Diversity Is All You Need for Contrastive Learning: Spectral Bounds on Gradient Magnitudes

We derive non-asymptotic spectral bands that bound the squared InfoNCE gradient norm via alignment, temperature, and batch spectrum, recovering the \(1/τ^{2}\) law and closely tracking batch-mean gradients on synthetic data and ImageNet. Using effective rank \(R_{\mathrm{eff}}\) as an anisotropy proxy, we design spectrum-aware batch selection, including a fast greedy builder. On ImageNet-100, Greedy-64 cuts time-to-67.5\% top-1 by 15\% vs.\ random (24\% vs.\ Pool--P3) at equal accuracy; CIFAR-10 shows similar gains. In-batch whitening promotes isotropy and reduces 50-step gradient variance by \(1.37\times\), matching our theoretical upper bound.