Basin-Preserving Discretizations of Modern Hopfield Retrieval Dynamics: Energy Cells, Dissipation, and the Attention Limit

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Original abstract

The retrieval dynamics of a modern Hopfield network is the gradient flow of a log-sum-exp energy, while the attention update is its exact difference-of-convex minimization step. We study which time discretizations preserve not only energy decay and equilibria but also basins of attraction. We introduce energy cells, connected components of sublevel sets containing one attractor and no other critical point. Our main theorem shows that every finite energy cell below the escape energy is contained simultaneously in the basin of the continuous flow, every relaxed attention map $Ψ_θ=(1-θ)\,\mathrm{id}+θ\,\mathrm{attention}$ for $0<θ<2$, and implicit Euler throughout its uniqueness regime. A parameter-uniform unit-curvature majorant yields unconditional dissipation and a monotone interpolation of each discrete step. We also derive explicit local contraction bounds near well-separated patterns, with a certified optimal slight overrelaxation; characterize proximal tunneling and overshoot beyond the preservation regimes; compare first-order error constants; establish an order barrier for scalar reparametrizations of the relaxed family; construct a second-order scalar-auxiliary-variable scheme; and extend cell preservation to damped difference-of-convex iterations in Bregman geometry, including a certified overrelaxed window under bounded asymmetry. Nine numerical campaigns test the bounds and failure mechanisms. In two-dimensional basin experiments, all observed disagreements between continuous and discrete retrieval occur above the attractor-specific numerically inferred escape level.

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