Legendre-Fenchel transforms and application in RL

Legendre-Frenchel transform is a generalization of the Legendre transform.

Legendre-Fenchel Transform [1]

Definition Consider a function . We define the Legendre-Fenchel (LF) transform of by the variational formula

supporting lines

We say that the function has or admits a supporting line at if there exists such that

for all . The parameter is the slope of the supporting line. We further say that a supporting line is strictly supporting at if

holds for all .

If admits a supporting line at and exists, then the slope of the supporting line must be equal to . In other words, for differentiable functions, a supporting line is also a tangent line.

Theorem if and only if admits a supporting line at .

Legendre-Fenchel transform [2]

Let be a strongly convex function. The Legendre-Fenchel transform (or convex conjugate) of is , defined as:

References

[1] @article{touchette2005legendre, title={Legendre-Fenchel transforms in a nutshell}, author={Touchette, Hugo} }

[2] @inproceedings{mensch2018differentiable, title={Differentiable dynamic programming for structured prediction and attention}, author={Mensch, Arthur and Blondel, Mathieu}, booktitle={International Conference on Machine Learning}, pages={3462--3471}, year={2018}, organization={PMLR} }