2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)最新文献

{"title":"A Characterization of Multiclass Learnability","authors":"Nataly Brukhim, Dan Carmon, Irit Dinur, S. Moran, A. Yehudayoff","doi":"10.1109/FOCS54457.2022.00093","DOIUrl":"https://doi.org/10.1109/FOCS54457.2022.00093","url":null,"abstract":"A seminal result in learning theory characterizes the PAC learnability of binary classes through the Vapnik-Chervonenkis dimension. Extending this characterization to the general multiclass setting has been open since the pioneering works on multiclass PAC learning in the late 1980s. This work resolves this problem: we characterize multiclass PAC learnability through the DS dimension, a combinatorial dimension defined by Daniely and Shalev-Shwartz, (2014). The classical characterization of the binary case boils down to empirical risk minimization. In contrast, our characterization of the multiclass case involves a variety of algorithmic ideas; these include a natural setting we call list PAC learning. In the list learning setting, instead of predicting a single outcome for a given unseen input, the goal is to provide a short menu of predictions. Our second main result concerns the Natarajan dimension, which has been a central candidate for characterizing multiclass learnability. This dimension was introduced by Natarajan (1988) as a barrier for PAC learning. He furthered showed that it is the only barrier, provided that the number of labels is bounded. Whether the Natarajan dimension characterizes PAC learnability in general has been posed as an open question in several papers since. This work provides a negative answer: we construct a non-learnable class with Natarajan dimension 1. For the construction, we identify a fundamental connection between concept classes and topology (i.e., colorful simplicial complexes). We crucially rely on a deep and involved construction of hyperbolic pseudo-manifolds by Januszkiewicz and Światkowski. It is interesting that hyperbolicity is directly related to learning problems that are difficult to solve although no obvious barriers exist. This is another demonstration of the fruitful links machine learning has with different areas in mathematics.","PeriodicalId":390222,"journal":{"name":"2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126169936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Maximum Flow and Minimum-Cost Flow in Almost-Linear Time","authors":"L. Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, M. Gutenberg, Sushant Sachdeva","doi":"10.1109/FOCS54457.2022.00064","DOIUrl":"https://doi.org/10.1109/FOCS54457.2022.00064","url":null,"abstract":"We give an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with m edges and polynomially bounded integral demands, costs, and capacities in $m^{1+o(1)}$ time. Our algorithm builds the flow through a sequence of $m^{1+o(1)}$ approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized $m^{o(1)}$ time using a new dynamic graph data structure. Our framework extends to algorithms running in $m^{1+o(1)}$ time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, p-norm flows, and p-norm isotonic regression on arbitrary directed acyclic graphs.","PeriodicalId":390222,"journal":{"name":"2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132399495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantum Tanner codes","authors":"Anthony Leverrier, Gilles Z'emor","doi":"10.1109/FOCS54457.2022.00117","DOIUrl":"https://doi.org/10.1109/FOCS54457.2022.00117","url":null,"abstract":"Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes. In this work we apply this prescription to the left-right Cayley complex that lies at the heart of the recent construction of a c3 locally testable code by Dinur et at. Specifically, we view this complex as two graphs that share the same set of edges. By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code. This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. This quantum code is closely related to the Dinur et at. code in more than one sense: indeed, we prove a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur et at. code.","PeriodicalId":390222,"journal":{"name":"2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)","volume":"158 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123748588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Classical Verification of Quantum Computations in Linear Time","authors":"Jiayu Zhang","doi":"10.1109/FOCS54457.2022.00012","DOIUrl":"https://doi.org/10.1109/FOCS54457.2022.00012","url":null,"abstract":"In the quantum computation verification problem, a quantum server wants to convince a client that the output of evaluating a quantum circuit C is some result that it claims. This problem is considered very important both theoretically and practically