Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
On the Fourier spectrum of monotone functions
Journal of the ACM (JACM)
Information Processing Letters
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On the noise sensitivity of monotone functions
Random Structures & Algorithms
Learning intersections and thresholds of halfspaces
Journal of Computer and System Sciences - Special issue on FOCS 2002
Noise stability of functions with low in.uences invariance and optimality
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Every decision tree has an in.uential variable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
On the fourier tails of bounded functions over the discrete cube
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Every Linear Threshold Function has a Low-Weight Approximator
Computational Complexity
Learning Monotone Decision Trees in Polynomial Time
SIAM Journal on Computing
Agnostically Learning Halfspaces
SIAM Journal on Computing
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Agnostically learning decision trees
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
On agnostic boosting and parity learning
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Learning Geometric Concepts via Gaussian Surface Area
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Randomly supported independence and resistance
Proceedings of the forty-first annual ACM symposium on Theory of computing
Improved Approximation of Linear Threshold Functions
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
Bounded Independence Fools Halfspaces
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
An invariance principle for polytopes
Proceedings of the forty-second ACM symposium on Theory of computing
Approximating the influence of monotone boolean functions in O(√n) query complexity
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Concentration and moment inequalities for polynomials of independent random variables
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Hardness results for agnostically learning low-degree polynomial threshold functions
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Approximating the Influence of Monotone Boolean Functions in O(√n) Query Complexity
ACM Transactions on Computation Theory (TOCT)
An invariance principle for polytopes
Journal of the ACM (JACM)
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We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of degree-d polynomial threshold functions (PTFs). These bounds hold both for PTFs over the Boolean hypercube {-1,1}n and for PTFs over Rn under the standard n-dimensional Gaussian distribution N(0,In). Our bound on the Boolean average sensitivity of PTFs represents progress towards the resolution of a conjecture of Gotsman and Linial [17], which states that the symmetric function slicing the middle d layers of the Boolean hypercube has the highest average sensitivity of all degree-d PTFs. Via the L1 polynomial regression algorithm of Kalai et al. [22], our bounds on Gaussian and Boolean noise sensitivity yield polynomial-time agnostic learning algorithms for the broad class of constant-degree PTFs under these input distributions. The main ingredients used to obtain our bounds on both average and noise sensitivity of PTFs in the Gaussian setting are tail bounds and anti-concentration bounds on low-degree polynomials in Gaussian random variables [20, 7]. To obtain our bound on the Boolean average sensitivity of PTFs, we generalize the "critical-index" machinery of [37] (which in that work applies to halfspaces, i.e. degree-1 PTFs) to general PTFs. Together with the "invariance principle" of [30], this lets us extend our techniques from the Gaussian setting to the Boolean setting. Our bound on Boolean noise sensitivity is achieved via a simple reduction from upper bounds on average sensitivity of Boolean PTFs to corresponding bounds on noise sensitivity.