A Convergent Method for Energy Optimization in Modern Hopfield Networks

Modern Hopfield networks are energy-based associative memory models whose performance critically depends on the structure and optimization of their energy functions. While recent formulations substantially improve storage capacity, the resulting non-convex energy landscapes are often optimized using heuristic update rules that can be sensitive to initialization and may not provide monotonic energy descent or rigorous convergence guarantees. In this work, we propose a new energy formulation for modern Hopfield networks together with a principled iterative optimization scheme. The proposed energy admits a natural decomposition that allows optimization via the concave–convex procedure (CCCP), yielding well-defined network dynamics with guaranteed energy descent beyond classical Hopfield updates. We establish fundamental theoretical properties of the proposed framework, including non-negativity, boundedness, and monotonic decrease in the energy along iterations. In particular, we prove that the induced dynamics converge to a stationary point of the energy function, providing explicit convergence guarantees for the resulting Hopfield-type model. We further evaluate the proposed approach on synthetic classification tasks and compare its optimization behavior with that of the original Hopfield network and several standard machine learning baselines. Experimental results demonstrate improved stability, convergence behavior, and competitive classification performance. We also validate the approach on real-world benchmark datasets to demonstrate utility beyond controlled experiments. Overall, this work provides a theoretically grounded energy-based optimization framework for modern Hopfield networks, clarifying the role of principled optimization in achieving stable and convergent associative memory dynamics.