Pub Date : 2023-03-10DOI: 10.48550/arXiv.2303.06001
V. Arvind, Pushkar S. Joglekar
Based on a theorem of Bergman we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely, we show the following: (1) In the white-box setting, given an n-variate noncommutative polynomial f in Fover a field F (either a finite field or the rationals) as an arithmetic circuit (or algebraic branching program), computing a complete factorization of f is deterministic polynomial-time reducible to white-box factorization of a noncommutative bivariate polynomial g in F; the reduction transforms f into a circuit for g (resp. ABP for g), and given a complete factorization of g the reduction recovers a complete factorization of f in polynomial time. We also obtain a similar deterministic polynomial-time reduction in the black-box setting. (2) Additionally, we show over the field of rationals that bivariate linear matrix factorization of 4 x 4 matrices is at least as hard as factoring square-free integers. This indicates that reducing noncommutative polynomial factorization to linear matrix factorization (as done in our recent work [AJ22]) is unlikely to succeed over the field of rationals even in the bivariate case. In contrast, multivariate linear matrix factorization for 3 x 3 matrices over rationals is in polynomial time.
{"title":"Multivariate to Bivariate Reduction for Noncommutative Polynomial Factorization","authors":"V. Arvind, Pushkar S. Joglekar","doi":"10.48550/arXiv.2303.06001","DOIUrl":"https://doi.org/10.48550/arXiv.2303.06001","url":null,"abstract":"Based on a theorem of Bergman we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely, we show the following: (1) In the white-box setting, given an n-variate noncommutative polynomial f in Fover a field F (either a finite field or the rationals) as an arithmetic circuit (or algebraic branching program), computing a complete factorization of f is deterministic polynomial-time reducible to white-box factorization of a noncommutative bivariate polynomial g in F; the reduction transforms f into a circuit for g (resp. ABP for g), and given a complete factorization of g the reduction recovers a complete factorization of f in polynomial time. We also obtain a similar deterministic polynomial-time reduction in the black-box setting. (2) Additionally, we show over the field of rationals that bivariate linear matrix factorization of 4 x 4 matrices is at least as hard as factoring square-free integers. This indicates that reducing noncommutative polynomial factorization to linear matrix factorization (as done in our recent work [AJ22]) is unlikely to succeed over the field of rationals even in the bivariate case. In contrast, multivariate linear matrix factorization for 3 x 3 matrices over rationals is in polynomial time.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"30 1","pages":"14:1-14:15"},"PeriodicalIF":0.0,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82737250","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}
Pub Date : 2023-03-09DOI: 10.48550/arXiv.2303.05044
Karthik Gajulapalli, Alexander Golovnev, Satyajeet Nagargoje, Sidhant Saraogi
Range Avoidance (AVOID) is a total search problem where, given a Boolean circuit $Ccolon{0,1}^nto{0,1}^m$, $m>n$, the task is to find a $yin{0,1}^m$ outside the range of $C$. For an integer $kgeq 2$, $mathrm{NC}^0_k$-AVOID is a special case of AVOID where each output bit of $C$ depends on at most $k$ input bits. While there is a very natural randomized algorithm for AVOID, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to $mathrm{NC}^0_4$-AVOID, thus establishing conditional hardness of the $mathrm{NC}^0_4$-AVOID problem. On the other hand, $mathrm{NC}^0_2$-AVOID admits polynomial-time algorithms, leaving the question about the complexity of $mathrm{NC}^0_3$-AVOID open. We give the first reduction of an explicit construction question to $mathrm{NC}^0_3$-AVOID. Specifically, we prove that a polynomial-time algorithm (with an $mathrm{NP}$ oracle) for $mathrm{NC}^0_3$-AVOID for the case of $m=n+n^{2/3}$ would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits. We also give deterministic polynomial-time algorithms for all $mathrm{NC}^0_k$-AVOID problems for $mgeq n^{k-1}/log(n)$. Prior work required an $mathrm{NP}$ oracle, and required larger stretch, $m geq n^{k-1}$.
