FSGD¶
Implements FSGD (Fractional Stochastic Gradient Descent), a matrix-factorization learner whose latent-factor updates add a Caputo fractional-derivative term to ordinary SGD.
FSGD targets latent-factor recommender systems, where the predicted rating is \(\hat{r}_{ui} = p_u^\top q_i\) for user factor \(p_u\) and item factor \(q_i\), and the error on an observed entry is \(e_{ui} = r_{ui} - p_u^\top q_i\). The idea is to enrich each stochastic gradient step with the fractional gradient of the squared-error loss: alongside the usual integer-order term, a second term proportional to the Caputo fractional derivative of order \(\alpha\) injects a memory of the current parameter magnitude through the factor \(|p_u|^{1-\alpha}\) (resp. \(|q_i|^{1-\alpha}\)). Tuning \(\alpha \in (0,1)\) interpolates between plain SGD (\(\alpha \to 1\)) and a more history-aware step, which the authors report improves convergence rate and prediction accuracy on rating data.
Per observed rating \((u,i)\) the latent factors are updated element-wise as
where \(p_u, q_i\) are the user and item latent-factor vectors, \(\eta\) is the integer-order learning rate, \(\eta_\alpha\) the fractional-order learning rate, \(\alpha \in (0,1)\) the fractional order, \(\Gamma\) the gamma function, \(\odot\) element-wise multiplication, and \(|\cdot|\) the element-wise absolute value. The two updates share the same error \(e_{ui}\) and differ only in the integer-order versus Caputo fractional-order gradient of the squared-error objective.
Reference: Zeshan Aslam Khan, Naveed Ishtiaq Chaudhary, Syed Zubair, "Fractional stochastic gradient descent for recommender systems", Electronic Markets 2019. https://doi.org/10.1007/s12525-018-0297-2