We examine the relation between the size of the id space and the number of rational agents in a network under which equilibrium in distributed algorithms is possible. When the number of agents in the network is not a-priori known, a single agent may duplicate to gain an advantage, pretending to be more than one agent. However, when the id space is limited, each duplication involves a risk of being caught. By comparing the risk against the advantage, given an id space of size L, we provide a method of calculating the minimal threshold t, the required number of agents in the network, such that the algorithm is in equilibrium. That is, it is the minimal value of t such that if agents a-priori know that n ≥ t then the algorithm is in equilibrium. We demonstrate this method by applying it to two problems, Leader Election and Knowledge Sharing, as well as providing a constant-time approximation t ≈ L 5 of the minimal threshold for Leader Election.
It is well known that for some tasks, labeled data sets may be hard to gather. Self-training, or pseudo-labeling, tackles the problem of having insufficient training data. In the self-training scheme, the classifier is first trained on a limited, labeled dataset, and after that, it is trained on an additional, unlabeled dataset, using its own predictions as labels, provided those predictions are made with high enough confidence. Using credible interval based on MC-dropout as a confidence measure, the proposed method is able to gain substantially better results comparing to several other pseudo-labeling methods and outperforms the former state-of-the-art pseudo-labeling technique by 7 % on the MNIST data-set. In addition to learning from large and static unlabeled datasets, the suggested approach may be more suitable than others as an online learning method where the classifier keeps getting new unlabeled data. The approach may be also applicable in the recent method of pseudo-gradients for training long sequential neural networks.
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