NeurIPS 2020

A Stochastic Path Integral Differential EstimatoR Expectation Maximization Algorithm

Review 1

Summary and Contributions: This paper studies a variant of online expectation-maximization(EM) algorithms, with the focus on the computational complexity in terms of the number of M-steps and per-sample conditional-expectation evaluations. They require O(n + \sqrt{n} \epsilon^{-1}) per-sample evaluations and O(\epsilon^{-1}) M-steps to reach the \epsilon first-order stationary point of the log-likelihood. This improves the computational complexity of existing online EM variants (e.g., sEM-vr, FIEM). They achieve this by integrating techniques developed in SPIDER algorithm into the online EM.

Strengths: This is a well-written paper that presents a solid improvement in the line of online EM research. The computational complexity of the proposed SPIDER-EM finds an \epsilon (1st order) stationary point using the optimal number of calls O(\sqrt{n} \epsilon^{-1}) to first-order oracles. Previous online EM variants that use variance reduction technique [6, 15] require O(n^{2/3} \epsilon^{-1}) calls. Backgrounds and related work are well-presented to involve readers in the context. Main intuitions are well explained, and technical references to existing works are frequently made when necessary. Numerical experiments are well designed to demonstrate the theoretical findings, and the presented results are easy to understand.

Weaknesses: I have a few questions about the tightness of the result. Q1. In Theorem 2, the dependency on the number of outer iterations k_out is given by 1/k_out. I would expect the geometric convergence in outer iterations for algorithms that use variance-reduction technique (as in sEM-vr). Can this be improved? It seems your experiment also implies that. Q2. I think the desired batch size b would be O(1) in a large dataset scenario, whereas the guarantee given by authors requires b = O(\sqrt{n}). Could the same computational complexity be guaranteed using b = O(1)? Q3. The analysis is in a very similar style to [1]. Does (or why not) your analysis fall in the framework of [1]? Q4. Can you say something on the second-order stationary point (as in SPIDER)? [1] Karimi et al., Non-asymptotic Analysis of Biased Stochastic Approximation Scheme.

Correctness: The analysis for the proposed algorithm looks correct. High-level intuition is well guiding readers to the details.

Clarity: This is a well-written paper that proposes an online variant of EM that achieves the state-of-the-art computational complexity.

Relation to Prior Work: This paper has a good literature review, and the comparison to prior art is well-presented.

Reproducibility: Yes

Additional Feedback: What would be the bottleneck if one try to show the local convergence (in a flavor of [1]) of online-EM + variance reduction algorithms? [1] Balakrishnan et al., "Statistical guarantees for the EM algorithm: From population to sample-based analysis" =============================================== EDIT: I read the author feedback and am satisfied with their responses.

Review 2

Summary and Contributions: This paper proposes SPIDER-EM algorithm, which is the combination of recently developed SPIDER estimator with Expectation Maximization (EM) algorithm. The paper also provides a unified framework of stochastic approximation (SA) within EM. The results of SPIDER-EM match the typical results of SPIDER in nonconvex optimization, i.e., O(\sqrt(n)) and improves the previous result on EM with SVRG O(n^{2/3}). Since it matches the typical results of SPIDER in nonconvex optimization, the obtained results should be correct. It is interesting that the SPIDER estimator can be applied to EM algorithms. On the other hand, since other variance reduction techniques (such as SVRG) have already been applied in the EM setting in the literature, the idea of SPIDER-EM is a combination of recent popular variance reduction algorithm SPIDER and the previous variance reduction EM algorithms, which is incremental. The theoretical improvement can be expected given the existing result in optimization, and the proof should not be hard to go through. Therefore, I found that the technical novelty of this paper is limited and is not sufficient for NeurIPS conference. Overall, due to the limited technical novelty, I suggest to weakly reject this paper. -------------------------- comments after rebuttal ------------------------------- The authors' feedback addressed my question on novelty to some extend by explaining what exactly they need to further develop beyond the given techniques for analyzing the performance. Since SVRG type of EM algorithms have been studied previously, I am still not very convinced that SPIDER-EM algorithms would cause significant challenge in analysis, given that SPIDER has been well analyzed in nonconvex optimization. Nevertheless, I tend to think that this paper does make a good contribution by improving the existing complexity bounds. So I raise my rating towards acceptance.

