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Rocco A Servedio - One of the best experts on this subject based on the ideXlab platform.
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efficient deterministic approximate counting for low Degree Polynomial threshold functions
Symposium on the Theory of Computing, 2014Co-Authors: Anindya De, Rocco A ServedioAbstract:We give a deterministic algorithm for approximately counting satisfying assignments of a Degree-d Polynomial threshold function (PTF). Given a Degree-d input Polynomial p(x) over Rn and a parameter e > 0, our algorithm approximates Pr [EQUATION] to within an additive ±e in time Od,e(1) · poly(nd). (Since it is NP-hard to determine whether the above probability is nonzero, any sort of efficient multiplicative approximation is almost certainly impossible even for randomized algorithms.) Note that the running time of our algorithm (as a function of nd, the number of coefficients of a Degree-d PTF) is a fixed Polynomial. The fastest previous algorithm for this problem [Kan12b], based on constructions of unconditional pseudorandom generators for Degree-d PTFs, runs in time [EQUATION] for all c > 0. The key novel technical contributions of this work are • A new multivariate central limit theorem, proved using tools from Malliavin calculus and Stein's Method. This new CLT shows that any collection of Gaussian Polynomials with small eigenvalues must have a joint distribution which is very close to a multidimensional Gaussian distribution. • A new decomposition of low-Degree multilinear Polynomials over Gaussian inputs. Roughly speaking we show that (up to some small error) any such Polynomial can be decomposed into a bounded number of multilinear Polynomials all of which have extremely small eigenvalues. We use these new ingredients to give a deterministic algorithm for a Gaussian-space version of the approximate counting problem, and then employ standard techniques for working with low-Degree PTFs (invariance principles and regularity lemmas) to reduce the original approximate counting problem over the Boolean hypercube to the Gaussian version. As an application of our result, we give the first deterministic fixed-parameter tractable algorithm for the following moment approximation problem: given a Degree-d Polynomial p(x1,..., xn) over {--1, 1}n, a positive integer k and an error parameter e, output a (1±e)-multiplicatively accurate estimate to [EQUATION]. Our algorithm runs in time Od,e,k(1) · poly(nd).
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efficient deterministic approximate counting for low Degree Polynomial threshold functions
arXiv: Computational Complexity, 2013Co-Authors: Anindya De, Rocco A ServedioAbstract:We give a deterministic algorithm for approximately counting satisfying assignments of a Degree-$d$ Polynomial threshold function (PTF). Given a Degree-$d$ input Polynomial $p(x_1,\dots,x_n)$ over $R^n$ and a parameter $\epsilon> 0$, our algorithm approximates $\Pr_{x \sim \{-1,1\}^n}[p(x) \geq 0]$ to within an additive $\pm \epsilon$ in time $O_{d,\epsilon}(1)\cdot \mathop{poly}(n^d)$. (Any sort of efficient multiplicative approximation is impossible even for randomized algorithms assuming $NP\not=RP$.) Note that the running time of our algorithm (as a function of $n^d$, the number of coefficients of a Degree-$d$ PTF) is a \emph{fixed} Polynomial. The fastest previous algorithm for this problem (due to Kane), based on constructions of unconditional pseudorandom generators for Degree-$d$ PTFs, runs in time $n^{O_{d,c}(1) \cdot \epsilon^{-c}}$ for all $c > 0$. The key novel contributions of this work are: A new multivariate central limit theorem, proved using tools from Malliavin calculus and Stein's Method. This new CLT shows that any collection of Gaussian Polynomials with small eigenvalues must have a joint distribution which is very close to a multidimensional Gaussian distribution. A new decomposition of low-Degree multilinear Polynomials over Gaussian inputs. Roughly speaking we show that (up to some small error) any such Polynomial can be decomposed into a bounded number of multilinear Polynomials all of which have extremely small eigenvalues. We use these new ingredients to give a deterministic algorithm for a Gaussian-space version of the approximate counting problem, and then employ standard techniques for working with low-Degree PTFs (invariance principles and regularity lemmas) to reduce the original approximate counting problem over the Boolean hypercube to the Gaussian version.
