## 1 Introduction
the strategy selection of the communication parties and the jammer in the multi-channel power allocation confrontation scenario can be modeled as a two-player zero-sum game [4], which is suitable for modeling and solving this problem using knowledge of game theory. For the multi-channel power allocation problem under a one-shot confrontation scenario, we focus on approximating the Nash equilibrium. In an actual confrontation scenario, there may be repeated confrontations between the communication parties and the jammer. In this scenario, it is not necessary for the communication parties to use Nash equilibrium as the adaptive optimal strategy when the jammer does not use Nash equilibrium. Therefore, our goal in this case is to minimize the expected regret of the communication parties after multiple confrontations. For the multi-channel power allocation problem under a one-shot confrontation scenario, the problem-solving methods are different depending on the optimization objective. For the case with the optimization objective as the communication capacity, the authors of [5] modeled the multi-channel power allocation anti-jamming problem as a Stackelberg game and proved the existence of the Stackelberg equilibrium. In [6], the authors studied the Nash equilibrium under a different channel model. For the AWGN (Additive White Gaussian Noise) channel model, the Nash equilibrium of opposing parties is to equally distribute power on all available channels, while for the frequency selective fading scenario, the Nash equilibrium of opposing parties can be solved by an iterative water-filling algorithm. So far, for multi-channel power allocation problems, the case of optimizing communication capacity has been well solved. However, they are not applicable in the actual communication system, because the actual communication system transmits information at a specific rate, with the SINR (Signal-to-Interference-plus-Noise Ratio) reaching the demodulation threshold, larger communication power cannot further improve the communication capacity. Therefore, the optimization objective of the communication parties should be the number of channels through which information is transmitted successfully. The multi-channel power allocation problem with the optimization objective of the number of successfully transmitted channels can be modeled as BG (Blotto Game) models [7; 8; 9]. In [7; 8], the authors construct the confrontation scenarios with the secondary user and attacker simultaneously being able to access all available channels in cognitive radio networks, and model the power allocation problem between the two parties as a BG model. They obtain the Nash equilibrium of the game by constructing a joint distribution that matches the expected marginal distribution and satisfies the power budget constraint. However, finding the specific Nash equilibrium determined by the joint probability distribution function is still a difficult task. In [9], the author modeled the multi-channel power allocation problem between attacker and defender in cognitive radio as a two-player constant BG model and solved the model using the basic concept of iterative Nash bargaining solution, with the objective of enabling IoT (Internet of Things) systems safety level to reach an effective and stable state. There are few works on multi-channel power allocation problems under a repeated confrontation scenario. We consider the online learning problem under the partially observable model studied in [10], which assumes that the learner is able to obtain the loss of the chosen actions, and observe the loss of some other actions. In [11], the authors modeled different types of bandit feedback information as a graph structure, and proposed definitions of strongly observable graphs, weakly observable graphs, and unobservable graphs, while the multi-channel power allocation problem corresponds to a weakly observable graph.