Overparameterization in large models

Improper learning, benign overfitting, double descent

April 4, 2018 — May 27, 2022

decision theory
feature construction
machine learning
model selection
neural nets
probabilistic algorithms
stochastic processes

Notes on the general weird behaviour of increasing the number of slack parameters we use, especially in machine learning, especially especially in neural nets. Most of these have far more parameters than we “need” which is a problem for classical models of learning, herein we learn to fear having to many parameters.

Figure 1

1 For making optimisation nice

Certainly, looking at how some classic non-convex optimization problems can be lifted in to convex problem by adding slack variables, we can imagine that something similar happens by analogy in neural nets. Is it enough to imagine that something similar happens in NN, perhaps not lifting them into convex problems _per_se but at least into better-behaved optimisations in some sense?

The combination of overparameterization and SGD is argued to be the secret to how deep learning works, by e.g. AllenZhuConvergence2018.

RJ Lipton discusses Arno van den Essen’s 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)? 🏗.

2 Double descent

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

Possibly this phenomenon relates to the concept of …

3 Data interpolation

a.k.a. benign overfitting. 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 necessary if one wants to interpolate the data smoothly. Namely we show that smooth interpolation 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.

I am not sure if this is a distinct thing from other double descent phenomena. Hastie et al. (2020) suggests perhaps not?

Interpolators — estimators that achieve zero training error — have attracted growing attention in machine learning, mainly because state-of-the art neural networks appear to be models of this type. In this paper, we study minimum \(\ell_2\) norm (“ridgeless”) interpolation in high-dimensional least squares regression. We consider two different models for the feature distribution: a linear model, where the feature vectors \(x_i \in {\mathbb R}^p\) are obtained by applying a linear transform to a vector of i.i.d. entries, \(x_i = \Sigma^{1/2} z_i\) (with \(z_i \in {\mathbb R}^p\)); and a nonlinear model, where the feature vectors are obtained by passing the input through a random one-layer neural network, \(x_i = \varphi(W z_i)\) (with \(z_i \in {\mathbb R}^d\), \(W \in {\mathbb R}^{p \times d}\) a matrix of i.i.d. entries, and \(\varphi\) an activation function acting componentwise on \(W z_i\)). We recover — in a precise quantitative way — several phenomena that have been observed in large-scale neural networks and kernel machines, including the “double descent” behavior of the prediction risk, and the potential benefits of overparametrization.

4 Lottery ticket hypothesis

Figure 2

The Lottery Ticket hypothesis (Frankle and Carbin 2019; Hayou et al. 2020) asserts something like “there is a good compact network hidden inside the overparameterized one you have”. 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?

5 In extremely large models

See neural nets at scale

6 In the wide-network limit

See Wide NNs.

7 Convex relaxation

See convex relaxation.

8 References

Allen-Zhu, Li, and Song. 2018. A Convergence Theory for Deep Learning via Over-Parameterization.”
Arora, Cohen, and Hazan. 2018. On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization.” arXiv:1802.06509 [Cs].
Bach. 2013. Convex Relaxations of Structured Matrix Factorizations.” arXiv:1309.3117 [Cs, Math].
Bahmani, and Romberg. 2014. Lifting for Blind Deconvolution in Random Mask Imaging: Identifiability and Convex Relaxation.” arXiv:1501.00046 [Cs, Math, Stat].
———. 2016. Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation.” arXiv:1610.04210 [Cs, Math, Stat].
Bartlett, Montanari, and Rakhlin. 2021. Deep Learning: A Statistical Viewpoint.” Acta Numerica.
Bubeck, and Sellke. 2021. A Universal Law of Robustness via Isoperimetry.” In.
Dziugaite, and Roy. 2017. Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters Than Training Data.” arXiv:1703.11008 [Cs].
Frankle, and Carbin. 2019. The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks.” arXiv:1803.03635 [Cs].
Głuch, and Urbanke. 2021. Noether: The More Things Change, the More Stay the Same.” arXiv:2104.05508 [Cs, Stat].
Goldstein, and Studer. 2016. PhaseMax: Convex Phase Retrieval via Basis Pursuit.” arXiv:1610.07531 [Cs, Math].
Hardt, Ma, and Recht. 2018. Gradient Descent Learns Linear Dynamical Systems.” The Journal of Machine Learning Research.
Hasson, Nastase, and Goldstein. 2020. Direct Fit to Nature: An Evolutionary Perspective on Biological and Artificial Neural Networks.” Neuron.
Hastie, Montanari, Rosset, et al. 2020. Surprises in High-Dimensional Ridgeless Least Squares Interpolation.”
Hayou, Ton, Doucet, et al. 2020. Pruning Untrained Neural Networks: Principles and Analysis.” arXiv:2002.08797 [Cs, Stat].
Hazan, Singh, and Zhang. 2017. Learning Linear Dynamical Systems via Spectral Filtering.” In NIPS.
Molchanov, Ashukha, and Vetrov. 2017. Variational Dropout Sparsifies Deep Neural Networks.” In Proceedings of ICML.
Nakkiran, Kaplun, Bansal, et al. 2019. Deep Double Descent: Where Bigger Models and More Data Hurt.” arXiv:1912.02292 [Cs, Stat].
Oliveira, and Skelton. 2001. Stability Tests for Constrained Linear Systems.” In Perspectives in Robust Control. Lecture Notes in Control and Information Sciences.
Ran, and Hu. 2017. Parameter Identifiability in Statistical Machine Learning: A Review.” Neural Computation.
Semenova, Rudin, and Parr. 2021. A Study in Rashomon Curves and Volumes: A New Perspective on Generalization and Model Simplicity in Machine Learning.” arXiv:1908.01755 [Cs, Stat].
Tropp. 2006. Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise.” IEEE Transactions on Information Theory.
You, Li, Xu, et al. 2019. Drawing Early-Bird Tickets: Toward More Efficient Training of Deep Networks.” In.
Zhang, Bengio, Hardt, et al. 2017. Understanding Deep Learning Requires Rethinking Generalization.” In Proceedings of ICLR.
———, et al. 2021. Understanding Deep Learning (Still) Requires Rethinking Generalization.” Communications of the ACM.