Deep learning models achieve high predictive accuracy in a broad spectrum of tasks, but rigorously quantifying their predictive uncertainty remains challenging. Usable estimates of predictive uncertainty should (1) cover the true prediction target with a high probability, and (2) discriminate between high-and low-confidence prediction instances. State-of-the-art methods for uncertainty quantification are based predominantly on Bayesian neural networks. However, Bayesian methods may fall short of (1) and (2) -i.e., Bayesian credible intervals do not guarantee frequentist coverage, and approximate posterior inference may undermine discriminative accuracy. To this end, this paper tackles the following question: can we devise an alternative frequentist approach for uncertainty quantification that satisfies (1) and (2)? To address this question, we develop the discriminative jackknife (DJ), a formal inference procedure that constructs predictive confidence intervals for a wide range of regression models, is easy to implement, and provides rigorous theoretical guarantees on (1) and (2). The DJ procedure uses higher-order influence functions (HOIFs) of the trained model parameters to construct a jackknife (leave-one-out) estimator of predictive confidence intervals. DJ computes HOIFs using a recursive formula that requires only oracle access to loss gradients and Hessian-vector products, hence it can be applied in a post-hoc fashion without compromising model accuracy or interfering with model training. Experiments demonstrate that DJ performs competitively compared to existing Bayesian and non-Bayesian baselines.
This paper studies how to construct confidence intervals for deep neural networks with guaranteed coverage. The authors propose an algorithm, “discriminative jackknife”, based on the standard jackknife confidence interval estimate which they augment by a “local uncertainty estimate” based on the variability of the n leave-one-out fitted versions of the underlying algorithm (n = # data points). The whole study is concluded with toy and real-world examples showing the proposed algorithm is competitive with existing methods while also achieving the desired coverage.
I am currently leaning towards recommending rejection of the paper. The main reasons for this are: (i) A potential failure to cite and acknowledge the prior contribution of [1] which seems to have non-trivial overlap with this paper (on arxiv since end of July which is more than 30 days before the ICLR submission deadline and thus should be treated as prior work); (ii) The claims of guaranteed frequentist coverage are not backed up as, according to thm.2, they only hold when n >> 0 and the number of influence functions used goes to infinity (ideally, the authors would provide non-asymptotic bounds as in [1], but at the very least, these limitations should have been clearly pointed out and their practical implications discussed). Finally, I would like to say that I am holding the paper to a higher standard due to its 10 page length as instructed by the guidelines.
Major comments:
- Can you please explain the relation of this work to [1]? It seems that [1] already proposes use of the higher order expansions, provides an efficient implementation based on forward mode autodiff (do you plan to release code?), and moreover provides non-asymptotic bounds which are not present in your work?! 
- On a related note, Giordano et al. provide a careful analysis and discussion of the assumptions in their sect.4. Can you please clarify which of the assumptions you also make, and why you don’t need the others (if any)? 
- Throughout the paper (e.g., in and around eq.1), you seem to assume that there exists a unique minimiser of the objective which generally won’t be true (especially in your application to deep neural networks). In [1], the technical assumptions ensure this is true but I don’t see how this is handled in your case?! Can you please help me understand how to interpret your work in case there are multiple (possibly local) minima, and whether this has any effect on the results in thm.2?
- I think it would be beneficial to the reader if you could please provide a discussion and/or formula for how large n has to be for Theorem 2 to apply.
- On p.8, you say “To ensure a fair comparison, the hyper-parameters of the model f(x; θ) were the same for all baselines.” I am not sure I agree this is a fair comparison. Commonly, one has the opportunity to select hyperparameters for their algorithm (e.g., using a validation set, cross-validation, etc.) so it seems it would have been fair to run each algorithm with its best hyperparameters. Can you please provide a discussion of how this would affect the reported results?
Minor comments:
- On p.3, you say “We do not pose any assumptions on how the loss function in (1) is optimized.” Do you mean to say that you do not assume anything **but** that the chosen optimiser reaches a (global?!) minimum of the loss function? It seems like you would want to exclude pathological optimisers (e.g., one that always outputs zero) but also more realistically think about the known pathologies of commonly used optimisers (see, e.g., [2]).
- Can you please clarify if and how the use of the algorithm from (Agarwal et al., 2016) for approximation of the Hessian products affects accuracy of your confidence intervals?
- In fig.3, it seems like some of the methods are not properly tuned. For example, MC-dropout should not have zero uncertainty around zero (did you by any chance set bias variance to zero?!), and BNN-SGLD does not seem to have converged (can you please provide plots providing some evidence that the MCMC sampler has mixed + information about how the hyperparameters were selected?).
- I am somewhat confused by the statement “The only hyper-parameter involved in our method is the number of HOIFs m — this was tuned by optimizing the evaluation metrics ...” on p.8. Wouldn’t thm.2 suggest that you should use as high m as possible?
- Also on p.8, can you please clarify how you selected the threshold for the evaluation of “discriminative power”? In particular, thm.2 seems to suggest that what one would desire is that the ranking based on width of predictive intervals is equivalent to the ranking based on the actual prediction error. This would suggest, for example, comparing these two rankings using Kendall’s tau coefficient (perhaps a binned modification where one would count only the number of times true error puts a point into a different bin than the width of its confidence interval). Note that I am not suggesting the above method is perfect either (it completely ignores the actual sizes of the intervals), but I’m currently having trouble interpreting the results you report, so it would be very helpful to understand how you selected this particular measure of “discriminative power” and why alternatives like the example above were discarded please.
References:
[1] Ryan Giordano, Michael I. Jordan, Tamara Broderick. A Higher-Order Swiss Army Infinitesimal Jackknife. https://arxiv.org/abs/1907.12116
[2] Ashia C. Wilson, Rebecca Roelofs, Mitchell Stern, Nathan Srebro, Benjamin Recht. The Marginal Value of Adaptive Gradient Methods in Machine Learning. https://arxiv.org/abs/1705.08292