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This research establishes a dynamic equivalence between Hyperball and Base optimizers, enabling the replication of standard optimizer dynamics through a novel geometric framework without direct execution. By deriving an inverse mapping, the authors demonstrate that these distinct paradigms yield nearly identical loss trajectories in scale-invariant networks, offering new flexibility in training strategies. The work advances theoretical understanding of optimization geometry and suggests potential for more efficient or constrained neural network training protocols.
HyperTransfer: Understanding the Equivalence between Base Optimizer and Hyperball
arXiv:2609.07017v1 Announce Type: new Abstract: Hyperball optimizers constrain parameter norms and update only their directions, establishing a distinct paradigm for neural network optimization. Although this geometry appears fundamentally different from that of conventional Base Optimizers, which update both parameter norms and directions, we show that the two paradigms are dynamically equivalent for scale-invariant networks. Building on this equivalence, we propose HyperTransfer, which constructs a Hyperball optimizer that reproduces the dynamics of a target Base Optimizer using only its initialization and learning-rate schedule, without running the target optimizer itself. We further derive the inverse mapping and extend the framework to non-scale-invariant networks. Experiments show that both HyperTransfer and the inverse mapping produce loss trajectories nearly identical to those of their targets, suggesting that Hyperball dynamics are governed primarily by the induced effective learning-rate schedule and optimizer state.
Abstract:Hyperball optimizers constrain parameter norms and update only their directions, establishing a distinct paradigm for neural network optimization. Although this geometry appears fundamentally different from that of conventional Base Optimizers, which update both parameter norms and directions, we show that the two paradigms are dynamically equivalent for scale-invariant networks. Building on this equivalence, we propose HyperTransfer, which constructs a Hyperball optimizer that reproduces the dynamics of a target Base Optimizer using only its initialization and learning-rate schedule, without running the target optimizer itself. We further derive the inverse mapping and extend the framework to non-scale-invariant networks. Experiments show that both HyperTransfer and the inverse mapping produce loss trajectories nearly identical to those of their targets, suggesting that Hyperball dynamics are governed primarily by the induced effective learning-rate schedule and optimizer state.
Machine Learning (cs.LG)
arXiv:2609.07017 [cs.LG]
(or arXiv:2609.07017v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.07017 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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