Wasserstein Distributionally Robust Kalman Filtering
We study a distributionally robust mean square error estimation problem over a nonconvex Wasserstein ambiguity set containing only normal distributions. We show that the optimal estimator and the least favorable distribution form a Nash equilibrium. Despite the non-convex nature of the ambiguity set, we prove that the estimation problem is equivalent to a tractable convex program. We further devise a Frank-Wolfe algorithm for this convex program whose direction-searching subproblem can be solved in a quasi-closed form. Using these ingredients, we introduce a distributionally robust Kalman filter that hedges against model risk.
Update: I have read the authors' response with interest; they have addressed most of my comments well. Regarding comment (2), I do not not see why one would need a formula for the Wasserstein distance between normal and arbitrary non-normal distributions. For any distribution Q, a normal distribution that has the same first and second moments as Q must be closer than Q to the nominal distribution in Wasserstein distance, as there exists a joint Gaussian distribution with the same second moment as the joint distribution attaining the Wasserstein distance between Q and the nominal distribution. Since a normal distribution with the same first and second moments will have linear estimation error identical to that of Q, this shows we may restrict our attention to normal distributions in the uncertainty set.  ---  The authors study robust estimation with a mean squared-error objective and uncertainty set defined by the Wasserstein distance with respect to a quadratic cost function. They show how to find the robust estimator by solving a convex program, give a Frank-Wolfe method for solving the program, and prove convergence guarantees for the method. Based on these developments, the authors propose a robust variant of the Kalman filter.   In my opinion this is a very good paper that contributes substantially to our knowledge on distributionally robust optimization with Wasserstein uncertainty, in which there seems to be fairly widespread interest among the NIPS community. As such, I am strongly in favor of accepting this paper.  Below I list some comments and suggestions regarding the paper.  1) While your proposal of a robust Kalman filter is quite appealing, I do not think it merits the title "minimax", that you attribute to it in the abstract and throughout the paper. To the best of my understanding the derivation of your filter is heuristic and meant to combine your robust MSE estimation formulation with the efficient recursive structure of the classical Kalman filter. As such, there is no uncertainty set for which the proposed filter minimizes the error of the worst-case instance, and therefore it is not "minimax". Please clarify this point.  2) In Section 2 you consider minimax estimators in the set of all measurable functions of y, and a nonconvex uncertainty set consisting of a Wasserstein ball restricted to  Gaussian distributions. This seems to give exactly the same result as restricting the set of estimators to affine functions of y, and letting the uncertainty set be simply the Wasserstein ball (considering all distributions). While equivalent, I believe the latter framing is better, as it makes Theorem 2.3 immediate from standard convex-concave optimization results, and it makes the proof of Theorem 2.5 substantially simpler. At the very least, the paper should comment on this alternative formulation.  3) It would be good if you could provide some additional illustration and discussion of what the worst case covariance S^* looks like. A natural test case is Wiener filtering (y = x + n, with n, x independent) - does the worst-case covariance maintain the independent noise structure? Does it have stronger noise, different input covariance, or both? It will be interesting to compare the result to robust Wiener filters with other uncertainty measures (see comment 5 below).   4) A straightforward approach to robust Kalman filtering is to treat estimation of x_t from Y_t simply as a linear estimation problem, and solve it with exactly the same robust formulation you develop in Sections 2 and 3. While far less computationally efficient than the robust filter you propose, this approach has the advantage of actually being minimax in some clear sense. As you seem to be able to run Algorithm 2 with d in the hundreds, it will be interesting to evaluate the simple approach described for the low dimensional test case considered in Section 5.2, and compare it to the proposed method.  5) The paper is missing some references on robust MSE estimation, which has been studied for many years; see "A competitive minimax approach to robust estimation of random parameters" by Eldar and Merhav (2004) and the references therein. These works consider various uncertainty sets for the input covariance, and can therefore be viewed as distributionally robust formulations. The authors might also consider citing more of the recent literature on Wasserstein-distributionally robust optimization, such as Sinha et al. (2018) and Blanchet et al. (2016).  Addressing the issues above above will, in my opinion, turn your paper from very good to excellent.  Below are a few additional minor comments:  - In line 108, replace "efficient" with a more conservative adjective, such as "tractable".  - \bar{sigma} is only defined after Algorithm 2 is stated, which is confusing; it should instead be an input to the algorithm.  - Using MMSE for "minimax mean squared error" (Section 5.1) is confusing, since MMSE is a commonly-used acronym for "minimum mean squared error". Consider using MMMSE or M3SE instead.  - In lines 222-225, \Delta and \Delta^\star appear to be used interchangeably.  - In line 97 of the Supplementary Material, I believe it should be Theorem 3.3 rather than Theorem 2.5.