Sample Amplification: Increasing Dataset Size even when Learning is Impossible

• URL: http://arxiv.org/abs/1904.12053v3
• Date: Sun, 25 Aug 2024 23:38:40 GMT
• Title: Sample Amplification: Increasing Dataset Size even when Learning is Impossible
• Authors: Brian Axelrod, Shivam Garg, Vatsal Sharan, Gregory Valiant,
• Abstract summary: Given data drawn from an unknown distribution, $D$, to what extent is it possible to amplify'' this dataset and output an even larger set of samples that appear to have been drawn from $D$? We formalize this question as follows: an $left(n, n + Theta(fracnsqrtk)right)$ amplifier exists, even though learning the distribution to small constant total variation distance requires $Theta(d)$ samples.
• Score: 15.864702679819544
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Given data drawn from an unknown distribution, $D$, to what extent is it possible to ``amplify'' this dataset and output an even larger set of samples that appear to have been drawn from $D$? We formalize this question as follows: an $(n,m)$ $\text{amplification procedure}$ takes as input $n$ independent draws from an unknown distribution $D$, and outputs a set of $m > n$ ``samples''. An amplification procedure is valid if no algorithm can distinguish the set of $m$ samples produced by the amplifier from a set of $m$ independent draws from $D$, with probability greater than $2/3$. Perhaps surprisingly, in many settings, a valid amplification procedure exists, even when the size of the input dataset, $n$, is significantly less than what would be necessary to learn $D$ to non-trivial accuracy. Specifically we consider two fundamental settings: the case where $D$ is an arbitrary discrete distribution supported on $\le k$ elements, and the case where $D$ is a $d$-dimensional Gaussian with unknown mean, and fixed covariance. In the first case, we show that an $\left(n, n + \Theta(\frac{n}{\sqrt{k}})\right)$ amplifier exists. In particular, given $n=O(\sqrt{k})$ samples from $D$, one can output a set of $m=n+1$ datapoints, whose total variation distance from the distribution of $m$ i.i.d. draws from $D$ is a small constant, despite the fact that one would need quadratically more data, $n=\Theta(k)$, to learn $D$ up to small constant total variation distance. In the Gaussian case, we show that an $\left(n,n+\Theta(\frac{n}{\sqrt{d}} )\right)$ amplifier exists, even though learning the distribution to small constant total variation distance requires $\Theta(d)$ samples. In both the discrete and Gaussian settings, we show that these results are tight, to constant factors. Beyond these results, we formalize a number of curious directions for future research along this vein.

Related papers

• Dimension-free Private Mean Estimation for Anisotropic Distributions [55.86374912608193]
  Previous private estimators on distributions over $mathRd suffer from a curse of dimensionality. We present an algorithm whose sample complexity has improved dependence on dimension.
  arXiv  Detail & Related papers  (2024-11-01T17:59:53Z)
• Outlier Robust Multivariate Polynomial Regression [27.03423421704806]
  We are given a set of random samples $(mathbfx_i,y_i) in [-1,1]n times mathbbR$ that are noisy versions of $(mathbfx_i,p(mathbfx_i)$. The goal is to output a $hatp$, within an $ell_in$-distance of at most $O(sigma)$ from $p$.
  arXiv  Detail & Related papers  (2024-03-14T15:04:45Z)
• TURF: A Two-factor, Universal, Robust, Fast Distribution Learning Algorithm [64.13217062232874]
  One of its most powerful and successful modalities approximates every distribution to an $ell$ distance essentially at most a constant times larger than its closest $t$-piece degree-$d_$. We provide a method that estimates this number near-optimally, hence helps approach the best possible approximation.
  arXiv  Detail & Related papers  (2022-02-15T03:49:28Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• The Sample Complexity of Robust Covariance Testing [56.98280399449707]
  We are given i.i.d. samples from a distribution of the form $Z = (1-epsilon) X + epsilon B$, where $X$ is a zero-mean and unknown covariance Gaussian $mathcalN(0, Sigma)$. In the absence of contamination, prior work gave a simple tester for this hypothesis testing task that uses $O(d)$ samples. We prove a sample complexity lower bound of $Omega(d2)$ for $epsilon$ an arbitrarily small constant and $gamma
  arXiv  Detail & Related papers  (2020-12-31T18:24:41Z)
• Sparse sketches with small inversion bias [79.77110958547695]
  Inversion bias arises when averaging estimates of quantities that depend on the inverse covariance. We develop a framework for analyzing inversion bias, based on our proposed concept of an $(epsilon,delta)$-unbiased estimator for random matrices. We show that when the sketching matrix $S$ is dense and has i.i.d. sub-gaussian entries, the estimator $(epsilon,delta)$-unbiased for $(Atop A)-1$ with a sketch of size $m=O(d+sqrt d/
  arXiv  Detail & Related papers  (2020-11-21T01:33:15Z)
• The Sparse Hausdorff Moment Problem, with Application to Topic Models [5.151973524974052]
  We give an algorithm for identifying a $k$-mixture using samples of $m=2k$ iid binary random variables. It suffices to know the moments to additive accuracy $w_mincdotzetaO(k)$.
  arXiv  Detail & Related papers  (2020-07-16T04:23:57Z)
• Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model [28.839136703139225]
  We study mean estimation for entangled single-sample Gaussians with a common mean but different unknown variances. We show that the method achieves error $O left(fracsqrtnln nmright)$ with high probability when $m=Omega(sqrtnln n)$. We further prove lower bounds, showing that the error is $Omegaleft(left(fracnm4right)1/6right)$ when $m$ is between $Omega(n
  arXiv  Detail & Related papers  (2020-07-10T18:25:38Z)
• Efficient Statistics for Sparse Graphical Models from Truncated Samples [19.205541380535397]
  We focus on two fundamental and classical problems: (i) inference of sparse Gaussian graphical models and (ii) support recovery of sparse linear models. For sparse linear regression, suppose samples $(bf x,y)$ are generated where $y = bf xtopOmega* + mathcalN(0,1)$ and $(bf x, y)$ is seen only if $y$ belongs to a truncation set $S subseteq mathbbRd$.
  arXiv  Detail & Related papers  (2020-06-17T09:21:00Z)
• Model-Free Reinforcement Learning: from Clipped Pseudo-Regret to Sample Complexity [59.34067736545355]
  Given an MDP with $S$ states, $A$ actions, the discount factor $gamma in (0,1)$, and an approximation threshold $epsilon > 0$, we provide a model-free algorithm to learn an $epsilon$-optimal policy. For small enough $epsilon$, we show an improved algorithm with sample complexity.
  arXiv  Detail & Related papers  (2020-06-06T13:34:41Z)
• Learning Mixtures of Spherical Gaussians via Fourier Analysis [0.5381004207943596]
  We find that a bound on the sample and computational complexity was previously unknown when $omega(1) leq d leq O(log k)$. These authors also show that the sample of complexity of a random mixture of gaussians in a ball of radius $d$ in $d$ dimensions, when $d$ is $Theta(sqrtd)$ in $d$ dimensions, when $d$ is at least $poly(k, frac1delta)$.
  arXiv  Detail & Related papers  (2020-04-13T08:06:29Z)

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.