Open weight models, which are ubiquitous, rarely provide access to their training data or loss function. This makes modifying such models for tasks such as pruning or unlearning, which are constrained by this unavailability, an active area of research. Existing techniques typically require gradients or ground-truth labels, rendering them infeasible in settings with limited computational resources. In this work, we investigate the fundamental question of identifying components that are critical to the model's predictive performance, without access to either gradients or the loss function, and with only distributional access such as synthetic data. We theoretically demonstrate that the global error is linearly bounded by local reconstruction errors for Lipschitz-continuous networks such as CNNs and well-trained Transformers (which, contrary to existing literature, we find exhibit Lipschitz continuity). This motivates using the locally reconstructive behavior of component subsets to quantify their global importance, via a metric that we term *Subset Fidelity*. In the uncorrelated features setting, selecting individual components based on their Subset Fidelity scores is optimal, which we utilize to propose **ModHiFi**, an algorithm for model modification that requires neither training data nor access to a loss function. **ModHiFi-P**, for structured pruning, achieves an 11\% speedup over the current state of the art on ImageNet models and competitive performance on language models. **ModHiFi-U**, for classwise unlearning, achieves complete unlearning on CIFAR-10 without fine-tuning and demonstrates competitive performance on Swin Transformers.

The success of the convex-concave procedure (CCCP), a widely used technique for non-convex optimization, crucially depends on finding a decomposition of the objective function as a difference of convex functions (dcds). Despite the widespread applicability of CCCP, finding such dcds has attracted little attention in machine learning. For graphical models with polynomial potentials, existing methods for finding dcds require solving a Sum-of-Squares (SOS) program, which is often prohibitively expensive. In this work, we leverage tools from algebraic geometry certifying the positivity of polynomials, to derive LP-based constructions of dcds of polynomials which are particularly suited for graphical model inference. Our experiments demonstrate that using our LP-based technique constructs dcds for polynomial potentials of Markov random fields significantly faster compared to SOS-based approaches used in previous works.