arXiv subjects

Justin Thaler

Publications and source records attributed to Justin Thaler.

At least 19 recordsLinked to original sources

arXiv2022

Order-Invariant Cardinality Estimators Are Differentially Private

We consider privacy in the context of streaming algorithms for cardinality estimation. We show that a large class of algorithms all satisfy $\epsilon$-differential privacy, so long as (a) the algorithm is combined with a simple down-sampling procedure, and (b) the cardinality of the input stream is $\Omega(k/\epsilon)$. Here, $k$ is a certain parameter of the sketch that is always at most the sketch size in bits, but is typically much smaller. We also show that, even with no modification, algorithms in our class satisfy $(\epsilon, \delta)$-differential privacy, where $\delta$ falls exponentially with the stream cardinality. Our analysis applies to essentially all popular cardinality estimation algorithms, and substantially generalizes and tightens privacy bounds from earlier works.

cs.DS↗

arXiv2021

Quantum Proofs of Proximity

We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property $\Pi$ and reject inputs that are $\varepsilon$-far from $\Pi$, while only probing a minuscule portion of their input. We investigate the complexity landscape of this model, showing that QMAPs can be exponentially stronger than both classical proofs of proximity and quantum testers. To this end, we extend the methodology of Blais, Brody, and Matulef (Computational Complexity, 2012) to prove quantum property testing lower bounds via reductions from communication complexity. This also resolves a question raised in 2013 by Montanaro and de Wolf (cf. Theory of Computing, 2016). Our algorithmic results include a purpose an algorithmic framework that enables quantum speedups for testing an expressive class of properties, namely, those that are succinctly decomposable. A consequence of this framework is a QMA algorithm to verify the Parity of an $n$-bit string with $O(n^{2/3})$ queries and proof length. We also propose a QMA algorithm for testing graph bipartitneness, a property that lies outside of this family, for which there is a quantum speedup.

quant-ph↗

arXiv2020

Streaming Verification for Graph Problems: Optimal Tradeoffs and Nonlinear Sketches

We study graph computations in an enhanced data streaming setting, where a space-bounded client reading the edge stream of a massive graph may delegate some of its work to a cloud service. We seek algorithms that allow the client to verify a purported proof sent by the cloud service that the work done in the cloud is correct. A line of work starting with Chakrabarti et al. (ICALP 2009) has provided such algorithms, which we call schemes, for several statistical and graph-theoretic problems, many of which exhibit a tradeoff between the length of the proof and the space used by the streaming verifier. This work designs new schemes for a number of basic graph problems---including triangle counting, maximum matching, topological sorting, and single-source shortest paths---where past work had either failed to obtain smooth tradeoffs between these two key complexity measures or only obtained suboptimal tradeoffs. Our key innovation is having the verifier compute certain nonlinear sketches of the input stream, leading to either new or improved tradeoffs. In many cases, our schemes in fact provide optimal tradeoffs up to logarithmic factors. Specifically, for most graph problems that we study, it is known that the product of the verifier's space cost $v$ and the proof length $h$ must be at least $\Omega(n^2)$ for $n$-vertex graphs. However, matching upper bounds are only known for a handful of settings of $h$ and $v$ on the curve $h \cdot v=\tilde{\Theta}(n^2)$. For example, for counting triangles and maximum matching, schemes with costs lying on this curve are only known for $(h=\tilde{O}(n^2), v=\tilde{O}(1))$, $(h=\tilde{O}(n), v=\tilde{O}(n))$, and the trivial $(h=\tilde{O}(1), v=\tilde{O}(n^2))$. A major message of this work is that by exploiting nonlinear sketches, a significant ``portion'' of costs on the tradeoff curve $h \cdot v = n^2$ can be achieved.

cs.DS↗

arXiv2020

Relative Error Streaming Quantiles

Estimating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of $n$ items from a data universe equipped with a total order, the task is to compute a sketch (data structure) of size polylogarithmic in $n$. Given the sketch and a query item $y$, one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than or equal to $y$. Most works to date focused on additive $\varepsilon n$ error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This paper investigates multiplicative $(1\pm\varepsilon)$-error approximations to the rank. Practical motivation for multiplicative error stems from demands to understand the tails of distributions, and hence for sketches to be more accurate near extreme values. The most space-efficient algorithms due to prior work store either $O(\log(\varepsilon^2 n)/\varepsilon^2)$ or $O(\log^3(\varepsilon n)/\varepsilon)$ universe items. We present a randomized sketch storing $O(\log^{1.5}(\varepsilon n)/\varepsilon)$ items that can $(1\pm\varepsilon)$-approximate the rank of each universe item with high constant probability; this space bound is within an $O(\sqrt{\log(\varepsilon n)})$ factor of optimal. Our algorithm does not require prior knowledge of the stream length and is fully mergeable, rendering it suitable for parallel and distributed computing environments.

