Certified methods enable precise training and inference for deep equilibrium networks
Certified Inference and Training for Deep Equilibrium Networks: A Continuation Framework with Polynomial Complexity Guarantees
Machine Learning
Summary
Deep equilibrium networks are a type of AI model that find a stable solution instead of running many layers. The authors develop a way to carefully and reliably track these stable solutions during both model use and learning. Their method comes with guarantees about accuracy and computational cost, making sure the process stays efficient and correct. They also use formal verification tools to check the core calculations and compare their method with other approaches.
What this means in practice
- •For machine learning engineers: Implement reliable and efficient training methods for deep equilibrium networks with guaranteed accuracy and controlled complexity.
- •For algorithm developers: Develop numerically stable algorithms for tracking equilibrium points in iterative models using certified continuation and error bounds.
Authors
Alex Borisevich
Abstract
We develop a certified continuation framework for equilibrium computation and for training deep equilibrium networks (DEQs), with training formulated as interpolation to accuracy $2^{-b}$. For inference, compact input homotopy selects a unique branch from a supplied start root, and a rounded Newton tracker follows it under certified boundary, conditioning, derivative, and tube-radius bounds. For training, we augment local-plus-low-rank recurrence with programmable dormant bilinear rank-one channels. Loaded Tikhonov solves diagnose a failed interpolation pass without spectral decomposition; an output-preserving repair aligned with the pass residual supplies the required direction. Training requires certified gate realization and column stability on each pass region, well-posed inference, and finite-update error budgets. With polynomial geometric, encoding, precision, and complete backend budgets, both certified inference and training have bit cost $O(\operatorname{poly}(L+b))$, where $L$ is the encoded instance length. The trainer uses $O(b+\ell)$ passes and reserve channels from an initial residual bounded by $2^\ell$. These guarantees concern a certified promise class. Lean 4 verifies the quantitative core and concrete inference backend; numerical comparisons illustrate the loaded mechanism.