Provable Gradient Editing of Deep Neural Networks

In explainable AI, DNN gradients are used to interpret the prediction; in safety-critical control systems, gradients could encode safety constraints; in scientific-computing 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 $\mathcal{N}$ with parameters $\theta$, and a set $\mathcal{S}$ of pairs $(\mathrm{x},\mathrm{Q})$ of input $\mathrm{x}$ and corresponding linear gradient constraints $\mathrm{Q}$, ProGrad finds new parameters ${\theta }^{\prime }$ such that ${\bigwedge }_{(\mathrm{x},\mathrm{Q})\in \mathcal{S}}\frac{\partial }{\partial \mathrm{x}}\mathcal{N}(\mathrm{x};{\theta }^{\prime })\in \mathrm{Q}$ while minimizing the changes $\Vert {\theta }^{\prime }-\theta \Vert$.

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

1. hard Grad-CAM constraints on ImageNet ResNet DNNs;
2. hard Integrated Gradients constraints on Llama 3 and Qwen 3 LLMs;
3. hard gradient constraints in training a DNN to approximate a target function 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.