  No more pesky learning rates , 2012, ICML.

[10]  Razvan Pascanu,et al.  Revisiting Natural Gradient for Deep Networks , 2013, ICLR.

[11]  Geoffrey E. Hinton,et al.  ImageNet classification with deep convolutional neural networks , 2012, Commun. ACM.

[12]  D. Ruppert,et al.  Efficient Estimations from a Slowly Convergent Robbins-Monro Process , 1988 .

[13]  Surya Ganguli,et al.  Fast large-scale optimization by unifying stochastic gradient and quasi-Newton methods , 2013, ICML.

[14]  Eric Moulines,et al.  Non-Asymptotic Analysis of Stochastic Approximation Algorithms for Machine Learning , 2011, NIPS.

[15]  Geoffrey E. Hinton,et al.  On the importance of initialization and momentum in deep learning , 2013, ICML.

[16]  Yoram Singer,et al.  Adaptive Subgradient Methods for Online Learning and Stochastic Optimization , 2011, J. Mach. Learn. Res..

[17]  Geoffrey E. Hinton Reducing the Dimensionality of Data with Neural , 2008 .

[18]  Nitish Srivastava,et al.  Improving neural networks by preventing co-adaptation of feature detectors , 2012, ArXiv.

[19]  Ning Qian,et al.  On the momentum term in gradient descent learning algorithms , 1999, Neural Networks.

[20]  Matthew D. Zeiler ADADELTA: An Adaptive Learning Rate Method , 2012, ArXiv.

[21]  Geoffrey E. Hinton,et al.  Speech recognition with deep recurrent neural networks , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[22]  Surya Ganguli,et al.  Identifying and attacking the saddle point problem in high-dimensional non-convex optimization , 2014, NIPS.

[23]  Geoffrey E. Hinton,et al.  Reducing the Dimensionality of Data with Neural Networks , 2006, Science.

[24]  Christopher D. Manning,et al.  Fast dropout training , 2013, ICML.

[25]  Tara N. Sainath,et al.  Deep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research Groups , 2012, IEEE Signal Processing Magazine.

[26]  Andrew W. Fitzgibbon,et al.  A fast natural Newton method , 2010, ICML.