范围回避(AVOID)是一个全搜索问题，其中给定一个布尔电路$Ccolon{0,1}^nto{0,1}^m$, $m>n$，任务是在$C$范围之外找到一个$yin{0,1}^m$。对于整数$kgeq 2$, $mathrm{NC}^0_k$ -AVOID是一种特殊情况，其中$C$的每个输出位最多依赖于$k$个输入位。虽然对于AVOID有一种非常自然的随机算法，但对于该问题的确定性算法将产生许多有趣的结果。Ren, Santhanam, and Wang (FOCS 2022)和Guruswami, Lyu, and Wang (RANDOM 2022)证明了高公式复杂度的函数，刚性矩阵和最优线性编码的显式构造可以减少到$mathrm{NC}^0_4$ -AVOID，从而建立了$mathrm{NC}^0_4$ -AVOID问题的条件硬度。另一方面，$mathrm{NC}^0_2$ -AVOID承认多项式时间算法，留下了关于$mathrm{NC}^0_3$ -AVOID复杂性的问题。我们给出一个显式结构问题的第一个约简为$mathrm{NC}^0_3$ -AVOID。具体地说，我们证明了$mathrm{NC}^0_3$ -AVOID的多项式时间算法(使用$mathrm{NP}$ oracle)对于$m=n+n^{2/3}$的情况将意味着刚性矩阵的显式构造，并且因此，对数深度电路大小的超线性下界。我们还给出了$mgeq n^{k-1}/log(n)$的所有$mathrm{NC}^0_k$ -AVOID问题的确定性多项式时间算法。之前的工作需要$mathrm{NP}$ oracle，并且需要更大的扩展$m geq n^{k-1}$。
{"title":"Range Avoidance for Constant-Depth Circuits: Hardness and Algorithms","authors":"Karthik Gajulapalli, Alexander Golovnev, Satyajeet Nagargoje, Sidhant Saraogi","doi":"10.48550/arXiv.2303.05044","DOIUrl":"https://doi.org/10.48550/arXiv.2303.05044","url":null,"abstract":"Range Avoidance (AVOID) is a total search problem where, given a Boolean circuit $Ccolon{0,1}^nto{0,1}^m$, $m>n$, the task is to find a $yin{0,1}^m$ outside the range of $C$. For an integer $kgeq 2$, $mathrm{NC}^0_k$-AVOID is a special case of AVOID where each output bit of $C$ depends on at most $k$ input bits. While there is a very natural randomized algorithm for AVOID, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to $mathrm{NC}^0_4$-AVOID, thus establishing conditional hardness of the $mathrm{NC}^0_4$-AVOID problem. On the other hand, $mathrm{NC}^0_2$-AVOID admits polynomial-time algorithms, leaving the question about the complexity of $mathrm{NC}^0_3$-AVOID open. We give the first reduction of an explicit construction question to $mathrm{NC}^0_3$-AVOID. Specifically, we prove that a polynomial-time algorithm (with an $mathrm{NP}$ oracle) for $mathrm{NC}^0_3$-AVOID for the case of $m=n+n^{2/3}$ would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits. We also give deterministic polynomial-time algorithms for all $mathrm{NC}^0_k$-AVOID problems for $mgeq n^{k-1}/log(n)$. Prior work required an $mathrm{NP}$ oracle, and required larger stretch, $m geq n^{k-1}$.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"5 1","pages":"65:1-65:18"},"PeriodicalIF":0.0,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86366917","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}
Pub Date : 2023-03-07DOI: 10.48550/arXiv.2303.03921
Mark Bun, N. Voronova
The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function. We introduce a framework for proving approximate degree lower bounds for certain oracle identification problems, where the goal is to recover a hidden binary string $x in {0, 1}^n$ given possibly non-standard oracle access to it. Our lower bounds apply to decision versions of these problems, where the goal is to compute the parity of $x$. We apply our framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of $Omega(n/log^2 n)$ for each. These lower bounds generalize to the weakly unbounded error setting, giving a new quantum query lower bound for the hidden string problem in this regime. Our lower bounds are driven by randomized communication upper bounds for the greater-than and equality functions.
