GFSGD

Implements GFSGD, a generalized fractional stochastic gradient descent for matrix-factorization recommender systems.

GFSGD augments the standard SGD update for the user and item latent factors with a Caputo-type fractional-derivative term. The fractional term adds a memory of the parameter's own magnitude through a power law, which captures the history of chaotic ratings and broadens the usable range of the fractional order. The fractional order \(\alpha\) controls the memory strength: at \(\alpha = 1\) the fractional term collapses to the ordinary gradient and GFSGD recovers plain SGD, while larger fractional contributions accelerate convergence.

For a rating \(r_{ij}\) with prediction error \(e_{ij} = r_{ij} - p_i^\top q_j\), the user factor \(p_i\) and item factor \(q_j\) are updated alternately by

\[ \begin{aligned} p_i &\leftarrow p_i + \gamma\, e_{ij}\, q_j + \frac{\gamma_f}{\Gamma(2-\alpha)}\, e_{ij}\, q_j \odot |p_i|^{\,1-\alpha}, \\ q_j &\leftarrow q_j + \gamma\, e_{ij}\, p_i + \frac{\gamma_f}{\Gamma(2-\alpha)}\, e_{ij}\, p_i \odot |q_j|^{\,1-\alpha}, \end{aligned} \]

where \(\gamma\) is the integer-order learning rate, \(\gamma_f\) is the fractional-order learning rate, \(\alpha\) is the fractional order, \(\Gamma\) is the gamma function, \(\odot\) is the elementwise product, and \(|\cdot|^{1-\alpha}\) is applied componentwise to the latent factor.

Reference: Zeshan Aslam Khan, Naveed Ishtiaq Chaudhary, Muhammad Asif Zahoor Raja, "Generalized fractional strategy for recommender systems with chaotic ratings behavior", Chaos, Solitons & Fractals 2022. https://doi.org/10.1016/j.chaos.2022.112204

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