We want to find two matrices $W$ and $H$ such that $V = WH$. Often a goal is to determine underlying patterns in the relationships between the concepts represented by each row and column. $W$ is some $m$ by $n$ matrix and we want the inner dimension of the factorization to be $r$. So 

$$\underbrace{V}_{m \times n} = \underbrace{W}_{m \times r} \underbrace{H}_{r \times n}$$

Let's consider an example matrix where of three customers (as rows) are associated with three movies (the columns) by a rating value.

$$
V = \left[\begin{array}{c c c}
5 & 4 & 1  \\\\
4 & 5 & 1 \\\\
2 & 1 & 5
\end{array}\right]
$$


We can decompose this into two matrices with $r = 1$. First lets do this without any non-negative constraint using an SVD reshaping matrices based on removing eigenvalues:


$$
W = \left[\begin{array}{c c c}
-0.656 \\\
 -0.652 \\\
 -0.379
\end{array}\right],
H = \left[\begin{array}{c c c}
-6.48 & -6.26 & -3.20\\\\
\end{array}\right]
$$

We can also decompose this into two matrices with $r = 1$ subject to the constraint that $w_{ij} \ge 0$ and  $h_{ij} \ge 0$. (Note: this is only possible when $v_{ij} \ge 0$):

$$
W = \left[\begin{array}{c c c}
0.388 \\\\
0.386 \\\\
0.224
\end{array}\right],
H = \left[\begin{array}{c c c}
11.22 & 10.57 & 5.41  \\\\
\end{array}\right]
$$

Both of these $r=1$ factorizations reconstruct matrix $V$ with the same error. 

$$
V \approx WH = \left[\begin{array}{c c c}
4.36 & 4.11 & 2.10 \\\
4.33 & 4.08 & 2.09 \\\
2.52 & 2.37 & 1.21 \\\
\end{array}\right]
$$


If they both yield the same reconstruction error then why is a non-negativity constraint useful? We can see above that it is easy to observe patterns in both factorizations such as similar customers and similar movies. `TODO: motivate why NMF is better`



#### Paper Contribution 

This paper discusses two approaches for iteratively creating a non-negative $W$ and $H$ based on random initial matrices. The paper discusses a multiplicative update rule where the elements of $W$ and $H$ are iteratively transformed by scaling each value such that error is not increased. 

The multiplicative approach is discussed in contrast to an additive gradient decent based approach where small corrections are iteratively applied. The multiplicative approach can be reduced to this by setting the learning rate ($\eta$) to a ratio that represents the magnitude of the element in $H$ to the scaling factor of $W$ on $H$.



### Still a draft

Nayebi and Ganguli propose saturating neural networks as defense against adversarial examples. The main observation driving this paper can be stated as follows: Neural networks are essentially based on linear sums of neurons (e.g. fully connected layers, convolutiona layers) which are then activated; by injecting a small amount of noise per neuron it is possible to shift the final sum by large values, thereby propagating the noisy through the network and fooling the network into misclassifying an example. To prevent the impact of these adversarial examples, the network should be trained in a manner to drive many neurons into a saturated regime – noisy will, so the argument, have less impact then. The authors also give a biological motivation, which I won't go into detail here.

Letting $\psi$ be the used activation function, e.g. sigmoid or ReLU, a regularizer is added to drive neurons into saturation. In particular, a penalty
$\lambda \sum_l \sum_i \psi_c(h_i^l)$
is added to the loss. Here, $l$ indexes the layer and $i$ the unit in the layer; $h_i^l$ then describes the input to the non-linearity computed for unit $i$ in layer $l$. $\psi_c$ is the complementary function defined as
$\psi_c(z) = \inf_{z': \psi'(z') = 0} |z – z'|$
It defines the distance of the point $z$ to the nearest saturated point $z'$ where $\psi'(z') = 0$. For ReLU activations, the complementary function is the ReLU function itself; for sigmoid activations, the complementary function is
$\sigma_c(z) = |\sigma(z)(1 - \sigma(z))|$.
In experiments, Nayebi and Ganguli show that training with the additional penalty yields networks with higher robustness against adversarial examples compared to adversarial training (i.e. training on adversarial examples). They also provide some insight, showing e.g. the activation and weight distribution of layers illustrating that neurons are indeed saturated in large parts. For details, see the paper.

I also want to point to a comment on the paper written by Brendel and Bethge [1] questioning the effectiveness of the proposed defense strategy. They discuss a variant of the fast sign gradient method (FSGM) with stabilized gradients which is able to fool saturated networks.

[1] W. Brendel, M. Behtge. Comment on “Biologically inspired protection of deep networks from adversarial attacks”, https://arxiv.org/abs/1704.01547.

