Paper ID: | 8196 |
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Title: | Hamiltonian descent for composite objectives |

To minimize a function f in a continuous time setting, this paper provides an ODE that converges linearly. This ODE is built from the Hamiltonian flow, well known in mechanics, where a perturbation is added. The Hamiltonian is used as a Lyapunov function and the perturbation is necessary, otherwise the Hamiltonian is conserved. However the perturbation is often intractable. Then the authors consider the case where $f$ is composite i.e $f = h \circ A + g$ and build an Hamiltonian from the duality gap of this problem. Using some tricks in the definition of the Hamiltonian, the Hamiltonian dynamics is now tractable. Therefore, the proposed dynamics provide an ODE that exhibits linear convergence of the duality gap to zero. Moreover, a careful discretization of this ODE leads to the well known ADMM. The derivations are elegant and the paper is pleasant to read. The approach of this paper is original and insightful. It provides an new way to look at primal dual algorithms, which are usually seen as discretization of monotone flows, see @article{peypouquet2009evolution, title={Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time}, author={Peypouquet, Juan and Sorin, Sylvain}, journal={arXiv preprint arXiv:0905.1270}, year={2009} } @article{bianchi2017constant, title={A constant step Forward-Backward algorithm involving random maximal monotone operators}, author={Bianchi, Pascal and Hachem, Walid and Salim, Adil}, journal={arXiv preprint arXiv:1702.04144}, year={2017} } or @article{condat2013primal, title={A primal--dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms}, author={Condat, Laurent}, journal={Journal of Optimization Theory and Applications}, volume={158}, number={2}, pages={460--479}, year={2013}, publisher={Springer} } ******************************************************************** I appreciate that the authors will cover PDHG algorithm in the camera-ready version. If PDHG can be covered, I expect that Vu-Condat can also be covered. Regarding the references provided in the review, please note that I'm not asking the authors to add all references to their paper (some may be relevant while some other may not). I pointed out these references for future works.

Although the paper contains a lot of reminders about the Hamiltonian, it proposes a new research direction that would deserve to be explored. The proposed Halmitonian descent seems indeed to provide better results than the gradient descent. Numerical results in two simple regression scenarios evidence this fact.

#### I'd like to thank the authors for their feedback. Since they also managed to obtain primal-dual hybrid gradient using hamiltonian, I increase my score from 4 to 5. For me it's very hard to say whether this work is really a good fit for neurips. To me it's more like pure math, but the other reviewers seems to be very confident that it addressed a very important problem. I'm partially convinced by them, so I put overall score 6 (i.e. an additional increase from 5 to 6 for importance). ### Thank you for your work and for clarity of writing. Unfortunately, I do not see any interesting message in this work. First, let's look at the hamiltonian descent. I agree with the authors that its importance depends on whether we can use it without knowing the optimum and without computing expensive gradient of the conjugate. Ideally, I want to see either a recipe how to do this for different problems or at least a significant number of examples. In this work we only see it for composite optimization, so I do not feel that the approach has generality. The next result, affine invariance, is quite nice as we expect a natural method to be invariant to the choice of basis. However, this property is not important per se and it doesn't seem that the authors use it in any way. There is a brief discussion of discretization. It's simple, clear and it's later used in the discussion of ADMM. Continuous time analysis usually comes with discretization schemes and it's mostly a technical detail. In appendix we see O(1/k) rate of convergence for the explicit scheme, which does not match what we see in the experiments. Then we come to the interesting part, which is composite minimization. Since the duality gap includes explicitly solutions in linear form, they cancel out with the corresponding terms in the hamiltonian descent dynamics. This is nice, because now we can indeed discretize it and obtain a method for composite minimization. This is where we come to ADMM as it turns out it can be obtained as a discretization of hamiltonian descent. To prove it, an extra trick was used, which is to add 0=(x(t) - x(t)) and then discretize it as x^{k+1} - x^k. It's not clear to me if a trick like that is always needed and how the authors came up with it. Probably this is natural to do if one wants to obtain the update of ADMM, but I feel that overall this makes this contribution smaller, because it requires knowledge of what we want to obtain. I don't see any discussion of what happens if we don't use the trick and it wasn't discussed in the section about discretization. And although it is interesting to see ADMM as a discretized version of Hamiltonian descent, we already have an ODE to obtain ADMM and its accelerated variant. There are whole two pages devoted to experiments and I don't think it's so important here. The main contribution is the theory and more theoretical results would be better. Moreover, the experiments do not validate the theory: the explicit scheme in the first experiment numerically converges linearly, while the theory only guarantees O(1/k) rate. The proposed approach outperforms an accelerated method as can be seen in Appendix in Figure 3, which is quite interesting, but there is no explanation for this phenomenon as the rate is not linear. A lot of details provided for experiment 1 could be moved to the appendix to create more space for theory. Second experiments would benefit from comparison to accelerated gradient method or a stochastic method. Let me make a conclusion. The presented theory is quite nice, but it definitely lacks depths. Probably this work is better suited for a mathematical journal, but I do not see how it can interest the NeurIPS community. This does not mean that the authors should give up on their work, there might be other methods that can be discovered using this approach. Alternatively, the authors might try to obtain more existing methods such as Primal-Dual Hybrid Gradient or Condat-Vu as a discretization of Hamiltonian Descent on some other problem. Or, another interesting question is what happens when primal-dual relation breaks because of nonconvexity. In any case, at the current stage the work does not go deep enough to be published.