07 regularization
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This document discusses various regularization techniques for deep learning models. It defines regularization as any modification to a learning algorithm intended to reduce generalization error without affecting training error. It then describes several specific regularization methods, including weight decay, norm penalties, dataset augmentation, early stopping, dropout, adversarial training, and tangent propagation. The goal of regularization is to reduce overfitting and improve generalizability of deep learning models.
07 regularization
- 1. Regularization for Deep Learning Lecture slides for Chapter 7 of Deep Learning www.deeplearningbook.org Ian Goodfellow 2016-09-27
- 2. (Goodfellow 2016) Definition • “Regularization is any modification we make to a learning algorithm that is intended to reduce its generalization error but not its training error.”
- 3. (Goodfellow 2016) Weight Decay as Constrained Optimization ARIZATION FOR DEEP LEARNING w1 w2 w⇤ ˜w Figure 7.1
- 4. (Goodfellow 2016) Norm Penalties • L1: Encourages sparsity, equivalent to MAP Bayesian estimation with Laplace prior • Squared L2: Encourages small weights, equivalent to MAP Bayesian estimation with Gaussian prior
- 6. (Goodfellow 2016) Multi-Task Learning network in figure 7.2. 2. Generic parameters, shared across all the tasks (which benefit from th pooled data of all the tasks). These are the lower layers of the neural networ in figure 7.2. h(1) h(1) h(2) h(2) h(3) h(3) y(1) y(1) y(2) y(2) h(shared) h(shared) xx Figure 7.2: Multi-task learning can be cast in several ways in deep learning frameworFigure 7.2
- 7. (Goodfellow 2016) Learning CurvesHAPTER 7. REGULARIZATION FOR DEEP LEARNING 0 50 100 150 200 250 Time (epochs) 0.00 0.05 0.10 0.15 0.20 Loss(negativelog-likelihood) Training set loss Validation set loss gure 7.3: Learning curves showing how the negative log-likelihood loss changes o Figure 7.3 Early stopping: terminate while validation set performance is better
- 8. (Goodfellow 2016) Early Stopping and Weight Decay R 7. REGULARIZATION FOR DEEP LEARNING w1 w2 w⇤ ˜w w1 w2 w⇤ ˜w Figure 7.4
- 9. (Goodfellow 2016) Sparse Representations HAPTER 7. REGULARIZATION FOR DEEP LEARNING 2 6 6 6 6 4 14 1 19 2 23 3 7 7 7 7 5 = 2 6 6 6 6 4 3 1 2 5 4 1 4 2 3 1 1 3 1 5 4 2 3 2 3 1 2 3 0 3 5 4 2 2 5 1 3 7 7 7 7 5 2 6 6 6 6 6 6 4 0 2 0 0 3 0 3 7 7 7 7 7 7 5 y 2 Rm B 2 Rm⇥n h 2 Rn (7.47) In the first expression, we have an example of a sparsely parametrized linear egression model. In the second, we have linear regression with a sparse representa- on h of the data x. That is, h is a function of x that, in some sense, represents he information present in x, but does so with a sparse vector. Representational regularization is accomplished by the same sorts of mechanisms hat we have used in parameter regularization. Norm penalty regularization of representations is performed by adding to the
- 10. (Goodfellow 2016) BaggingCHAPTER 7. REGULARIZATION FOR DEEP LEARNING 8 8 First ensemble member Second ensemble member Original dataset First resampled dataset Second resampled dataset Figure 7.5: A cartoon depiction of how bagging works. Suppose we train an 8 detector the dataset depicted above, containing an 8, a 6 and a 9. Suppose we make two differ resampled datasets. The bagging training procedure is to construct each of these data by sampling with replacement. The first dataset omits the 9 and repeats the 8. On t Figure 7.5
- 11. (Goodfellow 2016) Dropout CHAPTER 7. REGULARIZATION FOR DEEP LEARNING yy h1h1 h2h2 x1x1 x2x2 yy h1h1 h2h2 x1x1 x2x2 yy h1h1 h2h2 x2x2 yy h1h1 h2h2 x1x1 yy h2h2 x1x1 x2x2 yy h1h1 x1x1 x2x2 yy h1h1 h2h2 yy x1x1 x2x2 yy h2h2 x2x2 yy h1h1 x1x1 yy h1h1 x2x2 yy h2h2 x1x1 yy x1x1 yy x2x2 yy h2h2 yy h1h1 yy Base network Ensemble of subnetworks Figure 7.6
- 12. (Goodfellow 2016) Adversarial ExamplesCHAPTER 7. REGULARIZATION FOR DEEP LEARNING + .007 ⇥ = x sign(rxJ(✓, x, y)) x + ✏ sign(rxJ(✓, x, y)) y =“panda” “nematode” “gibbon” w/ 57.7% confidence w/ 8.2% confidence w/ 99.3 % confidence Figure 7.8: A demonstration of adversarial example generation applied to GoogLeNet (Szegedy et al., 2014a) on ImageNet. By adding an imperceptibly small vector whose elements are equal to the sign of the elements of the gradient of the cost function with respect to the input, we can change GoogLeNet’s classification of the image. Reproduced with permission from Goodfellow et al. (2014b). to optimize. Unfortunately, the value of a linear function can change very rapidly if it has numerous inputs. If we change each input by ✏, then a linear function with weights w can change by as much as ✏||w||1, which can be a very large amount if w is high-dimensional. Adversarial training discourages this highly sensitive locally linear behavior by encouraging the network to be locally constant Figure 7.8 Training on adversarial examples is mostly intended to improve security, but can sometimes provide generic regularization.
- 13. (Goodfellow 2016) Tangent Propagation ER 7. REGULARIZATION FOR DEEP LEARNING x1 x2 Normal Tangent 7.9: Illustration of the main idea of the tangent prop algorithm (Sima nd manifold tangent classifier (Rifai et al., 2011c), which both regul Figure 7.9