Strengths: 1. The paper explores new variance reduction technique (SPIDER) in EM algorithm. 2. The paper proposes SA scheme for EM algorithm. 3. The results improve the previous studies. In specific, the paper shows that the complexity bound for SPIDER-EM is given by K_opt = O(\epsilon^{-1}) and K_CE = n + \sqrt(n) \epsilon^{-1}, which match the typical result in nonconvex optimization.

Weaknesses: 1. The technical novelty is limited. Since the SPIDER [10, 21, 24] (the reference number follows from the number in this paper) estimator is well studied in nonconvex optimization, and the variance reduction algorithms have already been explored by [6, 15], it should not be hard to work out the proof of SPIDER-EM by combining the techniques based on previous studies. Overall, I think the technical novelty is limited. [6] J. Chen, J. Zhu, Y. Teh, and T. Zhang. Stochastic Expectation Maximization with Variance Reduction. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems 31, pages 7967–7977. Curran Associates, Inc., 2018. [10] C. Fang, C. Li, Z. Lin, and T. Zhang. SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path-Integrated Differential Estimator. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems 31, pages 689–699. Curran Associates, Inc., 2018.

Correctness: yes

Clarity: yes

Relation to Prior Work: yes

Reproducibility: Yes

Additional Feedback: 1. The authors may want to include a table in the paper to compare the complexity results of this paper with those of existing EM algorithms.

Review 3

Summary and Contributions: The paper extends the Stochastic Path-Integrated Differential EstimatoR technique to an EM algorithm that accelerates the convergence of the E-step and thus the overall convergence.

Strengths: The proposed algorithm has sound theoretical grounding as well as promising empirical verification.

Weaknesses: The work seems a bit incremental in that it is directly extending the SPIDER algorithm to stochastic approximation EM.

Correctness: Yes.

Clarity: Yes.

Relation to Prior Work: Yes.

Reproducibility: Yes

Additional Feedback: The paper is clearly written and provides thorough characterization of the proposed algorithm and with some simple empirical verification. The paper can be improved by highlighting and explaining the key contributions better. The current impression is that it is just extending SPIDER to stochastic EM case. The authors can explain more clearly what the technical difficulties are, both in terms of algorithm design and convergence analysis, that prevent straightforward application. ============================== The rebuttal addresses my concern of the novelty and contribution. I have raised the rating to 7.

Review 4

Summary and Contributions: This paper proposes to combine the SPIDER technique developed in [10] with the classical EM algorithm in the case of curved exponential family distributions. This combines two loops to accelerate the computation time while maintaining a control variate. The authors also provide a unified framework of stochastic approximation (SA) within EM algorithms including recently developed variants. They also prove the theoretical complexity bounds for this new algorithm, supported by numerical experiments.

Strengths: This paper addresses an important topic of computing penalized maximum likelihoods when it involves latent variables and computation of likelihoods of a large data set. The proposed solution, as a combination of efficient algorithms, is promising. The convergence analysis is given and a technical sketch of the proof provided. Numerical examples on both synthetic and MNIST data bases are supporting the results.

Weaknesses: The two points of view of the EM (in parameter or expectation space) are used and sometimes going back and forth from one to the other makes the paper difficult to read. Assumption H4.3 seems quite restrictive. Can the authors comment on it? Also for H5.

Correctness: The paper looks correct.

Clarity: The paper is sometimes hard to follow as it is really technical. Some remarks on the hypothesis would help see the potential of applications it can cover.

Relation to Prior Work: The literature is provided.

Reproducibility: Yes

Additional Feedback: