This paper proposes a neural architecture that allows to backpropagate gradients though a procedure that can go through a variable and adaptive number of iterations. These "iterations" for instance could be the number of times computations are passed through the same recurrent layer (connected to the same input) before producing an output, which is the case considered in this paper. 

This is essentially achieved by pooling the recurrent states and respective outputs computed by each iteration. The pooling mechanism is essentially the same as that used in the really cool Neural Stack architecture of Edward Grefenstette, Karl Moritz Hermann, Mustafa Suleyman and Phil Blunsom \cite{conf/nips/GrefenstetteHSB15}. It relies on the introduction of halting units, which are sigmoidal units computed at each iteration and which gives a soft weight on whether the computation should stop at the current iteration.

Crucially, the paper introduces a new ponder cost $P(x)$, which is a regularization cost that penalizes what is meant to be a smooth upper bound on the number of iterations $N(t)$ (more on that below).

The paper presents experiment on RNNs applied on sequences where, at each time step t (not to be confused with what I'm calling computation iterations, which are indexed by n) in the sequence the RNN can produce a variable number $N(t)$ of intermediate states and outputs. These are the states and outputs that are pooled, to produce a single recurrent state and output for the time step t. During each of the $N(t)$ iterations at time step t, the intermediate states are connected to the same time-step-t input. After the $N(t)$ iterations, the RNN pools the $N(t)$ intermediate states and outputs, and then moves to the next time step $t+1$. To mark the transitions between time steps, an extra binary input is appended, which is 1 only for the first intermediate computation iteration. 

Results are presented on a variety of synthetic problems and a character prediction problem.

Basically they observe a pattern they call The Filter Lottery (TFL) where the random seed causes a high variance  in the training accuracy:

![](http://i.imgur.com/5rWig0H.png)

They use the convolutional gradient norm ($CGN$) \cite{conf/fgr/LoC015} to determine how much impact a filter has on the overall classification loss function by taking the derivative of the loss function with respect each weight in the filter.

$$CGN(k) = \sum_{i} \left|\frac{\partial L}{\partial w^k_i}\right|$$

They use the CGN to evaluate the impact of a filter on error, and re-initialize filters when the gradient norm of its weights falls below a specific threshold.

Zahavy et al. introduce the concept of ensemble robustness and show that it can be used as indicator for generalization performance. In particular, the main idea is to lift he concept of robustness against adversarial examples to ensemble of networks – as trained, e.g. through Dropout or Bayes-by-Backprop. Letting $Z$ denote the sample set, a learning algorithm is $(K, \epsilon)$ robust if $Z$ can be divided into $K$ disjoint sets $C_1,\ldots,C_K$ such that for every training set $s_1,\ldots,s_n  \in Z$ it holds:

$\forall i, \forall z \in Z, \forall k = 1,\ldots, K$: if $s,z \in C_k$, then $l(f,s_i) – l(f,z)| \leq \epsilon(s_1,\ldots,s_n)$

where $f$ is the model produced by the learning algorithm, $l$ measures the loss and $\epsilon:Z^n \mapsto \mathbb{R}$. For ensembles (explicit or implicit) this definition is extended by considering the maximum generalization loss under the expectation of a randomized learning algorithm:

$\forall i, \forall k = 1,\ldots,K$: if $s \in C_k$, then $\mathbb{E}_f \max_{z \in C_k} |l(f,s_i) – l(f,z)| \leq \epsilon(s_1,\ldots,s_n)$

Here, the randomized learning algorithm computes a distribution over models given a training set.

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

TLDR; The authors introduce CopyNet, a variation on the seq2seq that incorporates a "copying mechanism". With this mechanism, the effective vocabulary is the union of the standard vocab and the words in the current source sentence. CopyNet predicts words based on a mixed probability of the standard attention mechanism and a new copy mechanism. The authors show empirically that on toy and summarization tasks CopNet behaves as expected: The decoder is dominated by copy mode when it tries to replicate something from the source.

This paper presents a novel neural network approach (though see [here](https://www.facebook.com/hugo.larochelle.35/posts/172841743130126?pnref=story) for a discussion on prior work) to density estimation, with a focus on image modeling. At its core, it exploits the following property on the densities of random variables. Let $x$ and $z$ be two random variables of equal dimensionality such that $x = g(z)$, where $g$ is some bijective and deterministic function (we'll note its inverse as $f = g^{-1}$). Then the change of variable formula gives us this relationship between the densities of $x$ and $z$:

$p_X(x) = p_Z(z) \left|{\rm det}\left(\frac{\partial g(z)}{\partial z}\right)\right|^{-1}$

Moreover, since the determinant of the Jacobian matrix of the inverse $f$ of a function $g$ is simply the inverse of the Jacobian of the function $g$, we can also write:

$p_X(x) = p_Z(f(x)) \left|{\rm det}\left(\frac{\partial f(x)}{\partial x}\right)\right|$

where we've replaced $z$ by its deterministically inferred value $f(x)$ from $x$.