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

This paper proposes a variant of Neural Turing Machine (NTM) for meta-learning or "learning to learn", in the specific context of few-shot learning (i.e. learning from few examples). Specifically, the proposed model is trained to ingest as input a training set of examples and improve its output predictions as examples are processed, in a purely feed-forward way. This is a form of meta-learning because the model is trained so that its forward pass effectively executes a form of "learning" from the examples it is fed as input.

During training, the model is fed multiples sequences (referred to as episodes) of labeled examples $({\bf x}_1, {\rm null}), ({\bf x}_2, y_1), \dots, ({\bf x}_T, y_{T-1})$, where $T$ is the size of the episode. For instance, if the model is trained to learn how to do 5-class classification from 10 examples per class, $T$ would be $5 \times 10 = 50$. Mainly, the paper presents experiments on the Omniglot dataset, which has 1623 classes. In these experiments, classes are separated into 1200 "training classes" and 423 "test classes", and each episode is generated by randomly selecting 5 classes (each assigned some arbitrary vector representation, e.g. a  one-hot vector that is consistent within the episode, but not across episodes) and constructing a randomly ordered sequence of 50 examples from within the chosen 5 classes. Moreover, the correct label $y_t$ of a given input ${\bf x}_t$ is always provided only at the next time step, but the model is trained to be good at its prediction of the label of ${\bf x}_t$ at the current time step. This is akin to the scenario of online learning on a stream of examples, where the label of an example is revealed only once the model has made a prediction.

The proposed NTM is different from the original NTM of Alex Graves, mostly in how it writes into its memory. The authors propose to focus writing to either the least recently used memory location or the most recently used memory location. Moreover, the least recently used memory location is reset to zero before every write (an operation that seems to be ignored when backpropagating gradients).

Intuitively, the proposed NTM should learn a strategy by which, given a new input, it looks into its memory for information from other examples earlier in the episode (perhaps similarly to what a nearest neighbor classifier would do) to predict the class of the new input.

The paper presents experiments in learning to do multiclass classification on the Omniglot dataset and regression based on functions synthetically generated by a GP. The highlights are that:

1. The proposed model performs much better than an LSTM and better than an NTM with the original write mechanism of Alex Graves (for classification).
2. The proposed model even performs better than a 1st nearest neighbor classifier.
3. The proposed model is even shown to outperform human performance, for the 5-class scenario.
4. The proposed model has decent performance on the regression task, compared to GP predictions using the groundtruth kernel.

**My two cents**

This is probably one of my favorite ICML 2016 papers. I really think meta-learning is a problem that deserves more attention, and this paper presents both an interesting proposal for how to do it and an interesting empirical investigation of it. 

Much like previous work [\[1\]][1] [\[2\]][2], learning is based on automatically generating a meta-learning training set. This is clever I think, since a very large number of such "meta-learning" examples (the episodes) can be constructed, thus transforming what is normally a "small data problem" (few shot learning) into a "big data problem", for which deep learning is more effective.

I'm particularly impressed by how the proposed model outperforms a 1-nearest neighbor classifier. That said, the proposed NTM actually performs 4 reads at each time step, which suggests that a fairer comparison might be with a 4-nearest neighbor classifier. I do wonder how this baseline would compare.

I'm also impressed with the observation that the proposed model surpassed humans.

The paper also proposes to use 5-letter words to describe classes, instead of one-hot vectors. The motivation is that this should make it easier for the model to scale to much more than 5 classes. However, I don't entirely follow the logic as to why one-hot vectors are problematic. In fact, I would think that arbitrarily assigning 5-letter words to classes would instead imply some similarity between classes that share letters that is arbitrary and doesn't reflect true class similarity. 

Also, while I find it encouraging that the performance for regression of the proposed model is decent, I'm curious about how it would compare with a GP approach that incrementally learns the kernel's hyper-parameter (instead of using the groundtruth values, which makes this baseline unrealistically strong).

Finally, I'm still not 100% sure how exactly the NTM is able to implement the type of feed-forward inference I'd expect to be required. I would expect it to learn a memory representation of examples that combines information from the input vector ${\bf x}_t$ *and* its label $y_t$. However, since the label of an input is presented at the following time step in an episode, it is not intuitive to me then how the read/write mechanisms are able to deal with this misalignment. My only guess is that since the controller is an LSTM, then it can somehow remember ${\bf x}_t$ until it gets $y_t$ and appropriately include the combined information into the memory. This could be supported by the fact that using a non-recurrent feed-forward controller is much worse than using an LSTM controller. But I'm not 100% sure of this either.

All the above being said, this is still a really great paper, which I hope will help stimulate more research on meta-learning. Hopefully code for this paper can eventually be released, which would help in popularizing the topic.