in quantum computation [1], [2], [3]. The client is considered to be limited in computational power, and one desirable property is that the client can be completely classical, which leads to the classical verification of quantum computation (CVQC) problem. In terms of the time complexity of server-side quantum computations (which typically dominate the total time complexity of both the client and the server), the fastest single-server CVQC protocol so far has complexity $O(mathrm{p}mathrm{o}mathrm{l}mathrm{y}(kappa)|C|^{3})$ where $|C|$ is the size of the circuit to be verified and $kappa$ is the security parameter, given by Mahadev [4]. This leads to a similar cubic time blowup in many existing protocols including multiparty quantum computation, zero knowledge and obfuscation [5], [6], [7], [8], [9], [10]. Considering the preciousness of quantum computation resources, this cubic complexity barrier could be a big obstacle for theoretical and practical development of protocols for these problems.In this work, by developing new techniques, we give a new CVQC protocol with complexity $O(mathrm{p}mathrm{o}mathrm{l}mathrm{y}(kappa)|C|)$ (in terms of the total time complexity of both the client and the server), which is significantly faster than existing protocols. Our protocol is secure in the quantum random oracle model [11] assuming the existence of noisy trapdoor claw-free functions [12], which are both extensively used assumptions in quantum cryptography. Along the way, we also give a new classical channel remote state preparation protocol for states in $displaystyle left{|+thetarangle=frac{1}{sqrt{2}}(|0rangle+e^{mathrm{i}thetapi/4}|1rangle) : thetainleft{0,1cdots 7right}right}$, another basic primitive in quantum cryptography. Our protocol allows for parallel verifiable preparation of L independently random states in this form (up to a constant overall error and a possibly unbounded server-side simulator), and runs in only O(poly($kappa$)L) time and constant rounds; for comparison, existing works (even for possibly simpler state families) all require very large or unestimated time and round complexities [13], [14], [15], [16].","PeriodicalId":390222,"journal":{"name":"2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114522255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved Optimal Testing Results from Global Hypercontractivity","authors":"T. Kaufman, Dor Minzer","doi":"10.1109/FOCS54457.2022.00017","DOIUrl":"https://doi.org/10.1109/FOCS54457.2022.00017","url":null,"abstract":"The problem of testing low-degree polynomials has received significant attention over the years due to its importance in theoretical computer science, and in particular in complexity theory. The problem is specified by three parameters: field size q, degree d and proximity parameter δ, and the goal is to design a tester making as few as possible queries to a given function, which is able to distinguish between the case the given function has degree at most d, and the case the given function is δ-far from any degree d function. With respect to these parameters, we say that a tester is optimal if it makes $O(q^{t}+1/delta)$ queries, where $t=t(d,q)$ is the testing dimension of d, q (defined as the minimum integer so that for all $g:mathbb{F}_{q}^{n}rightarrowmathbb{F}_{q}$ of degree more than d, there is a subspace of dimension t on which their restriction has degree exceeding d). For the field of size q, such tester was first given by Bhattacharyya et al. for q = 2, and later by Haramaty et al. [7] for all prime powers q. In fact, they showed that the natural t-flat tester is an optimal tester for the Reed-Muller code, for an appropriate t. Here, the t-flat tester is the tester that picks a uniformly random affine subspace A of dimension t, and checks that $operatorname{deg}(f|_{A})leqslant d$. Their analysis proves that the dependency of the t-flat tester on δ and d is optimal, however the dependency on the field size, i.e. the hidden constant in the O, is a tower-type function in q. We improve the result of Haramaty et al., showing that the dependency on the field size is polynomial. Our technique also applies in the more general setting of lifted affine invariant codes, and gives the same polynomial dependency on the field size. This answers a problem raised in [6]. Our approach significantly deviates from the strategy taken in earlier works [2], [7], [6], and is based on studying the structure of the collection of erroneous subspaces, i.e. subspaces A such that f|A has degree greater than d. Towards this end, we observe that these sets are poorly expanding in the affine version of the Grassmann graph and use that to establish structural results on them via global hypercontractivity. We then use this structure to perform local correction on