Explore the mathematical properties of the LogSumExp function or LSE, fundamental for energy functions in Transformers. Strict bounds are analyzed that relate LSE to the maximum function and the logarithmic mean, also demonstrating continuity and differentiability at any real point. Thanks to these lemmas, convexity and smoothness are guaranteed when optimizing energy functions with stable gradients
The LogSumExp function defines a lower and upper bound that bounds the difference with the maximum. Its global continuity property ensures that small changes in inputs produce small changes in the output, which is vital for training deep neural networks. Furthermore, the smoothing of the maximum through LSE allows for efficient numerical approximations
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