Phase Transitions in Learning Driven by Extremal Data

We study an extremal regression problem, an example of multiple instance learning (MIL), where the predictors are given as multisets, called bags. The bag elements (called instances) possess hidden labels which are pooled to produce an observed bag-level label. Max pooling is common in MIL applications, but gives rise to vanishing gradients. To handle this, modern MIL pipelines approximate the max with a softmax, treating the sharpness as a learnable parameter, denoted $λ$. When the max pooling model is the true data generating process, learning will be successful when $λ$ is increased during gradient descent, thereby driving the dynamics into regions of parameter space with vanishing gradients. To understand the conditions under which gradient descent breaks down, we study a minimal model and derive a low dimensional projection of the gradient flow in the joint limit of large bags $n$ and high dimension $d$ with $α=\ln n/d$ fixed. The flow has two distinct phase transitions: an intrinsic freezing transition related to the random energy model of disordered systems, and a retrieval transition occurring in modern Hopfield networks. Numerical simulation confirms both the phase diagram and the convergence of the finite-size training gradient to its predicted asymptotic form.