Guarded Block Transfers for Polytope Distance: Linear Convergence with Explicit Constants and Gap-Certified Screening for SVM Training

Mayank Gupta

Hard-margin SVM training is the distance problem between two convex hulls, which the Triangle Algorithm (TA) solves with a certified duality gap. We propose \emph{guarded block transfers}: $k$ Mitchell--Demyanov--Malozemov (MDM) transfers per scan are aggregated into one exact line search, guarded by a comparison with the single best transfer at a cost independent of the number of points $n$, and replaced by an away step when the leading transfer is capacity-clipped. On the Minkowski-difference polytope, where the objective is $1$-strongly convex, we prove global linear convergence with explicit constants and no combinatorial swap-step bounds; the per-step contraction is $k$-independent and only the drop-count bound grows with $k$, so aggregation is provably safe while the saving in scans is measured, not proved. Since the TA certificate is the Frank--Wolfe gap divided by the current distance, gap-safe screening applies, and we bound the iteration after which it has removed every zero-weight point off the optimal faces by $O\!\left(\tfrac{k}ρ\log\tfrac{1}τ\right)$ up to logarithmic factors in the geometry, $ρ$ the rate constant and $τ$ the optimal-face margin. On five LIBSVM…