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!

#### Introduction

* The paper explores the domain of conditional image generation by adopting and improving PixelCNN architecture.
* [Link to the paper](https://arxiv.org/abs/1606.05328)

#### Based on PixelRNN and PixelCNN

* Models image pixel by pixel by decomposing the joint image distribution as a product of conditionals.
* PixelRNN uses two-dimensional LSTM while PixelCNN uses convolutional networks.
* PixelRNN gives better results but PixelCNN is faster to train.

#### Gated PixelCNN

* PixelRNN outperforms PixelCNN due to the larger receptive field and because they contain multiplicative units, LSTM gates, which allow modelling more complex interactions.
* To account for these, deeper models and gated activation units (equation 2 in the [paper](https://arxiv.org/abs/1606.05328)) can be used respectively.
* Masked convolutions can lead to blind spots in the receptive fields.
* These can be removed by combining 2 convolutional network stacks:
    * Horizontal stack - conditions on the current row.
    * Vertical stack - conditions on all rows above the current row.
* Every layer in the horizontal stack takes as input the output of the previous layer as well as that of the vertical stack.
* Residual connections are used in the horizontal stack and not in the vertical stack (as they did not seem to improve results in the initial settings).

#### Conditional PixelCNN

* Model conditional distribution of image, given the high-level description of the image, represented using the latent vector h (equation 4 in the [paper](https://arxiv.org/abs/1606.05328))
* This conditioning does not depend on the location of the pixel in the image.
* To consider the location as well, map h to spatial representation $s = m(h)$ (equation 5 in the the [paper](https://arxiv.org/abs/1606.05328))

#### PixelCNN Auto-Encoders

* Start with a traditional auto-encoder architecture and replace the deconvolutional decoder with PixelCNN and train the network end-to-end.

#### Experiments

* For unconditional modelling, Gated PixelCNN either outperforms PixelRNN or performs almost as good and takes much less time to train.
* In the case of conditioning on ImageNet classes, the log likelihood measure did not improve a lot but the visual quality of the generated sampled was significantly improved.
* Paper also included sample images generated by conditioning on human portraits and by training a PixelCNN auto-encoder on ImageNet patches.

#### Very Brief Summary:

This paper combines stochastic variational inference with memory-augmented recurrent neural networks. The authors test 4 variants of their models against the Variational Recurrent Neural Network on 7 artificial tasks requiring long term memory. The reported log-likelihood lower bound is not obviously improved by the new models on all tasks but is slightly better on tasks requiring high capacity memory.

#### Slightly Less Brief Summary:

The authors propose a general class of generative models for time-series data with both deterministic and stochastic latents. The deterministic latents, $h_t$, evolve as a recurrent net with augmented memory and the stochastic latents, $z_t$ are gaussians whose mean and variance are a deterministic function of $h_t$. The observations at each time-step $x_t$ are also gaussians whose mean and variance are parametrised by a function of $h_{<t}, x_{<t}$.

#### Generative Temporal Models without Augmented Memory:

The family of generative temporal models is fairly broad and includes kalman filters, non-linear dynamical systems, hidden-markov models and switching state-space models. More recent non-linear models such as the variational RNN are most similar to the new models in this paper. In general all of the mentioned temporal models can be written as: 


$P_\theta(x_{\leq T}, z_{\leq T} ) = \prod_t P_\theta(x_t | f_x(z_{\leq t}, x_{\leq t}))P_\theta(z_t | f_z(z_{\leq t}, x_{\leq t}))$

The differences between models then come from the the exact forms of  $f_x$ and $f_z$ with most models making strong conditional independence assumptions and/or having linear dependence. For example in a Gaussian State Space model both $f_x$ and $f_z$ are linear, the latents form a first order Markov chain and the observations $x_t$ are conditionally independent of everything given $z_t$.

In the Variational Recurrent Neural Net (VRNN) an additional deterministic latent variable $h_t$ is introduced and at each time-step $x_t$ is the output of a VAE whose prior $z_t$ is conditioned on $h_t$. $h_t$ evolves as an RNN.

