wondering about your ubiased proof

[jack-pan-ai]

Hello, in your proof, Proposition 3.1. (Bias of WMF estimator) The bias of the estimator used in WMF is represented as follows

$\delta$ is as well concerned with \gamma (relevance probability).

The weighted matrix factorization loss function cannot be calculated directly without estimating \gamma (relevance probability). So, may I know how do you use this loss function in the real-world setup?

ps: sorry, I do not know how to type math formula in the reply

[usaito]

@PancheLone Thanks for your interest in our work! Please make sure that Eq. (6) does not depend on \gamma. It just uses implicit feedback Y (, which is observable) . Instead of using the unobservable quantity \gamma. How to estimate \gamma in the loss functions using only observable data is the key in our paper.

[jack-pan-ai]

Thanks for your prompt reply a lot! But according to your definition in Eq.(4), we have to know P(R_{ui}=1), which is \gamma, so that we can calculate the \delta in your Eq.(6).

Thanks for your help in advance, and looking forward to your reply!

[usaito]

@PancheLone In Eq.(6) (and the loss functions of some other methods), \delta^{(1)} and \delta^{(0)} are used, both of which do not depend on \gamma, right? For example, \delta^{(1)} = - \log (\hat{R}) where \hat{R} is a predicted value, not the true \gamma.

\Eq. (4) is the quantity that we want to maximize, and is not used in the training process.

[jack-pan-ai]

Ooooo, it's the predicted value!!!, It seems like I got your idea; predicted value means to use the learned embedding vector!!! Thank you very much for your kind explanation.

Moreover, I also followed your twitter, and hope for more valuable and interesting work from you in the future!

[usaito]

Great!