Title: Sobolev Training of End-to-End Optimization Proxies
Primary area: optimization
Keywords: Differentiable programming; Machine learning surrogate; Sobolev training; Multi-level Decision Making

Abstract: Optimization proxies—machine-learning models trained to approximate
the solution mapping of parametric optimization problems in a single
forward pass—offer dramatic reductions in inference time compared to
traditional iterative solvers.  This work investigates the integration
of solver sensitivities into such end-to-end proxies via a
Sobolev–training paradigm and does so in \emph{two distinct settings}:
(i) \emph{fully supervised} proxies, where exact solver outputs and
sensitivities are available, and (ii) \emph{self-supervised} proxies
that rely only on the objective and constraint structure of the
underlying optimization problem.  
By augmenting the standard training loss
with directional-derivative information extracted from the solver, the
proxy aligns both its predicted solutions \emph{and} local derivatives
with those of the optimizer.  Under Lipschitz-continuity
assumptions on the true solution mapping, matching
first-order sensitivities is shown to yield uniform approximation error
proportional to the training-set covering radius.  
Empirically, different impacts are observed in each studied setting.  
On three large Alternating Current
Optimal Power Flow benchmarks, supervised Sobolev training cuts mean-squared error
by up to 56 \% and the median worst-case constraint violation by up to
400 \% while keeping the optimality gap below 0.22 \%.  
For a mean–variance
portfolio task trained without labeled solutions, self-supervised Sobolev training
halves the average optimality gap in the medium-risk region
(i.e. standard deviation above $10\%$ of budget) and matches the baseline elsewhere.
Together, these results highlight Sobolev training—whether supervised or
self-supervised—as a path to fast, reliable surrogates for
safety-critical, large-scale optimization workloads.

TL;DR: Sobolev training—supervised or self-supervised—makes fast ML surrogates more accurate and reliable for large, safety-critical optimization tasks.