NeurIPS 2020

### Review 1

Summary and Contributions: The paper studies the fundamental problem of learning halfspaces under adversarial noise. It is shown that by running projected SGD on a non-convex function, one is always guaranteed with an near-optimal halfspace.

Strengths: - This work is technically sound. It gives a very simple formulation, i.e. minimizing a properly scaled sigmoid function, and shows that any stationary point serves as a good approximation to the desired halfspace. - Although similar idea has been explored in a very recent work for Massart noise, authors clarified that adversarial noise is a more involved model and requires extra treatments, e.g. the scaling parameter in the objective function cannot be too small. - The theoretical guarantee holds also for a family of heavy-tailed marginal distributions. This might be of independent interest.

Weaknesses: I like this paper but couple of concerns needs to be addressed. - The comparison to [ABL17] is not convincing to me. I do not really agree that [ABL17] is not practical. First of all, their algorithm runs in log(1/eps) epochs, and thus has an exponentially better dependence on 'eps' than this work, in terms of label complexity. Second, in each epoch, they optimize a convex program up to a constant accuracy. Therefore, the per-epoch computational complexity is very mild. - Technical contribution compared to the Massart noise [DKTZ20] needs more clarification. While I agree with the authors that adversarial noise is a different type from Massart noise, some prior works already showed that from the technical perspective, there might a towards unified way to analyze them. For example, in [A], it was shown that for both noise models, most of the analysis follows the same way: one only needs to slightly treat a concentration inequality for each. See Lemma 21 and Lemma 22 therein. My overall feeling is that this work presents a simpler formulation (which is appreciated), achieving similar excess error bound to [ABL17], but has worse label complexity, which makes the work not as exciting as [DKTZ20]. My another concern is that the technical contribution might be marginal in light of the published work of [DKTZ20]. I am happy to hear the author's response and reconsider the rating if they address my concerns. [A] Efficient active learning of sparse halfspaces, C. Zhang, COLT 2018 ---updates after author rebuttal--- Thank you for your response. It addressed some of my concerns. However, I still feel the contribution is marginal.

Correctness: Yes.

Clarity: Yes

Relation to Prior Work: Yes.

Reproducibility: Yes

### Review 2

Summary and Contributions: The paper is about agnostically learning homogeneous halfspaces in the distribution specific PAC model with adversarial label noise. It shows that SGD on a non-convex surrogate of the zero-one loss solves the problem with $O(opt) + \epsilon$ error in poly time (in d and \epsilon), where *opt* is the misclassification error of the best hyperplane. Interestingly, the authors also provide a lower bound result of showing that optimizing any convex surrogate will incur $\Omega(opt)$ error. Combining the first result, it means that applying SGD to the non-convex surrogate might be a better approach for solving the problem, compared to applying SGD to the convex surrogates.

Strengths: While there are some previous works studying the same problem (e.g. [Awasthi et al. 2017], the algorithms in those works seem to be complicated. One of the contributions of this paper is showing that standard SGD on a non-convex objective works for the problem, which is by building on the result of [Diakonikolas et al. 2020]. The *separateness* result between the non-convex surrogate and convex surrogates is novel and interesting. Overall, I think the contributions are significant and the proof/analysis seems to be correct.

Weaknesses: I understand that this is a theory paper. But it will be better if a proof-of-concept experiment is provided.

Correctness: Yes.

Clarity: Yes. But it seems to me that on line 419-422 and line 432 in the supplementary, you overload the notation 'C'. It looks like you use C to represent a quantity and its inverse simultaneously.

Relation to Prior Work: Yes, detailed comparison with the prior works is provided.

Reproducibility: Yes

### Review 3

Summary and Contributions: The paper studies the problem of agnostic learning of homogeneous halfspaces under structured distributions. For this problem, there exists algorithms that achieve O(opt) + epsilon error in time poly(d, 1/eps), where d=dim. of data, opt = error of best homogeneous halfspace and eps is an error tolerance. These algorithms are typically iterative in nature and in each iteration minimize a convex function. The current paper shows that the same results can be obtained via SGD on a non-convex objective. The paper builds upon a recent work of [DKTZ20] to show that a smooth approximation to the 0-1 loss based on a logistic form suffices. In particular, the main claim is Lemma 3.2 showing that stationary points of the loss function will be close to the true halfspaces, and hence will have small error. *******Post Rebuttal******** I've read the authors' rebuttal and my score remains unchanged.

Strengths: + shows that a recent approach designed for benign noise models such as the Massart noise models also works for agnostic learning.

Weaknesses: - Relies heavily on [DKTZ20]. Claim 3.4 is fairly standard and Lemma 3.2 essentially uses the analysis of [DKTZ20]. - It is unclear of the proposed algorithm is truly simpler since one needs to do a grid search on the value of opt (in order to set sigma).

Correctness: Yes, the claims are correct.

Clarity: Yes, the paper is largely well written.

Relation to Prior Work: Yes, related work is adequately discussed. However, as mentioned before, the techniques heavily overlap with [DKTZ20].

Reproducibility: Yes

Additional Feedback: The proof of Lemma 3.2 should be modified a bit since the noisy region S is a random set (since the label conditioned on x can also be a random variable).

### Review 4

Summary and Contributions: This paper proof that projected SGD on a non-convex loss function of homogeneous halfspace problem converges to a suboptimal solution, where the suboptimality gap is at most O(opt) and opt is the minimum of the loss. By proposing the novel non-convex loss function, the result of the paper simplifies previous algorithms on the aforementioned problem. The paper also shows that the proposed non-convex loss function outperforms all convex and non-decreasing loss function on the order of opt. The reviewer finds the result of the paper interesting and can provide inspiration to the community of optimization research.

Strengths: The paper shows that simple approach (PSGD) with a good choice of loss function, which is non-convex, will converge to a globally good solution in the problem of homogeneous halfspace. The result holds for a variety of distributions and the proof is solid.

Weaknesses: The challenge of analyzing convergence of SGD due to non-convexity is usually in the cases where the hypothesis h(x) is non-convex, e.g., a neural network, which implies that the inspiration of this paper on the study of gradient-base optimization methods in machine learning is limited.

Correctness: I believe the proof is correct, while I did not check all of them.

Clarity: The paper is well written and easy to follow.

Relation to Prior Work: The comparison with previous algorithm on the problem studied is not very clear.

Reproducibility: Yes

Additional Feedback: I hope the author can provide more explanantions on the reason why the non-convex loss is better than other convex loss function.