# Autoencoding beyond pixels using a learned similarity metric (ICML’16)

One of the key components of an autoencoder is a reconstruction error. This term measures how much useful information is compressed into a learned latent vector. The common reconstruction error is based on an element-wise measurement such as a binary cross entropy for a black-and-white image or a square error between a reconstructed image and the input image.

The authors think that an element-wise measurement is not an accurate indicator of goodness of a learned latent vector. Hence, they proposed to learn a similar metric via an adversarial training. Here is how they set up the objective functions:

They use VAE to learn a latent vector of the input image. There are two loss functions in a vanilla VAE: a KL loss and a negative log data-likelihood. They replace the second loss with a new reconstruction loss function. We will talk about this new loss function in the next paragraph. Then, they have a discriminator that tries to distinguish the real input data from the generated data from the decoder of VAE. The discriminator will encourage the VAE to learn stronger encoder and decoder.

The discriminator can be decomposed into 2 parts. If it has L + 1 layers, then the first L layers is a transform function that maps the input data into a new representation. The last layer is a binary classifier. This means that if we input any input through the first L layer, we will get a new representation that is easily classified by the last layer of the discriminator. When the discriminator is very good at detecting the real input, the representation at L layer is going to be much easier to classify compared to the input data. It means that a square error between a transformed input and its transformed reconstruction input should be somewhat small when these inputs are similar.

This model has trained the same fashion as GANs; simultaneously train VAE and GANs. This idea works well for an image because the square-error is not a good metric for an image quality. This idea may work on text dataset as well because we assess the quality of the reconstructed texts based on the whole input but not collectively evaluate one word at a time.

References:

https://arxiv.org/abs/1512.09300

Variational autoencoder (VAE) requires an expressive inference network in order to learn a complex posterior distribution. The more complex inference network will result in generating high-quality data.

This work utilizes an adversarial training to learn a function $T(x,z)$ that approximates $\log q_{\phi}(z|x) - \log p(z)$. The expectation of this term w.r.t $latex q_{\phi}(z|x)$ is in fact a KL-divergence term. Since the authors prove that the optimal $T^*(x, z) = \log q_{\phi}(z|x) - \log p(z)$, the ELBO becomes:

$E_{p_{D(x)}}E_{q_{\phi}(z|x)}[-T^*(x,z) + \log p_{\theta}(x|z)]$

In order to approximate $T^*(x, z)$, the discriminator needs to learn to distinguish between a sample from a prior $p_{D(x)}p(z)$ and the current inference model $p_{D(x)}q_{\phi}(z|x)$. Thus, the objective function for the discriminator is setup as:

$\max_T E_{p_{D(x)}}E_{q_{\phi(z|x)}} \log \sigma(T(x,z)) + E_{p_{D(x)}}E_{p(z)} \log(1 - \sigma(T(x,z)))$ (1)

Taking a gradient on $T(x,z)$ w.r.t. parameter $\phi$ can be problematic because the solution of this function depends on $q_{\phi}(z|x)$. But the author shows that the expectation of gradient of $T^*(x, z)$ w.r.t $\phi$ is 0. Thus, there is no gradient and no parameter update when taking a gradient of $T^*(x,z)$.

Since $T(x,z)$ requires sample z, the parametrization trick is applied and the ELBO becomes:

$E_{p_{D(x)}}E_{\epsilon}[-T^*(x, z_{\phi}(x, \epsilon) + \log p_{\theta}(x|z_{\phi}(x, \epsilon))]$ (2)

This step is crucial because now the sampling is just a transformation from a noise and let $T^*(x, z)$ to approximate the KL-divergence term. This made this model looks like a blackbox model because we do not explicitly define a distribution $q_{\phi}(z|x)$.

This model optimizes equation (1) and (2) using adversarial training. It optimizes eq.(1) several steps in order to keep $T(x, z)$ close to optimal while jointly optimizes eq. (2).

