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arxiv: 2604.05615 · v1 · submitted 2026-04-07 · 💻 cs.DS

Recognition: no theorem link

Classes Testable with O(1/ε) Queries for Small ε Independent of the Number of Variables

George Haddad, Nader H. Bshouty

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:10 UTC · model grok-4.3

classification 💻 cs.DS
keywords property testingBoolean functionsjunta testingFourier degreesparse polynomialsquery complexityexact learning
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The pith

k-juntas, low-degree functions, and sparse polynomials are testable with O(ψ + 1/ε) queries where ψ depends only on class parameters and is independent of n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that several natural classes of Boolean functions over n variables can be tested for membership using a query complexity that depends on the distance parameter ε and on a class-specific constant ψ, but not on n itself. For sufficiently small ε (at most 1/ψ), this simplifies to the optimal O(1/ε) bound. The result covers k-juntas, functions of Fourier degree at most d, s-sparse degree-d polynomials, and general s-sparse polynomials, and it also gives a general theorem that any class of functions on at most k variables that admits proper exact learning is testable in O(1/ε) queries for small ε. This extends earlier results known only for k-terms and k-parities.

Core claim

The classes k-junta, Fourier degree at most d, s-sparse polynomials of degree at most d, and s-sparse polynomials are all testable with O(ψ + 1/ε) queries, where ψ depends on the parameters of the class but is independent of the total number of variables n. When ε ≤ 1/ψ the complexity reduces to O(1/ε). Any class C of Boolean functions that depends on at most k variables and is properly exactly learnable is testable with O(1/ε) queries for ε < 1/ψ where ψ depends only on k.

What carries the argument

The exact learnability (or explicit structural properties such as sparsity and low degree) of k-variable subclasses, which is used to build testers whose query cost is bounded by a function of the class parameters plus 1/ε.

If this is right

  • Testing k-juntas requires only O(k + 1/ε) queries once ε is smaller than 1/k.
  • Testing any fixed-degree or fixed-sparsity polynomial class has query cost independent of the ambient dimension n.
  • Any properly exactly learnable k-variable class inherits the O(1/ε) testing bound for small ε.
  • The same testers work even when the input distribution is arbitrary, without extra assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that high-dimensional property testing can often be reduced to testing on the relevant variables once a constant-factor overhead is paid.
  • It may be possible to obtain similar n-independent testers for other classes that admit efficient exact learning algorithms, such as decision trees of bounded size.
  • The separation between ψ-dependent and 1/ε regimes highlights a regime where testing becomes as cheap as the information-theoretic lower bound.

Load-bearing premise

The structural properties of these classes, including exact learnability when restricted to k variables, suffice to build testers whose cost stays independent of n.

What would settle it

An explicit construction, for some fixed k and small ε, of a k-junta that requires Ω(n) queries to test would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.05615 by George Haddad, Nader H. Bshouty.

Figure 1
Figure 1. Figure 1: summarizes these results alongside previously known results. Class of Functions Query Complexity = O(1/ϵ) for Reference k-Junta O [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

In this paper, we study classes of Boolean functions that are testable with $O(\psi+1/\epsilon)$ queries, where $\psi$ depends on the parameters of the class (e.g., the number of terms, the number of relevant variables, etc.) but not on the total number of variables $n$. In particular, when $\epsilon\le 1/\psi$, the query complexity is $O(1/\epsilon)$, matching the known tight bound $\Omega(1/\epsilon)$. This result was previously known for classes of terms of size at most $k$ and exclusive OR functions of at most $k$ variables. In this paper, we extend this list to include the classes: $k$-junta, functions with Fourier degree at most $d$, $s$-sparse polynomials of degree at most $d$, and $s$-sparse polynomials. Additionally, we show that for any class $C$ of Boolean functions that depend on at most $k$ variables, if $C$ is properly exactly learnable, then it is testable with $O(1/\epsilon)$ queries for $\epsilon<1/\psi$, where $\psi$ depends on $k$ and independent of the total number of variables $n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that the classes of k-juntas, Fourier degree at most d functions, s-sparse degree-at-most-d polynomials, and s-sparse polynomials are testable with O(ψ + 1/ε) queries where ψ depends only on the class parameters (k, d, s) and is independent of n; for ε ≤ 1/ψ the bound simplifies to O(1/ε). It further claims a general result: any class C of Boolean functions depending on at most k variables that is properly exactly learnable is testable with O(1/ε) queries for ε < 1/ψ (ψ depending only on k). This extends prior results known for k-terms and k-XORs.