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hardness results for agnostically learning low Degree Polynomial threshold functions
Symposium on Discrete Algorithms, 2011Co-Authors: Ilias Diakonikolas, Ryan Odonnell, Rocco A Servedio, Yi WuAbstract:Hardness results for maximum agreement problems have close connections to hardness results for proper learning in computational learning theory. In this paper we prove two hardness results for the problem of finding a low Degree Polynomial threshold function (PTF) which has the maximum possible agreement with a given set of labeled examples in Rn x {−1, 1}. We prove that for any constants d ≥ 1, e > 0, • Assuming the Unique Games Conjecture, no Polynomial-time algorithm can find a Degree-d PTF that is consistent with a (1/2 + e) fraction of a given set of labeled examples in Rn x {−1, 1}, even if there exists a Degree-d PTF that is consistent with a 1 − e fraction of the examples. • It is NP-hard to find a Degree-2 PTF that is consistent with a (1/2 + e) fraction of a given set of labeled examples in Rn x {−1, 1}, even if there exists a half-space (Degree-1 PTF) that is consistent with a 1 − e fraction of the examples. These results immediately imply the following hardness of learning results: (i) Assuming the Unique Games Conjecture, there is no better-than-trivial proper learning algorithm that agnostically learns Degree-d PTFs under arbitrary distributions; (ii) There is no better-than-trivial learning algorithm that outputs Degree-2 PTFs and agnostically learns halfspaces (i.e. Degree-1 PTFs) under arbitrary distributions.
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hardness results for agnostically learning low Degree Polynomial threshold functions
arXiv: Learning, 2010Co-Authors: Ilias Diakonikolas, Ryan Odonnell, Rocco A Servedio, Yi WuAbstract:Hardness results for maximum agreement problems have close connections to hardness results for proper learning in computational learning theory. In this paper we prove two hardness results for the problem of finding a low Degree Polynomial threshold function (PTF) which has the maximum possible agreement with a given set of labeled examples in $\R^n \times \{-1,1\}.$ We prove that for any constants $d\geq 1, \eps > 0$, {itemize} Assuming the Unique Games Conjecture, no Polynomial-time algorithm can find a Degree-$d$ PTF that is consistent with a $(\half + \eps)$ fraction of a given set of labeled examples in $\R^n \times \{-1,1\}$, even if there exists a Degree-$d$ PTF that is consistent with a $1-\eps$ fraction of the examples. It is $\NP$-hard to find a Degree-2 PTF that is consistent with a $(\half + \eps)$ fraction of a given set of labeled examples in $\R^n \times \{-1,1\}$, even if there exists a halfspace (Degree-1 PTF) that is consistent with a $1 - \eps$ fraction of the examples. {itemize} These results immediately imply the following hardness of learning results: (i) Assuming the Unique Games Conjecture, there is no better-than-trivial proper learning algorithm that agnostically learns Degree-$d$ PTFs under arbitrary distributions; (ii) There is no better-than-trivial learning algorithm that outputs Degree-2 PTFs and agnostically learns halfspaces (i.e. Degree-1 PTFs) under arbitrary distributions.
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A Regularity Lemma, and Low-Weight Approximators, for Low-Degree Polynomial Threshold Functions
2010 IEEE 25th Annual Conference on Computational Complexity, 2010Co-Authors: Ilias Diakonikolas, Rocco A ServedioAbstract:We give a "regularity lemma" for Degree-d Polynomial threshold functions (PTFs) over the Boolean cube {-1,1}n. Roughly speaking, this result shows that every Degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a "regular" PTF is a PTF sign(p(x)) where the influence of each variable on the Polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for any constants d ≥ 1, ϵ > 0, every Degree-d PTF over n variables can be approximated to accuracy eps by a constant Degree PTF that has integer weights of total magnitude O(nd). This weight bound is shown to be optimal up to logarithmic factors.
