Lisbon Machine Learning Summer School (LxMLS) 2017

Last year I had the opportunity to attend this great summer school in the beautiful and lovely city of Lisbon. It was a great week together with a lot of interesting and intelligent people, all of them interested in the amazing and exciting area of machine learning and NLP. I liked it so much last year that I decided to come back this year to volunteer as an assistant in the summer school. Today was the -1 day, where we organized some of the registration stuff, welcomed some student and had some beers. Looks like it will be, again, a great time here in Lisbon

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A lower bound for expected value of log-sum

Lately, I have been working with Poisson Matrix Factorization models and
at some point a needed to work a lower bound for \text{E}_q[\log \sum_k X_k]. After seeing some people using this lower bound without a good explanation, I decided to write this blog post. Also, this is included as an appendix to my ECML-PKDD 2017 paper about poisson factorizatiom model for recommendation.
The function \log(.) is a concave function, which means that: \log(p_1 x_1+p_2 x_2) \geq p_1\log x_1+p_2 \log x_2, \forall p_1,p_2:p_1+p_2=1, p_1,p_2  \geq 0
By induction this property can be generalized to any convex combination of x_k (\sum_k p_k x_k with \sum_k p_k=1 and p_k \geq 0 ):

\log \sum_k p_k x_k \geq \sum_k p_k\log x_k

Now with the a random variable we can create a similar convex combination by multiplying and dividing each random variable X_k by p_k and apply the sum of of expectation property:
 \text{E}_q[\log \sum_k X_k] = \text{E}_q[\log \sum_k \frac{p_k X_k}{p_k}]
\log \sum_k p_k\frac{X_k}{p_k} \geq \sum_k p_k\log \frac{X_k}{p_k}
\Rightarrow\text{E}_q [\log \sum_k p_k\frac{X_k}{p_k}] \geq \sum_k p_k \text{E}_q[\log \frac{X_k}{p_k}]
\Rightarrow \text{E}_q [\log \sum_k X_k ] \geq \sum_k p_k \text{E}_q[\log X_k]- p_k\log p_k

If we want a tight lower bound we should use Lagrange multipliers to choose the set of p_k that maximize the lower-bound given that they should sum to 1.

L(p_1,\ldots,p_K) = \left(\sum_k p_k \text{E}_q[\log X_k]- p_k\log p_k\right)+\lambda \left(1-\sum_k p_k\right)
 \frac{\partial L}{\partial p_k} =\text{E}_q[\log X_k]-\log p_k-1-\lambda = 0
\frac{\partial L}{\partial \lambda} =1-\sum_k p_k = 0
 \Rightarrow \sum_k p_k = 1
 \Rightarrow\text{E}_q[\log X_k]=\log p_k+1+\lambda
\Rightarrow\text{E}_q[\log X_k]=\log p_k+1+\lambda
\Rightarrow \exp\text{E}_q[\log X_k]=p_k \exp(1+\lambda)
\Rightarrow \sum_k \exp\text{E}_q[\log X_k]=\exp(1+\lambda)\underbrace{\sum_k p_k}_{=1}
\Rightarrow p_k=\frac{\exp \{\text{E}_q[\log X_k]\}}{\sum_k \exp \{\text{E}_q[\log X_k]\}}

The final formula for p_k is exactly the same that we can find for the parameters of the the Multinomial distribution of the auxiliary variables in a Poisson model with rate parameter as sum of Gamma distributed latent variables. Also using this optimal p_k we can show a tight bound without the auxiliary variables.

\text{E}_q [\log \sum_k X_k ] \geq \sum_k \frac{\exp \{\text{E}_q[\log X_k]\}}{\sum_j \exp \{\text{E}_q[\log X_j]\}}\text{E}_q[\log X_k]- \frac{\exp \{\text{E}_q[\log X_k]\}}{\sum_j \exp \{\text{E}_q[\log X_j]\}}\log \frac{\exp \{\text{E}_q[\log X_k]\}}{\sum_j \exp \{\text{E}_q[\log X_j]\}}
= \sum_k \frac{\exp \{\text{E}_q[\log X_k]\}}{\sum_j \exp \{\text{E}_q[\log X_j]\}} \log \sum_j \exp \{\text{E}_q[\log X_j]\}
= \log \sum_j \exp \{\text{E}_q[\log X_j]\} \underbrace{ \sum_k \frac{\exp \{\text{E}_q[\log X_k]\}}{\sum_j \exp \{\text{E}_q[\log X_j]\}} }_{=1}
This results in:
\text{E}_q [\log \sum_k X_k ] \geq \log \sum_k \exp \{\text{E}_q[\log X_k]\}

Paper accepted at European Conference on Machine Learning (ECML-PKDD) 2017

We have a paper accepted at ECML-PKDD 2017: “Content-Based Social Recommendation with Poisson Matrix Factorization” (Eliezer de Souza da Silva, Helge Langseth and Heri Ramampiaro). This is our first full paper resulting from our research on Poisson factorization and integration of multiple sources of information in a single recommendation model. If you have interest on the paper please email me and I will be happy to discuss.

Also, I am uploading the supplement of the paper here (you can find it also on my publications page)

Supplementary material for: “Content-Based Social
Recommendation with Poisson Matrix Factorization”

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