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Life sciences · Preprint
arXiv · August 13, 2026
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This is an unrefereed preprint proposing a theoretical framework (AIM) and a new optimizer (RADAR) based on residual-penalty variable splitting and ADMM-inspired multiplier correction. The abstract claims convergence proofs and improvements over existing optimizers across vision, language, and reinforcement learning tasks, but provides no numerical results, statistical comparisons, or peer-reviewed validation.
Preprint.
AIM framework interprets momentum as multiplier-like correction driven by splitting residual RADAR combines relativistic adaptive geometry, decoupled residual correction, and second-order momentum filtering Stochastic convergence established through variance-perturbed Lyapunov drift analysis
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This is a methodological paper proposing new optimization theory and algorithms without empirical validation on standard benchmarks or clinical outcomes; it raises questions about momentum mechanisms rather than answering them with rigorous experimental evidence.
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Momentum-based optimizers are widely used in modern deep learning, yet the relations among momentum recursion, update geometry, and acceleration remain only partially understood. We develop an $\textbf{A}$DMM-$\textbf{I}$nspired $\textbf{M}$omentum (AIM) framework based on residual-penalty variable splitting, which interprets momentum as a multiplier-like correction driven by the splitting residual. AIM recovers the exponential moving average of gradients from an ADMM-style multiplier update and separates two mechanisms that are usually intertwined in practical optimizers: the residual penalty determines the update geometry, whereas the approximation of the objective-related subproblem determines the acceleration form. Building on AIM, we propose $\textbf{R}$elativistic $\textbf{A}$daptive gradient $\textbf{D}$escent with $\textbf{A}$ccelerated $\textbf{R}$esidual (RADAR), which combines relativistic adaptive geometry, decoupled residual correction, and second-order momentum filtering to improve the update direction and momentum estimation. We establish stochastic convergence through a variance-perturbed Lyapunov drift analysis. Experiments on supervised vision learning, language modeling, and reinforcement learning show that RADAR achieves consistent improvements over strong adaptive optimizer baselines.
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