Replica Symmetry Breaking and Algorithmic Thresholds in Empirical Risk Minimization under Multi-Index Model

Deep neural networks excel in high-dimensional problems, outperforming models such as kernel methods, which suffer from the curse of dimensionality. However, the theoretical foundations of this success remain poorly understood. We follow the idea that the compositional structure of the learning task is the key factor determining when deep networks outperform other approaches. Taking a step towards formalizing this idea, we consider a simple compositional model, namely the multi-index model (MIM). In this context, we introduce and study hyper-kernel ridge regression (HKRR), an approach blending neural networks and kernel methods. Our main contribution is a sample complexity result demonstrating that HKRR can adaptively learn MIM, overcoming the curse of dimensionality. Further, we exploit the kernel nature of the estimator to develop ad hoc optimization approaches. Indeed, we contrast alternating minimization and alternating gradient methods both theoretically and numerically. These numerical results complement and reinforce our theoretical findings.

We study a general class of optimization problems with decision variable $\boldsymbolΘ \in \mathbb{R}^{p \times k}$ and cost function which is the sum of $n$ terms, each dependent on $\boldsymbolΘ$ through the $k$-dimensional projection $\boldsymbolΘ^\top \boldsymbol{x}_i$, where $\boldsymbol{x}_i$, $i \leq n$ are i.i.d. random vectors. This setting is general enough to include examples of current interest in statistical physics, high-dimensional statistics, and statistical learning theory. We consider the proportional asymptotics $n, p \to \infty$, with $n/p = Θ(1)$, and prove that, whenever there exists a minimizer satisfying a suitable generalization of a "delocalization" condition, the minimum value is universal. Namely, (for subgaussian $\boldsymbol{x}_i$) it depends on the distribution of $\boldsymbol{x}_i$ only through its asymptotic mean and covariance. This delocalization condition is essentially necessary. Earlier universality results for such problems were limited to strongly convex loss functions. We derive applications of our theory to statistical learning and prove general universality results both for train and (under additional conditions) test error. In particular, we establish universality for vectors $\boldsymbol{x}_i$ generated by random 1-layer neural networks (random features models) and first-order Taylor approximations of 2-layer networks (neural tangent models). Finally, we establish that the delocalization property holds for a class of statistical learning problems under a condition that is easy to verify.

We study a prototypical situation when a learned predictor can discover useful low-dimensional structure in data, while using fewer samples than are needed for accurate prediction. Specifically, we consider the problem of recovering a multi-index polynomial $f^*(x)=h(Ux)$, with $U\in\mathbb{R}^{r\times d}$ and $r\ll d$, from finitely many data/label pairs. Importantly, the target function depends on input $x$ only through the projection onto an unknown $r$-dimensional central subspace. The algorithm we analyze is appealingly simple: fit kernel ridge regression (KRR) to the data and compute the Average Gradient Outer Product (AGOP) from the fitted predictor. Our main results show that under reasonable assumptions the top $r$-dimensional eigenspace of AGOP provably recovers the central subspace, even in regimes when the prediction error remains large. Specifically, if the target function $f^*$ has degree $p^*$, it is known that $n\asymp d^{p^*}$ samples are necessary for KRR to achieve accurate prediction. In contrast, we show that if a low degree $p$ component of $f^*$ already carries all relevant directions for prediction, subspace recovery occurs in the much lower sample regime $n\asymp d^{p+δ}$ for any $δ\in(0,1)$. Our results thus demonstrate a separation between prediction and representation, and provide an explanation for why iterative kernel methods such as Recursive Feature Machines (RFM) can be sample-efficient in practice.