Double Auction enables decentralized transfer of goods between multiple buyers and sellers, thus underpinning functioning of many online marketplaces. Buyers and sellers compete in these markets through bidding, but do not often know their own valuation a-priori. As the allocation and pricing happens through bids, the profitability of participants, hence sustainability of such markets, depends crucially on learning respective valuations through repeated interactions. We initiate the study of Double Auction markets under bandit feedback on both buyers' and sellers' side. We show with confidence bound based bidding, and'Average Pricing' there is an efficient price discovery among the participants.

A.1 Experimental design Figure 1 summarizes the experimental design used for our experiments. The participants that went through our experiments are users from the online platform Amazon Mechanical Turk (AMT). Through this platform, users stay anonymous, hence, we do not collect any sensitive personal information about them. We prioritized users with a Master qualification (which is a qualification attributed by AMT to users who have proven to be of excellent quality) or normal users with high qualifications (number of HIT completed = 10000and HIT accepted > 98%). Before going through the experiment, participants are asked to read and agree to a consent form, which specifies: the objective and procedure of the experiment, as well as the time expected to completion ( 5 - 8 min) with the reward associated ($1.4), and finally, the risk, benefits, and confidentiality of taking part in this study.

Many experiments compare count outcomes among treatment groups. Examples include the number of successful signups in conversion rate experiments or the number of errors produced by software versions in canary tests. Observations typically arrive in a sequence and practitioners wish to continuously monitor their experiments, sequentially testing hypotheses while maintaining Type I error probabilities under optional stopping and continuation. These goals are frequently complicated in practice by non-stationary time dynamics. We provide practical solutions through sequential tests of multinomial hypotheses, hypotheses about many inhomogeneous Bernoulli processes and hypotheses about many timeinhomogeneous Poisson counting processes. For estimation, we further provide confidence sequences for multinomial probability vectors, all contrasts among probabilities of inhomogeneous Bernoulli processes and all contrasts among intensities of time-inhomogeneous Poisson counting processes. Together, these provide an "anytime-valid" inference framework for a wide variety of experiments dealing with count outcomes, which we illustrate with several industry applications.

After completely excluding the temporal dimension from the model parameter space, we introduced the temporal conditioning operation to handle the temporal information. In particular, this operation enables memory-dependent processing as in biological coding circuits. Figure 6: Performances under di erent hidden state and latent space dimension settings on Movie 2 Retina 2 data. For hidden state experiments, the latent space dimension is set to 32. And for latent space experiments, the hidden state dimension is 64.

Graph neural networks (GNNs) have achieved superior performance in various applications, but training dedicated GNNs can be costly for large-scale graphs. Some recent work started to study the pre-training of GNNs. However, none of them provide theoretical insights into the design of their frameworks, or clear requirements and guarantees towards their transferability. In this work, we establish a theoretically grounded and practically useful framework for the transfer learning of GNNs. Firstly, we propose a novel view towards the essential graph information and advocate the capturing of it as the goal of transferable GNN training, which motivates the design of EGI (Ego-Graph Information maximization) to analytically achieve this goal. Secondly, when node features are structure-relevant, we conduct an analysis of EGI transferability regarding the difference between the local graph Laplacians of the source and target graphs. We conduct controlled synthetic experiments to directly justify our theoretical conclusions. Comprehensive experiments on two real-world network datasets show consistent results in the analyzed setting of direct-transfering, while those on large-scale knowledge graphs show promising results in the more practical setting of transfering with fine-tuning.1