So, the core of the proposed model is in proposing a design for bijective functions $g$ (actually, they design its inverse $f$, from which $g$ can be derived by inversion), that have the properties of being easily invertible and having an easy-to-compute determinant of Jacobian. Specifically, the authors propose to construct $f$ from various modules that all preserve these properties and allows to construct highly non-linear $f$ functions. Then, assuming a simple choice for the density $p_Z$ (they use a multidimensional Gaussian), it becomes possible to both compute $p_X(x)$ tractably and to sample from that density, by first samples $z\sim p_Z$ and then computing $x=g(z)$.

The building blocks for constructing $f$ are the following:

**Coupling layers**: This is perhaps the most important piece. It simply computes as its output $b\odot x + (1-b) \odot (x \odot \exp(l(b\odot x)) + m(b\odot x))$, where $b$ is a binary mask (with half of its values set to 0 and the others to 1) over the input of the layer $x$, while $l$ and $m$ are arbitrarily complex neural networks with input and output layers of equal dimensionality. 

In brief, for dimensions for which $b_i = 1$ it simply copies the input value into the output. As for the other dimensions (for which $b_i = 0$) it linearly transforms them as $x_i * \exp(l(b\odot x)_i) + m(b\odot x)_i$. Crucially, the bias ($m(b\odot x)_i$) and coefficient ($\exp(l(b\odot x)_i)$) of the linear transformation are non-linear transformations (i.e. the output of neural networks) that only have access to the masked input (i.e. the non-transformed dimensions). While this layer might seem odd, it has the important property that it is invertible and the determinant of its Jacobian is simply $\exp(\sum_i (1-b_i) l(b\odot x)_i)$. See Section 3.3 for more details on that.

**Alternating masks**: One important property of coupling layers is that they can be stacked (i.e. composed), and the resulting composition is still a bijection and is invertible (since each layer is individually a bijection) and has a tractable determinant for its Jacobian (since the Jacobian of the composition of functions is simply the multiplication of each function's Jacobian matrix, and the determinant of the product of square matrices is the product of the determinant of each matrix). This is also true, even if the mask $b$ of each layer is different. Thus, the authors propose using masks that alternate across layer, by masking a different subset of (half of) the dimensions. For images, they propose using masks with a checkerboard pattern (see Figure 3). Intuitively, alternating masks are better because then after at least 2 layers, all dimensions have been transformed at least once.

**Squeezing operations**: Squeezing operations corresponds to a reorganization of a 2D spatial layout of dimensions into 4 sets of features maps with spatial resolutions reduced by half (see Figure 3). This allows to expose multiple scales of resolutions to the model. Moreover, after a squeezing operation, instead of using a checkerboard pattern for masking, the authors propose to use a per channel masking pattern, so that "the resulting partitioning is not redundant with the previous checkerboard masking". See Figure 3 for an illustration.

Overall, the models used in the experiments usually stack a few of the following "chunks" of layers: 1) a few coupling layers with alternating checkboard masks, 2) followed by squeezing, 3) followed by a few coupling layers with alternating channel-wise masks. Since the output of each layers-chunk must technically be of the same size as the input image, this could become expensive in terms of computations and space when using a lot of layers. Thus, the authors propose to explicitly pass on (copy) to the very last layer ($z$) half of the dimensions after each layers-chunk, adding another chunk of layers only on the other half. This is illustrated in Figure 4b.

Experiments on CIFAR-10, and 32x32 and 64x64 versions of ImageNet show that the proposed model (coined the real-valued non-volume preserving or Real NVP) has competitive performance (in bits per dimension), though slightly worse than the Pixel RNN.

**My Two Cents**

The proposed approach is quite unique and thought provoking. Most interestingly, it is the only powerful generative model I know that combines A) a tractable likelihood, B) an efficient / one-pass sampling procedure and C) the explicit learning of a latent representation. While achieving this required a model definition that is somewhat unintuitive, it is nonetheless mathematically really beautiful!

I wonder to what extent Real NVP is penalized in its results by the fact that it models pixels as real-valued observations. First, it implies that its estimate of bits/dimensions is an upper bound on what it could be if the uniform sub-pixel noise was integrated out (see Equations 3-4-5 of [this paper](http://arxiv.org/pdf/1511.01844v3.pdf)). Moreover, the authors had to apply a non-linear transformation (${\rm logit}(\alpha + (1-\alpha)\odot x)$) to the pixels, to spread the $[0,255]$ interval further over the reals. Since the Pixel RNN models pixels as discrete observations directly, the Real NVP might be at a disadvantage.

I'm also curious to know how easy it would be to do conditional inference with the Real NVP. One could imagine doing approximate MAP conditional inference, by clamping the observed dimensions and doing gradient descent on the log-likelihood with respect to the value of remaining dimensions. This could be interesting for image completion, or for structured output prediction with real-valued outputs in general. I also wonder how expensive that would be.

In all cases, I'm looking forward to saying interesting applications and variations of this model in the future!