This paper derives an algorithm for passing gradients through a sample from a mixture of Gaussians. While the reparameterization trick allows to get the gradients with respect to the Gaussian means and covariances, the same trick cannot be invoked for the mixing proportions parameters (essentially because they are the parameters of a multinomial discrete distribution over the Gaussian components, and the reparameterization trick doesn't extend to discrete distributions).

One can think of the derivation as proceeding in 3 steps:

1. Deriving an estimator for gradients a sample from a 1-dimensional density $f(x)$ that is such that $f(x)$ is differentiable and its cumulative distribution function (CDF) $F(x)$ is tractable:

  $\frac{\partial \hat{x}}{\partial \theta} = - \frac{1}{f(\hat{x})}\int_{t=-\infty}^{\hat{x}} \frac{\partial f(t)}{\partial \theta} dt$

  where $\hat{x}$ is a sample from density $f(x)$ and $\theta$ is any parameter of $f(x)$ (the above is a simplified version of Equation 6). This is probably the most important result of the paper, and is based on a really clever use of the general form of the Leibniz integral rule.

2. Noticing that one can sample from a $D$-dimensional Gaussian mixture by decomposing it with the product rule $f({\bf x}) = \prod_{d=1}^D f(x_d|{\bf x}_{<d})$ and using ancestral sampling, where each $f(x_d|{\bf x}_{<d})$ are themselves 1-dimensional mixtures (i.e. with differentiable densities and tractable CDFs)

3. Using the 1-dimensional gradient estimator (of Equation 6) and the chain rule to backpropagate through the ancestral sampling procedure. This requires computing the integral in the expression for $\frac{\partial \hat{x}}{\partial \theta}$ above, where $f(x)$ is one of the 1D conditional Gaussian mixtures and $\theta$ is a mixing proportion parameter $\pi_j$. As it turns out, this integral has an analytical form (see Equation 22).

**My two cents**

This is a really surprising and neat result. The author mentions it could be applicable to variational autoencoders (to support posteriors that are mixtures of Gaussians), and I'm really looking forward to read about whether that can be successfully done in practice. 

The paper provides the derivation only for mixtures of Gaussians with diagonal covariance matrices. It is mentioned that extending to non-diagonal covariances is doable. That said, ancestral sampling with non-diagonal covariances would become more computationally expensive, since the conditionals under each Gaussian involves a matrix inverse.

Beyond the case of Gaussian mixtures, Equation 6 is super interesting in itself as its application could go beyond that case. This is probably why the paper also derived a sampling-based estimator for Equation 6, in Equation 9. However, that estimator might be inefficient, since it involves sampling from Equation 10 with rejection, and it might take a lot of time to get an accepted sample if $\hat{x}$ is very small. Also, a good estimate of Equation 6 might require *multiple* samples from Equation 10. 

Finally, while I couldn't find any obvious problem with the mathematical derivation, I'd be curious to see whether using the same approach to derive a gradient on one of the Gaussian mean or standard deviation parameters gave a gradient that is consistent with what the reparameterization trick provides.

Everyone has been thinking about how to apply GANs to discrete sequence data for the past year or so. This paper presents the model that I would guess most people thought of as the first-thing-to-try:

1. Build a recurrent generator model which samples from its softmax outputs at each timestep.
2. Pass sampled sequences to a recurrent discriminator model which distinguishes between sampled sequences and real-data sequences.
3. Train the discriminator under the standard GAN loss.
4. Train the generator with a REINFORCE (policy gradient) objective, where each trajectory is assigned a single episodic reward: the score assigned to the generated sequence by the discriminator.

Sounds hacky, right? We're learning a generator with a high-variance model-free reinforcement learning algorithm, in a very seriously non-stationary environment. (Here the "environment" is a discriminator being jointly learned with the generator.)

