The goal is to find vectors $x_u$ and $y_i$, both of the same length $f=64$, and to do this the following loss function is minimized:

$$\sum_{u, i} (1 + \alpha \cdot r_{ui})(p_{ui} - x_{u}^{T}y_i)^2$$

The vectors obtained are user-factors and item-factors, used for the purpose of rating calculation in a Recommender System. My goal is to find the optimum value of $\alpha$, and the initial thought was to try different values for $\alpha$, for which I would calculate the min of the above function; in the end, I would pick $\alpha$ for which the function has the lowest value. Is this a good idea? How could one find an optimal $\alpha$ value?

As for the $r_{ui}$ and $p_{ui}$ values, these would be the same for each user-item pair in each run of the minimization. Given that the function is used for a Recommender System, one way to identify good $\alpha$ would be to withhold a test set, and then check how the recommendations (top $x_u^{T}y_i$ values) are recovered.

  • $\begingroup$ Since x and y make a dot product and aren't associated with any other constraints why can't you simplify to using a single variable? $\endgroup$
    – Richard
    Aug 24 at 14:43
  • $\begingroup$ @Richard I was too quick when writing the loss function. Please have a look at (3) in this paper: yifanhu.net/PUB/cf.pdf I took the first part only $\endgroup$
    – kevin811
    Aug 24 at 14:53
  • $\begingroup$ A good question should be self-contained. The paper you link might disappear in a year or two rendering your question unintelligible. Please copy all relevant information into the question via the Edit button. $\endgroup$
    – Richard
    Aug 24 at 19:05
  • $\begingroup$ @Richard Ok, I think I've provided all the necessary info now. $\endgroup$
    – kevin811
    Aug 25 at 7:10
  • $\begingroup$ It still looks as though either x or y is redundant. $\endgroup$
    – Richard
    Aug 25 at 15:04

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