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Zac Kenton

Alumni

Publications

On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Length
Stanisław Jastrzębski
Zac Kenton
Nicolas Ballas
Asja Fischer
Yoshua Bengio
Amos Storkey
Stochastic Gradient Descent (SGD) based training of neural networks with a large learning rate or a small batch-size typically ends in well-… (voir plus)generalizing, flat regions of the weight space, as indicated by small eigenvalues of the Hessian of the training loss. However, the curvature along the SGD trajectory is poorly understood. An empirical investigation shows that initially SGD visits increasingly sharp regions, reaching a maximum sharpness determined by both the learning rate and the batch-size of SGD. When studying the SGD dynamics in relation to the sharpest directions in this initial phase, we find that the SGD step is large compared to the curvature and commonly fails to minimize the loss along the sharpest directions. Furthermore, using a reduced learning rate along these directions can improve training speed while leading to both sharper and better generalizing solutions compared to vanilla SGD. In summary, our analysis of the dynamics of SGD in the subspace of the sharpest directions shows that they influence the regions that SGD steers to (where larger learning rate or smaller batch size result in wider regions visited), the overall training speed, and the generalization ability of the final model.
2019-01-01
ICLR.cc/2019/Conference (poster)
openreview.net
openreview.net
Width of Minima Reached by Stochastic Gradient Descent is Influenced by Learning Rate to Batch Size Ratio
Stanisław Jastrzębski
Zac Kenton
Devansh Arpit
Nicolas Ballas
Asja Fischer
Yoshua Bengio
Amos Storkey
2018-09-27
Artificial Neural Networks and Machine Learning – ICANN 2018 (publié)
doi.org
Finding Flatter Minima with SGD
Stanisław Jastrzębski
Zac Kenton
Devansh Arpit
Nicolas Ballas
Asja Fischer
Yoshua Bengio
Amos Storkey
2018-02-12
International Conference on Learning Representations (publié)
dblp.uni-trier.de
SGD S MOOTHS THE S HARPEST D IRECTIONS
Stanisław Jastrzębski
Zac Kenton
Nicolas Ballas
Asja Fischer
Yoshua Bengio
Amos Storkey
Stochastic gradient descent (SGD) is able to find regions that generalize well, even in drastically over-parametrized models such as deep ne… (voir plus)ural networks. We observe that noise in SGD controls the spectral norm and conditioning of the Hessian throughout the training. We hypothesize the cause of this phenomenon is due to the dynamics of neurons saturating their non-linearity along the largest curvature directions, thus leading to improved conditioning.
2018-02-12
(publié)
openreview.net
openreview.net
LATTER M INIMA WITH SGD
Stanisław Jastrzębski
Zac Kenton
Devansh Arpit
Nicolas Ballas
Asja Fischer
Yoshua Bengio
Amos Storkey
It has been discussed that over-parameterized deep neural networks (DNNs) trained using stochastic gradient descent (SGD) with smaller batch… (voir plus) sizes generalize better compared with those trained with larger batch sizes. Additionally, model parameters found by small batch size SGD tend to be in flatter regions. We extend these empirical observations and experimentally show that both large learning rate and small batch size contribute towards SGD finding flatter minima that generalize well. Conversely, we find that small learning rates and large batch sizes lead to sharper minima that correlate with poor generalization in DNNs.
2018-01-01
(publié)
www.semanticscholar.org
LATTER M INIMA WITH SGD
Stanisław Jastrzębski
Zac Kenton
Devansh Arpit
Nicolas Ballas
Asja Fischer
Yoshua Bengio
Amos Storkey
Three Factors Influencing Minima in SGD
Stanisław Jastrzębski
Zac Kenton
Devansh Arpit
Nicolas Ballas
Asja Fischer
Yoshua Bengio
Amos Storkey
We study the statistical properties of the endpoint of stochastic gradient descent (SGD). We approximate SGD as a stochastic differential eq… (voir plus)uation (SDE) and consider its Boltzmann Gibbs equilibrium distribution under the assumption of isotropic variance in loss gradients.. Through this analysis, we find that three factors – learning rate, batch size and the variance of the loss gradients – control the trade-off between the depth and width of the minima found by SGD, with wider minima favoured by a higher ratio of learning rate to batch size. In the equilibrium distribution only the ratio of learning rate to batch size appears, implying that it’s invariant under a simultaneous rescaling of each by the same amount. We experimentally show how learning rate and batch size affect SGD from two perspectives: the endpoint of SGD and the dynamics that lead up to it. For the endpoint, the experiments suggest the endpoint of SGD is similar under simultaneous rescaling of batch size and learning rate, and also that a higher ratio leads to flatter minima, both findings are consistent with our theoretical analysis. We note experimentally that the dynamics also seem to be similar under the same rescaling of learning rate and batch size, which we explore showing that one can exchange batch size and learning rate in a cyclical learning rate schedule. Next, we illustrate how noise affects memorization, showing that high noise levels lead to better generalization. Finally, we find experimentally that the similarity under simultaneous rescaling of learning rate and batch size breaks down if the learning rate gets too large or the batch size gets too small.
2017-11-13
ArXiv (prépublication)
openreview.net
openreview.net