Due to privacy or commercial constraints, large pre-trained language models (PLMs) are often offered as black-box APIs. Fine-tuning such models to downstream tasks is challenging because one can neither access the model's internal representations nor propagate gradients through it. This paper addresses these challenges by developing techniques for adapting PLMs with only API access. Building on recent work on soft prompt tuning, we develop methods to tune the soft prompts without requiring gradient computation. Further, we develop extensions that in addition to not requiring gradients also do not need to access any internal representation of the PLM beyond the input embeddings. Moreover, instead of learning a single prompt, our methods learn a distribution over prompts allowing us to quantify predictive uncertainty. Ours is the first work to consider uncertainty in prompts when only having API access to the PLM. Finally, through extensive experiments, we carefully vet the proposed methods and find them competitive with (and sometimes even improving on) gradient-based approaches with full access to the PLM.

Likelihood-free (a.k.a. simulation-based) inference problems are inverse problems with expensive, or intractable, forward models. ODE inverse problems are commonly treated as likelihood-free, as their forward map has to be numerically approximated by an ODE solver. This, however, is not a fundamental constraint but just a lack of functionality in classic ODE solvers, which do not return a likelihood but a point estimate. To address this shortcoming, we employ Gaussian ODE filtering (a probabilistic numerical method for ODEs) to construct a local Gaussian approximation to the likelihood. This approximation yields tractable estimators for the gradient and Hessian of the (log-)likelihood. Insertion of these estimators into existing gradient-based optimization and sampling methods engenders new solvers for ODE inverse problems. We demonstrate that these methods outperform standard likelihood-free approaches on three benchmark-systems.