#### Types of Model with Augmented Memory:

This paper follows the same strategy as the VRNN but adds more structure to the underlying recurrent neural net. The authors motivate this by saying that the VRNN "scales poorly when higher capacity storage is required".

* "Introspective" Model: In the first augmented memory model, the deterministic latent M_t is simply a concatenation of the last $L$ latent stochastic variables $z_t$. A soft method of attention over the latent memory is used to generate a "memory context" vector at each time step. The observed output $x_t$ is a gaussian with mean and variance parameterised by the "memory context' and the stochastic latent $z_t$. Because this model does not learn to write to memory it is faster to train.

* In the later models the memory read and write operations are the same as those in the neural turing machine or differentiable neural computer.

#### My Two Cents:

In some senses this paper feels fairly inevitable since VAE's have already been married with RNNs and so it's a small leap to add augmented memory.

The actual read write operations introduced in the "introspective" model feel a little hacky and unprincipled.

The actual images generated are quite impressive.

I'd like to see how these kind of models do on language generation tasks and wether they can be adapted for question answering.

Frankle and Carbin discover so-called winning tickets, subset of weights of a neural network that are sufficient to obtain state-of-the-art accuracy. The lottery hypothesis states that dense networks contain subnetworks – the winning tickets – that can reach the same accuracy when trained in isolation, from scratch. The key insight is that these subnetworks seem to have received optimal initialization. Then, given a complex trained network for, e.g., Cifar, weights are pruned based on their absolute value – i.e., weights with small absolute value are pruned first. The remaining network is trained from scratch using the original initialization and reaches competitive performance using less than 10% of the original weights. As soon as the subnetwork is re-initialized, these results cannot be reproduced though. This suggests that these subnetworks obtained some sort of “optimal” initialization for learning.

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

This is a nice little empirical paper that does some investigation into which features get learned during the course of neural network training. To look at this, it uses a notion of "decodability", defined as the accuracy to which you can train a linear model to predict a given conceptual feature on top of the activations/learned features at a particular layer. This idea captures the amount of information about a conceptual feature that can be extracted from a given set of activations. 

They work with two synthetic datasets. 

1. Trifeature: Generated images with a color, shape, and texture, which can be engineered to be either entirely uncorrelated or correlated with each other to varying degrees. 
2. Navon: Generated images that are letters on the level of shape, and are also composed of letters on the level of texture 

The first thing the authors investigate is: to what extent are the different properties of these images decodable from their representations, and how does that change during training? In general, decodability is highest in lower layers, and lowest in higher layers, which makes sense from the perspective of the Information Processing Inequality, since all the information is present in the pixels, and can only be lost in the course of training, not gained. They find that decodability of color is high, even in the later layers untrained networks, and that the decodability of texture and shape, while much less high, is still above chance. When the network is trained to predict one of the three features attached to an image, you see the decodability of that feature go up (as expected), but you also see the decodability of the other features go down, suggesting that training doesn't just involve amplifying predictive features, but also suppressing unpredictive ones. This effect is strongest in the Trifeature case when training for shape or color; when training for texture, the dampening effect on color is strong, but on shape is less pronounced. 

https://i.imgur.com/o45KHOM.png

The authors also performed some experiments on cases where features are engineered to be correlated to various degrees, to see which of the predictive features the network will represent more strongly. In the case where two features are perfectly correlated (and thus both perfectly predict the label), the network will focus decoding power on whichever feature had highest decodability in the untrained network, and, interestingly, will reduce decodability of the other feature (not just have it be lower than the chosen feature, but decrease it in the course of training), even though it is equally as predictive. 
https://i.imgur.com/NFx0h8b.png

Similarly, the network will choose the "easy" feature (the one more easily decodable at the beginning of training) even if there's another feature that is slightly *more* predictive available. This seems quite consistent with the results of another recent paper, Shah et al, on the Pitfalls of Simplicity Bias in neural networks. The overall message of both of these experiments is that networks generally 'put all their eggs in one basket,' so to speak, rather than splitting representational power across multiple features. 

There were a few other experiments in the paper, and I'd recommend reading it in full - it's quite well written - but I think those convey most of the key insights for me.