Deeper networks should never have a higher **training** error than smaller ones. In the worst case, the layers should "simply" learn identities. It seems as this is not so easy with conventional networks, as they get much worse with more layers. So the idea is to add identity functions which skip some layers. The network only has to learn the **residuals**. 

Advantages:

* Learning the identity becomes learning 0 which is simpler
* Loss in information flow in the forward pass is not a problem anymore
    * No vanishing / exploding gradient
* Identities don't have parameters to be learned

## Evaluation

The learning rate starts at 0.1 and is divided by 10 when the error plateaus. Weight decay of 0.0001 ($10^{-4}$), momentum of 0.9. They use mini-batches of size 128.

* ImageNet ILSVRC 2015: 3.57% (ensemble)
* CIFAR-10: 6.43%
* MS COCO: 59.0% mAp@0.5 (ensemble)
* PASCAL VOC 2007: 85.6% mAp@0.5
* PASCAL VOC 2012: 83.8% mAp@0.5

## See also

* [DenseNets](http://www.shortscience.org/paper?bibtexKey=journals/corr/1608.06993)

In this tutorial paper, Carl E. Rasmussen gives an introduction to Gaussian Process Regression focusing on the definition, the hyperparameter learning and future research directions.

A Gaussian Process is completely defined by its mean function $m(\pmb{x})$ and its covariance function (kernel) $k(\pmb{x},\pmb{x}')$. The mean function $m(\pmb{x})$ corresponds to the mean vector $\pmb{\mu}$ of a Gaussian distribution whereas the covariance function $k(\pmb{x}, \pmb{x}')$ corresponds to the covariance matrix $\pmb{\Sigma}$. Thus, a Gaussian Process $f \sim \mathcal{GP}\left(m(\pmb{x}), k(\pmb{x}, \pmb{x}')\right)$ is a generalization of a Gaussian distribution over vectors to a distribution over functions. A random function vector $\pmb{\mathrm{f}}$ can be generated by a Gaussian Process through the following procedure:

1. Compute the components $\mu_i$ of the mean vector $\pmb{\mu}$ for each input $\pmb{x}_i$ using the mean function $m(\pmb{x})$
2. Compute the components $\Sigma_{ij}$ of the covariance matrix $\pmb{\Sigma}$ using the covariance function $k(\pmb{x}, \pmb{x}')$
3. A function vector $\pmb{\mathrm{f}} = [f(\pmb{x}_1), \dots, f(\pmb{x}_n)]^T$ can be drawn from the Gaussian distribution $\pmb{\mathrm{f}} \sim \mathcal{N}\left(\pmb{\mu}, \pmb{\Sigma} \right)$

Applying this procedure to regression, means that the resulting function vector $\pmb{\mathrm{f}}$ shall be drawn in a way that a function vector $\pmb{\mathrm{f}}$ is rejected if it does not comply with the training data $\mathcal{D}$. This is achieved by conditioning the distribution on the training data $\mathcal{D}$ yielding the posterior Gaussian Process $f \rvert \mathcal{D} \sim \mathcal{GP}(m_D(\pmb{x}), k_D(\pmb{x},\pmb{x}'))$ for noise-free observations with the posterior mean function $m_D(\pmb{x}) = m(\pmb{x}) + \pmb{\Sigma}(\pmb{X},\pmb{x})^T \pmb{\Sigma}^{-1}(\pmb{\mathrm{f}} - \pmb{\mathrm{m}})$  and the posterior covariance function $k_D(\pmb{x},\pmb{x}')=k(\pmb{x},\pmb{x}') - \pmb{\Sigma}(\pmb{X}, \pmb{x}')$ with $\pmb{\Sigma}(\pmb{X},\pmb{x})$ being a vector of covariances between every training case of $\pmb{X}$ and $\pmb{x}$.

Noisy observations $y(\pmb{x}) = f(\pmb{x}) + \epsilon$ with $\epsilon \sim \mathcal{N}(0,\sigma_n^2)$ can be taken into account with a second Gaussian Process with mean $m$ and covariance function $k$ resulting in $f \sim \mathcal{GP}(m,k)$ and $y \sim \mathcal{GP}(m, k + \sigma_n^2\delta_{ii'})$. The figure illustrates the cases of noisy observations (variance at training points) and of noise-free observationshttps://i.imgur.com/BWvsB7T.png (no variance at training points).

In the Machine Learning perspective, the mean and the covariance function are parametrised by hyperparameters and provide thus a way to include prior knowledge e.g. knowing that the mean function is a second order polynomial. To find the optimal hyperparameters $\pmb{\theta}$, 

1. determine the log marginal likelihood $L= \mathrm{log}(p(\pmb{y} \rvert \pmb{x}, \pmb{\theta}))$,
2. take the first partial derivatives of $L$ w.r.t. the hyperparameters, and
3. apply an optimization algorithm.

