  Gradient-based saliency maps are widely used to explain deep neural network
decisions. However, as models become deeper and more black-box, such as in
closed-source APIs like ChatGPT, computing gradients become challenging,
hindering conventional explanation methods. In this work, we introduce a novel
unified framework for estimating gradients in black-box settings and generating
saliency maps to interpret model decisions. We employ the likelihood ratio
method to estimate output-to-input gradients and utilize them for saliency map
generation. Additionally, we propose blockwise computation techniques to
enhance estimation accuracy. Extensive experiments in black-box settings
validate the effectiveness of our method, demonstrating accurate gradient
estimation and explainability of generated saliency maps. Furthermore, we
showcase the scalability of our approach by applying it to explain GPT-Vision,
revealing the continued relevance of gradient-based explanation methods in the
era of large, closed-source, and black-box models.