There's just one trick in this paper on top of that setup: for non-terminal states, the reward is defined as the *expectation* of the discriminator score after stochastically generating from that state forward. To restate using standard (somewhat sloppy) RL syntax, in different terms than the paper: (under stochastic sequential policy $\pi$, with current state $s_t$, trajectory $\tau_{1:T}$ and discriminator $D(\tau)$)

$$r_t = \mathbb E_{\tau_{t+1:T} \sim \pi(s_t)} \left[ D(\tau_{1:T}) \right]$$

The rewards are estimated via Monte Carlo — i.e., just take the mean of $N$ rollouts from each intermediate state. They claim this helps to reduce variance. That makes intuitive sense, but I don't see any results in the paper demonstrating the effect of varying $N$.

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Yep, so it turns out that this sort of works.. with a big caveat:

## The big caveat

Graph from appendix:

![](https://www.dropbox.com/s/5fqh6my63sgv5y4/Bildschirmfoto%202016-09-27%20um%2021.34.44.png?raw=1)

SeqGANs don't work without supervised pretraining. Makes sense — with a cold start, the generator just samples a bunch of nonsense and the discriminator overfits. Both the generator and discriminator are pretrained on supervised data in this paper (see Algorithm 1).

I think it must be possible to overcome this with the proper training tricks and enough sweat. But it's probably more worth our time to address the fundamental problem here of developing better RL for structured prediction tasks.

This recent paper, a collaboration involving some of the authors of MAML, proposes an intriguing application of techniques developed in the field of meta learning to the problem of unsupervised learning - specifically, the problem of developing representations without labeled data, which can then be used to learn quickly from a small amount of labeled data. As a reminder, the idea behind meta learning is that you train models on multiple different tasks, using only a small amount of data from each task, and update the model based on the test set performance of the model. 

The conceptual advance proposed by this paper is to adopt the broad strokes of the meta learning framework, but apply it to unsupervised data, i.e. data with no pre-defined supervised tasks. The goal of such a project is, as so often is the case with unsupervised learning, to learn representations, specifically, representations we believe might be useful over a whole distribution of supervised tasks. However, to apply traditional meta learning techniques, we need that aforementioned distribution of tasks, and we’ve defined our problem as being over unsupervised data.  How exactly are we supposed to construct the former out of the latter? This may seem a little circular, or strange, or definitionally impossible: how can we generate supervised tasks without supervised labels? 

https://i.imgur.com/YaU1y1k.png

The artificial tasks created by this paper are rooted in mechanically straightforward operations, but conceptually interesting ones all the same: it uses an off the shelf unsupervised learning algorithm to generate a fixed-width vector embedding of your input data (say, images), and then generates multiple different clusterings of the embedded data, and then uses those cluster IDs as labels in a faux-supervised task. It manages to get multiple different tasks, rather than just one - remember, the premise of meta learning is in models learned over multiple tasks - by randomly up and down-scaling dimensions of the embedding before clustering is applied. Different scalings of dimensions means different points close to one another, which means the partition of the dataset into different clusters. 

With this distribution of “supervised” tasks in hand, the paper simply applies previously proposed meta learning techniques - like MAML, which learns a model which can be quickly fine tuned on a new task, or prototypical networks, which learn an embedding space in which observations from the same class, across many possible class definitions are close to one another.

https://i.imgur.com/BRcg6n7.png

An interesting note from the evaluation is that this method - which is somewhat amusingly dubbed “CACTUs” - performs best relative to alternative baselines in cases where the true underlying class distribution on which the model is meta-trained is the most different from the underlying class distribution on which the model is tested. Intuitively, this makes reasonable sense: meta learning is designed to trade off knowledge of any given specific task against the flexibility to be performant on a new class division, and so it gets the most value from trade off where a genuinely dissimilar class split is seen during testing. 

One other quick thing I’d like to note is the set of implicit assumptions this model builds on, in the way it creates its unsupervised tasks. First, it leverages the smoothness assumptions of classes - that is, it assumes that the kinds of classes we might want our model to eventually perform on are close together, in some idealized conceptual space. While not a perfect assumption (there’s a reason we don’t use KNN over embeddings for all of our ML tasks) it does have a general reasonableness behind it, since rarely are the kinds of classes very conceptually heterogenous. Second, it assumes that a truly unsupervised learning method can learn a representation that, despite being itself sub-optimal as a basis for supervised tasks, is a well-enough designed feature space for the general heuristic of “nearby things are likely of the same class” to at least approximately hold. I find this set of assumptions interesting because they are so simplifying that it’s a bit of a surprise that they actually work: even if the “classes” we meta-train our model on are defined with simple Euclidean rules, optimizing to be able to perform that separation using little data does indeed seem to transfer to the general problem of “separating real world, messier-in-embedding-space classes using little data”.  