布尔函数的近似度数是实多项式逐点逼近的最小度数。对于任何布尔函数，其近似度作为其量子查询复杂度的下界，对于相关函数一般提升为量子通信复杂度的下界。我们引入了一个框架，用于证明某些oracle识别问题的近似度下界，其目标是在给定可能非标准oracle访问的情况下恢复隐藏的二进制字符串$x in {0, 1}^n$。我们的下界适用于这些问题的决策版本，其目标是计算$x$的奇偶性。我们将我们的框架应用于有序搜索和隐藏字符串问题，证明了它们的近似度下界近似$Omega(n/log^2 n)$。这些下界推广到弱无界错误设置中，给出了隐串问题的一个新的量子查询下界。我们的下界是由大于函数和相等函数的随机通信上界驱动的。
{"title":"Approximate degree lower bounds for oracle identification problems","authors":"Mark Bun, N. Voronova","doi":"10.48550/arXiv.2303.03921","DOIUrl":"https://doi.org/10.48550/arXiv.2303.03921","url":null,"abstract":"The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function. We introduce a framework for proving approximate degree lower bounds for certain oracle identification problems, where the goal is to recover a hidden binary string $x in {0, 1}^n$ given possibly non-standard oracle access to it. Our lower bounds apply to decision versions of these problems, where the goal is to compute the parity of $x$. We apply our framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of $Omega(n/log^2 n)$ for each. These lower bounds generalize to the weakly unbounded error setting, giving a new quantum query lower bound for the hidden string problem in this regime. Our lower bounds are driven by randomized communication upper bounds for the greater-than and equality functions.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"9 1","pages":"1:1-1:24"},"PeriodicalIF":0.0,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77915339","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}
Pub Date : 2023-02-14DOI: 10.48550/arXiv.2302.06984
Hervé Fournier, N. Limaye, Guillaume Malod, S. Srinivasan, Sébastien Tavenas
Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula. Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas. This improves on previous results in the regime when d is small (i.e., d<
{"title":"Towards Optimal Depth-Reductions for Algebraic Formulas","authors":"Hervé Fournier, N. Limaye, Guillaume Malod, S. Srinivasan, Sébastien Tavenas","doi":"10.48550/arXiv.2302.06984","DOIUrl":"https://doi.org/10.48550/arXiv.2302.06984","url":null,"abstract":"Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula. Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas. This improves on previous results in the regime when d is small (i.e., d<<s). In particular, for the setting of d=O(log s), along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas. This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this ``low-degree\"and ``low-depth\"setting. We also show that these results cannot be improved in the monotone setting, even for commutative formulas.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"1962 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91316240","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}
Pub Date : 2023-01-31DOI: 10.48550/arXiv.2301.13603
Ondrej Jezil
For an infinite class of finite graphs of unbounded size, we define a limit object, to be called wide limit, relative to some computationally restricted class of functions. The properties of the wide limit then reflect how a computationally restricted viewer"sees"a generic instance from the class. The construction uses arithmetic forcing with random variables [10]. We prove sufficient conditions for universal and existential sentences to be valid in the limit, provide several examples, and prove that such a limit object can then be expanded to a model of weak arithmetic. We then take the wide limit of all finite pointed paths to obtain a model of arithmetic where the problem OntoWeakPigeon is total but Leaf (the complete problem for $textbf{PPA}$) is not. This logical separation of the oracle classes of total NP search problems in our setting implies that Leaf is not reducible to OntoWeakPigeon even if some errors are allowed in the reductions.
