Randomised regression

January 14, 2017 — December 1, 2020

feature construction
functional analysis
linear algebra
probabilistic algorithms
sparser than thou
Figure 1

Tackling your regression, by using random embeddings of the predictors (and/or predictions?). Usually this means using low-dimensional projections, to reduce the dimensionality of a high dimensional regression. In this case it is not far from compressed sensing, except in how we handle noise. In this linear model case, this is of course random linear algebra, and may be a randomised matrix factorisation. You can do it the other way and prject something into a higher dimensional space, which is a popular trick for kernel approximation.

I am especially interested in seeing how this might be useful for dependent data, especially time series.

Brian McWilliams, Gabriel Krummenacher and Mario Lučić, Randomized Linear Regression: A brief overview and recent results. Gabriel implemented some of the algorithms mentioned, e.g.

  • Subsampled Randomized Fourier Transform.
  • SRHT: Uses the Subsampled Randomized Hadamard Transform (SRHT), equivalent to leverage based sampling.
  • aIWS, aRWS: Samples based on approximated statistical influence.
  • Uluru: SRHT with bias correction.

Martin Wainright, Statistics meets Optimization: Randomization and approximation for high-dimensional problems.

In the modern era of high-dimensional data, the interface between mathematical statistics and optimization has become an increasingly vibrant area of research. In this course, we provide some vignettes into this interface, including the following topics:

  • Dimensionality reduction via random projection. The naive idea of projecting high-dimensional data to a randomly chosen low-dimensional space is remarkably effective. We discuss the classical Johnson-Lindenstrauss lemma, as well as various modern variants that provide computationally-efficient embeddings with strong guarantees.

  • When is it possible to quickly obtain approximate solutions of large-scale convex programs? In practice, methods based on randomized projection can work very well, and arguments based on convex analysis and concentration of measure provide a rigorous underpinning to these observations.

  • Optimization problems with some form of nonconvexity arise frequently in statistical settings – for instance, in problems with latent variables, combinatorial constraints, or rank constraints. Nonconvex programs are known to be intractable in a complexity-theoretic sense, but the random ensembles arising in statistics are not adversarially constructed. Under what conditions is it possible to make rigorous guarantees about the behavior of simple iterative algorithms for such problems? We develop some general theory for addressing these questions, exploiting tools from both optimization theory and empirical process theory.

1 References

Bahmani, and Romberg. 2017. Anchored Regression: Solving Random Convex Equations via Convex Programming.” arXiv:1702.05327 [Cs, Math, Stat].
Choromanski, Rowland, and Weller. 2017. The Unreasonable Effectiveness of Random Orthogonal Embeddings.” arXiv:1703.00864 [Stat].
Cover. 1965. Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition.” IEEE Transactions on Electronic Computers.
Dezfouli, and Bonilla. 2015. Scalable Inference for Gaussian Process Models with Black-Box Likelihoods.” In Advances in Neural Information Processing Systems 28. NIPS’15.
Dhillon, Lu, Foster, et al. 2013. New Subsampling Algorithms for Fast Least Squares Regression.” In Advances in Neural Information Processing Systems.
Gilbert, Zhang, Lee, et al. 2017. Towards Understanding the Invertibility of Convolutional Neural Networks.” arXiv:1705.08664 [Cs, Stat].
Gottwald, and Reich. 2020. Supervised Learning from Noisy Observations: Combining Machine-Learning Techniques with Data Assimilation.” arXiv:2007.07383 [Physics, Stat].
Gribonval, Blanchard, Keriven, et al. 2017. Compressive Statistical Learning with Random Feature Moments.” arXiv:1706.07180 [Cs, Math, Stat].
Gupta, and Pensky. 2016. Solution of Linear Ill-Posed Problems Using Random Dictionaries.” arXiv:1605.07913 [Math, Stat].
Heinze, McWilliams, Meinshausen, et al. 2014. LOCO: Distributing Ridge Regression with Random Projections.” arXiv:1406.3469 [Stat].
Heinze, McWilliams, and Meinshausen. 2016. DUAL-LOCO: Distributing Statistical Estimation Using Random Projections.” In.
Krummenacher, McWilliams, Kilcher, et al. 2016. Scalable Adaptive Stochastic Optimization Using Random Projections.” In Advances in Neural Information Processing Systems 29.
Mahoney. 2010. Randomized Algorithms for Matrices and Data.
McWilliams, Balduzzi, and Buhmann. 2013. Correlated Random Features for Fast Semi-Supervised Learning.” In Advances in Neural Information Processing Systems 26.
McWilliams, Krummenacher, Lucic, et al. 2014. Fast and Robust Least Squares Estimation in Corrupted Linear Models.” In Advances in Neural Information Processing Systems.
Rahimi, and Recht. 2007. Random Features for Large-Scale Kernel Machines.” In Advances in Neural Information Processing Systems.
———. 2008. Uniform Approximation of Functions with Random Bases.” In 2008 46th Annual Allerton Conference on Communication, Control, and Computing.
Rosenfeld, and Tsotsos. 2018. “Intriguing Properties of Randomly Weighted Networks: Generalizing While Learning Next to Nothing.”
Scardapane, and Wang. 2017. Randomness in Neural Networks: An Overview.” Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery.
Sinha, and Duchi. 2016. Learning Kernels with Random Features.” In Advances in Neural Information Processing Systems 29.
Soni, and Mehdad. 2017. RIPML: A Restricted Isometry Property Based Approach to Multilabel Learning.” arXiv:1702.05181 [Cs, Stat].
Thanei, Heinze, and Meinshausen. 2017. Random Projections For Large-Scale Regression.” arXiv:1701.05325 [Math, Stat].
Wang, Zhu, and Ma. 2017. Optimal Subsampling for Large Sample Logistic Regression.” arXiv:1702.01166 [Stat].
Zhang, Wang, and Gu. 2017. Stochastic Variance-Reduced Gradient Descent for Low-Rank Matrix Recovery from Linear Measurements.” arXiv:1701.00481 [Stat].