Quasi-gradients of discrete parameters

2022-12-20 — 2024-05-17

calculus
classification
optimization
probabilistic algorithms
probability
score function
statistics

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Notes on taking gradients through things that look like they have no gradients because their arguments are discrete. There are a lot of loosely related concepts in here that may not reflect an actual common theme. TBC.

See also Polya-Gamma…

1 Stochastic gradients via REINFORCE

The classic generic REINFORCE/Score function method for estimating gradients of expectations of functions of random variables can be used to estimate gradients of functions of discrete random variables as a special case. There are particular extra tricks used for discrete random variables; see e.g. (Grathwohl et al. 2018; Liu et al. 2019; Mnih and Gregor 2014; Tucker et al. 2017).

2 Gumbel-(soft)max

A.k.a. the concrete distribution. See Gumbel-max.

3 Gradients of other weird things

Differentiable sorting? See, e.g. Grover et al. (2018) and Prillo and Eisenschlos (2020).

4 Avoiding the need for gradients

Famously, Expectation Maximization can handle some of the same optimisation problems as gradient-based methods, but without needing gradients. There are presumably more variants.

5 Other methods

What even are (Grathwohl et al. 2021; Zhang, Liu, and Liu 2022)? I think they work for quantised continuous vars, or possibly ordinal vars?

6 Examples

7 Incoming

8 References

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Grathwohl, Swersky, Hashemi, et al. 2021. “Oops I Took A Gradient: Scalable Sampling for Discrete Distributions.”
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