{"title":"Limits of structures and Total NP Search Problems","authors":"Ondrej Jezil","doi":"10.48550/arXiv.2301.13603","DOIUrl":"https://doi.org/10.48550/arXiv.2301.13603","url":null,"abstract":"For an infinite class of finite graphs of unbounded size, we define a limit object, to be called wide limit, relative to some computationally restricted class of functions. The properties of the wide limit then reflect how a computationally restricted viewer\"sees\"a generic instance from the class. The construction uses arithmetic forcing with random variables [10]. We prove sufficient conditions for universal and existential sentences to be valid in the limit, provide several examples, and prove that such a limit object can then be expanded to a model of weak arithmetic. We then take the wide limit of all finite pointed paths to obtain a model of arithmetic where the problem OntoWeakPigeon is total but Leaf (the complete problem for $textbf{PPA}$) is not. This logical separation of the oracle classes of total NP search problems in our setting implies that Leaf is not reducible to OntoWeakPigeon even if some errors are allowed in the reductions.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86753892","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}
Pub Date : 2023-01-25DOI: 10.48550/arXiv.2301.10617
J. Krajícek
For a finite set $cal F$ of polynomials over fixed finite prime field of size $p$ containing all polynomials $x^2 - x$ a Nullstellensatz proof of the unsolvability of the system $$ f = 0 , mbox{ all } f in {cal F} $$ in the field is a linear combination $sum_{f in {cal F}} h_f cdot f$ that equals to $1$ in the ring of polynomails. The measure of complexity of such a proof is its degree: $max_f deg(h_f f)$. We study the problem to establish degree lower bounds for some {em extended} NS proof systems: these systems prove the unsolvability of $cal F$ by proving the unsolvability of a bigger set ${cal F}cup {cal E}$, where set $cal E$ may use new variables $r$ and contains all polynomials $r^p - r$, and satisfies the following soundness condition: -- - Any $0,1$-assignment $overline a$ to variables $overline x$ can be appended by an assignment $overline b$ to variables $overline r$ such that for all $g in {cal E}$ it holds that $g(overline a, overline b) = 0$. We define a notion of pseudo-solutions of $cal F$ and prove that the existence of pseudo-solutions with suitable parameters implies lower bounds for two extended NS proof systems ENS and UENS defined in Buss et al. (1996/97). Further we give a combinatorial example of $cal F$ and candidate pseudo-solutions based on the pigeonhole principle.
对于大小为$p$的固定有限素数域上的多项式的有限集合$cal F$，其中包含所有多项式$x^2 - x$，一个Nullstellensatz证明系统$$ f = 0 , mbox{ all } f in {cal F} $$在该域中的不可解性是一个线性组合$sum_{f in {cal F}} h_f cdot f$，等于多项式环中的$1$。衡量这种证明的复杂程度就是它的程度:$max_f deg(h_f f)$。我们研究了一些{em扩展}NS证明系统的次下界的建立问题，这些系统通过证明一个更大集合${cal F}cup {cal E}$的不可解性来证明$cal F$的不可解性，其中集合$cal E$可以使用新的变量$r$并且包含所有多项式$r^p - r$，并且满足以下稳健性条件:对变量$overline x$的任何$0,1$赋值$overline a$都可以通过对变量$overline r$的赋值$overline b$来附加，这样对于所有$g in {cal E}$都持有$g(overline a, overline b) = 0$。我们定义了$cal F$伪解的概念，并证明了Buss et al.(1996/97)定义的两个扩展NS证明系统ENS和UENS具有合适参数的伪解的存在性意味着下界。进一步给出了一个基于鸽子洞原理的$cal F$和候选伪解的组合例子。
{"title":"Extended Nullstellensatz proof systems","authors":"J. Krajícek","doi":"10.48550/arXiv.2301.10617","DOIUrl":"https://doi.org/10.48550/arXiv.2301.10617","url":null,"abstract":"For a finite set $cal F$ of polynomials over fixed finite prime field of size $p$ containing all polynomials $x^2 - x$ a Nullstellensatz proof of the unsolvability of the system $$ f = 0 , mbox{ all } f in {cal F} $$ in the field is a linear combination $sum_{f in {cal F}} h_f cdot f$ that equals to $1$ in the ring of polynomails. The measure of complexity of such a proof is its degree: $max_f deg(h_f f)$. We study the problem to establish degree lower bounds for some {em extended} NS proof systems: these systems prove the unsolvability of $cal F$ by proving the unsolvability of a bigger set ${cal F}cup {cal E}$, where set $cal E$ may use new variables $r$ and contains all polynomials $r^p - r$, and satisfies the following soundness condition: -- - Any $0,1$-assignment $overline a$ to variables $overline x$ can be appended by an assignment $overline b$ to variables $overline r$ such that for all $g in {cal E}$ it holds that $g(overline a, overline b) = 0$. We define a notion of pseudo-solutions of $cal F$ and prove that the existence of pseudo-solutions with suitable parameters implies lower bounds for two extended NS proof systems ENS and UENS defined in Buss et al. (1996/97). Further we give a combinatorial example of $cal F$ and candidate pseudo-solutions based on the pigeonhole principle.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"394 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85007310","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}
Pub Date : 2023-01-20DOI: 10.48550/arXiv.2301.08486
N. Bshouty
Koch, Strassle, and Tan [SODA 2023], show that, under the randomized exponential time hypothesis, there is no distribution-free PAC-learning algorithm that runs in time $n^{tilde O(loglog s)}$ for the classes of $n$-variable size-$s$ DNF, size-$s$ Decision Tree, and $log s$-Junta by DNF (that returns a DNF hypothesis). Assuming a natural conjecture on the hardness of set cover, they give the lower bound $n^{Omega(log s)}$. This matches the best known upper bound for $n$-variable size-$s$ Decision Tree, and $log s$-Junta. In this paper, we give the same lower bounds for PAC-learning of $n$-variable size-$s$ Monotone DNF, size-$s$ Monotone Decision Tree, and Monotone $log s$-Junta by~DNF. This solves the open problem proposed by Koch, Strassle, and Tan and subsumes the above results. The lower bound holds, even if the learner knows the distribution, can draw a sample according to the distribution in polynomial time, and can compute the target function on all the points of the support of the distribution in polynomial time.
{"title":"Superpolynomial Lower Bounds for Learning Monotone Classes","authors":"N. Bshouty","doi":"10.48550/arXiv.2301.08486","DOIUrl":"https://doi.org/10.48550/arXiv.2301.08486","url":null,"abstract":"Koch, Strassle, and Tan [SODA 2023], show that, under the randomized exponential time hypothesis, there is no distribution-free PAC-learning algorithm that runs in time $n^{tilde O(loglog s)}$ for the classes of $n$-variable size-$s$ DNF, size-$s$ Decision Tree, and $log s$-Junta by DNF (that returns a DNF hypothesis). Assuming a natural conjecture on the hardness of set cover, they give the lower bound $n^{Omega(log s)}$. This matches the best known upper bound for $n$-variable size-$s$ Decision Tree, and $log s$-Junta. In this paper, we give the same lower bounds for PAC-learning of $n$-variable size-$s$ Monotone DNF, size-$s$ Monotone Decision Tree, and Monotone $log s$-Junta by~DNF. This solves the open problem proposed by Koch, Strassle, and Tan and subsumes the above results. The lower bound holds, even if the learner knows the distribution, can draw a sample according to the distribution in polynomial time, and can compute the target function on all the points of the support of the distribution in polynomial time.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"46 1","pages":"34:1-34:20"},"PeriodicalIF":0.0,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83733050","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}
Pub Date : 2023-01-13DOI: 10.48550/arXiv.2301.05658
Shachar Lovett, Jiapeng Zhang
Frequency estimation in data streams is one of the classical problems in streaming algorithms. Following much research, there are now almost matching upper and lower bounds for the trade-off needed between the number of samples and the space complexity of the algorithm, when the data streams are adversarial. However, in the case where the data stream is given in a random order, or is stochastic, only weaker lower bounds exist. In this work we close this gap, up to logarithmic factors. In order to do so we consider the needle problem, which is a natural hard problem for frequency estimation studied in (Andoni et al. 2008, Crouch et al. 2016). Here, the goal is to distinguish between two distributions over data streams with $t$ samples. The first is uniform over a large enough domain. The second is a planted model; a secret ''needle'' is uniformly chosen, and then each element in the stream equals the needle with probability $p$, and otherwise is uniformly chosen from the domain. It is simple to design streaming algorithms that distinguish the distributions using space $s approx 1/(p^2 t)$. It was unclear if this is tight, as the existing lower bounds are weaker. We close this gap and show that the trade-off is near optimal, up to a logarithmic factor. Our proof builds and extends classical connections between streaming algorithms and communication complexity, concretely multi-party unique set-disjointness. We introduce two new ingredients that allow us to prove sharp bounds. The first is a lower bound for an asymmetric version of multi-party unique set-disjointness, where players receive input sets of different sizes, and where the communication of each player is normalized relative to their input length. The second is a combinatorial technique that allows to sample needles in the planted model by first sampling intervals, and then sampling a uniform needle in each interval.
