Paper:
It has recently been shown that ReLU networks produce arbitrarily over-confident predictions far away from the training data. Thus, ReLU networks do not know when they don't know. However, this is a highly important property in safety critical applications. In the context of out-of-distribution detection (OOD) there have been a number of proposals to mitigate this problem but none of them are able to make any mathematical guarantees. In this paper we propose a new approach to OOD which overcomes both problems. Our approach can be used with ReLU networks and provides provably low confidence predictions far away from the training data as well as the first certificates for low confidence predictions in a neighborhood of an out-distribution point. In the experiments we show that stateof-the-art methods fail in this worst-case setting whereas our model can guarantee its performance while retaining state-of-the-art OOD performance.

Review:
Summary: This paper provides a method to train neural networks with guarantees on outputting probabilities/scores close to uniform on inputs that are out of domain.
Precisely, on some point x_0, we can obtain the largest radius such that on inputs in the ball around x_0 the classifier outputs some predetermined  maximum score. 
The paper combines a simple generative model (mixture of Gaussian) for modeling in-distribution vs. out of distribution. The simplicity allows to obtain guarantees on the probability of an input being considered out of class. The model output p(y | x) critically uses the probability of being out of distribution. Hence guarantees on being detected as out of distribution translate to guarantees on p(y|x) being close to uniform.
Decision: I vote for accepting the paper. The paper has some clear strengths. However, I have some concerns regarding experiments and comparison to previous work. It would be great if the authors could clarify. 
Strengths: The paper’s methodolgy is clearly written.  The modeling is clear and sound. Overall, this idea is a promising approach to obtain networks that are provably under-confident far from training examples. The training cost for this approach is comparable to standard training, and the approach seems scalable and broadly applicable in general. 
The experimental evaluation is also clearly described. Worst-case evaluation of OOD (out of domain) performance seems novel and the gains not this objective using the proposed approach of this paper are interesting and promising. 
Concerns: While the paper explains the proposed method well, the description of previous work and relation to previous work is inadequate. After spending some hours reading the cited paper, I am still confused about what’s the novelty of this work at a high level. 
This work uses p(x|i) and p(x|o) in the computation of p(y|x) during inference, where i and o are in distribution and out of distribution respectively. This is crucial to obtain guarantees one performance. However, how does this compare to other previous work that also uses some kind of generative modeling to model in/out distribution? A bunch of papers are cited in the introduction as doing this, but the relationship to the proposed work is unclear.
— I have a major experimental concern: When comparing against ACET, the baseline of performing adversarial style training on random noise inputs seems more appropriate since it’s closer to the evaluation metric (which picks random noise as out of domain and not 80M Tiny Images). What does the performance of this ACET baseline look like?
— During evaluation, how do you ensure that the radius is not too large such that it has images that the model should actually be confident on (close to in distribution samples). A flip-side is how sensitive the performance is to the score chosen for choosing the radius for computing the valid set of images for the adversary? It’s possible that the 11%  threshold is too high? What’s the minimum confidence of CCU on the test images?
Minor comments on writing: 
—K_i, K_0 are not defined when writing the expression for GMM