This paper presents a method to train a neural network to make predictions for *counterfactual* questions. In short, such questions are questions about what the result of an intervention would have been, had a different choice for the intervention been made (e.g. *Would this patient have lower blood sugar had she received a different medication?*).

One approach to tackle this problem is to collect data of the form $(x_i, t_i, y_i^F)$ where $x_i$ describes a situation (e.g. a patient), $t_i$ describes the intervention made (in this paper $t_i$ is binary, e.g. $t_i = 1$ if a new treatment is used while $t_i = 0$ would correspond to using the current treatment) and $y_i^F$ is the factual outcome of the intervention $t_i$ for $x_i$. From this training data, a predictor $h(x,t)$ taking the pair $(x_i, t_i)$ as input and outputting a prediction for $y_i^F$ could be trained. 

From this predictor, one could imagine answering counterfactual questions by feeding $(x_i, 1-t_i)$ (i.e. a description of the same situation $x_i$ but with the opposite intervention $1-t_i$) to our predictor and comparing the prediction $h(x_i, 1-t_i)$ with $y_i^F$. This would give us an estimate of the change in the outcome, had a different intervention been made, thus providing an answer to our counterfactual question.

The authors point out that this scenario is related to that of domain adaptation (more specifically to the special case of covariate shift) in which the input training distribution (here represented by inputs $(x_i,t_i)$) is different from the distribution of inputs that will be fed at test time to our predictor (corresponding to the inputs $(x_i, 1-t_i)$). If the choice of intervention $t_i$ is evenly spread and chosen independently from $x_i$, the distributions become the same. However, in observational studies, the choice of $t_i$ for some given $x_i$ is often not independent of $x_i$ and made according to some unknown policy. This is the situation of interest in this paper.

Thus, the authors propose an approach inspired by the domain adaptation literature. Specifically, they propose to have the predictor $h(x,t)$ learn a representation of $x$ that is indiscriminate of the intervention $t$ (see Figure 2 for the proposed neural network architecture). Indeed, this is a notion that is [well established][1] in the domain adaptation literature and has been exploited previously using regularization terms based on [adversarial learning][2] and [maximum mean discrepancy][3]. In this paper, the authors used instead a regularization (noted in the paper as $disc(\Phi_{t=0},\Phi_ {t=1})$) based on the so-called discrepancy distance of [Mansour et al.][4], adapting its use to the case of a neural network. 

As an example, imagine that in our dataset, a new treatment ($t=1$) was much more frequently used than not ($t=0$) for men. Thus, for men, relatively insufficient evidence for counterfactual inference is expected to be found in our training dataset. Intuitively, we would thus want our predictor to not rely as much on that "feature" of patients when inferring the impact of the treatment. 

In addition to this term, the authors also propose incorporating an additional regularizer where the prediction $h(x_i,1-t_i)$ on counterfactual inputs is pushed to be as close as possible to the target $y_{j}^F$ of the observation $x_j$ that is closest to $x_i$ **and** actually had the counterfactual intervention $t_j = 1-t_i$. 

The paper first shows a bound relating the counterfactual generalization error to the discrepancy distance. Moreover, experiments simulating counterfactual inference tasks are presented, in which performance is measured by comparing the predicted treatment effects (as estimated by the difference between the observed effect $y_i^F$ for the observed treatment and the predicted effect $h(x_i, 1-t_i)$ for the opposite treatment) with the real effect (known here because the data is simulated). The paper shows that the proposed approach using neural networks outperforms several baselines on this task.

**My two cents**

The connection with domain adaptation presented here is really clever and enlightening. This sounds like a very compelling approach to counterfactual inference, which can exploit a lot of previous work on domain adaptation.

The paper mentions that selecting the hyper-parameters (such as the regularization terms weights) in this scenario is not a trivial task. Indeed, measuring performance here requires knowing the true difference in intervention outcomes, which in practice usually cannot be known (e.g. two treatments usually cannot be given to the same patient once). In the paper, they somewhat "cheat" by using the ground truth difference in outcomes to measure out-of-sample performance, which the authors admit is unrealistic. Thus, an interesting avenue for future work would be to design practical hyper-parameter selection procedures for this scenario. I wonder whether the *reverse cross-validation* approach we used in our work on our adversarial approach to domain adaptation (see [Section 5.1.2][5]) could successfully be used here.

Finally, I command the authors for presenting such a nicely written description of counterfactual inference problem setup in general, I really enjoyed it!

[1]: https://papers.nips.cc/paper/2983-analysis-of-representations-for-domain-adaptation.pdf
[2]: http://arxiv.org/abs/1505.07818
[3]: http://ijcai.org/Proceedings/09/Papers/200.pdf
[4]: http://www.cs.nyu.edu/~mohri/pub/nadap.pdf
[5]: http://arxiv.org/pdf/1505.07818v4.pdf#page=16

Mask RCNN takes off from where Faster RCNN left, with some augmentations aimed at bettering instance segmentation (which was out of scope for FRCNN). Instance segmentation was achieved remarkably well in *DeepMask* , *SharpMask* and later *Feature Pyramid Networks* (FPN).

