Paper ID: | 5518 |
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Title: | A unified variance-reduced accelerated gradient method for convex optimization |

This paper proposes a novel randomized incremental gradient algorithm Varag for minimizing convex problems with finite sum structure. The algorithm efficiently incorporate the variance reduction technique in order to achieve the optimal convergence rate for convex or strongly convex problems. The algorithm is at the same time simple and rich, which admits many interesting features. For instance, the algorithm admits a non-uniform sampling strategy to handle the case that the Lipschitiz constant are unbalanced; also the algorithm is able to handle non-Euclidean case via Bregman distance. It is a great contribution to combine all these components together in a relatively simple algorithm. Moreover, extensions to stochastic setting and error bound condition are also included. Experiments on logistic regression or Lasso are conducted showing that the algorithm is competitive compared to existing accelerated algorithms. Overall, I am very positive about the paper, even though I have some minor concerns. One of the key feature of the proposed algorithm is to have a phase transition on the parameters. In particular, the parameters are more "aggressive" in the early iterations. And this leads to a very surprising result that the algorithm is converging linearly in the early iterations even in the convex but non-strongly convex setting. More precisely, the linear convergence holds roughly before all the individual function are visited once. Moreover, the convergence parameter in this phase only depends on m, the number individual functions. Could authors provide more intuitions/explanations on why a linear convergence rate is achievable in the early stages? Furthermore, in the early stages, the number of inner loop iterations T_s are increasing exponentially, this is quite similar to the stochastic minimization setting that we increase the sample size geometrically. Could authors provide some motivation of this choice? In the experiment, is the non-uniform sampling strategy used? How does it compared to the case of uniform sampling? EDIT after author's feedback: I thank authors for the clarification, I believe the paper deserves to be published.

Originality. The problem of finite-sum minimization is very popular and attains a substantial attention recent years. Despite to the number of successive works in this field, results on both accelerated and variance-reduced methods are still under investigation and required additional research. The authors proposes new accelerated stochastic gradient method with variance reduction with its global complexity analysis. This method achieves the currently known convergence bounds and has a number of nice features (among others): * Tackling both convex and strongly convex problems in an unified way; * Using only one composite (proximal) step per iteration; * Working with general Bregman distances; * Relatively easy in implementation and analysis. To the best of my knowledge, up-to-know there are no accelerated variance-reduced methods having all these advantages simultaneously. Quality. The paper contains a number of theoretical contributions with proofs. Two algorithms (Varag and Stochastic Varag) are proposed in a clear and self-contained way. A number of numerical experiments on logistic regression with other state-of-the-art optimization methods are made. Clarity. The results are written in a well-mannered and clear way. A small concern from my side would be to make the introduction part more structured. Especially, more concrete examples of the cases, when we can not handle strong convexity of $f$ (but still know the constant) would be interesting. It also would be good to add more discussion and examples to the text, related to "Stochastic finite-sum optimization" (Section 2.2.). Significance. The results of this work seems important and significant to the area of stochastic variance-reduced accelerated methods.

