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The complexity of joint computation

Scott Aaronson, Andrew Drucker

Published Jan 1, 2012• 3 citations
OpenAlex

Abstract

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.)

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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.)
Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting Q(f∘g)=Θ(Q(f)Q(g))Q(f\circ g)=Θ(Q(f)Q(g))Q(f∘g)=Θ(Q(f)Q(g)) for all Boolean functions fff and ggg; a direct sum theorem asserting that computing kkk copies of a function (or search problem) costs Ω(k)Ω(k)Ω(k) times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires Ω(k)Ω(k)Ω(k) times the cost of computing one copy to bounded error. However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution μμμ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing kkk copies of fff when the input comes from the product distribution μkμ^kμk instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions. In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the γ2γ_2γ2​ norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.
The "direct product problem" is a fundamental question in complexity theory which seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every T-query algorithm has success probability at most 1 - ε in computing the Boolean function f on input distribution μ, then for α ≤ 1, every αεTk-query algorithm has success probability at most (2αε(1 - ε))kin computing the fc-fold direct product f⊗kcorrectly 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 αR2(f)k-query randomized algorithm for f⊗kfalls 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 collec tion 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.
In Direct Sum problems |8|, one tries to show that for a given computational model, the complexity of computing a collection F = {f1(x1),hellipf1(x1)} of finite functions on independent inputs is approximately the sum of their individual complexities. In this paper, by contrast, we study the diversity of ways in which the joint computational complexity can behave when all the fiare evaluated on a common 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 small polynomial separation between deterministic and unambiguous query complexity shown by Savicky. We also pose a variant of the Direct Sum Conjecture of |8| which, if proved for a single family of functions, could yield an analogous result for models such as the communication model.
Query complexity studies the amount of information about an input that is needed to solve a computational problem. This thesis investigates query complexity in several settings, with applications to cryptography and property testing, as well as connections to communication complexity and other computational models. In property testing, we study tolerant junta testing and establish the first superpolynomial quantum advantage for this problem: within a certain parameter regime, a quantum algorithm uses polynomially many queries, whereas every classical randomised algorithm requires quasipolynomially many queries. In cryptography, we study the average-case complexity of search problems. We show that verifiable delay functions do not exist in the random oracle model, and strengthen known barriers to constructing average-case hard TFNP problems from one-way functions. Finally, we study lower bounds and their behaviour across computational models. We prove new direct sum results for randomised parity decision trees. We also establish a new separation between query complexity measures of total Boolean functions and, through query-to-communication lifting, derive new communication lower bounds and lower bounds for unambiguous finite automata.
We establish two new direct product theorems for the randomized query complexity of Boolean functions. The first shows that computing n copies of a function f, even with a small success probability of γn\gamma^{n}γn, requires Θ(n)\Theta(n)Θ(n) times the maximum distributional query complexity of f with success parameter γ\gammaγ. This result holds for all success parameters γ\gammaγ, even when γ\gammaγ is very close to 1/2 or to 1. As a result, it unifies and generalizes Drucker’s direct product theorem (2012) for γ\gammaγ bounded away from 12\frac{1}{2}21​ and 1 as well as the strong direct sum theorem of Blais and Brody (2019) for γ≈1−1/n\gamma \approx 1-1 / nγ≈1−1/n. The second establishes a general list decoding direct product theorem that captures many different variants of “partial computation” tasks related to the function fnf^{n}fn consisting of n copies of f. Notably, our list decoding direct product theorem yields a new threshold direct product theorem and other new variants such as the labelled-threshold direct product theorem. Both of these direct product theorems are obtained by taking a new approach. Instead of directly analyzing the query complexity of algorithms, we introduce a new measure of complexity of functions that we call discounted score. We show that this measure satisfies a number of useful structural properties, including tensorization, that make it particularly suitable for the study of direct product questions.
We establish two new direct product theorems for the randomized query complexity of Boolean functions. The first shows that computing nnn copies of a function fff, even with a small success probability of γnγ^nγn, requires Θ(n)Θ(n)Θ(n) times the "maximum distributional" query complexity of fff with success parameter γγγ. This result holds for all success parameters γγγ, even when γγγ is very close to 1/21/21/2 or to 111. As a result, it unifies and generalizes Drucker's direct product theorem (2012) for γγγ bounded away from 12\frac1221​ and 111 as well as the strong direct sum theorem of Blais and Brody (2019) for γ≈1−1/nγ\approx 1-1/nγ≈1−1/n. The second establishes a general "list decoding" direct product theorem that captures many different variants of partial computation tasks related to the function fnf^nfn consisting of nnn copies of fff. Notably, our list decoding direct product theorem yields a new threshold direct product theorem and other new variants such as the labelled-threshold direct product theorem. Both of these direct product theorems are obtained by taking a new approach. Instead of directly analyzing the query complexity of algorithms, we introduce a new measure of complexity of functions that we call "discounted score". We show that this measure satisfies a number of useful structural properties, including tensorization, that make it particularly suitable for the study of direct product questions.