Analog circuit sizing takes a significant amount of manual effort in a typical design cycle. With rapidly developing technology and tight schedules, bringing automated solutions for sizing has attracted great attention. This paper presents DNN-Opt, a Reinforcement Learning (RL) inspired Deep Neural Network (DNN) based black-box optimization framework for analog circuit sizing. The key contributions of this paper are a novel sample-efficient two-stage deep learning optimization framework leveraging RL actor-critic algorithms, and a recipe to extend it on large industrial circuits using critical device identification. Our method shows 5--30x sample efficiency compared to other black-box optimization methods both on small building blocks and on large industrial circuits with better performance metrics. To the best of our knowledge, this is the first application of DNN-based circuit sizing on industrial scale circuits.