### What is BN:
* Batch Normalization (BN) is a normalization method/layer for neural networks.
* Usually inputs to neural networks are normalized to either the range of [0, 1] or [-1, 1] or to mean=0 and variance=1. The latter is called *Whitening*.
* BN essentially performs Whitening to the intermediate layers of the networks.
### How its calculated:
* The basic formula is $x^* = (x - E[x]) / \sqrt{\text{var}(x)}$, where $x^*$ is the new value of a single component, $E[x]$ is its mean within a batch and `var(x)` is its variance within a batch.
* BN extends that formula further to $x^{**} = gamma * x^* +$ beta, where $x^{**}$ is the final normalized value. `gamma` and `beta` are learned per layer. They make sure that BN can learn the identity function, which is needed in a few cases.
* For convolutions, every layer/filter/kernel is normalized on its own (linear layer: each neuron/node/component). That means that every generated value ("pixel") is treated as an example. If we have a batch size of N and the image generated by the convolution has width=P and height=Q, we would calculate the mean (E) over `N*P*Q` examples (same for the variance).
### Theoretical effects:
* BN reduces *Covariate Shift*. That is the change in distribution of activation of a component. By using BN, each neuron's activation becomes (more or less) a gaussian distribution, i.e. its usually not active, sometimes a bit active, rare very active.
* Covariate Shift is undesirable, because the later layers have to keep adapting to the change of the type of distribution (instead of just to new distribution parameters, e.g. new mean and variance values for gaussian distributions).
* BN reduces effects of exploding and vanishing gradients, because every becomes roughly normal distributed. Without BN, low activations of one layer can lead to lower activations in the next layer, and then even lower ones in the next layer and so on.
### Practical effects:
* BN reduces training times. (Because of less Covariate Shift, less exploding/vanishing gradients.)
* BN reduces demand for regularization, e.g. dropout or L2 norm. (Because the means and variances are calculated over batches and therefore every normalized value depends on the current batch. I.e. the network can no longer just memorize values and their correct answers.)
* BN allows higher learning rates. (Because of less danger of exploding/vanishing gradients.)
* BN enables training with saturating nonlinearities in deep networks, e.g. sigmoid. (Because the normalization prevents them from getting stuck in saturating ranges, e.g. very high/low values for sigmoid.)
![MNIST and neuron activations](https://raw.githubusercontent.com/aleju/papers/master/neural-nets/images/Batch_Normalization__performance_and_activations.png?raw=true "MNIST and neuron activations")
*BN applied to MNIST (a), and activations of a randomly selected neuron over time (b, c), where the middle line is the median activation, the top line is the 15th percentile and the bottom line is the 85th percentile.*
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### Rough chapter-wise notes
* (2) Towards Reducing Covariate Shift
* Batch Normalization (*BN*) is a special normalization method for neural networks.
* In neural networks, the inputs to each layer depend on the outputs of all previous layers.
* The distributions of these outputs can change during the training. Such a change is called a *covariate shift*.
* If the distributions stayed the same, it would simplify the training. Then only the parameters would have to be readjusted continuously (e.g. mean and variance for normal distributions).
* If using sigmoid activations, it can happen that one unit saturates (very high/low values). That is undesired as it leads to vanishing gradients for all units below in the network.
* BN fixes the means and variances of layer inputs to specific values (zero mean, unit variance).
* That accomplishes:
* No more covariate shift.
* Fixes problems with vanishing gradients due to saturation.
* Effects:
* Networks learn faster. (As they don't have to adjust to covariate shift any more.)
* Optimizes gradient flow in the network. (As the gradient becomes less dependent on the scale of the parameters and their initial values.)
* Higher learning rates are possible. (Optimized gradient flow reduces risk of divergence.)
* Saturating nonlinearities can be safely used. (Optimized gradient flow prevents the network from getting stuck in saturated modes.)
* BN reduces the need for dropout. (As it has a regularizing effect.)
* How BN works:
* BN normalizes layer inputs to zero mean and unit variance. That is called *whitening*.
