Guo et al. study calibration of deep neural networks as post-processing step. Here, calibration means a correction of the predicted confidence scores as these are commonlz too overconfident in recent deep networks. They consider several state-of-the-art post-processing steps for calibration, but surprisingly, they show that a simple linear mapping, or even scaling, works surprisingly well. So if $z_i$ are the logits of the network, then (the network being fixed) a parameter $T$ is found such that

$\sigma(\frac{z_i}{T})$

is calibrated and minimized the NLL loss on a held-out validation set. Here, the temeratur $T$ either softens or roughens the probability distribution over classes. Interestingly, finding $T$ by optimizing the same training loss helps to reduce over-confidence.

Also find this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

Neelakantan et al. study gradient noise for improving neural network training. In particular, they add Gaussian noise to the gradients in each iteration:

$\tilde{\nabla}f = \nabla f + \mathcal{N}(0, \sigma^2)$

where the variance $\sigma^2$ is adapted throughout training as follows:

$\sigma^2 = \frac{\eta}{(1 + t)^\gamma}$

where $\eta$ and $\gamma$ are hyper-parameters and $t$ the current iteration. In experiments, the authors show that gradient noise has the potential to improve accuracy, especially given optimization.

Also find this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

Schmidt et al. theoretically and experimentally show that training adversarially robust models requires a higher sample complexity compared to regular generalization. Theoretically, they analyze two very simple families of datasets, e.g., consisting of two Gaussian distributions corresponding to a two-class problem. On such datasets, they proof that “robust generalization”, i.e., generalization to adversarial examples, required much higher sample complexity compared to regular generlization, i.e., generalization to the test set. These results are interesting because they suggest that the sample complexity might be even worse for more complex and realistic data distributions – as we commonly tackle in computer vision.

Experimentally, they show similar result son MNIST, CIFAR-10 and SVHN. Varying the size of the training set and plotting the accuracy on adversarially computed examples results in Figure 1. As can be seen, there seems to be a clear advantage of having larger training sets. Note that these models were trained using adversarial training using an $L_\infty$ adversary constrained by the given $\epsilon$.

https://i.imgur.com/SriBAt4.png
Figure 1: Training set size plotted against the adversarial test accuracy on MNIST, CIFAR-10 and SVHN. The models were trained using adversarial training.

Also find this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

The main contribution of [Understanding the difficulty of training deep feedforward neural networks](http://jmlr.org/proceedings/papers/v9/glorot10a/glorot10a.pdf) by Glorot et al. is a **normalized weight initialization**

$$W \sim U \left [  - \frac{\sqrt{6}}{\sqrt{n_j + n_{j+1}}}, \frac{\sqrt{6}}{\sqrt{n_j + n_{j+1}}} \right ]$$

where $n_j \in \mathbb{N}^+$ is the number of neurons in the layer $j$.

Showing some ways **how to debug neural networks** might be another reason to read the paper.

The paper analyzed standard multilayer perceptrons (MLPs) on a artificial dataset of $32 \text{px} \times 32 \text{px}$ images with either one or two of the 3 shapes: triangle, parallelogram and ellipse. The MLPs varied in the activation function which was used (either sigmoid, tanh or softsign).

However, no regularization was used and many mini-batch epochs were learned. It might be that batch normalization / dropout might change the influence of initialization very much.

Questions that remain open for me:

* [How is weight initialization done today?](https://www.reddit.com/r/MLQuestions/comments/4jsge9)
* Figure 4: Why is this plot not simply completely dependent on the data?
* Is softsign still used? Why not?
* If the only advantage of softsign is that is has the plateau later, why doesn't anybody use $\frac{1}{1+e^{-0.1 \cdot x}}$ or something similar instead of the standard sigmoid activation function?

If you were to survey researchers, and ask them to name the 5 most broadly influential ideas in Machine Learning from the last 5 years, I’d bet good money that Batch Normalization would be somewhere on everyone’s lists. Before Batch Norm, training meaningfully deep neural networks was an unstable process, and one that often took a long time to converge to success. When we added Batch Norm to models, it allowed us to increase our learning rates substantially (leading to quicker training) without the risk of activations either collapsing or blowing up in values. It had this effect because it addressed one of the key difficulties of deep networks: internal covariate shift. 

