Compared with Eq. 25 and Eq. 29, we find We can see the approximate solutions. The tasks are binary classification. We use the default train/val/test split with ratio 8:1:1. The experiments are easily influenced by the parameter initialization. The AUC-ROC results are reported in Tab. 2. H.4 We use the official splits and evaluation is conducted on validation set.

One can observe that the Sinkhorn potentials are not unique. Under Assumption A.2, for a fixed pair of Assume Assumptions A.2 and A.3, and denote In this section, we provide several lemmas to show the Lipschitz continuity (w.r.t. the underlying These lemmas will be used in the convergence analysis and the mean field analysis for SD . Please see the proof in Appendix C.3. It would be an interesting future work to remove this factor . W e note that the Lemma B.1 is strictly stronger than preexisting results: (1) Proposition 13 of Feydy et al. [2019] only shows that the dual potentials are continuous (not Lipschitz continuous) This is strictly weaker than (i) of Lemma B.1.

This work was partially supported by NIH R01AG027161 and R01EY032284. This research has also received funding from the program "Investissements d'avenir" ANR-10-IAIHU-06. This work was also funded in part by the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA

In the stochastic multi-armed bandit problem, a randomized probability matching policy called Thompson sampling (TS) has shown excellent performance in various reward models. In addition to the empirical performance, TS has been shown to achieve asymptotic problem-dependent lower bounds in several models. However, its optimality has been mainly addressed under light-tailed or one-parameter models that belong to exponential families. In this paper, we consider the optimality of TS for the Pareto model that has a heavy tail and is parameterized by two unknown parameters. Specifically, we discuss the optimality of TS with probability matching priors that include the Jeffreys prior and the reference priors. We first prove that TS with certain probability matching priors can achieve the optimal regret bound. Then, we show the suboptimality of TS with other priors, including the Jeffreys and the reference priors. Nevertheless, we find that TS with the Jeffreys and reference priors can achieve the asymptotic lower bound if one uses a truncation procedure. These results suggest carefully choosing noninformative priors to avoid suboptimality and show the effectiveness of truncation procedures in TS-based policies.

This paper presents a novel federated linear contextual bandits model, where individual clients face different $K$-armed stochastic bandits coupled through common global parameters. By leveraging the geometric structure of the linear rewards, a collaborative algorithm called Fed-PE is proposed to cope with the heterogeneity across clients without exchanging local feature vectors or raw data. Fed-PE relies on a novel multi-client G-optimal design, and achieves near-optimal regrets for both disjoint and shared parameter cases with logarithmic communication costs. In addition, a new concept called collinearly-dependent policies is introduced, based on which a tight minimax regret lower bound for the disjoint parameter case is derived. Experiments demonstrate the effectiveness of the proposed algorithms on both synthetic and real-world datasets.

Program verification offers a framework for ensuring program correctness and therefore systematically eliminating different classes of bugs. Inferring loop invariants is one of the main challenges behind automated verification of real-world programs which often contain many loops. In this paper, we present Continuous Logic Network (CLN), a novel neural architecture for automatically learning loop invariants directly from program execution traces. Unlike existing neural networks, CLNs can learn precise and explicit representations of formulas in Satisfiability Modulo Theories (SMT) for loop invariants from program execution traces. We develop a new sound and complete semantic mapping for assigning SMT formulas to continuous truth values that allows CLNs to be trained efficiently. We use CLNs to implement a new inference system for loop invariants, CLN2INV, that significantly outperforms existing approaches on the popular Code2Inv dataset. CLN2INV is the first tool to solve all 124 theoretically solvable problems in the Code2Inv dataset. Moreover, CLN2INV takes only 1.1 seconds on average for each problem, which is 40 times faster than existing approaches. We further demonstrate that CLN2INV can even learn 12 significantly more complex loop invariants than the ones required for the Code2Inv dataset.