Strengths: 1. This is a well written paper with strong theoretical results of wide appeal to the ML community in general and the NeurIPS community in particular. Accelerated methods are of particular importance for ill-conditioned problems. This makes Varag (like Katyusha) robust to ill-conditioned problems. 2. The Varag method takes a decisive (albeit by no means final) step in the direction of unifying certain algorithmic aspects of variance reduced methods for finite sum optimization. The results seamlesly cover both the strongly convex and convex regimes, the well and ill conditioned regimes, and the low and high accuracy regimes. Moreover, I like the additional support of a prox-function to better deal with the geometry of the constraint set, and support for linear convergence under and error bound condition, and support for infinite-sum (i.e., stochastic) optimization. 3. Varag achieves all results directly, unlike existing methods which in some cases rely on reformulation or perturbation. 4. To the best of my knowledge, Varag seems to be first accelerated variance reduced method which achieves a linear rate without strong convexity (and not for over-parameterized models): under an error bound condition. Such results have only recently been obtained for non-accelerated variance-reduced (Qian et al; SAGA with arbitrary sampling, ICML 2019); under a quadratic functional growth condition. Perhaps this can be mentioned. Issues and recommendations: 1. In Eq 2.6 one should have something else instead of x. Which optimal solution x^* should be used? This does not matter for the term, but matters for the second term. 2. Some explanations/statements are misleading. a. For instance, when explaining the results summarized in Table 2, the text in lines 74-78 mentioned linear convergence. However, in the regime m >= D_0/epsilon the rate O(m log 1/epsilon) is not linear in epsilon. The same issue is in lines 171-174 when describing the results of Theorem 1. b. The same issue again arises in lines 199-201 when describing the results of Theorem 2. Indeed, in the third case m can be replaced by D_0/epsilon, and this a sublinear rate and not a linear rate. 3. A further step towards unification would be achieved if support for minibatching was offered. Almost no one uses non-minibatch methods in practice currently. Surely, one can assume that each f_i depends in multiple data-points, forming an effective minibatch; and this would be covered by the provided results. But this only offers a limited flexibility for the formation of minibatches. 4. The abstract says: Varag is the “first of its kind” that benefits from the strong convexity of the data fidelity term. This is very vague as it is not at all clear what “first of its kind” means. Make this more clear. There exists many randomized variance-reduced methods that can utilize strong convexity present in the data fidelity term (e.g., the SAGA paper mentioned above, among many others). 5. A key issue with the Varag method is its reliance on an estimate of the strong convexity or error bound constant mu (\bar mu). Admittedly, all accelerated method suffer from this issue. One of the ways to handle this is via restarting, as done by, for instance, Fercoq and Qu (Adaptive restart of accelerated gradient methods under local quadratic growth condition, 2017; Restarting the accelerated coordinate descent method with a rough strong convexity estimate, 2018). This fact should be mentioned explicitly – I was surprised that this is left to the reader to discover; especially since strong claims are made about the unified nature of the method. The method is unified, but NOT adaptive to the strong convexity constants in the way other (non-accelerated) variance reduced methods are; e.g., SAGA. I would expect to see this difference in some experiments. That is, I suggest that experiments are performed with unknown mu, so that Varag has a hard time setting its parameters. Consider well and ill conditioned problems and compare Varag and SAGA or SVRG. 6. In practice, Loopless SVRG (Dmitry Kovalev et al, Don’t jump through hoops and remove those loops: SVRG and Katyusha are better without the outer loop, 2019) works better than SVRG. Also, it is better able to utilize information contained in the data (and, as SVRG, is adaptive to mu - even better so as it does not rely on mu to set the outer loop size). I wonder how this variant would compare in practice to Varag in the well conditioned / big data case (i.e., in the case when Varag is supposed switch to the non-accelerated regime). I would want to see a number of experiments on various artificial and real data, with the aim to judge whether Varag is able to do as well. I expect Varag to suffer as it needs to know the condition number to sets its parameters. 7. Can Varag be extended to a loopless variant, such as Loopless SVRG? This should lead to both a simpler analysis, and a better method in practice. 8. What happens with the results of Theorem 4 as one varies the outer and inner minibatch sizes b_s and B_s? A commentary is needed I think. 9. The experimental evaluation is not particularly strong. I made some recommendations above. Moreover, I do not see how were the various parameters set in the experiments. The text says: as in theory. But you do not know some of the constants, such as mu. How was mu computed? Was the calculation included in the runtime when comparing against SVRG? Small issues and typos: The paper contains a relatively large (but not excessively large) number small issues and typos, which need to be fixed. Some examples: 1. 3: conditional number -> condition number 2. 12: function -> functions 3. 13: smooth convex functions with -> smooth and convex 4. 17: necessarily -> necessary 5. 20: becomes -> became 6. 23: connect -> connected 7. 38: number of components -> number of components is 8. 88: rates -> rate 9. 97: class -> classes 10. 106: in -> to 11. 116: You may wish to use the words ceiling and floor instead. 12. 126: to simple -> to be simple 13. 142: log(1/eps -> log(1/eps) 14. 150: computational -> computationally 15. 154: updates -> update 16. 155: non-euclidean -> non-Euclidean 17. 162: its -> the 18. 199: smooth finite-sum -> smooth convex finite-sum 19. 199: strongly convex modulus -> strong convexity modulus 20. 222: an unified result as we shown in Theorem 2 -> a unified result, as shown in Theorem 2, 21. 223: conditional number -> condition number ----------------- I've read the other reviews and the rebuttal. I am keeping my score - this is a good paper.