Adaptive contrast technique is used to make $T(x, y)$ to be sufficiently close to the optimal. Basically, the KL term in ELBO is replaced by $KL(q_{\phi}(z|x), r_{\alpha}(z|x))$ where $r_{\alpha}(z|x)$ is an auxiliary distribution which could be a Gaussian distribution.

This model has a connection to variational autoencoder, adversarial autoencoder, f-GANs, and BiGANs. A new training method for VAE via adversarial training allows us to use a flexible inference that approximate a true distribution over the latent vectors.

References:

# LDA2Vec

This paper proposed an extension of word2vec by adding document and topic vectors inspired by LDA (Latent Dirichlet Allocation). The model can discover a linear relationship between words. For example, “California + technology = Silicon Valley”. The topic is interpretable by collecting all nearby word vectors to the selected topic vector. This work boosts word2vec with topic modeling via training in the similar fashion to word2vec.

The key difference of LDA2Vec is its loss function. There are 2 loss functions: the first one is Skipgram Negative Sampling Loss which is similar to Word2Vec. It wants to maximize the probability of predicting a target word $\vec w_j$ and non-target word (negative samples) given a context vector $\vec c_j$. This loss wants the model to distinguish a positive word (which related to the given context) from negative sampled words.

The innovation is the context vector $\vec c_j$. The intuition is that predict a nearby word given a pivot word also depends on the theme of the context. For example, if the document is about an airline when we want to predict nearby words given a word “Germany”, we will likely want to see word related to airlines but not country names. Thus, a context vector is a sum of a word vector and document vector. $\vec c_j = \vec w_j + \vec d_j$. Thus, LDA2vec attempts to capture both document-wide relationship and local interaction between words within its context window.

In order to learn a topic vector, the document is further decomposed as a linear combination of topic vectors. $\vec d_j = \sum_{k} p_{jk} \cdot \vec t_k$ where $p_{jk}$ is a probability of document j will be a topic k. Finally, the interpretability comes from a sparsity of topic assignment vector, $p_j$. One way to enforce sparsity is to design the loss function as:

$L^{d} = \lambda \sum_{jk} (\alpha - 1)\log p_{jk}$

When $\alpha < 1$, we encourage a topic assignment probability to put more mass on a small set of topics.

The results look interesting. This paper shows a simple way to combine topic modeling with word embedding. By embedding document vectors and topic vectors into the same semantic space as word vectors, we can learn a global semantic structure as well as word-level local interaction.

References:

[3] Skip-gram tutorial part1: http://mccormickml.com/2016/04/19/word2vec-tutorial-the-skip-gram-model/

[4] Skip-gram tutorial part2: http://mccormickml.com/2017/01/11/word2vec-tutorial-part-2-negative-sampling/

# One-Class Collaborative Filtering (ICDM’08)

Negative samples are very important in learning an effective collaborative filtering model. In an implicit feedback CF problem where we collect implicit data such as clicking or viewing by a user, those unclicked or non-viewed items can be either positive or negative sample. But when we train a CF model without carefully select negative samples, the model will be biased because we treat all missing data as if negative samples. Some unobserved positive samples would be negative samples as well.

This classic paper proposed two treatments on missing values. The first approach is to treat all missing values as negative samples. The key is to not penalize too much when the model mispredicts a negative sample. This approach introduces a weight parameter for each pair of user and item in the training data. The positive samples will have a weight of 1, but a negative sample will have a much smaller weight. This reflects our uncertainty on whether the given negative sample might be a positive sample.

The weight can be uniform for all missing data or can be user-based or item-based weighting. The user-based weighting assumes that once the current user has viewed a lot of items already, the chance of unobserved item to be a negative sample is very likely. The item-based weighting assumes that if a user has not viewed a popular item, it probably means that he/she does not like that item.

The above scheme is used to generate a richer dataset including negative samples and thus improve the MAP on the given CF model. However, since it trained by ALS, the computational cost is expensive. To alleviate this cost, sampling based scheme is utilized.

We can fix the number of samples to be somewhat small and sample negative items based on either uniform, user-based, or item-based assumptions. Then, we will train ALS to reconstruct an approximate rating matrix. Since the number of samples is much smaller, we need to approximate a lot of rating matrices. Finally, we average all approximate rating matrices to achieve the final predicted rating matrix.