It should be noted that a regularization term is not necessary for the log marginal likelihood $L$ because it already contains a complexity penalty term. Also, the tradeoff between data-fit and penalty is performed automatically.

Gaussian Processes provide a very flexible way for finding a suitable regression model. However, they require the high computational complexity $\mathcal{O}(n^3)$ due to the inversion of the covariance matrix. In addition, the generalization of Gaussian Processes to non-Gaussian likelihoods remains complicated. 

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.  

This paper presents an interpretation of dropout training as performing approximate Bayesian learning in a deep Gaussian process (DGP) model. This connection suggests a very simple way of obtaining, for networks trained with dropout, estimates of the model's output uncertainty. This estimate is based and computed from an ensemble of networks each obtained by sampling a new dropout mask.

#### My two cents

This is a really nice and thought provoking contribution to our understanding of dropout. Unfortunately, the paper in fact doesn't provide a lot of comparisons with either other ways of estimating the predictive uncertainty of deep networks, or to other approximate inference schemes in deep GPs (actually, see update below). The qualitative examples provided however do suggest that the uncertainty estimate isn't terrible.

Irrespective of the quality of the uncertainty estimate suggested here, I find the observation itself really valuable. Perhaps future research will then shed light on how useful that method is compared to other approaches, including Bayesian dark knowledge \cite{conf/nips/BalanRMW15}.

`Update: On September 27th`, the authors uploaded to arXiv a new version that now includes comparisons with 2 alternative Bayesian learning methods for deep networks, specifically the stochastic variational inference approach of Graves and probabilistic back-propagation of Hernandez-Lobato and Adams. Dropout actually does very well against these baselines and, across datasets, is almost always amongst the best performing method!

TLDR; The authors propose a new normalization scheme called "Layer Normalization" that works especially well for recurrent networks. Layer Normalization is similar to Batch Normalization, but only depends on a single training case. As such, it's well suited for variable length sequences or small batches. In Layer Normalization each hidden unit shares the same normalization term. The authors show through experiments that Layer Normalization converges faster, and sometimes to better solutions, than batch- or unnormalized RNNs. Batch normalization still performs better for CNNs.

This paper deals with the question what / how exactly CNNs learn, considering the fact that they usually have more trainable parameters than data points on which they are trained.

When the authors write "deep neural networks", they are talking about Inception V3, AlexNet and MLPs.

## Key contributions

* Deep neural networks easily fit random labels (achieving a training error of 0 and a test error which is just randomly guessing labels as expected). $\Rightarrow$Those architectures can simply brute-force memorize the training data.
* Deep neural networks fit random images (e.g. Gaussian noise) with 0 training error. The authors conclude that VC-dimension / Rademacher complexity, and uniform stability are bad explanations for generalization capabilities of neural networks
* The authors give a construction for a 2-layer network with $p = 2n+d$ parameters - where $n$ is the number of samples and $d$ is the dimension of each sample - which can easily fit any labeling. (Finite sample expressivity). See section 4.


## What I learned

* Any measure $m$ of the generalization capability of classifiers $H$ should take the percentage of corrupted labels ($p_c \in [0, 1]$, where $p_c =0$ is a perfect labeling and $p_c=1$ is totally random) into account: If $p_c = 1$, then $m()$ should be 0, too, as it is impossible to learn something meaningful with totally random labels.
* We seem to have built models which work well on image data in general, but not "natural" / meaningful images as we thought.


## Funny

> deep neural nets remain mysterious for many reasons

> Note that this is not exactly simple as the kernel matrix requires 30GB to store in memory. Nonetheless, this system can be solved in under 3 minutes in on a commodity workstation with 24 cores and 256 GB of RAM with a conventional LAPACK call.


## See also

* [Deep Nets Don't Learn Via Memorization](https://openreview.net/pdf?id=rJv6ZgHYg)

disclaimer: I'm the first author of the paper

## TL;DR 

We have made a lot of progress on catastrophic forgetting within the standard evaluation protocol,
i.e. sequentially learning a stream of tasks and testing our models' capacity to remember them all.

We think it's time a new approach to Continual Learning (CL), coined OSAKA, which is more aligned with real-life applications of CL. It brings CL closer to Online Learning and Open-World Learning.

main modifications we propose:
- bring CL closer to Online learning i.e. at test time, the model is continually learning and evaluated on its online predictions 
- it's fine to forget, as long as you can quickly remember (just like we humans do)
- we allow pretraining, (because you wouldn't deploy an untrained CL system, right?) but at test time, the model will have to quickly learn new out-of-distribution (OoD) tasks (because the world is full of surprises)
- the tasks distribution is actually a hidden Markov chain. This implies:
    - new and old tasks can re-occur (just like in real life). Better remember them quickly if you want to get a good total performance!
    - tasks have different lengths
    - and the tasks boundaries are unknown (task agnostic setting)

### Bonus:
We provide a unifying framework explaining the space of machine learning setting {supervised learning, meta learning, continual learning, meta-continual learning, continual-meta learning} in case it was starting to get confusing :p

## Motivation

We imagine an agent, embedded or not, first pre-trained in a controlled environment and later deployed in the real world, where it faces new or unexpected situations. This scenario is relevant for many applications. For instance, in robotics, the agent is pre-trained in a factory and deployed in homes or
in manufactures where it will need to adapt to new domains and maybe solve new tasks. Likewise, a virtual assistant can be pre-trained on static datasets and deployed in a user’s life to fit its personal needs. 

