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arXiv:2201.07736

The Query Complexity of Certification

Guy Blanc, Koch, Caleb, Jane Lange, Li-Yang Tan

Published Jan 19, 2022• 0 citations
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Abstract

We study the problem of {\sl certification}: given queries to a function $f : \{0,1\}^n \to \{0,1\}$ with certificate complexity $\le k$ and an input $x^\star$, output a size-$k$ certificate for $f$'s value on $x^\star$. This abstractly models a central problem in explainable machine learning, where we think of $f$ as a blackbox model that we seek to explain the predictions of. For monotone functions, a classic local search algorithm of Angluin accomplishes this task with $n$ queries, which we show is optimal for local search algorithms. Our main result is a new algorithm for certifying monotone functions with $O(k^8 \log n)$ queries, which comes close to matching the information-theoretic lower bound of $Ω(k \log n)$. The design and analysis of our algorithm are based on a new connection to threshold phenomena in monotone functions. We further prove exponential-in-$k$ lower bounds when $f$ is non-monotone, and when $f$ is monotone but the algorithm is only given random examples of $f$. These lower bounds show that assumptions on the structure of $f$ and query access to it are both necessary for the polynomial dependence on $k$ that we achieve.

DOI
10.48550/arxiv.2201.07736

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We study the problem of {\sl certification}: given queries to a function f:{0,1}n→{0,1}f : \{0,1\}^n \to \{0,1\}f:{0,1}n→{0,1} with certificate complexity ≤k\le k≤k and an input x⋆x^\starx⋆, output a size-kkk certificate for fff's value on x⋆x^\starx⋆. This abstractly models a central problem in explainable machine learning, where we think of fff as a blackbox model that we seek to explain the predictions of. For monotone functions, a classic local search algorithm of Angluin accomplishes this task with nnn queries, which we show is optimal for local search algorithms. Our main result is a new algorithm for certifying monotone functions with O(k8log⁡n)O(k^8 \log n)O(k8logn) queries, which comes close to matching the information-theoretic lower bound of Ω(klog⁡n)Ω(k \log n)Ω(klogn). The design and analysis of our algorithm are based on a new connection to threshold phenomena in monotone functions. We further prove exponential-in-kkk lower bounds when fff is non-monotone, and when fff is monotone but the algorithm is only given random examples of fff. These lower bounds show that assumptions on the structure of fff and query access to it are both necessary for the polynomial dependence on kkk that we achieve.
We study sublinear and local computation algorithms for decision trees, focusing on testing and reconstruction. Our first result is a tester that runs in poly(log⁡s,1/ε)⋅nlog⁡n\mathrm{poly}(\log s, 1/\varepsilon)\cdot n\log npoly(logs,1/ε)⋅nlogn time, makes poly(log⁡s,1/ε)⋅log⁡n\mathrm{poly}(\log s,1/\varepsilon)\cdot \log npoly(logs,1/ε)⋅logn queries to an unknown function fff, and: ∘\circ∘ Accepts if fff is ε\varepsilonε-close to a size-sss decision tree; ∘\circ∘ Rejects if fff is Ω(ε)\Omega(\varepsilon)Ω(ε)-far from decision trees of size sO~((log⁡s)2/ε2)s^{\tilde{O}((\log s)^2/\varepsilon^2)}sO~((logs)2/ε2). Existing testers distinguish size-sss decision trees from those that are ε\varepsilonε-far from from size-sss decision trees in poly(ss,1/ε)⋅n\mathrm{poly}(s^s,1/\varepsilon)\cdot npoly(ss,1/ε)⋅n time with O~(s/ε)\tilde{O}(s/\varepsilon)O~(s/ε) queries. We therefore solve an incomparable problem, but achieve doubly-exponential-in-sss and exponential-in-sss improvements in time and query complexities respectively. We obtain our tester by designing a reconstruction algorithm for decision trees: given query access to a function fff that is close to a small decision tree, this algorithm provides fast query access to a small decision tree that is close to fff. By known relationships, our results yield reconstruction algorithms for numerous other boolean function properties -- Fourier degree, randomized and quantum query complexities, certificate complexity, sensitivity, etc. -- which in turn yield new testers for these properties. Finally, we give a hardness result for testing whether an unknown function is ε\varepsilonε-close-to or Ω(ε)\Omega(\varepsilon)Ω(ε)-far-from size-sss decision trees. We show that an efficient algorithm for this task would yield an efficient algorithm for properly learning decision trees, a central open problem of learning theory. It has long been known that proper learning algorithms for any class H\mathcal{H}H yield property testers for H\mathcal{H}H; this provides an example of a converse.
Certificate complexity (C(f))(C(f ))(C(f)) is a fundamental measure of complexity of Boolean functions f which counts the number of bits of an input that need to be known in order for the value of the function to be determined. A certificate can be viewed as a partial assignment, or a boolean subcube where the function is constant. Certificate complexity is well understood for deterministic query (or decision tree) complexity (D)(D)(D) and other query models such as bounded-error randomized and quantum complexity (R,Q)(R, Q)(R,Q), but not as well for quantum zero-error (Q0)(Q_0)(Q0​) and exact query complexity (QE)(Q_E)(QE​), where there is no agreed-upon certificate 'object' (even for QQQ). Instead, we study an operational notion of certification and apply it to various query-based models, with a focus on zero-error and exact quantum query complexity, but also on polynomial degree measures. We give new characterizations of C,RCC, RCC,RC (randomized certificate complexity) and QCQCQC (quantum certificate complexity), in terms of various measures such as classical and quantum sabotage complexity, unambiguous certificate complexity, and variants of polynomial degree. Certification complexity also gives rise to new lower bounds on QEQ_EQE​ and Q0Q_0Q0​, the quantum analogues of DDD and R0R_0R0​, complexity measures for which few lower bound techniques are known which are not already lower bounds for two-sided error quantum query complexity. We exhibit a total Boolean function for which our certification complexity measure gives a tight lower bound for Q0Q_0Q0​, but rational degree and QQQ are asymptotically smaller.