Significance. If the claims hold, the work is significant for enlarging the list of classes admitting n-independent testers that achieve the optimal O(1/ε) bound for sufficiently small ε. The general reduction from proper exact learnability to testability could facilitate identifying further such classes. Credit is given for treating multiple natural classes (k-juntas, low-degree, sparse polynomials) with separate arguments rather than relying solely on the general statement.

major comments (2)
  1. [Abstract and general theorem] Abstract and the general theorem (likely §3 or §4): The claim that proper exact learnability of any k-variable class C implies an n-independent tester with O(1/ε) queries for small ε is load-bearing for the general result. Proper exact learning requires identifying an unknown support of size ≤k; information-theoretically this needs Ω(k log n) queries in the worst case to distinguish among binom(n,k) possibilities, as each oracle query returns one bit. The manuscript does not exhibit an n-independent exact learner or show how the reduction removes the log n factor. This must be clarified or the general theorem restricted to classes with known support.
  2. [k-junta section] Section on k-juntas (likely §2): The tester for k-juntas is asserted to achieve the stated bound, but the reduction from the exact learner (or direct construction) must be shown to avoid any hidden dependence on n when the relevant variables are unknown. Without the explicit query-complexity analysis or error bounds in the provided text, it is unclear whether the O(ψ + 1/ε) claim holds for the standard definition of juntas over n variables.
minor comments (2)
  1. [Notation and definitions] The parameter ψ is referenced throughout but its exact functional dependence on k, d, s is not collected in one place; a single displayed equation or table summarizing ψ for each class would improve clarity.
  2. [Introduction] The abstract states the results but supplies no proof sketches or references to the relevant theorems; adding one-sentence pointers to the main theorems in the introduction would help readers locate the arguments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review, and for acknowledging the significance of extending the list of classes testable with n-independent O(1/ε) query complexity for small ε. We address the two major comments point by point below. Where the presentation or analysis requires expansion, we will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and general theorem] Abstract and the general theorem (likely §3 or §4): The claim that proper exact learnability of any k-variable class C implies an n-independent tester with O(1/ε) queries for small ε is load-bearing for the general result. Proper exact learning requires identifying an unknown support of size ≤k; information-theoretically this needs Ω(k log n) queries in the worst case to distinguish among binom(n,k) possibilities, as each oracle query returns one bit. The manuscript does not exhibit an n-independent exact learner or show how the reduction removes the log n factor. This must be clarified or the general theorem restricted to classes with known support.

    Authors: We thank the referee for raising this important point regarding the general theorem. The result is stated in Section 3 as a black-box reduction from proper exact learnability of any class C depending on at most k variables to an n-independent tester. While the high-level reduction is given, we agree that the manuscript does not provide a fully expanded query-complexity analysis that explicitly addresses the information-theoretic concern and demonstrates removal of any log n dependence. In the revised version we will supply a complete proof of the reduction. The argument proceeds by using the exact learner only to produce a candidate hypothesis on a randomly sampled set of O(k) variables (whose size is independent of n) and then verifying consistency with the tester; for ε ≪ 1/ψ the failure probability can be bounded without enumerating all possible supports. Should the expanded proof reveal that the log n factor cannot be avoided in full generality, we will restrict the theorem to classes whose support is known in advance, as the referee suggests. revision: yes

  2. Referee: [k-junta section] Section on k-juntas (likely §2): The tester for k-juntas is asserted to achieve the stated bound, but the reduction from the exact learner (or direct construction) must be shown to avoid any hidden dependence on n when the relevant variables are unknown. Without the explicit query-complexity analysis or error bounds in the provided text, it is unclear whether the O(ψ + 1/ε) claim holds for the standard definition of juntas over n variables.

    Authors: We appreciate the referee’s request for explicit analysis in the k-juntas section. The tester is obtained by reducing to an exact learner for k-juntas and then using the learned hypothesis to test closeness. The current draft sketches the reduction but omits the full error-probability and query-count calculation. In the revision we will insert a detailed lemma that bounds the total number of queries by O(ψ(k) + 1/ε), with all constants independent of n; the argument relies on a preliminary sampling step that identifies a superset of the relevant variables of size O(k) with probability 1−δ using O(k log(1/δ)) queries (still independent of n once δ is fixed) followed by the exact learner on that superset. This will make the claimed bound fully rigorous for the standard definition of k-juntas over n variables. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on standard reductions and class-specific structure without self-referential reduction

full rationale

The paper's central claims extend prior results on specific classes (k-juntas, low-degree functions, sparse polynomials) via learnability-to-testability reductions and structural properties that bound query complexity by a function of class parameters (k, d, s) and ε alone. The general theorem assumes proper exact learnability of k-variable classes as an external premise and derives the O(1/ε) tester bound for small ε from that assumption plus standard testing definitions; no equation or step equates the claimed bound to a fitted parameter, renames a known result, or reduces via a self-citation chain that itself depends on the target result. Prior mentions of results for terms and XOR functions are non-load-bearing extensions rather than foundational justifications. The derivation remains self-contained against external benchmarks of property testing.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates entirely within the standard model of Boolean function property testing and exact learning; no new free parameters, invented entities, or non-standard axioms are introduced in the abstract.

axioms (2)
  • domain assumption Standard definition of ε-testing with query access to a Boolean function on n variables
    The entire query-complexity analysis presupposes the usual property-testing model.
  • domain assumption Proper exact learnability of a k-variable class implies the existence of an O(1/ε) tester
    This is the load-bearing general theorem stated in the abstract.

pith-pipeline@v0.9.0 · 5535 in / 1462 out tokens · 70645 ms · 2026-05-10T19:10:32.111895+00:00 · methodology

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