The experimental results show that User-based assumption performs slightly better than uniform and item-based assumption. The reason why uniform assumption still perform as good as item-based because most missing or unlabelled items are negative samples anyway.

References:

[1] Pan, Rong, et al. “One-class collaborative filtering.” Data Mining, 2008. ICDM’08. Eighth IEEE International Conference on. IEEE, 2008.

http://rongpan.net/publications/pan-oneclasscf.pdf

# Neural Collaborative Filtering (WWW’17)

This is another paper that applies deep neural network for collaborative filtering problem. Although there are a few outstanding deep learning models for CF problems such as CF-NADE and AutoRec, the author claims that those models are solving for explicit feedback and positioned this work to solve for ‘implicit feedback CF’ problem.

The model is straightforward and similar to Neural Factorization Machine. The idea is to find embedding vectors for users and items and model their interaction as a non-linear function via a multi-layers neural network. A non-linear interaction has been commonly used in many recent works such as Neural Factorization Machine [2], Deep Relevant Matching Model [3].

The authors proposed 3 models incrementally: (1) Generalized Matrix Factorization – which is basically MF with additional non-linear transformation. In this case, they use sigmoid function; (2) Multi-layer Perceptron – which concatenate user and item embedded vectors and transform them by a learned non-linear function; (3) Fusion model can be either shared the same embedding vectors and add them up at the last layers or learned separate embedding vectors and concatenate them at the output layer.

Since they want to predict either the given item is preferable or not, it is a binary classification problem. Then the loss function can be a binary cross-entropy. To me, implicit feedback seems to be an easier problem than rating prediction problem because we only need to make a binary prediction.

The baseline seems okay but I wish the authors include more recent deep learning models such as [4]. The AutoRec model is also applicable for an implicit feedback by forcing the output to be a binary output. Regardless of their baseline, the extensive experiments tried to convince readers that deep neural network can model a complex interaction and will improve the performance.

Speak of the performance, the author uses hit-ratio and NDCG. Basically, there is one test sample for each user. The model tries to give a high score for that sample. This metric is more practical than MSE and RMSE since the dataset is implicit feedback dataset.

Non-linear interaction is a simple extension to a traditional MF model. This author shows that this simple extension does work for MovieLens and Pinterest dataset.

Reference:

[1] He, Xiangnan, et al. “Neural collaborative filtering.” Proceedings of the 26th International Conference on World Wide Web. International World Wide Web Conferences Steering Committee, 2017.

http://www.comp.nus.edu.sg/~xiangnan/papers/ncf.pdf

[2] Xiangnan He, Tat-Seng Chua, “Neural Factorization Machines for Sparse Predictive Analytics”, ACM SIGIR 2017.

[3] Guo, Jiafeng, et al. “A deep relevance matching model for ad-hoc retrieval.” Proceedings of the 25th ACM International on Conference on Information and Knowledge Management. ACM, 2016.

[4] Zheng, Yin, et al. “Neural Autoregressive Collaborative Filtering for Implicit Feedback.” Proceedings of the 1st Workshop on Deep Learning for Recommender Systems. ACM, 2016. APA

# IRGAN: A Minimax Game for Unifying Generative and Discriminative Information Retrieval Models (SIGIR’17)

This paper uses GANs framework to combine generative and discriminative information retrieval model. It shows a promising result on web search, item recommendations, and Q/A tasks.

Typically, many relevant models are classified into 2 types:

• Generative retrieval model
• It generates a document given a query and possibly relevant score. The model is $p(d|q, r)$.
• Discriminative retrieval model
• It computes a relevance score for the given query and document pair. The model is $p(r| d, q)$.

The generative model tried to find a connection between document and query. On the other hand, the discriminative model attempts to model the interaction between query and document based on relevance scores.

