  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 $O(\sqrt{T\log{T}})$. By showing that the regret is
lower bounded by $\Omega(\sqrt{T})$ for any strategy, we conclude that DPDS is
order optimal up to a $\sqrt{\log{T}}$ 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.