To understand this, imagine the smaller problem, of a one-layer model that’s trying to classify based on a set of input features. Now, imagine that, over the course of training, the input distribution of features moved around, so that, perhaps, a value that was at the 70th percentile of the data distribution initially is now at the 30th. We have an obvious intuition that this would make the model quite hard to train, because it would learn some mapping between feature values and class at the beginning of training, but that would become invalid by the end. This is, fundamentally, the problem faced by higher layers of deep networks, since, if the distribution of activations in a lower layer changed even by a small amount, that can cause a “butterfly effect” style outcome, where the activation distributions of higher layers change more dramatically. Batch Normalization - which takes each feature “channel” a network learns, and normalizes [normalize = subtract mean, divide by variance] it by the mean and variance of that feature over spatial locations and over all the observations in a given batch - helps solve this problem because it ensures that, throughout the course of training, the distribution of inputs that a given layer sees stays roughly constant, no matter what the lower layers get up to. 

On the whole, Batch Norm has been wildly successful at stabilizing training, and is now canonized - along with the likes of ReLU and Dropout - as one of the default sensible training procedures for any given network. However, it does have its difficulties and downsides. One salient one of these comes about when you train using very small batch sizes - in the range of 2-16 examples per batch. Under these circumstance, the mean and variance calculated off of that batch are noisy and high variance (for the general reason that statistics calculated off of small sample sizes are noisy and high variance), which takes away from the stability that Batch Norm is trying to provide. 

One proposed alternative to Batch Norm, that didn’t run into this problem of small sample sizes, is Layer Normalization. This operates under the assumption that the activations of all feature “channels” within a given layer hopefully have roughly similar distributions, and, so, you an normalize all of them by taking the aggregate mean over all channels, *for a given observation*, and use that as the mean and variance you normalize by. Because there are typically many channels in a given layer, this means that you have many “samples” that go into the mean and variance. However, this assumption - that the distributions for each feature channel are roughly the same - can be an incorrect one. 

A useful model I have for thinking about the distinction between these two approaches is the idea that both are calculating approximations of an underlying abstract notion: the in-the-limit mean and variance of a single feature channel, at a given point in time. Batch Normalization is an approximation of that insofar as it only has a small sample of points to work with, and so its estimate will tend to be high variance. Layer Normalization is an approximation insofar as it makes the assumption that feature distributions are aligned across channels: if this turns out not to be the case, individual channels will have normalizations that are biased, due to being pulled towards the mean and variance calculated over an aggregate of channels that are different than them. 

Group Norm tries to find a balance point between these two approaches, one that uses multiple channels, and normalizes within a given instance (to avoid the problems of small batch size), but, instead of calculating the mean and variance over all channels, calculates them over a group of channels that represents a subset. The inspiration for this idea comes from the fact that, in old school computer vision, it was typical to have parts of your feature vector that - for example - represented a histogram of some value (say: localized contrast) over the image. Since these multiple values all corresponded to a larger shared “group” feature. If a group of features all represent a similar idea, then their distributions will be more likely to be aligned, and therefore you have less of the bias issue. 

One confusing element of this paper for me was that the motivation part of the paper strongly implied that the reason group norm is sensible is that you are able to combine statistically dependent channels into a group together. However, as far as I an tell, there’s no actually clustering or similarity analysis of channels that is done to place certain channels into certain groups; it’s just done so semi-randomly based on the index location within the feature channel vector. So, under this implementation, it seems like the benefits of group norm are less because of any explicit seeking out of dependant channels, and more that just having fewer channels in each group means that each individual channel makes up more of the weight in its group, which does something to reduce the bias effect anyway.

The upshot of the Group Norm paper, results-wise, is that Group Norm performs better than both Batch Norm and Layer Norm at very low batch sizes. This is useful if you’re training on very dense data (e.g. high res video), where it might be difficult to store more than a few observations in memory at a time. However, once you get to batch sizes of ~24, Batch Norm starts to do better, presumably since that’s a large enough sample size to reduce variance, and you get to the point where the variance of BN is preferable to the bias of GN.