Tree-Projected Gradient Descent for Estimating Gradient-Sparse Parameters on Graphs

• URL: http://arxiv.org/abs/2006.01662v1
• Date: Sun, 31 May 2020 20:08:13 GMT
• Title: Tree-Projected Gradient Descent for Estimating Gradient-Sparse Parameters on Graphs
• Authors: Sheng Xu, Zhou Fan, Sahand Negahban
• Abstract summary: We study estimation of a gradient-sparse parameter vector $boldsymboltheta* in mathbbRp$. We show that, under suitable restricted strong convexity and smoothness assumptions for the loss, the resulting estimator achieves the squared-error risk $fracs*n log (1+fracps*)$ up to a multiplicative constant that is independent of $G$.
• Score: 10.846572437131872
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study estimation of a gradient-sparse parameter vector $\boldsymbol{\theta}^* \in \mathbb{R}^p$, having strong gradient-sparsity $s^*:=\|\nabla_G \boldsymbol{\theta}^*\|_0$ on an underlying graph $G$. Given observations $Z_1,\ldots,Z_n$ and a smooth, convex loss function $\mathcal{L}$ for which $\boldsymbol{\theta}^*$ minimizes the population risk $\mathbb{E}[\mathcal{L}(\boldsymbol{\theta};Z_1,\ldots,Z_n)]$, we propose to estimate $\boldsymbol{\theta}^*$ by a projected gradient descent algorithm that iteratively and approximately projects gradient steps onto spaces of vectors having small gradient-sparsity over low-degree spanning trees of $G$. We show that, under suitable restricted strong convexity and smoothness assumptions for the loss, the resulting estimator achieves the squared-error risk $\frac{s^*}{n} \log (1+\frac{p}{s^*})$ up to a multiplicative constant that is independent of $G$. In contrast, previous polynomial-time algorithms have only been shown to achieve this guarantee in more specialized settings, or under additional assumptions for $G$ and/or the sparsity pattern of $\nabla_G \boldsymbol{\theta}^*$. As applications of our general framework, we apply our results to the examples of linear models and generalized linear models with random design.

Related papers

• Conditional regression for the Nonlinear Single-Variable Model [4.565636963872865]
  We consider a model $F(X):=f(Pi_gamma):mathbbRdto[0,rmlen_gamma]$ where $Pi_gamma: [0,rmlen_gamma]tomathbbRd$ and $f:[0,rmlen_gamma]tomathbbR1$. We propose a nonparametric estimator, based on conditional regression, and show that it can achieve the $one$-dimensional optimal min-max rate
  arXiv  Detail & Related papers  (2024-11-14T18:53:51Z)
• Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit [75.4661041626338]
  We study the problem of gradient descent learning of a single-index target function $f_*(boldsymbolx) = textstylesigma_*left(langleboldsymbolx,boldsymbolthetarangleright)$ under isotropic Gaussian data. We prove that a two-layer neural network optimized by an SGD-based algorithm learns $f_*$ of arbitrary link function with a sample and runtime complexity of $n asymp T asymp C(q) cdot d
  arXiv  Detail & Related papers  (2024-06-03T17:56:58Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priors [33.0212223058894]
  The problem of recovering a signal from a quadratic system $y_i=boldsymbol xtopboldsymbol A_iboldsymbol x, i=1,ldots,m$ with full-rank matrices $boldsymbol A_i$ frequently arises in applications such as unassigned distance geometry and sub-wavelength imaging. This paper addresses the high-dimensional case where $mll n$ incorporating by prior knowledge of $boldsymbol x$.
  arXiv  Detail & Related papers  (2023-09-16T16:00:07Z)
• Estimating the minimizer and the minimum value of a regression function under passive design [72.85024381807466]
  We propose a new method for estimating the minimizer $boldsymbolx*$ and the minimum value $f*$ of a smooth and strongly convex regression function $f$. We derive non-asymptotic upper bounds for the quadratic risk and optimization error of $boldsymbolz_n$, and for the risk of estimating $f*$.
  arXiv  Detail & Related papers  (2022-11-29T18:38:40Z)
• Structure Learning in Graphical Models from Indirect Observations [17.521712510832558]
  This paper considers learning of the graphical structure of a $p$-dimensional random vector $X in Rp$ using both parametric and non-parametric methods. Under mild conditions, we show that our graph-structure estimator can obtain the correct structure.
  arXiv  Detail & Related papers  (2022-05-06T19:24:44Z)
• High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation [89.21686761957383]
  We study the first gradient descent step on the first-layer parameters $boldsymbolW$ in a two-layer network. Our results demonstrate that even one step can lead to a considerable advantage over random features.
  arXiv  Detail & Related papers  (2022-05-03T12:09:59Z)
• Random Graph Matching in Geometric Models: the Case of Complete Graphs [21.689343447798677]
  This paper studies the problem of matching two complete graphs with edge weights correlated through latent geometries. We derive an approximate maximum likelihood estimator, which provably achieves, with high probability, perfect recovery of $pi*$. As a side discovery, we show that the celebrated spectral algorithm of [Ume88] emerges as a further approximation to the maximum likelihood in the geometric model.
  arXiv  Detail & Related papers  (2022-02-22T04:14:45Z)
• Universality of empirical risk minimization [12.764655736673749]
  Consider supervised learning from i.i.d. samples where $boldsymbol x_i inmathbbRp$ are feature vectors and $y in mathbbR$ are labels. We study empirical risk universality over a class of functions that are parameterized by $mathsfk.
  arXiv  Detail & Related papers  (2022-02-17T18:53:45Z)
• Fast Graph Sampling for Short Video Summarization using Gershgorin Disc Alignment [52.577757919003844]
  We study the problem of efficiently summarizing a short video into several paragraphs, leveraging recent progress in fast graph sampling. Experimental results show that our algorithm achieves comparable video summarization as state-of-the-art methods, at a substantially reduced complexity.
  arXiv  Detail & Related papers  (2021-10-21T18:43:00Z)
• Agnostic Learning of a Single Neuron with Gradient Descent [92.7662890047311]
  We consider the problem of learning the best-fitting single neuron as measured by the expected square loss. For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$. For the ReLU activation, our population risk guarantee is $O(mathsfOPT1/2)+epsilon$.
  arXiv  Detail & Related papers  (2020-05-29T07:20:35Z)

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.