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Q2: active manifold M is dependent on x [...] A: The manifold M in Theorem 3.1 is the manifold associated to x^*, to clarify this, we will denote it as M_{x^*}. Q3: why applying the Baillon-Haddad theorem [...] A: The subdifferential partial Phi is a set-valued operator and *cannot* be non-expansive in the classical sense. It is indeed non-expansiveness of G_k that we need, and this is a consequence of Baillon-Haddad theorem which states that \nabla F is firmly non-expansive, and thus that G_k is \alpha_k-averaged, hence non-expansive, for the prescribed range of \gamma_k.
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Review for NeurIPS paper: Cross-validation Confidence Intervals for Test Error
Weaknesses: Some major comments: 1) The connection to algorithmic stability is interesting, but I am not convinced that this can deliver as strong results as we would like beyond what can already be achieved through standard results/analysis. More specifically, algorithmic stability has mostly shown O(1/n) results for ERM or SGD, but this is just a rehashing of standard results, essentially following from iid-ness, that is, that every datapoint contributes the same information on average. This is not a problem with the current paper per se, but more a critique of algorithmic stability analysis. Rather, my concern for the current paper is twofold: a) the connection to algorithmic stability cannot deliver, as far as I understand, any stronger results than what is already possible through standard methods; b) and thus a basic CLT for CV error is attainable through a more standard analysis. Indeed, the path to asymptotic normality is pretty straightforward in the paper, since all important steps are more-or-less assumed: Square integrability of mean loss \bar h_n, song convexity of such loss function which guarantees O(1/n) rates, etc. 2) The experimental setup is very confusing to me.
Review for NeurIPS paper: Upper Confidence Primal-Dual Reinforcement Learning for CMDP with Adversarial Loss
Weaknesses: (W1): As such the high-level outline of the proof strategy follows previous procedures for drift analysis in (Yu et al. 2017) and MDP analysis in (Neu et al. 2012 and Rosenberg et al. 2019). Lemma B.2 is very similar to Lemma 4 in Neu et al. 2012 and Lemma B.2 in Rosenberg et al. 2019. Lemma 5.2 mirrors Lemma 8 in Yu et al. 2017. Technical lemmas for stochastic analysis are also from the previous paper: (Lemma B.6 and B.7 are Lemma 5 and 9 in Yu et al. 2017). The main lemma, Lemma 5.3, has the same goal as Lemma 7 in Yu et al. 2017, which is to show Q_t satisfies the drift condition stated in Lemma 5 in Yu et al. 2017. Lemma 5.6 is also exact as Lemma 3 in Yu et al. 2017.
Reviews: A Communication Efficient Stochastic Multi-Block Alternating Direction Method of Multipliers
This paper considers a communication efficient distributed optimization algorithm based on ADMM for stochastic optimization. The main idea is to perform multiple steps (can be timevarying) of stochastic gradient updates before the agents communicate, and therefore improving the communication efficiency. The proposed algorithm is shown to converge (in objective value & constraint violation) under a general non-smooth, non-strongly convex settings with O(1/eps) communication rounds and O(1/eps 2) unbiased gradient oracle calls. Other setting such as smooth strongly convex and non-smooth strongly convex are also analyzed and presented. Using multiple steps is however not novel.
Reviews: Byzantine Stochastic Gradient Descent
The paper studies stochastic convex optimization in a distributed master/workers framework, where on each round each machine out of m produces a stochastic gradient and sends it to the master, which aggregates these into a mini-batch. In this paper the authors allow a fraction of alpha of the machines to be Byzantine, i.e., they do not need to report valid stochastic gradients but may produce arbitrary vectors, even in an adversarial manner. The goal is to aggregate the reports of the machines and to converge to an optimal solution of the convex objective despite the malicious Byzantine machines. The authors present a novel variant of minibatch-SGD which tackles the difficulty the dealing with Byzantine machines. They prove upper-bounds on the convergence and nearly optimal matching lower-bounds on any algorithm working in such framework, and in this sense the results are quite satisfactory.