Training materials through periodic drive allows us to endow materials and structures with complex elastic functions. As a result of the driving, the system explores the high-dimensional space of structures, ultimately converging to a structure with the desired response. However, increasing the complexity of the desired response results in ultraslow convergence and degradation. Here, we show that by constraining the search space, we are able to increase robustness, extend the maximal capacity, train responses that previously did not converge, and in some cases accelerate convergence by many orders of magnitude. We identify the geometrical constraints that prevent the formation of spurious low-frequency modes, which are responsible for failure. We argue that these constraints are analogous to regularization used in machine learning. We propose a unified relationship between complexity, degradation, convergence, and robustness.

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• Dynamics of networks
• Plastic deformation
• Smart materials