Notes on the general technique of increasing the number of slack parameters you have, especially in machine learning, especially especially in neural nets.

## For smoothness

This insight is fresh. Bubeck and Sellke (2021) argue

Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in the current practice of deep learning is that models are trained with many more parameters than what this classical theory would suggest. We propose a theoretical explanation for this phenomenon. We prove that for a broad class of data distributions and model classes, overparametrization is

necessaryif one wants to interpolate the datasmoothly. Namely we show thatsmoothinterpolation requires \(d\) times more parameters than mere interpolation, where \(d\) is the ambient data dimension. We prove this universal law of robustness for any smoothly parametrized function class with polynomial size weights, and any covariate distribution verifying isoperimetry. In the case of two-layers neural networks and Gaussian covariates, this law was conjectured in prior work by Bubeck, Li and Nagaraj. We also give an interpretation of our result as an improved generalization bound for model classes consisting of smooth functions.

## In the wide-network limit

See Wide NNs.

## For making optimisation nice

The combination of overparameterization and SGD is argued to be the secret to how deep learning works, by Zeyuan Allen-Zhu, Yuanzhi Li and Zhao Song. Certainly, looking at how some classic optimizations can be lifted in to convex problems, we can imagine that something similar happens by analaogy here.

RJ Lipton discusses Arno van den Essenβs incidental work on stabilisation methods of polynomials, which relates. AFAICT, to transfer-function-type stability.
Does this connect to the overparameterization of rational transfer function analysis of Hardt, Ma, and Recht (2018)?
*π*.

## Double descent

When adding data (or parameters?) can make the model worse. E.g. Deep Double Descent

## Lottery ticket hypothesis

The Lottery Ticket hypothesis (Frankle and Carbin 2019; Hayou et al. 2020) asserts something like βthere is a good compact network hidden inside the overparameterized one you haveβ. Intuitively it is computationally hard to find the hidden optimal network. I am interested in computational bounds for this; How much cheaper is it to calculate with a massive network than to find the tiny networks that does better? The calculus here is altered by SIMD architectures such as GPUs, which change the relative cost (although not the scaling) of certain types of calculations, which is arguably how we got to the modern form of neural net obsession.

## References

*arXiv:1802.06509 [Cs]*, February.

*arXiv:1309.3117 [Cs, Math]*, September.

*arXiv:1501.00046 [Cs, Math, Stat]*, December.

*arXiv:1610.04210 [Cs, Math, Stat]*, October.

*arXiv:1703.11008 [Cs]*, October.

*arXiv:1803.03635 [Cs]*, March.

*arXiv:2104.05508 [Cs, Stat]*, April.

*arXiv:1610.07531 [Cs, Math]*, October.

*The Journal of Machine Learning Research*19 (1): 1025β68.

*Neuron*105 (3): 416β34.

*arXiv:2002.08797 [Cs, Stat]*, June.

*NIPS*.

*Proceedings of ICML*.

*arXiv:1912.02292 [Cs, Stat]*, December.

*Perspectives in Robust Control*, 241β57. Lecture Notes in Control and Information Sciences. Springer, London.

*Neural Computation*29 (5): 1151β1203.

*arXiv:1908.01755 [Cs, Stat]*, April.

*IEEE Transactions on Information Theory*52 (3): 1030β51.

*Proceedings of ICLR*.

*Communications of the ACM*64 (3): 107β15.

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