Regularising neural networks

Generalisation for street fighters

February 12, 2017 — September 24, 2021

machine learning
model selection
neural nets
sparser than thou
Figure 1

TBD: I have not examined this stuff for a long time and it is probably out of date.

How do we get generalisation from neural networks? As in all ML it is probably about controlling overfitting to the training set by some kind of regularization.

1 Early stopping

e.g. (Prechelt 2012). Don’t keep training your model. The regularisation method that actually makes learning go faster, because you don’t bother to do as much of it. Interesting connection to NN at scale

2 Stochastic weight averaging

Izmailov et al. (2018) Pytorch’s introduction to Stochastic Weight Averaging has all the diagrams and references we could want. Also this ends up having some interesting connection to Bayesian posterior uncertainty.

3 Noise layers

See NN ensembles,

3.1 Input perturbation

Parametric noise applied to the data.

4 Weight penalties

\(L_1\), \(L_2\), dropout… Seems to be applied to weights, but rarely to actual neurons.

See Compressing neural networks for that latter use.

This is attractive but has a potentially expensive hyperparameter to choose. Also, should we penalize each weight equally, or are there some expedient normalization schemes? For that, see the next section:

5 Normalization

Mario Lezcano, in the PyTorch Tutorials mentions

Regularizing deep-learning models is a surprisingly challenging task. Classical techniques such as penalty methods often fall short when applied on deep models due to the complexity of the function being optimized. This is particularly problematic when working with ill-conditioned models. Examples of these are RNNs trained on long sequences and GANs. A number of techniques have been proposed in recent years to regularize these models and improve their convergence. On recurrent models, it has been proposed to control the singular values of the recurrent kernel for the RNN to be well-conditioned. This can be achieved, for example, by making the recurrent kernel orthogonal. Another way to regularize recurrent models is via “weight normalization”. This approach proposes to decouple the learning of the parameters from the learning of their norms. To do so, the parameter is divided by its Frobenius norm and a separate parameter encoding its norm is learnt. A similar regularization was proposed for GANs under the name of “spectral normalization”. This method controls the Lipschitz constant of the network by dividing its parameters by their spectral norm, rather than their Frobenius norm.

5.1 Weight Normalization

Pragmatically, controlling for variability in your data can be very hard in, e.g. deep learning, so you might normalise it by the batch variance. Salimans and Kingma (Salimans and Kingma 2016) have a more satisfying approach to this.

We present weight normalization: a reparameterisation of the weight vectors in a neural network that decouples the length of those weight vectors from their direction. By reparameterizing the weights in this way we improve the conditioning of the optimization problem and we speed up convergence of stochastic gradient descent. Our reparameterisation is inspired by batch normalization but does not introduce any dependencies between the examples in a minibatch. This means that our method can also be applied successfully to recurrent models such as LSTMs and to noise-sensitive applications such as deep reinforcement learning or generative models, for which batch normalization is less well suited. Although our method is much simpler, it still provides much of the speed-up of full batch normalization. In addition, the computational overhead of our method is lower, permitting more optimization steps to be taken in the same amount of time.

They provide an open implementation for keras, Tensorflow and lasagne.

5.2 Adversarial training

See GANS for one type of this.