* Naive method: Train on a batch. Update model parameters. Then normalize. Doesn't work: Leads to exploding biases while distribution parameters (mean, variance) don't change.
* A proper method has to include the current example *and* all previous examples in the normalization step.
* This leads to calculating in covariance matrix and its inverse square root. That's expensive. The authors found a faster way.
* (3) Normalization via Mini-Batch Statistics
* Each feature (component) is normalized individually. (Due to cost, differentiability.)
* Normalization according to: `componentNormalizedValue = (componentOldValue - E[component]) / sqrt(Var(component))`
* Normalizing each component can reduce the expressitivity of nonlinearities. Hence the formula is changed so that it can also learn the identiy function.
* Full formula: `newValue = gamma * componentNormalizedValue + beta` (gamma and beta learned per component)
* E and Var are estimated for each mini batch.
* BN is fully differentiable. Formulas for gradients/backpropagation are at the end of chapter 3 (page 4, left).
* (3.1) Training and Inference with Batch-Normalized Networks
* During test time, E and Var of each component can be estimated using all examples or alternatively with moving averages estimated during training.
* During test time, the BN formulas can be simplified to a single linear transformation.
* (3.2) Batch-Normalized Convolutional Networks
* Authors recommend to place BN layers after linear/fully-connected layers and before the ninlinearities.
* They argue that the linear layers have a better distribution that is more likely to be similar to a gaussian.
* Placing BN after the nonlinearity would also not eliminate covariate shift (for some reason).
* Learning a separate bias isn't necessary as BN's formula already contains a bias-like term (beta).
* For convolutions they apply BN equally to all features on a feature map. That creates effective batch sizes of m\*pq, where m is the number of examples in the batch and p q are the feature map dimensions (height, width). BN for linear layers has a batch size of m.
* gamma and beta are then learned per feature map, not per single pixel. (Linear layers: Per neuron.)
* (3.3) Batch Normalization enables higher learning rates
* BN normalizes activations.
* Result: Changes to early layers don't amplify towards the end.
* BN makes it less likely to get stuck in the saturating parts of nonlinearities.
* BN makes training more resilient to parameter scales.
* Usually, large learning rates cannot be used as they tend to scale up parameters. Then any change to a parameter amplifies through the network and can lead to gradient explosions.
* With BN gradients actually go down as parameters increase. Therefore, higher learning rates can be used.
* (something about singular values and the Jacobian)
* (3.4) Batch Normalization regularizes the model
* Usually: Examples are seen on their own by the network.
* With BN: Examples are seen in conjunction with other examples (mean, variance).
* Result: Network can't easily memorize the examples any more.
* Effect: BN has a regularizing effect. Dropout can be removed or decreased in strength.
* (4) Experiments
* (4.1) Activations over time
** They tested BN on MNIST with a 100x100x10 network. (One network with BN before each nonlinearity, another network without BN for comparison.)
** Batch Size was 60.
** The network with BN learned faster. Activations of neurons (their means and variances over several examples) seemed to be more consistent during training.
** Generalization of the BN network seemed to be better.
* (4.2) ImageNet classification
** They applied BN to the Inception network.
** Batch Size was 32.
** During training they used (compared to original Inception training) a higher learning rate with more decay, no dropout, less L2, no local response normalization and less distortion/augmentation.
** They shuffle the data during training (i.e. each batch contains different examples).
** Depending on the learning rate, they either achieve the same accuracy (as in the non-BN network) in 14 times fewer steps (5x learning rate) or a higher accuracy in 5 times fewer steps (30x learning rate).
** BN enables training of Inception networks with sigmoid units (still a bit lower accuracy than ReLU).
** An ensemble of 6 Inception networks with BN achieved better accuracy than the previously best network for ImageNet.
* (5) Conclusion
** BN is similar to a normalization layer suggested by Gülcehre and Bengio. However, they applied it to the outputs of nonlinearities.
** They also didn't have the beta and gamma parameters (i.e. their normalization could not learn the identity function).

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