Sequential Minimal Optimization for $\varepsilon$-SVR with MAPE Loss and Sample-Dependent Box Constraints

• URL: http://arxiv.org/abs/2605.01446v2
• Date: Thu, 07 May 2026 18:38:49 GMT
• Title: Sequential Minimal Optimization for $\varepsilon$-SVR with MAPE Loss and Sample-Dependent Box Constraints
• Abstract summary: We derive a Sequential Minimal Optimization (SMO) algorithm for the quadratic dual problem arising from $varepsilon$-SVRciteVapnik1995, Drucker1997, Smola2004 modified to minimize the Mean Absolute Percentage Error (MAPE)citeMakridakis1993, Hyndman2006.<n>An implementation is available in the open-source textttpsvr R packageciteBenavidesHerrera2026Rpsvr.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: We derive a Sequential Minimal Optimization (SMO) algorithm for the quadratic dual problem arising from $\varepsilon$-SVR~\cite{Vapnik1995, Drucker1997, Smola2004} modified to minimize the Mean Absolute Percentage Error (MAPE)~\cite{Makridakis1993, Hyndman2006} directly in the loss function~\cite{benavides2025support}. This formulation is part of a broader family of SVR models with percentage-error losses that also includes least-squares variants~\cite{Suykens2002} and symmetric-kernel extensions~\cite{Espinoza2005}, whose unified structure is studied in~\cite{benavides2026unified}. The key structural difference from standard $\varepsilon$-SVR is that the box constraints become \emph{sample-dependent}: $α_k, α_k^* \in [0,\, 100C/y_k]$. We show that this modification affects only (i) the feasibility sets $\Iup$ and $\Idown$ in the working-set selection and (ii) the clipping bounds in the analytic two-variable update, while leaving the curvature formula and gradient update structurally identical to the standard SMO~\cite{Platt1998, Platt1999, Fan2005}. A shrinking heuristic adapted to the sample-dependent bounds is derived and shown to introduce an asymmetry between $α$- and $α^*$-variables controlled by the gap $2y_k\varepsilon/100$. The same solver applies to the symmetric-kernel variant (m2) by replacing $Ω$ with $Ω_s = \tfrac{1}{2}(Ω+ aΩ^*)$~\cite{Espinoza2005}. Numerical validation against an interior-point QP reference solver confirms solution agreement to within solver termination tolerance across ten synthetic configurations spanning both kernel variants and symmetry types. An implementation is available in the open-source \texttt{psvr} R package~\cite{BenavidesHerrera2026Rpsvr}.

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