NeurIPS 2020

Finding Second-Order Stationary Points Efficiently in Smooth Nonconvex Linearly Constrained Optimization Problems

Review 1

Summary and Contributions: In this paper, the authors consider computing approximate Second Order Stationary Points (SOSPs) of problems with the assumption that the objective is generally non-convex, and constrains are linear. The authors then propose two methods to solve this optimization problem. One is to use the Successive Negative-curvature grAdient Projection (SNAP), which is a second-order algorithm. The other one is based on SNAP but only uses first-order information. Both of these two methods are proved to have polynomial per-iteration complexity and global sublinear rates.

Strengths: 1. The technical contributions of this work are solid. Two proposed methods have convergence guarantees. More specifically, the per-iteration complexity of the proposed methods are polynomial-time and they can converge to a good quality with high probability after certain iterations. Especially, the per-iterations are polynomial time is new compared with previous works which have exponential complexity. 2. The problem considered is an generic optimization problem which could be useful in a wide range of real-world applications. 3. The authors conduct experiments on the valuation of these two methods from simulation and real-world datasets. The empirical results show that SNAP and SNAP+ converge faster and better in most cases.

Weaknesses: 1. There is a lack of discussion of how these objectives (especially considered in experiments such as the NMF and Two Layer Non-linear Neural Networks.) can fit into the assumption 1 w.r.t. parameters L1 and L2. 2. The datasets used both in the paper and supplementary are small-scale. There is a lack of performance comparison on the large-scale datasets.

Correctness: Claims and methods are correct. The empirical methodology is also correct.

Clarity: The presentation is clear. The paper is well written.

Relation to Prior Work: Yes. The authors clearly discussed the related work in the Introduction section.

Reproducibility: No

Additional Feedback: To summarize, the authors consider the problem of finding approximate Second Order Stationary Points (SOSPs). Compared with other works, the authors assume that the objective is generally non-convex and constrains are linear. Two methods are designed to solve this optimization problem. Both of these two methods are proved to have polynomial per-iteration complexity and global sublinear rates. In general, the problem is both theoretically and empirically interesting. The presentation is clear and the thectical contributions are solid. However, my only concern is how these proposed methods can efficiently solve the real-world problems in large-scale since the per-iteration complexity is still high. Based on this concern, I have the following questions: Q1. The authors conduct experiments on training nonnegative neural networks in Section E.3. It would be helpful to discuss how the equation (138) satisfies the assumption 1. Any concrete examples of L1 and L2 in this case? Will these parameters be related to the dimensionality and data size? Q2. What is the performance between the proposed method and methods in [1] (Second Order Frank-Wolfe) and [2] (Trust region). Although, there is no constraint considered in these papers, it seems that it can solve the problem at some sense (see Section 4 of [2]). Q3. There are a lot of methods that can solve the NMF problem, it would helpful to make a comparison between these methods and the proposed. Or the authors can make some discussion on this. [1] Nouiehed, Maher, Jason D. Lee, and Meisam Razaviyayn. "Convergence to second-order stationarity for constrained non-convex optimization." arXiv preprint arXiv:1810.02024 (2018). [2] Nouiehed, Maher, and Meisam Razaviyayn. "A trust region method for finding second-order stationarity in linearly constrained non-convex optimization." arXiv preprint arXiv:1904.06784 (2019). After rebuttal: After I read the rebuttal and other reviewers' comments, I think the technical contribution of this paper is solid. I will keep my score unchanged.

Review 2

Summary and Contributions: This paper considers the problem of computing second order stationary points (SOSP) of smooth nonconvex objectives with linear constraint. Two algorithms are proposed, namely, SNAP and SNAP+.

Strengths: I like this paper very much. Finding second order stationary points for general nonconvex problem is NP hard. This paper identify the situations in which finding second order stationary points is easy, and efficient algorithm is proposed and analyzed. Numerical simulations validates the findings. The theoretical findings are solid, and the numerical results are convincing.

Weaknesses: My major concern is the computational costs of checking SOSP1, in which the smallest eigenvalue of the restricted Hessian is required. Here, by "restricted", I mean the space is the null space of A_A(x^*) rather than the whole space. In addition, such an operation is required repeatedly in the algorithm. For large scale problem, the cost of this step can be overwhelming. How to deal with such a problem needs some discussions/investigations.

Correctness: Yes. But I didn't check the proof.

Clarity: Yes. This paper is well organized and the results are clear.

Relation to Prior Work: Yes

Reproducibility: Yes

Additional Feedback:

Review 3

Summary and Contributions: This paper studies escaping from saddle in polytope constrianed nonconvex optimization. When SC condition is satisfied, the rate is polynomial wrt parameters. =========================================== Rebuttal is clear and makes sense. Raised score to 8. BTW two papers about NMF

Strengths: This paper proposes the concrete rate and thourough analysis with the constrained escape from saddle problem.

Weaknesses: I don't see major weaknesses, but I have some questions, raised in "additional feedback" section.

Correctness: I have read through the proof and think it's correct (with minor questions that do not hurt). The experiments are reasonable.

Clarity: Yes. Minor questions are below.

Relation to Prior Work: I think they've done great work in literature review which covers the most important papers such as Ge et al, Jin et al, Mokhtari et al., Nouiehed et al. etc. as well as other important related extensions. The logic is very clear and reasonable.

Reproducibility: Yes

Additional Feedback: I'd like to raise my scores if the questions are addressed. 1. Can you compare with this paper as well? Avdiukhin et al., Escaping Saddle Points with Inequality Constraints via Noisy Sticky Projected Gradient Descent, 2. One drawback might be that, when people talk about GD applied to escaping from saddle, they prefer the simplicity of the oracles and similarity with what people have already applied to convex optimization. That's why literature discuss GD with perturbation or trust region, so the Hessian estimation or line search steps here might weaken that (note the above paper's algo is fairly simple as well). I believe that due to the hardness of constraint some sophistication can be necessary, but I'd suggest trying to make the algorithm simple, or discuss about its simplicity. 3. I guess the SC condition is required everywhere, if so the author should mention it in all the theorems. 4. Due to the SC condition, the result can be limited and is motivated by data driven problem (so randomness guarantees the condition). But I'm wondering if such second order stationaries are really the solution people search for (i.e., what are the qualities of the optimizers)? People have shown that many problems such as matrix factorization are strict saddle functions and have no spurious local minimum, thus unconstrained escaping from saddle works. Is there any instance that the quality of second order minimum is guaranteed in constrained regime (say all local mins are global)? But I think theoretically GD cannot do anything with local min and it's still of value just to point out the convergence rate. 5. Some algorithms are in the appendix, so if they're quoted in the main paper, make sure to say "see appendix section ..." 6. State/Refer to how to choose v in Algorithm 1 line 7. 7. Question: to discuss the active constraint, the iterate has to be exactly on the boundary/hypersurface of the constraint polytope. Is it caused by exactly solving (23) in algorithm 2? Seems proximal operation or projection are not used in the algorithms. 8. A minor comment (doesn't hurt to ignore this): to be more inclusive, the authors can discuss the case when some active constraints are linearly dependent, or MFCQ. 9. (Minor comment) Prop 2,3: I'm not sure why the authors have to discuss \epsilon = 0 or limit case. Is it required for the analysis or practical instances? 10. Lemma 7: "Algorithm 1 will stop..." is it referred to algorithm 1 or only NCD step? And in algorithm 1, once a constraint turns active, is it always active in the following iterates?