Paper: Fast Stochastic Kernel Approximation by Dual Wasserstein Distance Method (ICLR-2024)

Reviewer comment (Theory):
The authors solve the mixed-integer linear optimization formulation (11) by relaxing the boolean variable constraint $\gamma\in\{0,1\}$ with $\gamma\in[0,1]$. However, such a relaxation will usually induce large optimization error. Besides, the authors claim that "However, for large dimensions, the gap is minimal, as our experience demonstrates." When I look at the numerical results in Table 2, the optimization gap actually increases when the data dimension increases. Therefore, I think Section 3 does not present a reasonable approach for optimization.

Author response:
As the Area Chair prompted us, we are responding to the criticism of B8LX.
Indeed, our approach does not solve the mixed-integer optimization problem (11) exactly, because our dual approach solves its linear programming relaxation. This is our main idea. Its advantage is that the dual subgradient method of section 3 scales very well; it does not even involve matrix-vector multiplication. We openly admit a minor negative: relaxation is solved. However, the number of representative points $M_{t+1}$ in (11e) is not a hard modeling parameter; if we use 1003 points instead of 1000, the model is equally good. This, after the dual method converges, we can see how many $\gamma$'s were set to 1, and increase $M_{t+1}$ accordingly. Then, with the new $M_{t+1}$, the gap is zero and our solution from Algorithm 1 is optimal. But to satisfy the curiosity of the Reviewer, we estimated in Table 3 the gap in the following way. We reset to 0 the "surplus" $\gamma$'s for which the expression in braces {...} in (13) is closest to 0, and we reassigned other variables as described below (13). This provides a feasible integer solution and an upper bound. The gap is minimal. Why? Because of very many points, the probability mass reassigned is small, and the neighboring points are almost as good as the deleted "surplus" points.

Reviewer comment (Theory):
Equation (9) presents a variation of the Wasserstein barycenter problem. However, the manuscript lacks a comprehensive discussion of the Wasserstein barycenter, and it would be beneficial to explicitly address this concept. Additionally, a comparison of the proposed method with relevant baselines is warranted.

Author response:
Equation (9) does not represent a variation of the Wasserstein barycenter problem. While our research may bear a superficial resemblance to the Wasserstein barycenter problem, our focus differs dramatically. Wasserstein barycenters offer a geometrically meaningful approach to aggregating probability distributions, grounded in optimal transport theory. However, our methodology diverges by not aiming to identify an aggregated measure that best captures multiple distributions, but rather a simplified kernel whose values approximate the values of the original kernel.  As demonstrated by Theorem 2.2 and our numerical example, our approach is directly applicable to Markov systems with multiple time stages, a scenario where the Wasserstein barycenter problem falls short. Although we cite papers related to the Wasserstein barycenter problem, they are only tangentially related to our goals.

Reviewer comment (Writing):
Numerous typographical errors are present in the manuscript. For instance, on page 1, there is a repeated "the the." Equation (6) is lacking the variable $p," and these issues need prompt correction for the manuscript to maintain clarity and accuracy.

Author response:
The two typos have been corrected. Thank you for bringing them to our attention. Regarding Equation (6), we maintain that the variable "p" was not missing; it features in the definition of the matrix $D$.