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--- data_mining:neural_network:regularization [2017/08/19 20:14] – [More regularization techniques] phreazer
+++ data_mining:neural_network:regularization [2017/08/19 20:24] (current) – gelöscht phreazer
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-====== Regularization ======
-Solves Variance problems (See [[data_mining:error_analysis|Error Analysis]])
-==== Overfitting (How well does the network generalize?) ====
-  * Target values unreliable?
-  * Sampling errors (accidental regularities of particular training cases)
-Regularization Methods:
-  * Weight decay (small weights, simpler model)
-  * Weight-sharing (same weights)
-  * Early-stopping (Fake testset, when performance gets worse, stop training)
-  * Model-Averaging
-  * Bayes fitting (like model averaging)
-  * Dropout (Randomly ommit hidden units)
-  * Generative pre-training)
-=== $L_1$ and $L_2$ regularization ===
-== Example for logistic regression: ==
-$L_2$ regularization:
-E.g. for Logistic regression, add to cost function $J$: $\dots + \frac{\lambda}{2m} ||w||^2_2 = \dots + \sum_{j=i}^{n_x} w_j^2 = \dots + w^T w$
-$L_1$ regularization:
-$\frac{\lambda}{2m} ||w||_1$
-$w$ will be sparse.
-Use hold-out test set to set hyperparameter.
-== Neural Network ==
-Cost function
-$J(\dots) = 1/m \sum_{i=1}^n L(\hat{y}^{i}, y^{i}) + \frac{\lambda}{2m} \sum_{l=1}^L ||W^{[l]}||_F^2$
-Frobenius Norm: $||W^{[l]}||_F^2$
-For gradient descent:
-$dW^{[l]} = \dots + \frac{\lambda}{m} W^{[l]}$
-Called **weight decay** (additional multiplication with weights)
-Large $\lambda$: Every layer ~ linear; z small range of values (in case of tanh activation fct)
-===== Dropout =====
-Ways to combine output of multiple models:
-  * MIXTURE: Combine models by averaging their output probabilities.
-  * PRODUCT: by geometric mean (typically less than one) $\sqrt{x*y}/ \sum$
-NN with one hidden layer.
-Randomly omit each hidden unit with probability 0.5, for each training sample.
-Randomly sampling from 2^H architextures.
-Sampling form 2^H models, and each model only gets one training example (extreme bagging)
-Sharing of the weights means that every model is very strongly regularized.
-What to do at test time?
-Use all hidden units, but halve their outgoing weights. This exactly computes the geometric mean of the predictions of all 2^H models.
-What if we have more hidden Layers?
-* Use dropout of 0.5 in every layer.
-* At test time, use mean net, that has all outgoing weights halved. Not the same, as averaging all separate dropped out models, but approximation.
-Dropout prevents overfitting.
-For each training example: For each node toss a coin, e.g. with prob 0.5 and eleminate nodes.
-==== Inverted dropout ====
-Layer $l=3$.
-$keep.prob = 0.8$
-$d3 = np.random.rand(a3.shape[0], a3.shape[i]) < keep.prob$
-$a3 = np.multiply(a3,d3)$
-$a3 /= keep.prob$ // e.g. 50 units => 10 units shut off
-$Z = Wa+b$ // reduced by 20% => standardize with 0.8 => expected value stays the same
-Making predictions at test time: No drop out
-**Why does it work?**
-Intuition: Can't rely on any one feature, so spread out of weights => shrink weights.
-Different keep.probs can be set for different layers (e.g. layers with a lot of parameters).
-Used in computer vision.
-Downside: J is not longer well-defined. Performance check problematic (e.g. can set keepprob to one).