We study the problem of learning to bid when the bidder's value is dynamic, i.e., when the current value depends on past outcomes. Specifically, we consider a bidder participating in repeated second-price auctions whose value depends on the time elapsed since their last successful bid, with auctions arriving in continuous time and only aggregated feedback revealed at the end of the horizon. Such a bidder must (1) balance the immediate benefit of winning the current auction against its impact on future values and (2) learn unknown environmental parameters. We derive regret bounds for a class of learning methods that combine plug-in estimators with a differential-equation characterization of the optimal policy, and show that a specific confidence bound algorithm learns the optimal policy with a near optimal regret of $\widetilde{O}(\log N)$ for piecewise linear primitives, and $\widetilde{O}(N^{1/3})$ for general, smooth primitives, achieving these regrets without explicit randomization. These theoretical results are supported by numerical experiments.

For our local hypergraph clustering experiments, we inserted SPARSECARD as a subroutine into the method HYPERLOCAL, which finds a cluster S in a hypergraph H = (V,E) that is localized around an input set Z V. It does so by minimizing the following ratio cut objective: φ(S) = cutH(S) vol(Z S) βvol( Z S), subject to vol( Z S) 0. (35) Here, Z = V\Z denotes the complement set of Z. For a node set T V, vol(T) denotes volume of T, i.e., the sum of node degrees. The term vol(Z S) in the denominator rewards a high overlap between the output cluster S and the input set Z. The second term βvol( Z S) is a penalty for including too many nodes outside the input set Z. This is tuned by a locality parameter β > 0. For smaller values of β, the algorithm will explore a larger region in the hypergraph in search for good clusters.

Event cameras are neuromorphic sensors that summarize the evolving world as a stream of events . Each event describes the pixel coordinates, time, and polarity of an intensity change.

Membership in Ris already proven by Abrahamsen, Kleist and Miltzow in [3]. Thealgorithm then needs to verify that the neural network described byΘ fits all data points inD with a total error at mostγ. The goal of this appendix is to build a geometric understanding off(,Θ). We point the interested reader to these articles [6, 26, 49, 66, 92] investigating the set of functions exactly represented by different architecturesofReLUnetworks. To see that this observation is true, consider the following construction.

The task is to i) predict the unknown parameters, then ii) solve the optimization problem using the predicted parameters, such that the resulting solutions are good even under true parameters.