数据流中的频率估计是流算法中的经典问题之一。经过大量的研究，当数据流是对抗性的时，对于样本数量和算法的空间复杂度之间的权衡，现在几乎有匹配的上限和下限。然而，在数据流以随机顺序给定的情况下，或者是随机的，只存在较弱的下界。在这项工作中，我们缩小了这个差距，直到对数因子。为了做到这一点，我们考虑了针问题，这是在(Andoni et al. 2008, Crouch et al. 2016)中研究的频率估计的自然难题。这里的目标是区分具有$t$样本的数据流上的两个分布。第一种在足够大的范围内是均匀的。第二种是被植入的模型;统一选择一个秘密“针”，然后流中的每个元素以概率$p$等于针，否则从域中统一选择。设计使用空间$s 约1/(p^2 t)$来区分分布的流算法很简单。目前尚不清楚这是否严格，因为现有的下限较弱。我们缩小了这个差距，并表明权衡接近最优，达到对数因子。我们的证明建立并扩展了流算法和通信复杂性之间的经典联系，具体来说是多方唯一集不连接。我们引入两种新的成分，使我们能够证明尖锐的界限。第一个是多方唯一集不连接的非对称版本的下界，其中玩家接收不同大小的输入集，并且每个玩家的通信相对于他们的输入长度进行规范化。第二种是一种组合技术，允许在种植模型中通过第一次采样间隔对针进行采样，然后在每个间隔中对均匀的针进行采样。
{"title":"Streaming Lower Bounds and Asymmetric Set-Disjointness","authors":"Shachar Lovett, Jiapeng Zhang","doi":"10.48550/arXiv.2301.05658","DOIUrl":"https://doi.org/10.48550/arXiv.2301.05658","url":null,"abstract":"Frequency estimation in data streams is one of the classical problems in streaming algorithms. Following much research, there are now almost matching upper and lower bounds for the trade-off needed between the number of samples and the space complexity of the algorithm, when the data streams are adversarial. However, in the case where the data stream is given in a random order, or is stochastic, only weaker lower bounds exist. In this work we close this gap, up to logarithmic factors. In order to do so we consider the needle problem, which is a natural hard problem for frequency estimation studied in (Andoni et al. 2008, Crouch et al. 2016). Here, the goal is to distinguish between two distributions over data streams with $t$ samples. The first is uniform over a large enough domain. The second is a planted model; a secret ''needle'' is uniformly chosen, and then each element in the stream equals the needle with probability $p$, and otherwise is uniformly chosen from the domain. It is simple to design streaming algorithms that distinguish the distributions using space $s approx 1/(p^2 t)$. It was unclear if this is tight, as the existing lower bounds are weaker. We close this gap and show that the trade-off is near optimal, up to a logarithmic factor. Our proof builds and extends classical connections between streaming algorithms and communication complexity, concretely multi-party unique set-disjointness. We introduce two new ingredients that allow us to prove sharp bounds. The first is a lower bound for an asymmetric version of multi-party unique set-disjointness, where players receive input sets of different sizes, and where the communication of each player is normalized relative to their input length. The second is a combinatorial technique that allows to sample needles in the planted model by first sampling intervals, and then sampling a uniform needle in each interval.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78367821","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}
Pub Date : 2023-01-13DOI: 10.4230/LIPIcs.ICALP.2023.17
P. Beame, Niels Kornerup
Cumulative memory -- the sum of space used per step over the duration of a computation -- is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many multiple instances of an algorithm are interleaved in parallel. We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits. Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution. The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight. Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions. Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least $1/text{poly}(n)$ requires cumulative memory $tilde Omega(n^2)$, any classical matrix multiplication algorithm requires cumulative memory $Omega(n^6/T)$, any quantum sorting circuit requires cumulative memory $Omega(n^3/T)$, and any quantum circuit that finds $k$ disjoint collisions in a random function requires cumulative memory $Omega(k^3n/T^2)$.