f.","PeriodicalId":390222,"journal":{"name":"2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128839751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Shortest Paths without a Map, but with an Entropic Regularizer","authors":"Sébastien Bubeck, Christian Coester, Y. Rabani","doi":"10.1109/FOCS54457.2022.00036","DOIUrl":"https://doi.org/10.1109/FOCS54457.2022.00036","url":null,"abstract":"In a 1989 paper titled “shortest paths without a map”, Papadimitriou and Yannakakis introduced an online model of searching in a weighted layered graph for a target node, while attempting to minimize the total length of the path traversed by the searcher. This problem, later called layered graph traversal, is parametrized by the maximum cardinality k of a layer of the input graph. It is an online setting for dynamic programming, and it is known to be a rather general and fundamental model of online computing, which includes as special cases other acclaimed models. The deterministic competitive ratio for this problem was soon discovered to be exponential in k, and it is now nearly resolved: it lies between $Omega(2^{k})$ and $O(k2^{k})$. Regarding the randomized competitive ratio, in 1993 Ramesh proved, surprisingly, that this ratio has to be at least $Omega(k^{2}/log^{1+varepsilon}k)$ (for any constant $varepsilongt0)$. In the same paper, Ramesh also gave an $O(k^{13})$-competitive randomized online algorithm. Since 1993, no progress has been reported on the randomized competitive ratio of layered graph traversal. In this work we show how to apply the mirror descent framework on a carefully selected evolving metric space, and obtain an $O(k^{2})$ competitive randomized online algorithm, nearly matching the known lower bound on the randomized competitive ratio.","PeriodicalId":390222,"journal":{"name":"2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123388705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cut Query Algorithms with Star Contraction","authors":"Simon Apers, Yuval Efron, Paweł Gawrychowski, Troy Lee, Sagnik Mukhopadhyay, Danupon Nanongkai","doi":"10.1109/FOCS54457.2022.00055","DOIUrl":"https://doi.org/10.1109/FOCS54457.2022.00055","url":null,"abstract":"We study the complexity of determining the edge connectivity of a simple graph with cut queries. We show that (i) there is a bounded-error randomized algorithm that computes edge connectivity with $O(n)$ cut queries, and (ii) there is a bounded-error quantum algorithm that computes edge connectivity with $tilde{O}(sqrt{}$n) cut queries. To prove these results we introduce a new technique, called star contraction, to randomly contract edges of a graph while preserving non-trivial minimum cuts. In star contraction vertices randomly contract an edge incident on a small set of randomly chosen “center” vertices. In contrast to the related 2-out contraction technique of Ghaffari, Nowicki, and Thorup [SODA’20], star contraction only contracts vertex-disjoint star subgraphs, which allows it to be efficiently implemented via cut queries. The $O(n)$ bound from item (i) was not known even for the simpler problem of connectivity, and it improves the $O(nlog^{3}n)$ upper bound by Rubinstein, Schramm, and Weinberg [ITCS’18]. The bound is tight under the reasonable conjecture that the randomized communication complexity of connectivity is $Omega(nlog n)$, an open question since the seminal work of Babai, Frankl, and Simon [FOCS’86]. The bound also excludes using edge connectivity on simple graphs to prove a superlinear randomized query lower bound for minimizing a symmetric submodular function. The quantum algorithm from item (ii) gives a nearlyquadratic separation with the randomized complexity, and addresses an open question of Lee, Santha, and Zhang [SODA’21]. The algorithm can alternatively be viewed as computing the edge connectivity of a simple graph with $tilde{O}(sqrt{}$n) matrix-vector multiplication queries to its adjacency matrix. Finally, we demonstrate the use of star contraction outside of the cut query setting by designing a one-pass semi-streaming algorithm for computing edge connectivity in the complete vertex arrival setting. This contrasts with the edge arrival setting where two passes are required.","PeriodicalId":390222,"journal":{"name":"2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127811868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Estimating the Longest Increasing Subsequence in Nearly Optimal Time","authors":"Alexandr Andoni, Negev Shekel Nosatzki, S. Sinha, C. Stein","doi":"10.1109/FOCS54457.2022.00073","DOIUrl":"https://doi.org/10.1109/FOCS54457.2022.00073","url":null,"abstract":"Longest Increasing Subsequence (LIS) is a fundamental statistic of a sequence, and has been studied for decades. While the LIS of a sequence of length n can be computed