6 References

Bach. 2014. Breaking the Curse of Dimensionality with Convex Neural Networks.” arXiv:1412.8690 [Cs, Math, Stat].
Bahadori, Chalupka, Choi, et al. 2017. Neural Causal Regularization Under the Independence of Mechanisms Assumption.” arXiv:1702.02604 [Cs, Stat].
Ba, Kiros, and Hinton. 2016. Layer Normalization.”
Baldi, Sadowski, and Lu. 2016. Learning in the Machine: Random Backpropagation and the Learning Channel.” arXiv:1612.02734 [Cs].
Bardes, Ponce, and LeCun. 2022. VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning.”
Bartlett, Montanari, and Rakhlin. 2021. Deep Learning: A Statistical Viewpoint.” Acta Numerica.
Baydin, and Pearlmutter. 2014. Automatic Differentiation of Algorithms for Machine Learning.” arXiv:1404.7456 [Cs, Stat].
Belkin. 2021. Fit Without Fear: Remarkable Mathematical Phenomena of Deep Learning Through the Prism of Interpolation.” Acta Numerica.
Belkin, Hsu, Ma, et al. 2019. Reconciling Modern Machine-Learning Practice and the Classical Bias–Variance Trade-Off.” Proceedings of the National Academy of Sciences.
Belkin, Ma, and Mandal. 2018. To Understand Deep Learning We Need to Understand Kernel Learning.” In International Conference on Machine Learning.
Bengio. 2000. Gradient-Based Optimization of Hyperparameters.” Neural Computation.
Dasgupta, Yoshizumi, and Osogami. 2016. Regularized Dynamic Boltzmann Machine with Delay Pruning for Unsupervised Learning of Temporal Sequences.” arXiv:1610.01989 [Cs, Stat].
Domingos. 2020. Every Model Learned by Gradient Descent Is Approximately a Kernel Machine.” arXiv:2012.00152 [Cs, Stat].
Finlay, Jacobsen, Nurbekyan, et al. n.d. “How to Train Your Neural ODE: The World of Jacobian and Kinetic Regularization.” In ICML.
Gal, and Ghahramani. 2016. A Theoretically Grounded Application of Dropout in Recurrent Neural Networks.” In arXiv:1512.05287 [Stat].
Golowich, Rakhlin, and Shamir. 2017. Size-Independent Sample Complexity of Neural Networks.” arXiv:1712.06541 [Cs, Stat].
Graves. 2011. Practical Variational Inference for Neural Networks.” In Proceedings of the 24th International Conference on Neural Information Processing Systems. NIPS’11.
Hardt, Recht, and Singer. 2015. Train Faster, Generalize Better: Stability of Stochastic Gradient Descent.” arXiv:1509.01240 [Cs, Math, Stat].
Immer, Bauer, Fortuin, et al. 2021. Scalable Marginal Likelihood Estimation for Model Selection in Deep Learning.” In Proceedings of the 38th International Conference on Machine Learning.
Im, Tao, and Branson. 2016. An Empirical Analysis of the Optimization of Deep Network Loss Surfaces.” arXiv:1612.04010 [Cs].
Ioffe, and Szegedy. 2015. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift.”
Izmailov, Podoprikhin, Garipov, et al. 2018. Averaging Weights Leads to Wider Optima and Better Generalization.”
Kawaguchi, Kaelbling, and Bengio. 2017. Generalization in Deep Learning.” arXiv:1710.05468 [Cs, Stat].
Kelly, Bettencourt, Johnson, et al. 2020. Learning Differential Equations That Are Easy to Solve.” In.
Klambauer, Unterthiner, Mayr, et al. 2017. Self-Normalizing Neural Networks.” In Proceedings of the 31st International Conference on Neural Information Processing Systems.
Koch, and Corso. 2016. Sparse Factorization Layers for Neural Networks with Limited Supervision.” arXiv:1612.04468 [Cs, Stat].
Lee, Xiao, Schoenholz, et al. 2019. Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent.” In Advances in Neural Information Processing Systems.
Lobacheva, Chirkova, and Vetrov. 2017. Bayesian Sparsification of Recurrent Neural Networks.” In Workshop on Learning to Generate Natural Language.
Loog, Viering, Mey, et al. 2020. A Brief Prehistory of Double Descent.” Proceedings of the National Academy of Sciences.
Maclaurin, Duvenaud, and Adams. 2015. Gradient-Based Hyperparameter Optimization Through Reversible Learning.” In Proceedings of the 32nd International Conference on Machine Learning.
Molchanov, Ashukha, and Vetrov. 2017. Variational Dropout Sparsifies Deep Neural Networks.” In Proceedings of ICML.
Nøkland. 2016. Direct Feedback Alignment Provides Learning in Deep Neural Networks.” In Advances In Neural Information Processing Systems.
Pan, Dong, and Guo. 2016. DropNeuron: Simplifying the Structure of Deep Neural Networks.” arXiv:1606.07326 [Cs, Stat].
Papyan, Romano, Sulam, et al. 2017. “Convolutional Dictionary Learning via Local Processing.” In Proceedings of the IEEE International Conference on Computer Vision.
Perez. 2016. Deep Learning: The Unreasonable Effectiveness of Randomness.” Medium (blog).
Prechelt. 2012. Early Stopping — But When? In Neural Networks: Tricks of the Trade. Lecture Notes in Computer Science 7700.
Salimans, and Kingma. 2016. Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks.” In Advances in Neural Information Processing Systems 29.
Santurkar, Tsipras, Ilyas, et al. 2019. How Does Batch Normalization Help Optimization? arXiv:1805.11604 [Cs, Stat].
Scardapane, Comminiello, Hussain, et al. 2016. Group Sparse Regularization for Deep Neural Networks.” arXiv:1607.00485 [Cs, Stat].
Srinivas, and Babu. 2016. Generalized Dropout.” arXiv:1611.06791 [Cs].
Srivastava, Hinton, Krizhevsky, et al. 2014. Dropout: A Simple Way to Prevent Neural Networks from Overfitting.” The Journal of Machine Learning Research.
Taheri, Xie, and Lederer. 2020. Statistical Guarantees for Regularized Neural Networks.” arXiv:2006.00294 [Cs, Math, Stat].
Xie, Liang, and Song. 2016. Diversity Leads to Generalization in Neural Networks.” arXiv:1611.03131 [Cs, Stat].
Xuan Vinh, Erfani, Paisitkriangkrai, et al. 2016. Training Robust Models Using Random Projection.” In.
You, Ye, Li, et al. 2018. Adversarial Noise Layer: Regularize Neural Network By Adding Noise.” arXiv:1805.08000 [Cs].
Zhang, Bengio, Hardt, et al. 2017. Understanding Deep Learning Requires Rethinking Generalization.” In Proceedings of ICLR.