Faster RCNN was not designed for pixel-to-pixel alignment between network inputs and outputs. This is most evident in how RoIPool , the de facto core operation for attending to instances, performs coarse spatial quantization for feature extraction. Mask RCNN fixes that by introducing RoIAlign in place of RoIPool.

#### Methodology

Mask RCNN retains most of the architecture of Faster RCNN. It adds the a third branch for segmentation. The third branch takes the output from RoIAlign layer and predicts binary class masks for each class.

##### Major Changes and intutions

**Mask prediction**

Mask prediction segmentation predicts a binary mask for each RoI using fully convolution - and the stark difference being usage of *sigmoid* activation for predicting final mask instead of *softmax*, implies masks don't compete with each other. This *decouples* segmentation from classification. The class prediction branch is used for class prediction and for calculating loss, the mask of predicted loss is used calculating Lmask.

Also, they show that a single class agnostic mask prediction works almost as effective as separate mask for each class, thereby supporting their method of decoupling classification from segmentation

**RoIAlign**

RoIPool first quantizes a floating-number RoI to the discrete granularity of the feature map, this quantized RoI is then subdivided into spatial bins which are themselves quantized, and finally feature values covered by each bin are aggregated (usually by max pooling). Instead of  quantization of the RoI boundaries
or bin bilinear interpolation is used to compute the exact values of the input features at four regularly sampled locations in each RoI bin, and aggregate the result (using max or average).

**Backbone architecture**

Faster RCNN uses a VGG like structure for extracting features from image, weights of which were shared among RPN and region detection layers. Herein, authors experiment with 2 backbone architectures - ResNet based VGG like in FRCNN and ResNet based [FPN](http://www.shortscience.org/paper?bibtexKey=journals/corr/LinDGHHB16) based. FPN uses convolution feature maps from previous layers and recombining them to produce pyramid of feature maps to be used for prediction instead of single-scale feature layer (final output of conv layer before connecting to fc layers was used in Faster RCNN) 

**Training Objective**

The training objective looks like this 
![](https://i.imgur.com/snUq73Q.png)

Lmask is the addition from Faster RCNN. The method to calculate was mentioned above

#### Observation

Mask RCNN performs significantly better than COCO instance segmentation winners *without any bells and whiskers*. Detailed results are available in the paper

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?

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!

The goal of this work is to perform transfer learning among numerous tasks and to discover visual relationships among them. Specifically, while we intiutively might guess the depth of an image and surface normals are related, this work takes a step forward and discovers a beneficial relationship among 26 tasks in terms of task transferability - many of them are not obvious. This is important for scenarios when an insufficient budget is available for target task for annotation, thus, learned representation from the 'cheaper' task could be used along with small dataset for the target task to reach sufficient performance on par with fully supervised training on a large dataset.

The basis of the approach is to compute an affinity matrix among tasks based on whether the solution for one task can be sufficiently easily used for another task. This approach does not impose human intuition about the task relationships and chooses task transferability based on the quality of a transfer operation in a fully computational manner.

The task taxonomy (i.e. **taskonomy**) is a computationally found directed hypergraph that captures the notion of task transferability over any given task dictionary. It performed using a four-step process depicted in the figure below: ![Process overview. The steps involved in creating the taxonomy.](http://taskonomy.stanford.edu/img/svg/Process.svg)

-  In stage I (**Task-specific Modelling**), a task-specific network is trained in a fully supervised manner. The network is composed of the encoder (modified ResNet-50), and fully convolutional decoder for pixel-to-pixel tasks, or 2-3 FC layers for low-dimensional tasks. Dataset consists of 4 million images of indoor scenes from about 600 buildings; every image has an annotation for every task.

- In stage II (**Transfer modeling**), all feasible transfers between sources and targets are trained (multiple inputs task to single target transfer is also considered). Specifically, after the task-specific networks are trained in stage I, the weights of an encoder are fixed (frozen network is used to extract representations only) and the representation from the encoder is used to train a small readout network (similar to a decoder from stage I) with a new task as a target (i.e. ground truth is available). In total, about 3000 transfer possibilities are trained.

- In stage III (**Taxonomy solver**), the task affinities acquired from the transfer functions performance are normalized. This is needed because different tasks lie in different spaces and transfer function scale. This is performed using ordinal normalization - Analytical Hierarchy Process (details are in the paper - Section 3.3). This results in an affinity matrix where a complete graph of relationships is completely normalized and this graph quantifies a pair-wise set of tasks evaluated in terms of a transfer function (i.e. task dependency).

- In stage IV (**Computed Taxonomy**), a hypergraph which can predict the performance of any transfer policy and optimize for the optimal one is synthesized. This is solved using Binary Integer Program as a subgraph selection problem where tasks are nodes and transfers are edges. After the optimization process, the solution devices a connectivity that solves all target tasks, maximizes their collective performance while using only available source tasks under user-specified constraints (e.g. budget).

So, if you want to train your network on an unseen task, you can obtain pretrained weights for existing tasks from the [project page](https://github.com/StanfordVL/taskonomy/tree/master/taskbank), train readout functions against each task (as well as combination of multiple inputs), build an affinity matrix to know where your task is positioned against the other ones, and through subgraph selection procedure observe what tasks have favourable influence on your task. Consequently, you can train your task with much less data by utilizing representations from the existing tasks which share visual significance with your task. Magnificent!