LogSumExp Function Properties: Lemmas for Energy Functions

Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Related Work 2 Related Work 3 Model and 3.1 Associative memories 3 Model and 3.1 Associative memories 3.2 Transformer blocks 3.2 Transformer blocks 4 A New Energy Function 4 A New Energy Function 4.1 The layered structure 4.1 The layered structure 5 Cross-Entropy Loss 5 Cross-Entropy Loss 6 Empirical Results and 6.1 Empirical evaluation of the radius 6 Empirical Results and 6.1 Empirical evaluation of the radius 6.2 Training GPT-2 6.2 Training GPT-2 6.3 Training Vanilla Transformers 6.3 Training Vanilla Transformers 7 Conclusion and Acknowledgments 7 Conclusion and Acknowledgments Appendix A. Deferred Tables Appendix A. Deferred Tables Appendix B. Some Properties of the Energy Functions Appendix B. Some Properties of the Energy Functions Appendix C. Deferred Proofs from Section 5 Appendix C. Deferred Proofs from Section 5 Appendix D. Transformer Details: Using GPT-2 as an Example Appendix D. Transformer Details: Using GPT-2 as an Example References References Appendix B. Some Properties of the Energy Functions We introduce some useful properties of the LogSumExp function defined below. This is particularly useful because The softmax function, widely utilized in the Transformer models, is the gradient of the LogSumExp function. As shown in (Grathwohl et al., 2019), the LogSumExp corresponds to the energy function of the a classifier. Lemma 1 LogSumExp(x) is convex. Lemma 1 is convex. Proof Proof Consequently, we have the following smooth approximation for the min function. B.1 Proof of Proposition 2 Authors: (1) Xueyan Niu, Theory Laboratory, Central Research Institute, 2012 Laboratories, Huawei Technologies Co., Ltd.; (2) Bo Bai baibo (8@huawei.com); (3) Lei Deng (deng.lei2@huawei.com); (4) Wei Han (harvey.hanwei@huawei.com). Authors: Authors: (1) Xueyan Niu, Theory Laboratory, Central Research Institute, 2012 Laboratories, Huawei Technologies Co., Ltd.; (2) Bo Bai baibo (8@huawei.com); (3) Lei Deng (deng.lei2@huawei.com); (4) Wei Han (harvey.hanwei@huawei.com). This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. available on arxiv available on arxiv