[1]: http://snowedin.net/tmp/Hochreiter2001.pdf
[2]: http://www.thespermwhale.com/jaseweston/ram/papers/paper_16.pdf

_Objective:_ Design Feed-Forward Neural Network (fully connected) that can be trained even with very deep architectures.

*   _Dataset:_ [MNIST](yann.lecun.com/exdb/mnist/), [CIFAR10](https://www.cs.toronto.edu/%7Ekriz/cifar.html), [Tox21](https://tripod.nih.gov/tox21/challenge/) and [UCI tasks](https://archive.ics.uci.edu/ml/datasets/optical+recognition+of+handwritten+digits).
*   _Code:_ [here](https://github.com/bioinf-jku/SNNs)

## Inner-workings:

They introduce a new activation functio the Scaled Exponential Linear Unit (SELU) which has the nice property of making neuron activations converge to a fixed point with zero-mean and unit-variance.  
They also demonstrate that upper and lower bounds and the variance and mean for very mild conditions which basically means that there will be no exploding or vanishing gradients.

The activation function is:  
[![screen shot 2017-06-14 at 11 38 27 am](https://user-images.githubusercontent.com/17261080/27125901-1a4f7276-50f6-11e7-857d-ebad1ac94789.png)](https://user-images.githubusercontent.com/17261080/27125901-1a4f7276-50f6-11e7-857d-ebad1ac94789.png)  
With specific parameters for alpha and lambda to ensure the previous properties. The tensorflow impementation is:

    def selu(x):
        alpha = 1.6732632423543772848170429916717
        scale = 1.0507009873554804934193349852946
        return scale*np.where(x>=0.0, x, alpha*np.exp(x)-alpha)
    

They also introduce a new dropout (alpha-dropout) to compensate for the fact that [![screen shot 2017-06-14 at 11 44 42 am](https://user-images.githubusercontent.com/17261080/27126174-e67d212c-50f6-11e7-8952-acad98b850be.png)](https://user-images.githubusercontent.com/17261080/27126174-e67d212c-50f6-11e7-8952-acad98b850be.png)

## Results:

Batch norm becomes obsolete and they are also able to train deeper architectures. This becomes a good choice to replace shallow architectures where random forest or SVM used to be the best results. They outperform most other techniques on small datasets.  
[![screen shot 2017-06-14 at 11 36 30 am](https://user-images.githubusercontent.com/17261080/27125798-bd04c256-50f5-11e7-8a74-b3b6a3fe82ee.png)](https://user-images.githubusercontent.com/17261080/27125798-bd04c256-50f5-11e7-8a74-b3b6a3fe82ee.png)

Might become a new standard for fully-connected activations in the future.

This paper introduces a deep universal word embedding based on using a bidirectional LM (in this case, biLSTM). First words are embedded with a CNN-based, character-level, context-free, token embedding into $x_k^{LM}$ and then each sentence is parsed using a biLSTM, maximizing the log-likelihood of a word given it's forward and backward context (much like a normal language model). 

The innovation is in taking the output of each layer of the LSTM ($h_{k,j}^{LM}$ being the output at layer $j$) 
$$
\begin{align}
R_k &= \{x_k^{LM}, \overrightarrow{h}_{k,j}^{LM}, \overleftarrow{h}_{k,j}^{LM} | j = 1 \ldots L \} \\
&= \{h_{k,j}^{LM} | j = 0 \ldots L \}
\end{align}
$$
and allowing the user to learn a their own task-specific weighted sum of these hidden states as the embedding:
$$
ELMo_k^{task} = \gamma^{task} \sum_{j=0}^L s_j^{task} h_{k,j}^{LM}
$$

The authors show that this weighted sum is better than taking only the top LSTM output (as in their previous work or in CoVe) because it allows capturing syntactic information in the lower layer of the LSTM and semantic information in the higher level. Table below shows that the second layer is more useful for the semantic task of word sense disambiguation, and the first layer is more useful for the syntactic task of POS tagging.

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

On other benchmarks, they show it is also better than taking the average of the layers (which could be done by setting $\gamma = 1$)
https://i.imgur.com/f78gmKu.png

To add the embeddings to your supervised model,  ELMo is concatenated with your context-free embeddings, $\[ x_k; ELMo_k^{task} \]$. It can also be concatenated with the output of your RNN model $\[ h_k; ELMo_k^{task} \]$ which can show improvements on the same benchmarks

https://i.imgur.com/eBqLe8G.png

Finally, they show that adding ELMo to a competitive but simple baseline gets SOTA (at the time) on very many NLP benchmarks

https://i.imgur.com/PFUlgh3.png

It's all open-source and there's a tutorial [here](https://github.com/allenai/allennlp/blob/master/tutorials/how_to/elmo.md)