cs.DS↗

arXiv2020

Improved Approximate Degree Bounds For k-distinctness

An open problem that is widely regarded as one of the most important in quantum query complexity is to resolve the quantum query complexity of the k-distinctness function on inputs of size N. While the case of k=2 (also called Element Distinctness) is well-understood, there is a polynomial gap between the known upper and lower bounds for all constants k>2. Specifically, the best known upper bound is O(N^{(3/4)-1/(2^{k+2}-4)}) (Belovs, FOCS 2012), while the best known lower bound for k >= 2 is Omega(N^{2/3} + N^{(3/4)-1/(2k)}) (Aaronson and Shi, J.~ACM 2004; Bun, Kothari, and Thaler, STOC 2018). For any constant k >= 4, we improve the lower bound to Omega(N^{(3/4)-1/(4k)}). This yields, for example, the first proof that 4-distinctness is strictly harder than Element Distinctness. Our lower bound applies more generally to approximate degree. As a secondary result, we give a simple construction of an approximating polynomial of degree O(N^{3/4}) that applies whenever k <= polylog(N).

quant-ph↗

arXiv2019

Vanishing-Error Approximate Degree and QMA Complexity

The $ε$-approximate degree of a function $f\colon X \to \{0, 1\}$ is the least degree of a multivariate real polynomial $p$ such that $|p(x)-f(x)| \leq ε$ for all $x \in X$. We determine the $ε$-approximate degree of the element distinctness function, the surjectivity function, and the permutation testing problem, showing they are $Θ(n^{2/3} \log^{1/3}(1/ε))$, $\tildeΘ(n^{3/4} \log^{1/4}(1/ε))$, and $Θ(n^{1/3} \log^{2/3}(1/ε))$, respectively. Previously, these bounds were known only for constant $ε.$ We also derive a connection between vanishing-error approximate degree and quantum Merlin--Arthur (QMA) query complexity. We use this connection to show that the QMA complexity of permutation testing is $Ω(n^{1/4})$. This improves on the previous best lower bound of $Ω(n^{1/6})$ due to Aaronson (Quantum Information & Computation, 2012), and comes somewhat close to matching a known upper bound of $O(n^{1/3})$.

cs.CC↗

arXiv2019

The Polynomial Method Strikes Back: Tight Quantum Query Bounds via Dual Polynomials

The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. Approximate degree is known to be a lower bound on quantum query complexity. We resolve or nearly resolve the approximate degree and quantum query complexities of the following basic functions: $\bullet$ $k$-distinctness: For any constant $k$, the approximate degree and quantum query complexity of $k$-distinctness is $Ω(n^{3/4-1/(2k)})$. This is nearly tight for large $k$ (Belovs, FOCS 2012). $\bullet$ Image size testing: The approximate degree and quantum query complexity of testing the size of the image of a function $[n] \to [n]$ is $\tildeΩ(n^{1/2})$. This proves a conjecture of Ambainis et al. (SODA 2016), and it implies the following lower bounds: $-$ $k$-junta testing: A tight $\tildeΩ(k^{1/2})$ lower bound, answering the main open question of Ambainis et al. (SODA 2016). $-$ Statistical Distance from Uniform: A tight $\tildeΩ(n^{1/2})$ lower bound, answering the main question left open by Bravyi et al. (STACS 2010 and IEEE Trans. Inf. Theory 2011). $-$ Shannon entropy: A tight $\tildeΩ(n^{1/2})$ lower bound, answering a question of Li and Wu (2017). $\bullet$ Surjectivity: The approximate degree of the Surjectivity function is $\tildeΩ(n^{3/4})$. The best prior lower bound was $Ω(n^{2/3})$. Our result matches an upper bound of $\tilde{O}(n^{3/4})$ due to Sherstov, which we reprove using different techniques. The quantum query complexity of this function is known to be $Θ(n)$ (Beame and Machmouchi, QIC 2012 and Sherstov, FOCS 2015). Our upper bound for Surjectivity introduces new techniques for approximating Boolean functions by low-degree polynomials. Our lower bounds are proved by significantly refining techniques recently introduced by Bun and Thaler (FOCS 2017).

