in Periodic Zero-Sum
in Periodic Zero-Sum
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周期性零和
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通讯作者：
H. C. Yadav
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作者：
S. Gulati;H. C. Yadav
本附录的结构和内容如下。附录A涵盖了其他相关工作，我们在文档末尾提供了单独的参考书目。在附录A之后，本文中的理论结果的证明按照它们出现的顺序给出。具体地说，附录B包含了4.1节中关于GDA动力学和递归的结果的证明。这包括命题1、引理1、引理2和定理1的证明。在附录C中，我们提供了4.2节中关于GDA时间平均行为的命题2的证明。附录D包含了5.1节关于FTRL动力学和递归的分析，包括引理3、引理4和定理2的证明。附录E涵盖了FTRL动力学的时间平均行为，如第5.2节所述。具体地，定理3和命题3的证明在附录E中给出。最后，附录F包含了额外的实验结果。
The organization and contents of this appendix is as follows. Appendix A covers additional related work, which we provide a separate bibliography for at the end of the document. Following Appendix A, proofs for the theoretical results in the paper are presented in the order that they appeared. Specifically, Appendix B contains the proofs for the results presented in Section 4.1 on GDA dynamics and recurrence. This includes the proofs of Proposition 1, Lemma 1, Lemma 2, and Theorem 1. In Appendix C, we provide the proof of Proposition 2 from Section 4.2 on the time-average behavior of GDA. Appendix D contains the analysis for Section 5.1 on FTRL dynamics and recurrence, including the proofs of Lemma 3, Lemma 4, and Theorem 2. Appendix E covers the time-average behavior of FTRL dynamics as presented in Section 5.2. Specifically, the proofs of Theorem 3 and Proposition 3 are given in Appendix E. Finally, Appendix F contains additional experimental results.