NeurIPS 2020

Sharper Generalization Bounds for Pairwise Learning

Review 1

Summary and Contributions: This paper gives new generalization bounds for learning problems where the loss function depends on pairs of examples. This setting comes up naturally in ranking problems and in metric learning. The approach is through uniform stability, and a natural average case variation. While uniform stability arguments have been sharpened in recent years (e.g. the work of Feldman and Vondrak) in the pointwise learning setting, there were many gaps between the lower and upper bounds in the pairwise setting. This paper manages to close many of them, in some cases improving the generalization bound by as much as a \sqrt{n} factor where n is the number of examples.

Strengths: There is a ton of work on generalization bounds, and yet this paper still manages to provide a fresh angle. First, they highlight an important class of problems, namely pairwise learning, where obtaining nearly tight bounds is considerably harder, in large part because the objective function is no longer the sum of iid random variables. Second, they give a general methodology for obtaining sharper connections between stability and generalization in this setting. If an algorithm is \gamma uniformly stable they give a generalization bound of \gamma \log n that holds with high probability, which improves upon the previously best known bound of \gamma \sqrt{n}. Third they introduce a refined notion of on-average stability that allows them to prove better bounds in optimistic cases (i.e. when the best model has small error). Fourth give compelling applications to generalization bounds for ranking and metric learning. A notable highlight is that these include the first generalization bounds for SGD in pairwise learning. Existing bounds could not obtain non-trivial guarantees.

Weaknesses: I don't think the paper has any particular weaknesses.

Correctness: I have not verified all the proofs, but the ones I have looked through are all correct.

Clarity: The paper is well-written. There are a lot of results in the paper, but nevertheless it is easy to navigate given how the results build on each other.

Relation to Prior Work: The paper does a nice job of surveying the relevant generalization literature. It also gives some nice intuitions about the relative strengths and weaknesses of different approaches (e.g. the ability of stability approaches to give dimension independent guarantees).

Reproducibility: Yes

Additional Feedback: I have read the author response. I agree with the comment pointed out by another reviewer that the results are not quite as optimal as they were claimed, however I still think this is a compelling paper with both an interesting overarching message and neat theoretical applications.

Review 2

Summary and Contributions: This paper provided a refined stability analysis by developing generalization bounds, and further apply these results to ranking and metric learning. Main contributions: This paper extended the stablity analysis of pointwise learning [4] to pairwise learning, which can be $\sqrt{n}$-times faster than the existing results.They further apply the above stablity-based results to ranking and metric learning.

Strengths: 1: Theoretical: This paper derived a refined stability analysis for pairwise learning, which is faster than the existing results. 2: They apply the refined stablity-based results for ranking, SGD-based ranking and metric learning. They give the first high-probability generalization bound for SGD in pairwise learning.

Weaknesses: The main contribution of this manuscript is the extension of the stablity analysis of pointwise learning to pairwise learning. However, the proof techniques is mainly based on [4], which limits the novelty of manuscript.

Correctness: Some claims are not rigorous. The proofs seems to be correct, I do not check the proof details lines by lines.

Clarity: This paper is well wirtten.

Relation to Prior Work: Yes

Reproducibility: Yes

Additional Feedback: In lines 208-210, the authors claims that "the bound $O(n^{-1/2}\log(n))$ proposed is mimimax optimal". This claim is not correct, because of the conditions of the upper bound is different from the low bound. To obtian the upper bound of $O(n^{-1/2}\log(n))$, the loss function should be strongly convex, but for the lower bound is not needed. If you claim a bound is mimimax optimal, the conditions of upper and lower bounds must the same. The condition of $\|w_R^\ast\|=O(1)$ may be too strict. Usually, the norm of w is dependent on the dimension of w. In this manuscript, they claims that they don't require the conditoin of the "bounded" loss function, but for most of the results the loss should be strongly convex. Minor comments: Lines 130: What's the "RRM" means? Lines 197: $\lambda$->$\sigma$

Review 3

Summary and Contributions: The paper adresses the generalization of pairwise learning (minimization of an expected loss depending on independent data pairs) under the perspective of algorithmic stability. Pairwise learning is relevant to ranking and metric learning. Theorem 1, the first bound on the generalization gap on which most other results in the paper are based, relies on uniform stability of the training algorithm, as introduced by Boulsquet/Elisseeff in 2002. The bound is of O(\gamma log n + n^{-1/2}, where \gamma is the stability parameter and n the sample size, and it also replaces the frequenttly assumed boundedness of the loss function by a uniform bound on the sample-expectation of the loss of the algorithms output. Theorem 3 then considers regularized algorithms minimizing a strongly convex objective under Lipschitz assumptions, and second-moment assumptions on the minimizer of the regularized risk. Uniform stability is verified and corresponding bounds are derived from Theorem 1. Under second-moment assumptions on the minimizer of the true risk there is also a bound on the excess risk. Theorem 4 gives a generalization bound for stochastic gradient descent. Under the condition of \alpha-smoothness of the loss function uniform stability is demonstrated, depending on the number of training iterations. Again Theorem 1 is invoked to give the result. Theorem 6 then gives a bound on the excess risk in expectation, of order \sqrt{R/n} + 1/n, where R is the risk of the true-risk-minimizer. A final section applies the above results to ranking and metric learning. -----------------Update---------------------------------------------------- I have read the authors response and the other reviews and I keep my score at 7

Strengths: The paper extends state of the art results in the theory of algorithmic stability from pointwise to pairwise learning. The bounds given are powerful and, in the context of pairwise learning, novel. The proof techniques are sophisticated, but presented with sufficient clarity.

Weaknesses: I have very little substantial criticism. Perhaps the second paragraphs of section 2 exaggerates the merits of algorithmic stability a little bit. Clearly uniform bounds can also provide very valuable information in non-convex settings when exact minimization is impossible and algorithmic stability is limited to the results in [21]. I also think reference [7] (Bousquet et al 2019) could have been acknowledged a bit more. Apart from the tricky decomposition of U-statistics, the proof of Theorem 1 is a fairly direct application of the elegant techniques of [7].

Correctness: I checked the proofs in appendices A and B. It all seemed sound and well explained, but I studied the proof of Theorem 4 (Appendix C) only superficially, and I didn't read the proof of Theorem 6.

Clarity: Well written and polished with very few typos. l116 \subseteq should replace \in l140 insenstitive to perturbations l182 arg inf should be arg min (shouldn't \cal{W} be closed if it is bounded?) l197 \lambda should be \sigma l198 shouldn't the "mild assumption" in l122 (Appendix B) be stated here? l226 "and any" should be "and for any"

Relation to Prior Work: Generally quite well, but see my remark above w.r.t. reference [7]

Reproducibility: Yes

Additional Feedback: