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Instance-Optimality in I/O-Efficient Sampling and Sequential Estimation

Shyam Narayanan, Václav Rozhoň, Jakub Tětek, Mikkel Thorup

TL;DR

This paper provides non-parametric instance-optimal results for several fundamental problems: mean and quantile estimation, as well as learning mixture distributions with respect to $\ell_{\infty}$ and the so-called Kolmogorov-Smirnov distance.

Abstract

Suppose we have a memory storing $0$s and $1$s and we want to estimate the frequency of $1$s by sampling. We want to do this I/O-efficiently, exploiting that each read gives a block of $B$ bits at unit cost; not just one bit. If the input consists of uniform blocks: either all 1s or all 0s, then sampling a whole block at a time does not reduce the number of samples needed for estimation. On the other hand, if bits are randomly permuted, then getting a block of $B$ bits is as good as getting $B$ independent bit samples. However, we do not want to make any such assumptions on the input. Instead, our goal is to have an algorithm with instance-dependent performance guarantees which stops sampling blocks as soon as we know that we have a probabilistically reliable estimate. We prove our algorithms to be instance-optimal among algorithms oblivious to the order of the blocks, which we argue is the strongest form of instance optimality we can hope for. We also present similar results for I/O-efficiently estimating mean with both additive and multiplicative error, estimating histograms, quantiles, as well as the empirical cumulative distribution function. We obtain our above results on I/O-efficient sampling by reducing to corresponding problems in the so-called sequential estimation. In this setting, one samples from an unknown distribution until one can provide an estimate with some desired error probability. We then provide non-parametric instance-optimal results for several fundamental problems: mean and quantile estimation, as well as learning mixture distributions with respect to $\ell_\infty$ and the so-called Kolmogorov-Smirnov distance.

Instance-Optimality in I/O-Efficient Sampling and Sequential Estimation

TL;DR

This paper provides non-parametric instance-optimal results for several fundamental problems: mean and quantile estimation, as well as learning mixture distributions with respect to and the so-called Kolmogorov-Smirnov distance.

Abstract

Suppose we have a memory storing s and s and we want to estimate the frequency of s by sampling. We want to do this I/O-efficiently, exploiting that each read gives a block of bits at unit cost; not just one bit. If the input consists of uniform blocks: either all 1s or all 0s, then sampling a whole block at a time does not reduce the number of samples needed for estimation. On the other hand, if bits are randomly permuted, then getting a block of bits is as good as getting independent bit samples. However, we do not want to make any such assumptions on the input. Instead, our goal is to have an algorithm with instance-dependent performance guarantees which stops sampling blocks as soon as we know that we have a probabilistically reliable estimate. We prove our algorithms to be instance-optimal among algorithms oblivious to the order of the blocks, which we argue is the strongest form of instance optimality we can hope for. We also present similar results for I/O-efficiently estimating mean with both additive and multiplicative error, estimating histograms, quantiles, as well as the empirical cumulative distribution function. We obtain our above results on I/O-efficient sampling by reducing to corresponding problems in the so-called sequential estimation. In this setting, one samples from an unknown distribution until one can provide an estimate with some desired error probability. We then provide non-parametric instance-optimal results for several fundamental problems: mean and quantile estimation, as well as learning mixture distributions with respect to and the so-called Kolmogorov-Smirnov distance.

Paper Structure

This paper contains 61 sections, 34 theorems, 130 equations, 2 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Additive errors.
  3. Sequential estimation.
  4. I/O-efficient sampling.
  5. Experimental evaluation: our approach is practical.
  6. How to read this paper.
  7. Our results and techniques
  8. Results on sequential estimation
  9. Results on I/O-efficient estimation
  10. Our techniques
  11. Mean estimation upper bound (\ref{['res:mean']}).
  12. Quantile estimation upper bound (\ref{['res:quantile']})
  13. Quantile estimation lower bound (\ref{['res:quantile']})
  14. Equivalence between the I/O and sequential estimation settings.
  15. Defining and motivating order-oblivious instance-optimality
  16. ...and 46 more sections

Key Result

Theorem 1.5

The following holds under mild technical conditions. Whenever we have an instance-optimal algorithm for a problem on mixture distributions in the setting of sequential estimation, we can also get an order-oblivious instance-optimal algorithm for the same problem in the model of I/O efficient algorit

Figures (2)

  • Figure 1: Our results and the implications between them. We use the equivalence between the I/O and sequential estimation settings (see \ref{['thm:io_transfer']}) to get the implications going from the top (in the sequential estimation setting) to the bottom (on I/O-efficient sampling).
  • Figure 2: Estimating the frequency of the letter 'e' in the corpus of English Wikipedia.

Theorems & Definitions (84)

  • Theorem 1.5: Informal version of \ref{['thm:io_transfer']}
  • Definition 1.6: Informal definition of Instance-optimality and Order-oblivious instance optimality
  • Claim 1.7
  • proof : Proof sketch
  • Claim 1.8
  • proof : Proof sketch.
  • Definition 2.1: Mixture distribution
  • Lemma 2.3: Chebyshev's inequality
  • Lemma 2.4: stackexchange_variance
  • Lemma 2.5: Chernoff bound
  • ...and 74 more