Paper:
COLA: Decentralized Linear Learning Lie He
Decentralized machine learning is a promising emerging paradigm in view of global challenges of data ownership and privacy. We consider learning of linear classification and regression models, in the setting where the training data is decentralized over many user devices, and the learning algorithm must run ondevice, on an arbitrary communication network, without a central coordinator. We propose COLA, a new decentralized training algorithm with strong theoretical guarantees and superior practical performance. Our framework overcomes many limitations of existing methods, and achieves communication efficiency, scalability, elasticity as well as resilience to changes in data and allows for unreliable and heterogeneous participating devices.

Review:
The authors consider the problem learning a (regularized) linear model over a graph of agents. At each iteration, the agents exchange with their neighbors and perform an (approximate) proximal gradient step. They provide linear and sublinear rates depending on the strong convexity of the functions.  Despite interesting ideas, including the rate derivations, this papers lack connections with the literature and some clarifications in its present form.  [Distributed Setup] The distributed setup is rather well explained although it is not exactly standard as the data matrix is split by the *columns* (i.e. the features) among the agents; usually, the lines/examples are split. This difference should be more clearly mentioned and motivated in a practical context (near l64 page 2). As the simulations are performed on a single machine, one would expect a justification for considering this kind of data splitting and network topology. * Then, the communications are done using a doubly-stochastic mixing matrix as commonly accepted in Gossip theory. The paragraph before Assumption 1 is unclear: it is well known that as soon as an (undirected) graph is connected, the spectral gap is positive, this would make assumption 1 more readable.   [Optimization algorithm] As the authors mention (Sec 1.2, e.g. 108), the proposed algorithm falls into the primal-dual methods.  * At first glance, it seems close (as the matrix A cannot be inverted) to Condat-Vu's algorithm (Condat, L. (2013). A primal–dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. Journal of Optimization Theory and Applications, 158(2), 460-479.) Decentralized algorithms based on this algorithm were proposed, see e.g. Bianchi, P., Hachem, W., & Iutzeler, F. (2016). A coordinate descent primal-dual algorithm and application to distributed asynchronous optimization. IEEE Transactions on Automatic Control, 61(10), 2947-2957. The synchronous version (DADMM+ in their paper) looks indeed close to what you propose and should definitively be compared to the proposed method.  * Line 5 in the algorithm you approximatively solve problem (2), which leads to two concerns: (i) why: minimizing (2) seems equivalent to a proximity operation on the corresponding g_i on a point that is a gradient step on the coordinates of f with stepsize tau/sigma'; this seems to be computable exactly quite cheaply especially for the Lasso and Ridge problems considered in the experiments; (ii) how: I don't get how to obtain approximate solutions of this problem apart from not computing some coordinates but then theta can be arbitrarily bad and so are the rates. * I do not see a discussion on how to choose gamma. Furthermore, it seems rather explicit that the gradient descent is performed with the inverse of the associated Lipschitz constant.  [Convergence rates] The derivations seem valid but their significance is kind of obfuscated. * In Theorem 1, the assumption that g has a bounded support (or rather that the iterates stay in a bounded subspace) is rather strong but ok, what bothers me is that (i) it is hard to decipher the rate, a simple case with theta = 0, gamma =1 might help. In addition, the assumption on rho also would require some explanations. * In Theorem 2, it look that we actually have the usual conditioning/T rate but clarifications are also welcome * In Theorem 3, two conditions bother me: (i) mu tau >= D_1 seem like a minimal conditioning of the problem to have a linear rate; (ii) the condition of beta seems to look like a condition on how well connected the graph has to be to get the linear rate. Again, this is not discussed.   Minor comments: * l91 "bridgin"G * l232: the authors say "CoLa is free of tuning parameters" but the gradient descent is actually natively performed with the inverse Lipschitz constant which is a possibility in most algorithms of the literature (and can be local as extended in Apx E1). Furthermore, there is also the parameter gamma to tune, theta to adapt...   ============ After Rebuttal ============  After reading the author feedback and the other reviews, I upgraded my score to 6. The main reasons for that upgrade are: * the improvement of their main theorem to any connected graph * a misconception on my side for the difficulty of pb. (2)  However, it still seems important to me to clarify the non-triviality of (2), mention the relations between CoCoA and ADMM, and clarify that gamma and Theta are not hyper-parameter (cf l. 12-14 of the feedback).