Optimal VC Dimension of Contrastive Learning With Margin

Contrastive learning is a successful paradigm for learning $d$-dimensional geometric representations from a collection of “anchor–positive–negative” triplets $(i,j^{+},k^{-})$, indicating that “item $i$ is closer to $j$ than to $k$.” Despite its success, understanding why contrastive learning leads to representations of high *generalization* quality—beyond the often pessimistic predictions from PAC-learning—remains a central question. Recently, Alon et al. (2024) proved that, for PAC-learning $d$-dimensional Euclidean representations of $n$-point datasets, $\Theta (\min (nd,n^2))$ triplets are necessary and sufficient, while they posed as an open question whether their VC dimension bounds for the more realistic setting of *contrastive learning with a margin* can be improved. For a margin parameter $\alpha >0$, a triplet $(i,j^{+},k^{-}{)}_{\alpha }$ is satisfied by the embedding $\varphi :[n]\to {\mathbb{R}}^d$, if $\Vert \varphi (i)-\varphi (k){\Vert }_2>(1+\alpha )\cdot \Vert \varphi (i)-\varphi (j){\Vert }_2$. In this work, we resolve their question by proving that the VC dimension of contrastive learning under any margin $\alpha \in (0,1)$ is in fact $O(n/{\alpha }^2)$, improving on the previous bound of $O(n\log (n)/{\alpha }^2)$. We also establish that the bounds are optimal up to constant factors, by providing a matching lower bound of $\Omega (\frac{n}{{\alpha }^2})$ (the previously known lower bound was $\Omega (\frac{n}{\alpha })$), for $\alpha \ge \max (n^{-1/2},d^{-1/2})$.