Ilias Diakonikolas - One of the best experts on this subject based on the ideXlab platform.
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hardness results for agnostically learning low Degree Polynomial threshold functions
Symposium on Discrete Algorithms, 2011Co-Authors: Ilias Diakonikolas, Ryan Odonnell, Rocco A Servedio, Yi WuAbstract:Hardness results for maximum agreement problems have close connections to hardness results for proper learning in computational learning theory. In this paper we prove two hardness results for the problem of finding a low Degree Polynomial threshold function (PTF) which has the maximum possible agreement with a given set of labeled examples in Rn x {−1, 1}. We prove that for any constants d ≥ 1, e > 0, • Assuming the Unique Games Conjecture, no Polynomial-time algorithm can find a Degree-d PTF that is consistent with a (1/2 + e) fraction of a given set of labeled examples in Rn x {−1, 1}, even if there exists a Degree-d PTF that is consistent with a 1 − e fraction of the examples. • It is NP-hard to find a Degree-2 PTF that is consistent with a (1/2 + e) fraction of a given set of labeled examples in Rn x {−1, 1}, even if there exists a half-space (Degree-1 PTF) that is consistent with a 1 − e fraction of the examples. These results immediately imply the following hardness of learning results: (i) Assuming the Unique Games Conjecture, there is no better-than-trivial proper learning algorithm that agnostically learns Degree-d PTFs under arbitrary distributions; (ii) There is no better-than-trivial learning algorithm that outputs Degree-2 PTFs and agnostically learns halfspaces (i.e. Degree-1 PTFs) under arbitrary distributions.
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hardness results for agnostically learning low Degree Polynomial threshold functions
arXiv: Learning, 2010Co-Authors: Ilias Diakonikolas, Ryan Odonnell, Rocco A Servedio, Yi WuAbstract:Hardness results for maximum agreement problems have close connections to hardness results for proper learning in computational learning theory. In this paper we prove two hardness results for the problem of finding a low Degree Polynomial threshold function (PTF) which has the maximum possible agreement with a given set of labeled examples in $\R^n \times \{-1,1\}.$ We prove that for any constants $d\geq 1, \eps > 0$, {itemize} Assuming the Unique Games Conjecture, no Polynomial-time algorithm can find a Degree-$d$ PTF that is consistent with a $(\half + \eps)$ fraction of a given set of labeled examples in $\R^n \times \{-1,1\}$, even if there exists a Degree-$d$ PTF that is consistent with a $1-\eps$ fraction of the examples. It is $\NP$-hard to find a Degree-2 PTF that is consistent with a $(\half + \eps)$ fraction of a given set of labeled examples in $\R^n \times \{-1,1\}$, even if there exists a halfspace (Degree-1 PTF) that is consistent with a $1 - \eps$ fraction of the examples. {itemize} These results immediately imply the following hardness of learning results: (i) Assuming the Unique Games Conjecture, there is no better-than-trivial proper learning algorithm that agnostically learns Degree-$d$ PTFs under arbitrary distributions; (ii) There is no better-than-trivial learning algorithm that outputs Degree-2 PTFs and agnostically learns halfspaces (i.e. Degree-1 PTFs) under arbitrary distributions.
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A Regularity Lemma, and Low-Weight Approximators, for Low-Degree Polynomial Threshold Functions
2010 IEEE 25th Annual Conference on Computational Complexity, 2010Co-Authors: Ilias Diakonikolas, Rocco A ServedioAbstract:We give a "regularity lemma" for Degree-d Polynomial threshold functions (PTFs) over the Boolean cube {-1,1}n. Roughly speaking, this result shows that every Degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a "regular" PTF is a PTF sign(p(x)) where the influence of each variable on the Polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for any constants d ≥ 1, ϵ > 0, every Degree-d PTF over n variables can be approximated to accuracy eps by a constant Degree PTF that has integer weights of total magnitude O(nd). This weight bound is shown to be optimal up to logarithmic factors.