Further motivations can be found in time series forecasting, e.g., market prediction,
game playing, autonomous customer service, recommendation systems, autonomous driving, to name a few. In this scenario, we are interested in the cumulative performance of the agent throughout its lifetime. Differently, standard CL reports the agent’s final performance on all tasks at the end of its life. In order
to succeed in this scenario, agents need the ability to learn new tasks as well as quickly remembering old ones.

## Unifying Framework

We propose a unifying framework explaining the space of machine learning setting {supervised learning, meta learning, continual learning, meta-continual learning, continual-meta learning} with meta learning terminology. 

https://i.imgur.com/U16kHXk.png

(easier to digest with accompanying text)

## OSAKA

The main features of the evaluation framework are
- task agnosticism
- pre-training is allowed, but OoD tasks at test time
- task revisiting
- controllable non-stationarity
- online evaluation

(see paper for the motivations of the features)

## Continual-MAML: an initial baseline

A simple extension of MAML that is better suited than previous methods in the proposed setting.

https://i.imgur.com/C86WUc8.png

Features are:
- Fast Adapatation
- Dynamic Representation
- Task boundary detection
- Computational efficiency


## Experiments

We provide a suite of 3 benchmarks to test algorithms in the new setting.

The first includes the Omniglot, MNIST and FashionMNIST dataset. 

The second and third use the Synbols (Lacoste et al. 2018) and TieredImageNet datasets, respectively.

The first set of experiments shows that the baseline outperforms previous approaches, i.e., supervised learning, meta learning, continual learning, meta-continual learning, continual-meta learning, in the new setting.

https://i.imgur.com/IQ1WYTp.png

The second and third experiments lead us to similar conclusions

code: https://github.com/ElementAI/osaka

Hosseini and Poovendran propose semantic adversarial examples by randomly manipulating hue and saturation of images. In particular, in an iterative algorithm, hue and saturation are randomly perturbed and projected back to their valid range. If this results in mis-classification the perturbed image is returned as the adversarial example and the algorithm is finished; if not, another iteration is run. The result is shown in Figure 1. As can be seen, the structure of the images is retained while hue and saturation changes, resulting in mis-classified images.

https://i.imgur.com/kFcmlE3.jpg
Figure 1: Examples of the computed semantic adversarial examples.

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

During the past few years, sellers have increasingly offered discounted or free products to selected reviewers of ecommerce platforms in exchange for their reviews. Such incentivized (and often very positive) reviews can improve the rating of a product which in turn sways other users’ opinions about the product. 

Here, we examine the problem of detecting and characterizing incentivized reviews in two primary categories of Amazon products.  We show that the key features of EIRs and normal reviews exhibit different characteristics. 

https://i.imgur.com/1BdwsK4.png

Furthermore, we illustrate how the prevalence of EIRs has evolved and been affected by Amazon’s ban. 

https://i.imgur.com/1XsSaxX.png

Our examination of the temporal pattern of submitted reviews for sample products reveals promotional campaigns by the corresponding sellers and their effectiveness in attracting other users. 
https://i.imgur.com/TvQiDlc.png

Finally, we demonstrate that a classifier that is trained by EIRs (without explicit keywords) and normal reviews can accurately detect other EIRs as well as implicitly incentivized reviews. Overall, this analysis sheds an insightful light on the impact of EIRs on Amazon products and users.

Bafna et al. show that iterative hard thresholding results in $L_0$ robust Fourier transforms. In particular, as shown in Algorithm 1, iterative hard thresholding assumes a signal $y = x + e$ where $x$ is assumed to be sparse, and $e$ is assumed to be sparse. This translates to noise $e$ that is bounded in its $L_0$ norm, corresponding to common adversarial attacks such as adversarial patches in computer vision. Using their algorithm, the authors can provably reconstruct the signal, specifically the top-$k$ coordinates for a $k$-sparse signal, which can subsequently be fed to a neural network classifier. In experiments, the classifier is always trained on sparse signals, and at test time, the sparse signal is reconstructed prior to the forward pass. This way, on MNIST and Fashion-MNIST, the algorithm is able to recover large parts of the original accuracy.

https://i.imgur.com/yClXLoo.jpg
Algorithm 1 (see paper for details): The iterative hard thresholding algorithm resulting in provable robustness against $L_0$ attack on images and other signals.

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