Provable Gradient Editing of Deep Neural Networks

In explainable AI, DNN gradients are used to interpret the prediction; in safetycritical control systems, gradients could encode safety constraints; in scientificcomputing applications, gradients could encode physical invariants. While recent work on provable editing of DNNs has focused on input-output constraints, the problem of enforcing hard constraints on DNN gradients remains unaddressed. We present ProGrad, the first efficient approach for editing the parameters of a DNN to provably enforce hard constraints on the DNN gradients. Given a DNN N with parameters θ, and a set S of pairs (x x x, Q) of input x x x and corresponding linear gradient constraints Q, ProGrad finds new parameters θ θ θ such that (x x x,Q)∈S ∂ ∂x x x N(x x x; θ θ θ) ∈ Q while minimizing the changes ∥ θ θ θ -θ∥. The key contribution is a novel conditional variable gradient of DNNs, which relaxes the NP-hard provable gradient editing problem to a linear program (LP), enabling ProGrad to use an LP solver to efficiently and effectively enforce the gradient constraints. We experimentally evaluated ProGrad via enforcing (i) hard Grad-CAM constraints on IMAGENET ResNet DNNs; (ii) hard Integrated Gradients constraints on Llama 3 and Qwen 3 LLMs; (iii) hard gradient constraints in training a function-approximation DNN as a proxy for safety constraints in control systems and physical invariants in scientific applications. The results highlight the unique capability of ProGrad in enforcing hard constraints on DNN gradients. class: stingray 1 (a) Original image. Cosine: 0% IoU: 0% 1 (b) Reference Grad-CAM on the original image. misclassified: coral reef 1 (c) Misclassified Gaussian-noise corrupted image. Cos: 34.66% IoU: 4.35% 1 (d) Deviated Grad-CAM on the corrupted image. Cos: 100% IoU: 100%