quant-ph↗

arXiv2019

Approximate degree, secret sharing, and concentration phenomena

The $ε$-approximate degree $deg_ε(f)$ of a Boolean function $f$ is the least degree of a real-valued polynomial that approximates $f$ pointwise to error $ε$. The approximate degree of $f$ is at least $k$ iff there exists a pair of probability distributions, also known as a dual polynomial, that are perfectly $k$-wise indistinguishable, but are distinguishable by $f$ with advantage $1 - ε$. Our contributions are: We give a simple new construction of a dual polynomial for the AND function, certifying that $deg_ε(f) \geq Ω(\sqrt{n \log 1/ε})$. This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the $1/3$-approximate degree of any read-once DNF is $Ω(\sqrt{n})$. We show that any pair of symmetric distributions on $n$-bit strings that are perfectly $k$-wise indistinguishable are also statistically $K$-wise indistinguishable with error at most $K^{3/2} \cdot \exp(-Ω(k^2/K))$ for all $k < K < n/64$. This implies that any symmetric function $f$ is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-$K$ coalitions with statistical error $K^{3/2} \exp(-Ω(deg_{1/3}(f)^2/K))$ for all values of $K$ up to $n/64$ simultaneously. Previous secret sharing schemes required that $K$ be determined in advance, and only worked for $f=$ AND. Our analyses draw new connections between approximate degree and concentration phenomena. As a corollary, we show that for any $d < n/64$, any degree $d$ polynomial approximating a symmetric function $f$ to error $1/3$ must have $\ell_1$-norm at least $K^{-3/2} \exp({Ω(deg_{1/3}(f)^2/d)})$, which we also show to be tight for any $d > deg_{1/3}(f)$. These upper and lower bounds were also previously only known in the case $f=$ AND.

cs.CC↗

arXiv2019

Quantum Lower Bounds for Approximate Counting via Laurent Polynomials

We study quantum algorithms that are given access to trusted and untrusted quantum witnesses. We establish strong limitations of such algorithms, via new techniques based on Laurent polynomials (i.e., polynomials with positive and negative integer exponents). Specifically, we resolve the complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set $S \subseteq [N]$, in two natural generalizations of quantum query complexity. Our first result holds in the standard Quantum Merlin--Arthur ($\mathsf{QMA}$) setting, in which a quantum algorithm receives an untrusted quantum witness. We show that, if the algorithm makes $T$ quantum queries to $S$, and also receives an (untrusted) $m$-qubit quantum witness, then either $m = \Omega(|S|)$ or $T=\Omega \bigl(\sqrt{N/\left| S\right| } \bigr)$. This is optimal, matching the straightforward protocols where the witness is either empty, or specifies all the elements of $S$. As a corollary, this resolves the open problem of giving an oracle separation between $\mathsf{SBP}$, the complexity class that captures approximate counting, and $\mathsf{QMA}$. In our second result, we ask what if, in addition to a membership oracle for $S$, a quantum algorithm is also given "QSamples" -- i.e., copies of the state $\left| S\right\rangle = \frac{1}{\sqrt{\left| S\right| }} \sum_{i\in S}|i\rangle$ -- or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either $\Theta \bigl(\sqrt{N/\left| S\right| }\bigr)$ queries or else $\Theta \bigl(\min \bigl\{\left| S\right| ^{1/3}, \sqrt{N/\left| S\right| }\bigr\}\bigr)$ QSamples or accesses to the unitary. Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways.

quant-ph↗

arXiv2019

Sign-Rank Can Increase Under Intersection

The communication class $\mathbf{UPP}^{\text{cc}}$ is a communication analog of the Turing Machine complexity class $\mathbf{PP}$. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds. For a communication problem $f$, let $f \wedge f$ denote the function that evaluates $f$ on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem $f$ with $\mathbf{UPP}(f)= O(\log n)$, and $\mathbf{UPP}(f \wedge f) = Θ(\log^2 n)$. This is the first result showing that $\mathbf{UPP}$ communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that $\mathbf{UPP}^{\text{cc}}$, the class of problems with polylogarithmic-cost $\mathbf{UPP}$ communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on $n$ bits has dimension complexity $n^{Ω(\log n)}$. This matches an upper bound of (Klivans, O'Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time.

cs.CC↗

arXiv2018

Quantum algorithms and approximating polynomials for composed functions with shared inputs

