Understanding the Curse of Unrolling

Bilevel optimization (BO) is widely applicable to many machine learning problems. Scaling BO, however, requires repeatedly computing hypergradients, which involves solving inverse Hessian-vector products (IHVPs). In practice, these operations are often approximated using crude surrogates such as one-step gradient unrolling or identity/short Neumann expansions, which discard curvature information. We build on implicit function theorem-based algorithms and propose to incorporate Kronecker-factored approximate curvature (KFAC), yielding curvature-aware hypergradients with a better performance efficiency trade-off than Conjugate Gradient (CG) or Neumann methods and consistently outperforming unrolling. We evaluate this approach across diverse tasks, including meta-learning and AI safety problems. On models up to BERT, we show that curvature information is valuable at scale, and KFAC can provide it with only modest memory and runtime overhead. Our implementation is available at https://github.com/liaodisen/NeuralBo.

Deep unfolding (DU) accelerates iterative optimizers by introducing learnable components and training them through unrolled iterations, but extending DU to the large-scale semidefinite programs (SDPs) common in robotics has remained limited. Unrolling a full-update conic solver such as COSMO exposes two obstacles that prior work on learned conic solvers has not: backpropagating through the per-iteration linear-system solve incurs memory quadratic in the problem size once the coefficient matrix is formed explicitly, and backpropagating through the positive semidefinite (PSD) cone projection becomes numerically unstable when eigenvalues coincide. We address the first obstacle with a matrix-free implicit differentiation rule that operates entirely through matrix-vector products, reducing memory from $O(n^2)$ to $O(n)$ and enabling backpropagation at scales where direct factorization runs out of memory. We address the second with a backward rule based on the Dalečkii--Krein representation of the Fréchet derivative, which remains well-defined under repeated eigenvalues. Together these make it possible to learn lightweight hyperparameter policies and warm-starts for a full-update conic solver. We evaluate on nonlinear covariance steering problems solved via sequential convex programming (SCP), as well as standalone SDPs and second-order cone programs ranging from max-cut and Lovász $\vartheta$ SDPs to robust estimation and control problems. The learned policies outperform state-of-the-art solvers across all problems, and can provide up to a 50$\times$ speedup depending on the class. When used as a subroutine in SCP, the learned approach delivers over a 30$\times$ speedup compared to COSMO.

Bilevel optimization underpins many machine learning applications, including hyperparameter optimization, meta-learning, neural architecture search, and reinforcement learning. While hypergradient-based methods have advanced significantly, a gap persists between theoretical guarantees and practical single-loop implementations required for efficiency. We bridge this gap by establishing sharper convergence results for single-loop approximate implicit differentiation (AID) and iterative differentiation (ITD) methods, leveraging our proposed analytical framework, decoupled norm analysis (DNA). For AID, we improve the convergence rate from $\mathcal{O}(κ^6/K)$ to $\mathcal{O}(κ^5/K)$, where $κ$ is the condition number of the inner-level problem. For ITD, we prove that the asymptotic error is $\mathcal{O}(κ^2)$, exactly matching the known lower bound and improving upon the previous $\mathcal{O}(κ^3)$ guarantee. Numerical experiments on synthetic and real tasks corroborate our theoretical findings.