This is only for the ease of visualization. For linear MDP,In the generative model setting, Agarwal et al. [2020] shows model-based approach is still minimax optimal O((1 γ) 3SA/2)byusing as-absorbing MDP construction andthismodelbased technique is later reused for other more general settings (e.g. Itrequires high probability guarantee for learning optimal policyforany reward function, which is strictly stronger than the standard learning task that one only needs to learn to optimal policy for a fixed reward. B.2 GeneralabsorbingMDP The general absorbing MDP is defined as follows: for a fixed states and a sequence {ut}Ht=1, MDPMs,{ut}Ht=1 is identical toM for all states excepts, and state s is absorbing in the sense PMs,{ut}Ht=1(s|s,a) = 1 for all a, and the instantaneous reward at timet is rt(s,a) = ut for all a A. Also,weusetheshorthand notationVπ{s,ut} forVπs,Ms,{u We focus on the first claim. Later we shall remove the conditional onN (see SectionB.7). We use the singleton-absorbing MDPMs,{u?t}Ht=1 to handle the case (recallu?t

First, note that we shall use the same MDPs defined in Appendix D.1 as follows

In addition, we define1 to be a vector with all the entries being 1, andI be the identity matrix. Suppose thatδ > 0andε (0,(1 γ) 1/2]. The remainder of this section is devotedtoprovingTheorem3. VT) to be the policy (resp. The remainder of this section is devotedtoprovingTheorem4.

We first introduce some additional notations for convenience. Our proof mainly consists of the following steps: 1. Helper lemmas and a crude bound. See A.2, and more precisely, Lemmas A.9 and A.10. 3. Final bound for null -approximate NE value. See A.3. 4. Final bounds for null -NE policy. See A.5. 14 A.1 Important Lemmas We start with the component-wise error bounds.

The $\ell_p$-norm objectives for correlation clustering present a fundamental trade-off between minimizing total disagreements (the $\ell_1$-norm) and ensuring fairness to individual nodes (the $\ell_\infty$-norm). Surprisingly, in the offline setting it is possible to simultaneously approximate all $\ell_p$-norms with a single clustering. Can this powerful guarantee be achieved in an online setting? This paper provides the first affirmative answer. We present a single algorithm for the online-with-a-sample (AOS) model that, given a small constant fraction of the input as a sample, produces one clustering that is simultaneously $O(\log^4 n)$-competitive for all $\ell_p$-norms with high probability, $O(\log n)$-competitive for the $\ell_\infty$-norm with high probability, and $O(1)$-competitive for the $\ell_1$-norm in expectation. This work successfully translates the offline "all-norms" guarantee to the online world. Our setting is motivated by a new hardness result that demonstrates a fundamental separation between these objectives in the standard random-order (RO) online model. Namely, while the $\ell_1$-norm is trivially $O(1)$-approximable in the RO model, we prove that any algorithm in the RO model for the fairness-promoting $\ell_\infty$-norm must have a competitive ratio of at least $Ω(n^{1/3})$. This highlights the necessity of a different beyond-worst-case model. We complement our algorithm with lower bounds, showing our competitive ratios for the $\ell_1$- and $\ell_\infty$- norms are nearly tight in the AOS model.

They are designed to have a game-theoretic equilibrium where everyone reveals their private information truthfully.

First, note that we shall use the same MDPs defined in Appendix D.1 as follows

We first introduce some additional notations for convenience. Our proof mainly consists of the following steps: 1. Helper lemmas and a crude bound. See A.2, and more precisely, Lemmas A.9 and A.10. 3. Final bound for null -approximate NE value. See A.3. 4. Final bounds for null -NE policy. See A.5. 14 A.1 Important Lemmas We start with the component-wise error bounds.

This work proposes a bid shading strategy for first-price auctions as a measure-valued optimization problem. We consider a standard parametric form for bid shading and formulate the problem as convex optimization over the joint distribution of shading parameters. After each auction, the shading parameter distribution is adapted via a regularized Wasserstein-proximal update with a data-driven energy functional. This energy functional is conditional on the context, i.e., on publisher/user attributes such as domain, ad slot type, device, or location. The proposed algorithm encourages the bid distribution to place more weight on values with higher expected surplus, i.e., where the win probability and the value gap are both large. We show that the resulting measure-valued convex optimization problem admits a closed form solution. A numerical example illustrates the proposed method.