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let $f$ be an $m$-bit Boolean function and consider an $n$-bit function $F$ obtained by applying $f$ to conjunctions of possibly overlapping subsets of $n$ variables. If $f$ has quantum query complexity $Q(f)$, we give an algorithm for evaluating $F$ using $\tilde{O}(\sqrt{Q(f) \cdot n})$ quantum queries. This improves on the bound of $O(Q(f) \cdot \sqrt{n})$ that follows by treating each conjunction independently, and our bound is tight for worst-case choices of $f$. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of $f$. By recursively applying our composition theorems, we obtain a nearly optimal $\tilde{O}(n^{1-2^{-d}})$ upper bound on the quantum query complexity and approximate degree of linear-size depth-$d$ AC$^0$ circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC$^0$ circuits. As an additional consequence, we show that AC$^0 \circ \oplus$ circuits of depth $d+1$ require size $\tilde{\Omega}(n^{1/(1- 2^{-d})}) \geq \omega(n^{1+ 2^{-d}} )$ to compute the Inner Product function even on average. The previous best size lower bound was $\Omega(n^{1+4^{-(d+1)}})$ and only held in the worst case (Cheraghchi et al., JCSS 2018).

quant-ph↗

arXiv2017

A High-Performance Algorithm for Identifying Frequent Items in Data Streams

Estimating frequencies of items over data streams is a common building block in streaming data measurement and analysis. Misra and Gries introduced their seminal algorithm for the problem in 1982, and the problem has since been revisited many times due its practicality and applicability. We describe a highly optimized version of Misra and Gries' algorithm that is suitable for deployment in industrial settings. Our code is made public via an open source library called DataSketches that is already used by several companies and production systems. Our algorithm improves on two theoretical and practical aspects of prior work. First, it handles weighted updates in amortized constant time, a common requirement in practice. Second, it uses a simple and fast method for merging summaries that asymptotically improves on prior work even for unweighted streams. We describe experiments confirming that our algorithms are more efficient than prior proposals.

cs.DS↗

arXiv2017

On the Power of Statistical Zero Knowledge

We examine the power of statistical zero knowledge proofs (captured by the complexity class SZK) and their variants. First, we give the strongest known relativized evidence that SZK contains hard problems, by exhibiting an oracle relative to which SZK (indeed, even NISZK) is not contained in the class UPP, containing those problems solvable by randomized algorithms with unbounded error. This answers an open question of Watrous from 2002 [Aar]. Second, we "lift" this oracle separation to the setting of communication complexity, thereby answering a question of Göös et al. (ICALP 2016). Third, we give relativized evidence that perfect zero knowledge proofs (captured by the class PZK) are weaker than general zero knowledge proofs. Specifically, we exhibit oracles relative to which SZK is not contained in PZK, NISZK is not contained in NIPZK, and PZK is not equal to coPZK. The first of these results answers a question raised in 1991 by Aiello and Håstad (Information and Computation), and the second answers a question of Lovett and Zhang (2016). We also describe additional applications of these results outside of structural complexity. The technical core of our results is a stronger hardness amplification theorem for approximate degree, which roughly says that composing the gapped-majority function with any function of high approximate degree yields a function with high threshold degree.

cs.CC↗

arXiv2017

A Nearly Optimal Lower Bound on the Approximate Degree of AC$^0$

The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function $f$ with approximate degree $d$ into a function $F$ on $O(n \cdot \operatorname{polylog}(n))$ variables with approximate degree at least $D = Ω(n^{1/3} \cdot d^{2/3})$. In particular, if $d= n^{1-Ω(1)}$, then $D$ is polynomially larger than $d$. Moreover, if $f$ is computed by a polynomial-size Boolean circuit of constant depth, then so is $F$. By recursively applying our transformation, for any constant $δ> 0$ we exhibit an AC$^0$ function of approximate degree $Ω(n^{1-δ})$. This improves over the best previous lower bound of $Ω(n^{2/3})$ due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of $n$ that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant $δ> 0$, an $Ω(n^{1-δ})$ lower bound on the quantum communication complexity of a function in AC$^0$. * A Boolean function $f$ with approximate degree at least $C(f)^{2-o(1)}$, where $C(f)$ is the certificate complexity of $f$. This separation is optimal up to the $o(1)$ term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC$^0$.