Contrastive learning works by performing augmentations on a batch of images, and training a network to match the representations of the two augmented parts of a pair together, and push the representations of images not in a pair farther apart. Historically, these algorithms have benefitted from using stronger augmentations, which has the effect of making the two positive elements in a pair more visually distinct from one another. This paper tries to build on that success, and, beyond just using a strong augmentation, tries to learn a way to perturb images that adversarially increases contrastive loss. As with adversarial training in normal supervised setting, the thinking here is that examples which push loss up the highest are the hardest and thus most informative for the network to learn from 

While the concept of this paper made some sense, I found the notation and the explanation of mechanics a bit confusing, particularly when it came to choice to frame a contrastive loss as a cross-entropy loss, with the "weights" of the dot product in the the cross-entropy loss being, in fact, the projection by the learned encoder of various of the examples in the batch. 

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

This notion of the learned representations being "weights" is just odd and counter-intuitive, and the process of trying to wrap my mind around it isn't one I totally succeeded at. I think the point of using this frame is because it provides an easy analogue to the Fast Gradient Sign Method of normal supervised learning adversarial examples, even though it has the weird effect that, as the authors say "your weights vary by batch...rather than being consistent across training," 

Notational weirdness aside, my understanding is that the method of this paper: 

- Runs a forward pass of normal contrastive loss (framed as cross-entropy loss) which takes augmentations p and q and runs both forward through an encoder.
- Calculates a delta to apply to each input image in the q that will increase the loss most, taken over all the images in the p set
    - I think the delta is per-image in q, and is just aggregated over all images in p, but I'm not fully confident of this, as a result of notational confusion. It could also be one delta applied for all all images in q.
- Calculate the loss that results when you run forward the adversarially generated q against the normal p
- Train a combined loss that is a weighted combination of the normal p/q contrastive part and the adversarial p/q contrastive part

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

The authors show a small but relatively consistent improvement to performance using their method. Notably, this improvement is much stronger when using larger encoders (presumably because they have more capacity to learn from harder examples). One frustration I have with the empirics of the paper is that, at least in the main paper, they don't discuss the increase in training time required to calculate these perturbations, which, a priori, I would imagine to be nontrivial.

The main contribution of this paper is introducing a new transformation that the authors call Batch Normalization (BN). The need for BN comes from the fact that during the training of deep neural networks (DNNs) the distribution of each layer’s input change. This phenomenon is called internal covariate shift (ICS).

#### What is BN?
Normalize each (scalar) feature independently with respect to the mean and variance of the mini batch. Scale and shift the normalized values with two new parameters (per activation) that will be learned. The BN consists of making normalization part of the model architecture.

#### What do we gain?
According to the author, the use of BN provides a great speed up in the training of DNNs. In particular, the gains are greater when it is combined with higher learning rates. In addition, BN works as a regularizer for the model which allows to use less dropout or less L2 normalization. Furthermore, since the distribution of the inputs is normalized, it also allows to use sigmoids as activation functions without the saturation problem.

#### What follows?
This seems to be specially promising for training recurrent neural networks (RNNs). The vanishing and exploding gradient problems \cite{journals/tnn/BengioSF94} have their origin in the iteration of transformation that scale up or down the activations in certain directions (eigenvectors). It seems that this regularization would be specially useful in this context since this would allow the gradient to flow more easily. When we unroll the RNNs, we usually have ultra deep networks.

#### Like
* Simple idea that seems to improve training.
* Makes training faster.
* Simple to implement. Probably.
* You can be less careful with initialization.

#### Dislike
* Does not work with stochastic gradient descent (minibatch size = 1).
* This could reduce the parallelism of the algorithm since now all the examples in a mini batch are tied.
* Results on ensemble of networks for ImageNet makes it harder to evaluate the relevance of BN by itself. (Although they do mention the performance of a single model).