mF-SGD

Implements mF-SGD, a momentum-accelerated fractional stochastic gradient descent for matrix-factorization recommender systems.

The method factorizes the rating matrix into user features \(a_u\) and item features \(b_i\) by minimizing the squared prediction error. Beyond the ordinary gradient, it adds a Riemann–Liouville fractional-order gradient term: the fractional derivative of the squared error contributes a factor \(|\theta|^{1-\nu}/\Gamma(2-\nu)\), giving an extra search direction controlled by the fractional order \(\nu\). A momentum (velocity) accumulation over the combined integer- and fractional-order gradients then accelerates convergence relative to plain fractional SGD.

For each observed entry the error is \(E_{ui} = C_{ui} - a_u^\top b_i\). Writing \(g_t\) for the combined gradient and \(v_t\) for the velocity, the per-feature updates are

\[ \begin{aligned} g^{(a)}_t &= \eta\, E_{ui}\, b_i + \frac{\eta_{fr}}{\Gamma(2-\nu)}\, E_{ui}\, b_i \odot |a_u|^{\,1-\nu}, \\ g^{(b)}_t &= \eta\, E_{ui}\, a_u + \frac{\eta_{fr}}{\Gamma(2-\nu)}\, E_{ui}\, a_u \odot |b_i|^{\,1-\nu}, \\ v^{(a)}_t &= \beta\, v^{(a)}_{t-1} + g^{(a)}_t, \qquad a_u \leftarrow a_u + v^{(a)}_t, \\ v^{(b)}_t &= \beta\, v^{(b)}_{t-1} + g^{(b)}_t, \qquad b_i \leftarrow b_i + v^{(b)}_t, \end{aligned} \]

where \(\eta\) is the integer-order learning rate, \(\eta_{fr}\) the fractional learning rate, \(\nu \in (0,1)\) the fractional order, \(\Gamma(\cdot)\) the Gamma function, \(\beta \in (0,1)\) the momentum weight, \(\odot\) elementwise product, and \(E_{ui}\) the prediction error on entry \((u,i)\). The signs are additive because the error gradient \(\partial E_{ui}^2\) is negated toward the minimum.

Reference: Zeshan Aslam Khan, Syed Zubair, Hani Alquhayz, Muhammad Azeem, Allah Ditta, "Design of Momentum Fractional Stochastic Gradient Descent for Recommender Systems", IEEE Access 2019. https://doi.org/10.1109/ACCESS.2019.2954859

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