On the Fine-Grained Hardness of Inverting Generative Models

TL;DR

This paper investigates the computational hardness of inverting generative models, establishing new lower bounds under various complexity hypotheses for both exact and approximate inversion problems.

Contribution

It provides the first fine-grained complexity bounds for generative model inversion, strengthening existing results and covering different norms and approximation settings.

Findings

Exact inversion is NP-hard under SETH with exponential lower bounds.

Approximate inversion with odd p norms has exponential lower bounds under SETH.

Even p norms for approximate inversion are hard under ETH, with similar exponential bounds.

Abstract

The objective of generative model inversion is to identify a size-$n$ latent vector that produces a generative model output that closely matches a given target. This operation is a core computational primitive in numerous modern applications involving computer vision and NLP. However, the problem is known to be computationally challenging and NP-hard in the worst case. This paper aims to provide a fine-grained view of the landscape of computational hardness for this problem. We establish several new hardness lower bounds for both exact and approximate model inversion. In exact inversion, the goal is to determine whether a target is contained within the range of a given generative model. Under the strong exponential time hypothesis (SETH), we demonstrate that the computational complexity of exact inversion is lower bounded by $\Omega(2^n)$ via a reduction from $k$-SAT; this is a strengthening of known results. For the more practically relevant problem of approximate inversion, the goal is to determine whether a point in the model range is close to a given target with respect to the $\ell_p$-norm. When $p$ is a positive odd integer, under SETH, we provide an $\Omega(2^n)$ complexity lower bound via a reduction from the closest vectors problem (CVP). Finally, when $p$ is even, under the exponential time hypothesis (ETH), we provide a lower bound of $2^{\Omega (n)}$ via a reduction from Half-Clique and Vertex-Cover.