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

Key points

• Hard-margin SVM training is the distance problem between two convex hulls, which the Triangle Algorithm (TA) solves with a certified duality gap.1
• 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.1
• 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.1
• But our full-cache sequential minimal optimisation (SMO) code ties or beats it on kernel $L2$ margins, LIBLINEAR's dual solver is up to $4.5\times$ faster on the separable $L2$ margin, and our solver is one to three orders of magnitude slower than LIBLINEAR's Newton solver on dense-support $L2$ margins and than LIBSVM on a nearly touching kernel hull.1

Sources (1)

• [1]Guarded Block Transfers for Polytope Distance: Linear Convergence with Explicit Constants and Gap-Certified Screening for SVM Training
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 11, 06:32 PM

  We propose 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.

  Hard-margin SVM training is the distance problem between two convex hulls, which the Triangle Algorithm (TA) solves with a certified duality gap.

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