NOVA: Discovering Well-Conditioned Winograd Transforms through Numerical Optimization of Vandermonde Arithmetic

TL;DR

NOVA introduces a novel numerical optimization framework that discovers stable, fractional Winograd transforms, significantly improving numerical conditioning and enabling high-accuracy, efficient low-precision convolution inference without retraining.

Contribution

NOVA breaks the traditional integer constraint in Winograd transforms by treating point selection as a continuous optimization problem, uncovering stable fractional configurations that enhance numerical stability.

Findings

Improves F(8,3) conditioning by 415x in 1D

Achieves 172,484x better conditioning in 2D convolution

Restores full accuracy in FP16 ImageNet inference without retraining

Abstract

Winograd convolution is the standard algorithm for efficient inference, reducing arithmetic complexity by 2.25x for 3x3 kernels. However, it faces a critical barrier in the modern era of low precision computing: numerical instability. As tiles scale to maximize efficiency (e.g., F(6,3), F(8,3)), the condition numbers of standard integer based transforms explode, reaching kappa = 2 x 10^5 for F(8,3), rendering them unusable in FP16 or Int8. We introduce NOVA (Numerical Optimization of Vandermonde Arithmetic), a discovery framework that breaks the decades old convention of integer interpolation. Treating Winograd point selection as a continuous optimization problem, NOVA searches the manifold R^n-1 via Evolution Strategy, snaps candidates to simple rationals, and guarantees correctness via symbolic verification. This process uncovers a hidden landscape of stable, fractional configurations such as {+-5/6, +-7/6, +-3/5} that defy traditional vocabulary constraints. The impact is transformative: NOVA improves the conditioning of F(8,3) by 415x in 1D, which squares to a 172,484x improvement for 2D convolution. In real world FP16 ImageNet inference, where standard transforms collapse to random chance (e.g., 4.7 percent accuracy on VGG16), NOVA's points restore full accuracy (75 to 78 percent), recovering over 70 percentage points without retraining, calibration, or learned parameters. These discovered transforms act as drop in replacements, effectively unlocking the efficiency of large tile Winograd convolution for next generation hardware.