Simulating, Fast and Slow: Learning Policies for Black-Box Optimization

Simulators based on differential equations are ubiquitous in science and engineering. They are often used in simulation-based inference to evaluate the posterior distribution of the input parameters based on real-world observations of the simulator outputs. However, inference becomes challenging when individual simulator evaluations are computationally expensive. In such cases, a Bayesian optimization-based active learning approach with Gaussian process surrogate models has been used to maximize the information obtained from a limited simulation budget. Recently, gradients of simulator outputs with respect to input parameters have become increasingly available, yet they are rarely exploited for inference. Even though we only need to learn the simulator input-output relationship, gradient information can provide an additional valuable signal to guide the active learning procedure. This is of particular interest in the case of expensive simulators, when sample efficiency is crucial. In this paper, we demonstrate how incorporating gradient information into the Gaussian process surrogate accelerates Bayesian optimization-based inference under a limited simulation budget. Our results show significant improvement in convergence speed from using gradient information. For reverse-mode differentiation, the inference efficiency gains are maintained when accounting for the additional computational cost. In contrast, for forward-mode differentiation, the inference speed-up does not outweigh the computational costs. These results indicate that gradient-enhanced surrogates are beneficial primarily in problems where the number of parameters exceeds the output dimensionality, where reverse-mode differentiation is efficient.

Diffusion models have demonstrated strong power in generative modeling tasks across multiple domains, exhibiting a remarkable capability of learning complex distributions from samples. In this paper, we leverage such capability to design an efficient universal diffusion-based solver for parameterized black-box optimizations (BBO), where the optimizer has only black-box access to queries of the objective function at the learning stage, yet is able to reduce the additional computational cost at the inference stage for each BBO instance while also capturing the potential multi-modal landscape of non-convex objectives. To cast our formulation as a compatible generative modeling task, we introduce the notion of minimization policy as a new solution concept, which defines a sampling distribution over the solutions that should concentrate around the minimizer set for each BBO instance. We then propose Diffusion-based Iterative Minimization Policy Learning (D-IMPL), a practical generative-model-based solver for solving parameterized BBOs that employs diffusion models to learn a minimization policy, whose density is proportional to the exponential of the negated objective values, thereby amortizing the computational costs across different BBO instances. Furthermore, we demonstrate the performance of our D-IMPL algorithm by establishing a sample complexity guarantee showing that a $δ$-approximate minimization policy can be effectively learned within $O(\log(1/δ))$ iterations, and by extensive empirical evaluations over a range of constrained and unconstrained BBO tasks.

Exact stochastic simulation of continuous-time Markov chains (CTMCs) is essential when discreteness and noise drive system behavior, but the hard categorical event selection in Gillespie-type algorithms blocks gradient-based learning. We eliminate this constraint by decoupling forward simulation from backward differentiation, with hard categorical sampling generating exact trajectories and gradients propagating through a continuous massively-parallel Gumbel-Softmax straight-through surrogate. Our approach enables accurate optimization at parameter scales over four orders of magnitude beyond existing simulators. We validate for accuracy, scalability, and reliability on a reversible dimerization model (0.09% error), a genetic oscillator (1.2% error), a 203,796-parameter gene regulatory network achieving 98.4% MNIST accuracy (a prototypical deep-learning multilayer perceptron benchmark), and experimental patch-clamp recordings of ion channel gating (R^2 = 0.987) in the single-channel regime. Our GPU implementation delivers 1.9 billion steps per second, matching the scale of non-differentiable simulators. By making exact stochastic simulation massively parallel and autodiff-compatible, our results enable high-dimensional parameter inference and inverse design across systems biology, chemical kinetics, physics, and related CTMC-governed domains.