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a regularity lemma and low weight approximators for low Degree Polynomial threshold functions
Conference on Computational Complexity, 2010Co-Authors: Ilias Diakonikolas, Rocco A ServedioAbstract:We give a "regularity lemma" for Degree-d Polynomial threshold functions (PTFs) over the Boolean cube {−1,1}^n. Roughly speaking, this result shows that every Degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a "regular" PTF is a PTF sign(p(x)) where the influence of each variable on the Polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for any constants d >= 1, eps > 0, every Degree-d PTF over n variables can be approximated to accuracy eps by a constant Degree PTF that has integer weights of total magnitude O(n^d). This weight bound is shown to be optimal up to logarithmic factors.
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a regularity lemma and low weight approximators for low Degree Polynomial threshold functions
arXiv: Computational Complexity, 2009Co-Authors: Ilias Diakonikolas, Rocco A ServedioAbstract:We give a "regularity lemma" for Degree-d Polynomial threshold functions (PTFs) over the Boolean cube {-1,1}^n. This result shows that every Degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a "regular PTF is a PTF sign(p(x)) where the influence of each variable on the Polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for any constants d \geq 1, \eps \geq 0, every Degree-d PTF over n variables has can be approximated to accuracy eps by a constant-Degree PTF that has integer weights of total magnitude O(n^d). This weight bound is shown to be optimal up to constant factors.
Mehrdad Hosseini Zadeh - One of the best experts on this subject based on the ideXlab platform.
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VTC Fall - Automatic Vehicle Parallel Parking Design Using Fifth Degree Polynomial Path Planning
2011 IEEE Vehicular Technology Conference (VTC Fall), 2011Co-Authors: Shuwen Zhang, Mehrdad Simkani, Mehrdad Hosseini ZadehAbstract:Automatic vehicle parallel parking design and its related concerns about safety improvement remain some of the heated problems for automatic land vehicular control. This paper presents the calculation process of a parallel parking car's path planning and the algorithm development for its motion design based on a fifth-Degree Polynomial curve. In addition to the proposed algorithm for automatic vehicle parking, the minimum horizontal distance allowed for parking between a car and a parking spot is also investigated. The preliminary results show that the fifth Degree Polynomial path planning and the algorithm are well applied to the automatic parallel parking problem.
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Automatic Vehicle Parallel Parking Design Using Fifth Degree Polynomial Path Planning
2011 IEEE Vehicular Technology Conference (VTC Fall), 2011Co-Authors: Shuwen Zhang, Mehrdad Simkani, Mehrdad Hosseini ZadehAbstract:Automatic vehicle parallel parking design and its related concerns about safety improvement remain some of the heated problems for automatic land vehicular control. This paper presents the calculation process of a parallel parking car's path planning and the algorithm development for its motion design based on a fifth-Degree Polynomial curve. In addition to the proposed algorithm for automatic vehicle parking, the minimum horizontal distance allowed for parking between a car and a parking spot is also investigated. The preliminary results show that the fifth Degree Polynomial path planning and the algorithm are well applied to the automatic parallel parking problem.
Ma. Aracelia Alcorta G. - One of the best experts on this subject based on the ideXlab platform.
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Optimal risk-sensitive control for third Degree Polynomial systems
Proceedings of the 2010 American Control Conference, 2010Co-Authors: Ma. Aracelia Alcorta G., Michael Basin, Sonia Anguiano G. R., Yosefat Nava A.Abstract:The optimal exponential-quadratic control problem is considered for stochastic Gaussian systems with Polynomial third Degree drift terms and intensity parameters multiplying diffusion terms in the state equation. The closed-form optimal control algorithm is obtained using a quadratic value function as a solution to the corresponding Hamilton-Jacobi-Bellman equation. The performance of the obtained risk-sensitive regulator for stochastic third Degree Polynomial systems is verified in a numerical example, through comparing the exponential-quadratic criteria values for the optimal risk-sensitive control and third Degree control algorithms. The simulation results reveal strong advantages in favor of the designed risk-sensitive algorithm in regard to the final criteria values for all values of the parameter ε.
