In this paper, we specifically investigate the deep equilibrium model (DEQ), an infinite-depth neural network with shared weight matrices across layers.

We propose ODER as a new strategy for improving the efficiency of DEQ through stochastic approximations of the measurement models. We theoretically analyze ODER giving insights intoits ability to approximate the traditional DEQ approach for solving inverse problems.

Deep equilibrium models (DEQs) refrain from the traditional layer-stacking paradigm and turn tofind the fixed point ofasingle layer.

Implicit models such as Deep Equilibrium Models (DEQs) have emerged as promising alternative approaches for building deep neural networks. Their certified robustness has gained increasing research attention due to security concerns. Existing certified defenses for DEQs employing interval bound propagation and Lipschitz-bounds not only offer conservative certification bounds but also are restricted to specific forms of DEQs. In this paper, we provide the first randomized smoothing certified defense for DEQs to solve these limitations. Our study reveals that simply applying randomized smoothing to certify DEQs provides certified robustness generalized to large-scale datasets but incurs extremely expensive computation costs. To reduce computational redundancy, we propose a novel Serialized Randomized Smoothing (SRS) approach that leverages historical information. Additionally, we derive a new certified radius estimation for SRS to theoretically ensure the correctness of our algorithm. Extensive experiments and ablation studies on image recognition demonstrate that our algorithm can significantly accelerate the certification of DEQs by up to 7x almost without sacrificing the certified accuracy. The implementation will be publicly available upon the acceptance of this work.

Neural networks with wide layers have attracted significant attention due to their equivalence to Gaussian processes, enabling perfect fitting of training data while maintaining generalization performance, known as benign overfitting. However, existing results mainly focus on shallow or finite-depth networks, necessitating a comprehensive analysis of wide neural networks with infinite-depth layers, such as neural ordinary differential equations (ODEs) and deep equilibrium models (DEQs). In this paper, we specifically investigate the deep equilibrium model (DEQ), an infinite-depth neural network with shared weight matrices across layers. Our analysis reveals that as the width of DEQ layers approaches infinity, it converges to a Gaussian process, establishing what is known as the Neural Network and Gaussian Process (NNGP) correspondence. Remarkably, this convergence holds even when the limits of depth and width are interchanged, which is not observed in typical infinite-depth Multilayer Perceptron (MLP) networks. Furthermore, we demonstrate that the associated Gaussian vector remains non-degenerate for any pairwise distinct input data, ensuring a strictly positive smallest eigenvalue of the corresponding kernel matrix using the NNGP kernel. These findings serve as fundamental elements for studying the training and generalization of DEQs, laying the groundwork for future research in this area.