cs.CC↗

arXiv2017

Time-Optimal Interactive Proofs for Circuit Evaluation

Recently, researchers have been working toward the development of practical general-purpose protocols for verifiable computation. These protocols enable a computationally weak verifier to offload computations to a powerful but untrusted prover, while providing the verifier with a guarantee that the prover performed the computations correctly. Despite substantial progress, existing implementations are not yet practical. The main bottleneck is typically the extra effort required by the prover to return an answer with a guarantee of correctness, compared to returning an answer with no guarantee. We describe a refinement of a powerful interactive proof protocol originally due to Goldwasser, Kalai, and Rothblum. Cormode, Mitzenmacher, and Thaler show how to implement the prover in this protocol in time O(S log S), where S is the size of an arithmetic circuit computing the function of interest. Our refinements apply to circuits whose wiring pattern is sufficiently "regular"; for these circuits, we bring the runtime of the prover down to O(S). That is, our prover can evaluate the circuit with a guarantee of correctness, with only a constant-factor blowup in work compared to evaluating the circuit with no guarantee. We argue that our refinements capture a large class of circuits, and prove some theorems formalizing this. Experimentally, our refinements yield a 200x speedup for the prover over the implementation of Cormode et al., and our prover is less than 10x slower than a C++ program that simply evaluates the circuit. Along the way, we describe a special-purpose protocol for matrix multiplication that is of interest in its own right. Our final contribution is a protocol targeted at general data parallel computation. Compared to prior work, this protocol can more efficiently verify complicated computations as long as that computation is applied independently to many pieces of data.

cs.CR↗

arXiv2016

Reliably Learning the ReLU in Polynomial Time

We give the first dimension-efficient algorithms for learning Rectified Linear Units (ReLUs), which are functions of the form $\mathbf{x} \mapsto \max(0, \mathbf{w} \cdot \mathbf{x})$ with $\mathbf{w} \in \mathbb{S}^{n-1}$. Our algorithm works in the challenging Reliable Agnostic learning model of Kalai, Kanade, and Mansour (2009) where the learner is given access to a distribution $\cal{D}$ on labeled examples but the labeling may be arbitrary. We construct a hypothesis that simultaneously minimizes the false-positive rate and the loss on inputs given positive labels by $\cal{D}$, for any convex, bounded, and Lipschitz loss function. The algorithm runs in polynomial-time (in $n$) with respect to any distribution on $\mathbb{S}^{n-1}$ (the unit sphere in $n$ dimensions) and for any error parameter $ε= Ω(1/\log n)$ (this yields a PTAS for a question raised by F. Bach on the complexity of maximizing ReLUs). These results are in contrast to known efficient algorithms for reliably learning linear threshold functions, where $ε$ must be $Ω(1)$ and strong assumptions are required on the marginal distribution. We can compose our results to obtain the first set of efficient algorithms for learning constant-depth networks of ReLUs. Our techniques combine kernel methods and polynomial approximations with a "dual-loss" approach to convex programming. As a byproduct we obtain a number of applications including the first set of efficient algorithms for "convex piecewise-linear fitting" and the first efficient algorithms for noisy polynomial reconstruction of low-weight polynomials on the unit sphere.

cs.LG↗

arXiv2016

Determining Tournament Payout Structures for Daily Fantasy Sports

With an exploding global market and the recent introduction of online cash prize tournaments, fantasy sports contests are quickly becoming a central part of the social gaming and sports industries. For sports fans and online media companies, fantasy sports contests are an opportunity for large financial gains. However, they present a host of technical challenges that arise from the complexities involved in running a web-scale, prize driven fantasy sports platform. We initiate the study of these challenges by examining one concrete problem in particular: how to algorithmically generate contest payout structures that are 1) economically motivating and appealing to contestants and 2) reasonably structured and succinctly representable. We formalize this problem and present a general two-staged approach for producing satisfying payout structures given constraints on contest size, entry fee, prize bucketing, etc. We then propose and evaluate several potential algorithms for solving the payout problem efficiently, including methods based on dynamic programming, integer programming, and heuristic techniques. Experimental results show that a carefully designed heuristic scales very well, even to contests with over 100,000 prize winners. Our approach extends beyond fantasy sports -- it is suitable for generating engaging payout structures for any contest with a large number of entrants and a large number of prize winners, including other massive online games, poker tournaments, and real-life sports tournaments.

cs.DS↗

arXiv2016

Space Lower Bounds for Itemset Frequency Sketches

Given a database, computing the fraction of rows that contain a query itemset or determining whether this fraction is above some threshold are fundamental operations in data mining. A uniform sample of rows is a good sketch of the database in the sense that all sufficiently frequent itemsets and their approximate frequencies are recoverable from the sample, and the sketch size is independent of the number of rows in the original database. For many seemingly similar problems there are better sketching algorithms than uniform sampling. In this paper we show that for itemset frequency sketching this is not the case. That is, we prove that there exist classes of databases for which uniform sampling is a space optimal sketch for approximate itemset frequency analysis, up to constant or iterated-logarithmic factors.

cs.DS↗