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Sub-optimal risk-sensitive filtering for third Degree Polynomial stochastic systems
2009 IEEE Control Applications (CCA) & Intelligent Control (ISIC), 2009Co-Authors: Ma. Aracelia Alcorta G., Michael Basin, Sonia G. Anguiano, Juan J. MaldonadoAbstract:The risk-sensitive filter design problem with respect to the exponential mean-square criterion is considered for stochastic Gaussian systems with Polynomial drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form suboptimal filtering algorithm is obtained by linearizing a nonlinear third Degree Polynomial system at the operating point and reducing the original problem to the optimal filter design for a first Degree Polynomial system. The reduced filtering problem is solved using quadratic value functions as solutions to the corresponding Fokker-Planck-Kolmogorov equation. The performance of the obtained risk-sensitive filter for stochastic third Degree Polynomial systems is verified in a numerical example against the mean-square optimal third Degree Polynomial filter and extended Kalman-Bucy filter, through comparing the exponential mean-square criteria values. The simulation results reveal strong advantages in favor of the designed risk-sensitive algorithm for large values of the intensity parameters.
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CCA/ISIC - Sub-optimal risk-sensitive filtering for third Degree Polynomial stochastic systems
2009 IEEE International Conference on Control Applications, 2009Co-Authors: Ma. Aracelia Alcorta G., Michael Basin, Sonia G. Anguiano, Juan J. MaldonadoAbstract:The risk-sensitive filter design problem with respect to the exponential mean-square criterion is considered for stochastic Gaussian systems with Polynomial drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form suboptimal filtering algorithm is obtained by linearizing a nonlinear third Degree Polynomial system at the operating point and reducing the original problem to the optimal filter design for a first Degree Polynomial system. The reduced filtering problem is solved using quadratic value functions as solutions to the corresponding Fokker-Planck-Kolmogorov equation. The performance of the obtained risk-sensitive filter for stochastic third Degree Polynomial systems is verified in a numerical example against the mean-square optimal third Degree Polynomial filter and extended Kalman-Bucy filter, through comparing the exponential mean-square criteria values. The simulation results reveal strong advantages in favor of the designed risk-sensitive algorithm for large values of the intensity parameters.
Paula Castro-tinttori - One of the best experts on this subject based on the ideXlab platform.
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EUSIPCO - Implementation of unbiased FIR filters with low-Degree Polynomial gains
2010Co-Authors: Oscar Ibarra-manzano, Yuriy S. Shmaliy, Nasser Kehtarnavaz, Issa Panahi, Paula Castro-tinttoriAbstract:We discuss implementation of the unbiased finite impulse response (FIR) filters. The transfer function and general block-diagram are presented for the l-Degree Polynomial FIR filter along with its fundamental properties in the z-transform domain. As a special results, we show a fundamental identity that is uniquely featured to such filters and can serve as an indicator of unbiasedness in filter design. For low-Degree gains, the transfer function is represented in simple closed forms and compact block-diagrams. An example of applications is given for filtering of time errors in a crystal clock.
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Implementation of unbiased FIR filters with low-Degree Polynomial gains
2010 18th European Signal Processing Conference, 2010Co-Authors: Oscar Ibarra-manzano, Yuriy S. Shmaliy, Nasser Kehtarnavaz, Issa Panahi, Paula Castro-tinttoriAbstract:We discuss implementation of the unbiased finite impulse response (FIR) filters. The transfer function and general block-diagram are presented for the l-Degree Polynomial FIR filter along with its fundamental properties in the z-transform domain. As a special results, we show a fundamental identity that is uniquely featured to such filters and can serve as an indicator of unbiasedness in filter design. For low-Degree gains, the transfer function is represented in simple closed forms and compact block-diagrams. An example of applications is given for filtering of time errors in a crystal clock.