Online Learning of Optimal Bidding Strategy in Repeated Multi-Commodity Auctions
We study the online learning problem of a bidder who participates in repeated auctions. With the goal of maximizing his T-period payoff, the bidder determines the optimal allocation of his budget among his bids for K goods at each period. As a bidding strategy, we propose a polynomial-time algorithm, inspired by the dynamic programming approach to the knapsack problem. The proposed algorithm, referred to as dynamic programming on discrete set (DPDS), achieves a regret order of O( √ T log T ). By showing that the regret is lower bounded by Ω( √ T ) for any strategy, we conclude that DPDS is order optimal up to a √ log T term. We evaluate the performance of DPDS empirically in the context of virtual trading in wholesale electricity markets by using historical data from the New York market. Empirical results show that DPDS consistently outperforms benchmark heuristic methods that are derived from machine learning and online learning approaches.
This paper  studies the online learning (stochastic and full-information) problem of bidding in multi commodity first price auctions. The paper introduces a polynomial time algorithm that achieves a regret of \sqrt{T log(T)} that has a near optimal dependence on T.
The main challenge that the paper has to deal with is to find a computationally efficient algorithm for computing the best biding strategy given a known distribution.The authors first demonstrate that natural approaches for solving this problem exactly are not computationally efficient (this is not a formal np-hardness proof). Then, they provide a FPTAS for solving the problem using dynamic programming. Once they have a FPTAS for the offline problem, their results hold for the stochastic online setting using existing reductions. I haven’t carefully looked in to the details of their analysis of the dynamic programming, but I think the effectiveness of it here is interesting and surprising — specially given that the variation of this problem for the second price auctions is hard to approximate. 
As for the weaknesses, I find the theorems to be hard to interpret. Particularly because there are a lot of variables in them. The authors should provide one simple variation of theorem 1 with fewer parameters. For example, when l = 0, and u = 1, the lipschitz conditions holds with L = 1 and p = 1, and B = K. Assigning \alpha = \sqrt{t} gives a regret of \tilde O(\sqrt{t \log(T)}).
As mentioned above the results mentioned here hold for the full-information online stochastic setting, that is, the valuation and the clearing price are drawn from a the same distribution at every step. I find this to be a weird choice of a setting. In most cases, full information settings are accompanied by adversarial arrival of the actions, rather than stochastic. This is indeed due to the fact that the statistical and computational aspects of online stochastic full-information setting is very similar to the offline stochastic full-information setting. So, I think a much more natural setting would have been to consider the adversarial online setting.
In this regard, there are two very relevant works on no-regret learning in auctions that should be cited here. Daskalakis and Syrgkanis FOCS’16 and later on Dudik et al. FOCS’17 study the problem of online learning in Simultaneous Second Price auctions (SiSPA). This is also the problem of optimal bidding in a repeated multi-commodity auction where each item is assigned based on a second price auctions, rather than a first price auction that is considered in this paper. In particular, the latter paper discusses computationally efficient mechanisms for obtaining a no-regret algorithm in the adversarial setting when one can solve the offline problem efficiently. Their methods seem to apply to the problem discussed in this paper and might be specially useful since this paper provides an approximation method for solving the offline problem. It is worth seeing whether the machinery used in their paper can directly strengthen your result so that it holds for the adversarial online setting. 
After rebuttal:
I suggest that the authors take a closer look at the works of Dudik et al. and Daskalakis and Syrgkanis. In their response, the authors mention that those works are inherently for discrete spaces. From a closer look at the work of Dudik et al. it seems that they also work with some auctions specifically in the continuous space by first showing that they can discretize the auction space. This discretization step is quite common in no-regret learning in auctions. Given that the current submission also shows that one can discretize the auctions space first, I think the results of Dudik et al. can be readily applied here. This would strengthen the results of this paper and make the setting more robust. I suggest that the authors take a closer look at these works when preparing their discussion.