{"title":"Cumulative Memory Lower Bounds for Randomized and Quantum Computation","authors":"P. Beame, Niels Kornerup","doi":"10.4230/LIPIcs.ICALP.2023.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.ICALP.2023.17","url":null,"abstract":"Cumulative memory -- the sum of space used per step over the duration of a computation -- is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many multiple instances of an algorithm are interleaved in parallel. We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits. Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution. The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight. Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions. Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least $1/text{poly}(n)$ requires cumulative memory $tilde Omega(n^2)$, any classical matrix multiplication algorithm requires cumulative memory $Omega(n^6/T)$, any quantum sorting circuit requires cumulative memory $Omega(n^3/T)$, and any quantum circuit that finds $k$ disjoint collisions in a random function requires cumulative memory $Omega(k^3n/T^2)$.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87095246","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}
Pub Date : 2023-01-05DOI: 10.48550/arXiv.2301.01924
N. Alon, O. Bousquet, Kasper Green Larsen, S. Moran, S. Moran
We study several variants of a combinatorial game which is based on Cantor's diagonal argument. The game is between two players called Kronecker and Cantor. The names of the players are motivated by the known fact that Leopold Kronecker did not appreciate Georg Cantor's arguments about the infinite, and even referred to him as a"scientific charlatan". In the game Kronecker maintains a list of m binary vectors, each of length n, and Cantor's goal is to produce a new binary vector which is different from each of Kronecker's vectors, or prove that no such vector exists. Cantor does not see Kronecker's vectors but he is allowed to ask queries of the form"What is bit number j of vector number i?"What is the minimal number of queries with which Cantor can achieve his goal? How much better can Cantor do if he is allowed to pick his queries emph{adaptively}, based on Kronecker's previous replies? The case when m=n is solved by diagonalization using n (non-adaptive) queries. We study this game more generally, and prove an optimal bound in the adaptive case and nearly tight upper and lower bounds in the non-adaptive case.
{"title":"Diagonalization Games","authors":"N. Alon, O. Bousquet, Kasper Green Larsen, S. Moran, S. Moran","doi":"10.48550/arXiv.2301.01924","DOIUrl":"https://doi.org/10.48550/arXiv.2301.01924","url":null,"abstract":"We study several variants of a combinatorial game which is based on Cantor's diagonal argument. The game is between two players called Kronecker and Cantor. The names of the players are motivated by the known fact that Leopold Kronecker did not appreciate Georg Cantor's arguments about the infinite, and even referred to him as a\"scientific charlatan\". In the game Kronecker maintains a list of m binary vectors, each of length n, and Cantor's goal is to produce a new binary vector which is different from each of Kronecker's vectors, or prove that no such vector exists. Cantor does not see Kronecker's vectors but he is allowed to ask queries of the form\"What is bit number j of vector number i?\"What is the minimal number of queries with which Cantor can achieve his goal? How much better can Cantor do if he is allowed to pick his queries emph{adaptively}, based on Kronecker's previous replies? The case when m=n is solved by diagonalization using n (non-adaptive) queries. We study this game more generally, and prove an optimal bound in the adaptive case and nearly tight upper and lower bounds in the non-adaptive case.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"2016 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86292643","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}
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