exactly in time $O(nlog n)$, the complexity of estimating the (length of the) LIS in sublinear time, especially when LIS $ll n$, is still open. We show that for any $ninmathbb{N}$ and $lambda=o(1)$, there exists a (randomized) non-adaptive algorithm that, given a sequence of length n with LIS $geqlambda n$, approximates the LIS up to a factor of $1/lambda^{o(1)}$ in $ n^{o(1)}/lambda$ time. Our algorithm improves upon prior work substantially in terms of both approximation and run-time: (i) we provide the first sub-polynomial approximation for LIS in sub-linear time; and (ii) our run-time complexity essentially matches the trivial sample complexity lower bound of $Omega(1/lambda)$, which is required to obtain any non-trivial approximation of the LIS. As part of our solution, we develop two novel ideas which may be of independent interest. First, we define a new Genuine-LIS problem, in which each sequence element may be either genuine or corrupted. In this model, the user receives unrestricted access to the actual sequence, but does not know a priori which elements are genuine. The goal is to estimate the LIS using genuine elements only, with the minimal number of tests for genuineness. The second idea, Precision Tree, enables accurate estimations for composition of general functions from “coarse” (sub-)estimates. Precision Tree essentially generalizes classical precision sampling, which works only for summations. As a central tool, the Precision Tree is pre-processed on a set of samples, which thereafter is repeatedly used by multiple components of the algorithm, improving their amortized complexity.","PeriodicalId":390222,"journal":{"name":"2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123785983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Implicit Graph Conjecture is False","authors":"Hamed Hatami, P. Hatami","doi":"10.1109/FOCS54457.2022.00109","DOIUrl":"https://doi.org/10.1109/FOCS54457.2022.00109","url":null,"abstract":"An efficient implicit representation of an n-vertex graph G in a family $mathcal{F}$ of graphs assigns to each vertex of G a binary code of length O(log n) so that the adjacency between every pair of vertices can be determined only as a function of their codes. This function can depend on the family but not on the individual graph. Every family of graphs admitting such a representation contains at most $2^{O(nlog(n))}$ graphs on n vertices, and thus has at most factorial speed of growth. The Implicit Graph Conjecture states that, conversely, every hereditary graph family with at most factorial speed of growth admits an efficient implicit representation. We refute this conjecture by establishing the existence of hereditary graph families with factorial speed of growth that require codes of length $n^{Omega(1)}$.","PeriodicalId":390222,"journal":{"name":"2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131967584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal","authors":"Elazar Goldenberg, T. Kociumaka, Robert Krauthgamer, B. Saha","doi":"10.1109/FOCS54457.2022.00070","DOIUrl":"https://doi.org/10.1109/FOCS54457.2022.00070","url":null,"abstract":"We study the problem of approximating edit distance in sublinear time. This is formalized as the $(k, k^{mathrm{c}})$-GAP EDIT DISTANCE problem, where the input is a pair of strings $X, mathrm{Y}$ and parameters $k, cgt 1$, and the goal is to return YES if ED(X, Y) $leq k$, NO if ED(X, Y) $gt k^{mathrm{c}}$, and an arbitrary answer when $klt $ ED(X, Y) $leq k^{mathrm{c}}$. Recent years have witnessed significant interest in designing sublinear-time algorithms for GAP EDIT DISTANCE.In this work, we resolve the non-adaptive query complexity of GAP EDIT DISTANCE for the entire range of parameters, improving over a sequence of previous results. Specifically, we design a non-adaptive algorithm with query complexity $tilde{O}(n/k^{mathrm{c}-mathrm{O}.5})$, and we further prove that this bound is optimal up to polylogarithmic factors.Our algorithm also achieves optimal time complexity $tilde{O}(n/k^{mathrm{c}-mathrm{O}.5})$ whenever $ cgeq$ 1.5. For $1 lt clt $ 1.5, the running time of our algorithm is $tilde{O}(n/k^{2mathrm{c}-2})$. In the restricted case of $k^{mathrm{c}}=Omega(n)$, this matches a known result [Batu, Ergün, Kilian, Magen, Raskhodnikova, Rubinfeld, and Sami; STOC 2003], and in all other (nontrivial) cases, our running time is strictly better than all previous algorithms, including the adaptive ones. However, independent work of Bringmann, Cassis, Fischer, and Nakos [STOC 2022] provides an adaptive algorithm that bypasses the non-adaptive lower bound, but only for small enough k and c.","PeriodicalId":390222,"journal":{"name":"2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126148086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}