Algorithmic Gaussianization through Sketching: Converting Data into Sub-gaussian Random Designs

• URL: http://arxiv.org/abs/2206.10291v2
• Date: Thu, 27 Jul 2023 17:48:41 GMT
• Title: Algorithmic Gaussianization through Sketching: Converting Data into Sub-gaussian Random Designs
• Authors: Micha{\l} Derezi\'nski
• Abstract summary: We provide an algorithmic framework for gaussianizing data distributions via averaging. We show that it is possible to efficiently construct data sketches that are nearly indistinguishable from sub-gaussian random designs.
• Score: 22.925108493465363
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Algorithmic Gaussianization is a phenomenon that can arise when using randomized sketching or sampling methods to produce smaller representations of large datasets: For certain tasks, these sketched representations have been observed to exhibit many robust performance characteristics that are known to occur when a data sample comes from a sub-gaussian random design, which is a powerful statistical model of data distributions. However, this phenomenon has only been studied for specific tasks and metrics, or by relying on computationally expensive methods. We address this by providing an algorithmic framework for gaussianizing data distributions via averaging, proving that it is possible to efficiently construct data sketches that are nearly indistinguishable (in terms of total variation distance) from sub-gaussian random designs. In particular, relying on a recently introduced sketching technique called Leverage Score Sparsified (LESS) embeddings, we show that one can construct an $n\times d$ sketch of an $N\times d$ matrix $A$, where $n\ll N$, that is nearly indistinguishable from a sub-gaussian design, in time $O(\text{nnz}(A)\log N + nd^2)$, where $\text{nnz}(A)$ is the number of non-zero entries in $A$. As a consequence, strong statistical guarantees and precise asymptotics available for the estimators produced from sub-gaussian designs (e.g., for least squares and Lasso regression, covariance estimation, low-rank approximation, etc.) can be straightforwardly adapted to our sketching framework. We illustrate this with a new approximation guarantee for sketched least squares, among other examples.

Related papers

• Computational-Statistical Gaps in Gaussian Single-Index Models [77.1473134227844]
  Single-Index Models are high-dimensional regression problems with planted structure. We show that computationally efficient algorithms, both within the Statistical Query (SQ) and the Low-Degree Polynomial (LDP) framework, necessarily require $Omega(dkstar/2)$ samples.
  arXiv  Detail & Related papers  (2024-03-08T18:50:19Z)
• Computational-Statistical Gaps for Improper Learning in Sparse Linear Regression [4.396860522241307]
  We show that an efficient learning algorithm for sparse linear regression can be used to solve sparse PCA problems with a negative spike. We complement our reduction with low-degree and statistical query lower bounds for the sparse problems from which we reduce.
  arXiv  Detail & Related papers  (2024-02-21T19:55:01Z)
• Sparse Gaussian Graphical Models with Discrete Optimization: Computational and Statistical Perspectives [8.403841349300103]
  We consider the problem of learning a sparse graph underlying an undirected Gaussian graphical model. We propose GraphL0BnB, a new estimator based on an $ell_0$-penalized version of the pseudolikelihood function. Our numerical experiments on real/synthetic datasets suggest that our method can solve, to near-optimality, problem instances with $p = 104$.
  arXiv  Detail & Related papers  (2023-07-18T15:49:02Z)
• Sketched Gaussian Model Linear Discriminant Analysis via the Randomized Kaczmarz Method [7.593861427248019]
  We present sketched linear discriminant analysis, an iterative randomized approach to binary-class Gaussian model linear discriminant analysis (LDA) for very large data. We harness a least squares formulation and mobilize the descent gradient framework. We present convergence guarantees for the sketched predictions on new data within a fixed number of iterations.
  arXiv  Detail & Related papers  (2022-11-10T18:29:36Z)
• Adaptive Sketches for Robust Regression with Importance Sampling [64.75899469557272]
  We introduce data structures for solving robust regression through gradient descent (SGD) Our algorithm effectively runs $T$ steps of SGD with importance sampling while using sublinear space and just making a single pass over the data.
  arXiv  Detail & Related papers  (2022-07-16T03:09:30Z)
• Convergence for score-based generative modeling with polynomial complexity [9.953088581242845]
  We prove the first convergence guarantees for the core mechanic behind Score-based generative modeling. Compared to previous works, we do not incur error that grows exponentially in time or that suffers from a curse of dimensionality. We show that a predictor-corrector gives better convergence than using either portion alone.
  arXiv  Detail & Related papers  (2022-06-13T14:57:35Z)
• Newton-LESS: Sparsification without Trade-offs for the Sketched Newton Update [88.73437209862891]
  In second-order optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration. We show that the Gaussian sketching matrix can be drastically sparsified, significantly reducing the computational cost of sketching. We prove that Newton-LESS enjoys nearly the same problem-independent local convergence rate as Gaussian embeddings.
  arXiv  Detail & Related papers  (2021-07-15T17:33:05Z)
• Estimating Stochastic Linear Combination of Non-linear Regressions Efficiently and Scalably [23.372021234032363]
  We show that when the sub-sample sizes are large then the estimation errors will be sacrificed by too much. To the best of our knowledge, this is the first work that and guarantees for the lineartext+Stochasticity model.
  arXiv  Detail & Related papers  (2020-10-19T07:15:38Z)
• Breaking the Sample Size Barrier in Model-Based Reinforcement Learning with a Generative Model [50.38446482252857]
  This paper is concerned with the sample efficiency of reinforcement learning, assuming access to a generative model (or simulator) We first consider $gamma$-discounted infinite-horizon Markov decision processes (MDPs) with state space $mathcalS$ and action space $mathcalA$. We prove that a plain model-based planning algorithm suffices to achieve minimax-optimal sample complexity given any target accuracy level.
  arXiv  Detail & Related papers  (2020-05-26T17:53:18Z)
• Learning Gaussian Graphical Models via Multiplicative Weights [54.252053139374205]
  We adapt an algorithm of Klivans and Meka based on the method of multiplicative weight updates. The algorithm enjoys a sample complexity bound that is qualitatively similar to others in the literature. It has a low runtime $O(mp2)$ in the case of $m$ samples and $p$ nodes, and can trivially be implemented in an online manner.
  arXiv  Detail & Related papers  (2020-02-20T10:50:58Z)
• Optimal Iterative Sketching with the Subsampled Randomized Hadamard Transform [64.90148466525754]
  We study the performance of iterative sketching for least-squares problems. We show that the convergence rate for Haar and randomized Hadamard matrices are identical, andally improve upon random projections. These techniques may be applied to other algorithms that employ randomized dimension reduction.
  arXiv  Detail & Related papers  (2020-02-03T16:17:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.