Both models have their shortcoming. Many generative models require a predefined data generating story. The wrong assumption will lead to the poor performance. The generative model is usually trying to fit the data to its model without external guidance. Meanwhile, the discriminative model requires a lot of labeled data to be effective, especially for a deep neural network model.

By train both models using GANs framework, it is now possible to solve their shortcoming. The generative model is now adaptive because the discriminator will reward the generative model when it can create or select good samples. This adaptive guidance from the discriminator is unique in GANs framework and will help the generator learns to pick good samples from the data distribution. At the same time, the discriminator can receive even more training data from the generative model. This is similar to semi-supervised learning where unlabeled data are utilized. Adversarial training allows us to improve both generative and discriminative models via jointly learning through the

Adversarial training allows us to improve both generative and discriminative models via jointly learning through the minimax training. The traditional training based on maximum likelihood does not have principle way to allow both models to give each other feedbacks.

The paper seems to be promising and their results on 3 information retrieval tasks are really good. But I notice that their training procedure requires pretraining. This made me wonder if pre-training is part of the performance boost during testing. I don’t find the part in the paper that explains the benefit of pretraining in their settings.

The discriminative model is straight forward. It is a sigmoid function. The discriminator basically gives a high probability when the given document-query pair is relevant. The generative model is more interesting. In the standard GANs, the generator will create a sample from a simple distribution, but IRGAN does not generate a new document-query pair. Instead, the author chose to let the generator select the sample from the document pool. In my opinion, this approach is simpler than creating a new data because the sample is realistic. Also, IRGAN cares about finding a function to compute a relevance score so it is unnecessary to generate a completely new data.

However, the cost function for the generator is an expectation over all documents in the corpus. The Monte Carlo approximation will have a high variance. Thus, they use policy gradient to reduce the variance so that the model can learn a useful representation. Although $p(d|q,r)$ is a discrete distribution,  the backprop is applicable because we pre-sample all documents from $p(d|q,r)$ beforehand. Thus, eq. 5 is differentiable. The extra care may need in order to reduce variance further. They use an advantage function. (Please look at the reference on Reinforcement Learning [2]).

Generating positive and negative samples are still confusing in this paper. It seems to be application specific. The author mentioned about using softmax with temperature hyper-parameter to put more or less focus on top documents. My guess is when we put less focus on top documents, the generator has more chance to pick up more negative samples. After I read the paper again, it seems that all samples selected by the generator model are negative samples. This part remains unclear and I need to ask the author for more details.

In conclusion, I like this paper because it tried to combine generative and discriminative retrieval models via GANs framework. I would not be appreciated if they simply applied the IR task to GANs blindly. Instead, they explained their motivation and discussed the advantage of jointly train both models. It seems adversarial training is useful for IR tasks as well.

References:

[1] Jun Wang, Lantao Yu, Weinan Zhang, Yu Gong, Yinghui Xu, Benyou Wang, Peng Zhang, and Dell Zhang. 2017. IRGAN: A Minimax Game for Unifying Generative and Discriminative Information Retrieval Models. In Proceedings of SIGIR’17, Shinjuku, Tokyo, Japan, August 7-11, 2017, 10 pages.

[2] Richard S Sutton, David A McAllester, Satinder P Singh, Yishay Mansour, and others. 1999. Policy Gradient Methods for Reinforcement Learning with Function Approximation. In NIPS.

[3] IRGAN code (http://lantaoyu.com/publications/IRGAN)

# My takes on GANs (Generative Adversarial Nets)

What is the fuss with GANs? Everybody loves GANs now. The joke I heard from my advisor is “If your current model does not work, try it aGAN !” (pun-intended).  I realize that it comes down to its adversarial objective function. Some people may think it is sexier than VAE and NADE’s objective function in which they just maximize the log-probability – the likelihood that the model will maximize the given data samples. For VAE, we will maximize the evidence lower-bound (ELBO) while NADE will maximize the conditional log probability. GANs simultaneously maximize and minimize two cost functions.