Joint computation is the ubiquitous scenario in which a computer is presented with not one, but many computational tasks to perform. A fundamental question arises: when can we cleverly combine computations, to perform them with greater efficiency or reliability than by tackling them separately? This thesis investigates the power and, especially, the limits of efficient joint computation, in several computational models: query algorithms, circuits, and Turing machines. We significantly improve and extend past results on limits to efficient joint computation for multiple independent tasks; identify barriers to progress towards better circuit lower bounds for multiple-output operators; and begin an original line of inquiry into the complexity of joint computation. In more detail, we make contributions in the following areas: Improved product theorems for randomized query complexity: The direct product problem seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following product theorem (DPT) for query complexity: if every T-query algorithm has success probability at most 1 – e in computing the Boolean function f on input distribution μ, then for α ≤ 1, every αeTk-query algorithm has success probability at most (2αe(1 – e))k in computing the k-fold product f ⊗k correctly on k independent inputs from μ. In light of examples due to Shaltiel, this statement gives an essentially optimal tradeoff between the query bound and the error probability. Using this DPT, we show that for an absolute constant α > 0, the worst-case success probability of any αR 2(f)k-query randomized algorithm for f⊗k falls exponentially with k. The best previous statement of this type, due to Klauck, Spalek, and de Wolf, required a query bound of O( bs(f)k). Our proof technique involves defining and analyzing a collection of martingales associated with an algorithm attempting to solve f ⊗k. Our method is quite general and yields a new XOR lemma and threshold DPT for the query model, as well as DPTs for the query complexity of learning tasks, search problems, and tasks involving interaction with dynamic entities. We also give a version of our DPT in which decision tree size is the resource of interest. Joint complexity in the Decision Tree Model: We study the diversity of possible behaviors of the joint computational complexity of a collection f1, …, fk of Boolean functions over a shared input. We focus on the deterministic decision tree model, with depth as the complexity measure; in this model, we prove a result to the effect that the obvious constraints on joint computational complexity are essentially the only ones. The proof uses an intriguing new type of cryptographic data structure called a mystery bin, which we construct using a polynomial separation between deterministic and unambiguous query complexity shown by Savický. We also pose a conjecture in the communication model which, if proved, would extend our result to that model. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.) (Abstract shortened by UMI.)
In the noisy query model, the (binary) return value of every query (possibly repeated) is independently flipped with some fixed probability p∈(0,1/2)p \in (0, 1/2)p∈(0,1/2). In this paper, we obtain tight bounds on the noisy query complexity of several fundamental problems. Our first contribution is to show that any Boolean function with total influence Ω(n)Ω(n)Ω(n) has noisy query complexity Θ(nlog⁡n)Θ(n\log n)Θ(nlogn). Previous works often focus on specific problems, and it is of great interest to have a characterization of noisy query complexity for general functions. Our result is the first noisy query complexity lower bound of this generality, beyond what was known for random Boolean functions [Reischuk and Schmeltz, FOCS 1991]. Our second contribution is to prove that Graph Connectivity has noisy query complexity Θ(n2log⁡n)Θ(n^2 \log n)Θ(n2logn). In this problem, the goal is to determine whether an undirected graph is connected using noisy edge queries. While the upper bound can be achieved by a simple algorithm, no non-trivial lower bounds were known prior to this work. Last but not least, we determine the exact number of noisy queries (up to lower order terms) needed to solve the kkk-Threshold problem and the Counting problem. The kkk-Threshold problem asks to decide whether there are at least kkk ones among nnn bits, given noisy query access to the bits. We prove that (1±o(1))nlog⁡(min⁡{k,n−k+1}/δ)(1−2p)log⁡1−pp(1\pm o(1)) \frac{n\log (\min\{k,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}(1±o(1))(1−2p)logp1−p​nlog(min{k,n−k+1}/δ)​ queries are both sufficient and necessary to achieve error probability δ=o(1)δ= o(1)δ=o(1). Previously, such a result was only known when min⁡{k,n−k+1}=o(n)\min\{k,n-k+1\}=o(n)min{k,n−k+1}=o(n) [Wang, Ghaddar, Zhu and Wang, arXiv 2024]. We also show a similar (1±o(1))nlog⁡(min⁡{k+1,n−k+1}/δ)(1−2p)log⁡1−pp(1\pm o(1)) \frac{n\log (\min\{k+1,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}(1±o(1))(1−2p)logp1−p​nlog(min{k+1,n−k+1}/δ)​ bound for the Counting problem, where one needs to count the number of ones among nnn bits given noisy query access and kkk denotes the answer.
We introduce a notion of the quantum query complexity of a certificate structure. This is a formalisation of a well-known observation that many quantum query algorithms only require the knowledge of the disposition of possible certificates in the input string, not the precise values therein. Next, we derive a dual formulation of the complexity of a non-adaptive learning graph, and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure and such that a learning graph gives an optimal quantum query algorithm for it. For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays, and generalizes the quantum query lower bound for the kkk-sum problem derived recently in arXiv:1206.6528. Finally, we use these results to show that the learning graph for the triangle problem from arXiv:1210.1014 is almost optimal in these settings. This also gives a quantum query lower bound for the triangle-sum problem.