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

TL;DR

This paper introduces an SMO algorithm tailored for $ ext{ε}$-SVR with MAPE loss, featuring sample-dependent box constraints, and validates its effectiveness through numerical experiments and an open-source R package.

Contribution

The paper develops a novel SMO algorithm for $ ext{ε}$-SVR with MAPE loss incorporating sample-dependent constraints, extending standard SMO to this new setting.

Findings

Solution matches interior-point solver within tolerance

Algorithm adapts to sample-dependent bounds and kernel variants

Open-source implementation available in R package

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}: $\alpha_k, \alpha_k^* \in [0,\, 100C/y_k]$. 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 $\alpha$- and $\alpha^*$-variables controlled by the gap $2y_k\varepsilon/100$. The same solver applies to the symmetric-kernel variant (m2) by replacing $\Omega$ with $\Omega_s = \tfrac{1}{2}(\Omega + a\Omega^*)$~\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}.