GANs’ framework has two components: data generator and discriminator. The generator will attempt to create a sample that is hopefully similar to sample from the actual data while the discriminator needs to guess if the given sample is fake or real. This process can be viewed as a competitive game where a generator tries to fool the discriminator. The game is complete when the discriminator is unable to distinguish a generated sample from a real one. On the other hand, we can also look at GANs as a cooperating game where discriminator helps the generator to create a more realistic sample by providing a feedback.

There are 2 cost functions: one for the generator and another for the discriminator. The cost function for the discriminator is a binary cross entropy:

$J^{(D)}(\theta^{(D)}, \theta^{(G)}) = -\frac{1}{2}E_{x \sim p_{\text{data}}}[\log D(x)] - \frac{1}{2}E_z[\log(1 - D(G(z)))]$

There are two set of mini-batches: one from the real samples and one coming from the generator. The optimal discriminator will predict the chance of being a true sample as $\frac{p_{\text{data}}}{p_{\text{data}} + p_{\text{model}}}$.

There are many cost functions for a generator. The original paper proposed 2 cost functions. If we construct this learning procedure as a zero-sum game, then the cost function for the generator is simply, $J^{(G)} = -J^{(D)}$. We simply want to minimize the likelihood of the discriminator for being correct.

Goodfellow [1] mentioned that the above cost function has a vanish gradient problem when the sample is likely to be fake. Hence, to overcome this, he proposed a new cost function for the generator by maximizing the likelihood of the discriminator for being wrong. The new cost function is:

$J^{(G)} = -\frac{1}{2}E_z[\log D(G(z))]$

The new cost function is more useful in practice but the former function is useful for theoretical analysis, in which we will discuss next. For now, the objective function is:

$V(D,G) = \min_G \max_D J^{(D)}(\theta^{(D)}, \theta^{(G)})$

Goodfellow shows that learning the zero-sum game is the same as minimizing the Jensen-Shannon divergence. Given that we have an optimal discriminator, the cost function is now:

$C(G) = \max_D V(D,G) = E_{x \sim p_{\text{data}}}[\log D^*(x)] + E_z[\log(1 - D^*(G(z)))]$

Since we know the optimal discriminator, we now have:

$C(G) = \max_D V(D,G) = E_{x \sim p_{\text{data}}}[\log (\frac{p_{\text{data}}}{p_{\text{data}} + p_{\text{model}}})] + E_z[\log(\frac{p_{\text{model}}}{p_{\text{data}} + p_{\text{model}}})]$

Then, simplify further:

$C(G) = \text{KL}(p_{\text{data}}||\frac{p_{\text{data}} + p_{\text{model}}}{2}) + \text{KL}(p_{\text{model}}||\frac{p_{\text{data}} + p_{\text{model}}}{2}) + \log({\frac{1}{4}})$

At the global minimum, we will have $latex p_{\text{data}} = p_{\text{model}}$, hence:

$C^*(G) = \log({\frac{1}{4}})$

People believed that minimizing the Jensen-Shannon divergence is the reason why GANs can produce a sharp image while VAE produces blurry image since it minimizes KL divergence. However, this belief is no longer true [2].

However, GANs has received a lot of criticism. For one instance, Schmidhuber mentioned that Goodfellow should have cited Schmidhuber’s 1992 paper [3]. (I found Reddit’s article on the public attack during NIPS2016 tutorial here: https://www.reddit.com/r/MachineLearning/comments/5go4sa/n_whats_happening_at_nips_2016_jurgen_schmidhuber/ ). Interestingly, Schmidhuber is one of the reviewers on GANs original paper and he rejected it.

While many researchers claim that GANs generates a realistic sample, a few NLP researchers do not believe so [4]. Meanwhile,  the standard dataset for some particular tasks is necessary because many machine learning research paper is really depending on the empirical results.

References:

[1] Goodfellow, Ian, et al. “Generative adversarial nets.” Advances in neural information processing systems. 2014.

[2] Goodfellow, Ian. “NIPS 2016 Tutorial: Generative Adversarial Networks.” arXiv preprint arXiv:1701.00160 (2016).

[3] ftp://ftp.idsia.ch/pub/juergen/factorial.pdf