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Lower Bounds for Convexity Testing

Xi Chen, Anindya De, Shivam Nadimpalli, Rocco A. Servedio, Erik Waingarten

Abstract

We consider the problem of testing whether an unknown and arbitrary set $S \subseteq \mathbb{R}^n$ (given as a black-box membership oracle) is convex, versus $\varepsilon$-far from every convex set, under the standard Gaussian distribution. The current state-of-the-art testing algorithms for this problem make $2^{\tilde{O}(\sqrt{n})\cdot \mathrm{poly}(1/\varepsilon)}$ non-adaptive queries, both for the standard testing problem and for tolerant testing. We give the first lower bounds for convexity testing in the black-box query model: - We show that any one-sided tester (which may be adaptive) must use at least $n^{Ω(1)}$ queries in order to test to some constant accuracy $\varepsilon>0$. - We show that any non-adaptive tolerant tester (which may make two-sided errors) must use at least $2^{Ω(n^{1/4})}$ queries to distinguish sets that are $\varepsilon_1$-close to convex versus $\varepsilon_2$-far from convex, for some absolute constants $0<\varepsilon_1<\varepsilon_2$. Finally, we also show that for any constant $c>0$, any non-adaptive tester (which may make two-sided errors) must use at least $n^{1/4 - c}$ queries in order to test to some constant accuracy $\varepsilon>0$.

Lower Bounds for Convexity Testing

Abstract

We consider the problem of testing whether an unknown and arbitrary set (given as a black-box membership oracle) is convex, versus -far from every convex set, under the standard Gaussian distribution. The current state-of-the-art testing algorithms for this problem make non-adaptive queries, both for the standard testing problem and for tolerant testing. We give the first lower bounds for convexity testing in the black-box query model: - We show that any one-sided tester (which may be adaptive) must use at least queries in order to test to some constant accuracy . - We show that any non-adaptive tolerant tester (which may make two-sided errors) must use at least queries to distinguish sets that are -close to convex versus -far from convex, for some absolute constants . Finally, we also show that for any constant , any non-adaptive tester (which may make two-sided errors) must use at least queries in order to test to some constant accuracy .

Paper Structure

This paper contains 49 sections, 39 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our Results and Discussion
  3. Techniques
  4. The Nazarov Body
  5. One-Sided Adaptive Lower Bound
  6. Two-Sided Non-Adaptive Tolerant Lower Bound
  7. Two-Sided Non-Adaptive Bound
  8. Related Work
  9. Preliminaries
  10. Geometry
  11. Gaussian and Chi-Squared Random Variables
  12. Property Testing and Tolerant Property Testing
  13. Nazarov's Body
  14. Useful Estimates
  15. One-Sided Adaptive Lower Bound
  16. ...and 34 more sections

Key Result

Theorem 0

For some absolute constant $\varepsilon>0$, any one-sided (potentially adaptive) $\varepsilon$-tester for convexity over $N(0,I_n)$ must use $n^{\Omega(1)}$ queries.

Figures (3)

  • Figure 1: A depiction of $\boldsymbol{B}$ (in green) sampled from ${\mathrm{Naz}}(r,N)$.
  • Figure 2: A depiction of ${\cal D}_{\mathrm{yes}}$. We identify the control subspace $\boldsymbol{C} \cong \mathbb{R}^n$. The annulus defined by the boundary of $\mathrm{Ball}(\sqrt{n})$ and the dotted circle corresponds to points $x$ which satisfy $x \in {\mathrm{Shell}}_{n+1}$ and $\|x_{\boldsymbol{C}}\| \leq \sqrt{n}$. Finally, the red region in the action subspace $\boldsymbol{A} \cong \mathbb{R}$ corresponds to $\mathrm{Curb}$.
  • Figure 3: A depiction of ${\cal D}_{\mathrm{no}}$. Our conventions are as in \ref{['fig:tolerant-yes']}.

Theorems & Definitions (67)

  • Theorem 0: One-sided adaptive lower bound
  • Theorem 0: Two-sided non-adaptive tolerant testing lower bound
  • Theorem 0: Two-sided non-adaptive lower bound
  • Lemma 1
  • Proposition 2: Theorem 1.2.6 of durrett_2019 or Equation 2.58 of TAILBOUND
  • Proposition 3: Section 4.1 of laurent2000
  • Definition 4: Property testers and tolerant property testers
  • Theorem 5: Yao's principle
  • Definition 6: Nazarov's body
  • Lemma 7
  • ...and 57 more