Li-Yang Tan

New Algorithms and Lower Bounds for Monotonicity Testing

We consider the problem of testing whether an unknown Boolean function f : {- 1, 1}ⁿ → {-1, 1} is monotone versus ε-far from every monotone function. The two main results of this paper are a new lower bound and a new algorithm for this well-studied problem. Lower bound: We prove an Ω̅(n^1/5) lower bound on the query complexity of any non-adaptive two-sided error algorithm for testing whether an unknown Boolean function f is monotone versus constant-far from monotone. This gives an exponential improvement on the previous lower bound of Ω(log n) due to Fischer et al. [1]. We show that the same lower bound holds for monotonicity testing of Boolean-valued functions over hypergrid domains {1,···, m}ⁿ for all m ≥ 2. Upper bound: We present an O(n^5/6) poly(1/ε)-query algorithm that tests whether an unknown Boolean function f is monotone versus ε-far from monotone. Our algorithm, which is non-adaptive and makes one-sided error, is a modified version of the algorithm of Chakrabarty and Seshadhri[2], which makes O(n^7/8) poly(1/ε) queries.

Average Sensitivity and Noise Sensitivity of Polynomial Threshold Functions

We give the first nontrivial upper bounds on the Boolean average sensitivity and noise sensitivity of degree-

An Average-Case Depth Hierarchy Theorem for Boolean Circuits

d

Learning Circuits with few Negations.

polynomial threshold functions (PTFs). Our bound on the Boolean average sensitivity of PTFs represents the first progress toward the resolution of a conjecture of Gotsman and Linial [Combinatorica, 14 (1994), pp. 35--50], which states that the symmetric function slicing the middle

Adaptivity helps for testing juntas

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A composition theorem for the fourier entropy-influence conjecture

layers of the Boolean hypercube has the highest average sensitivity of all degree-

Near-optimal small-depth lower bounds for small distance connectivity

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Poly-logarithmic Frege depth lower bounds via an expander switching lemma

PTFs. Via the

Settling the Query Complexity of Non-Adaptive Junta Testing

L_1

What Circuit Classes Can Be Learned with Non-Trivial Savings?

polynomial regression algorithm of Kalai et al. [SIAM J. Comput., 37 (2008), pp. 1777--1805], our bound on Boolean noise sensitivity yields the first polynomial-time agnostic learning algorithm for the broad class of constant-degree PTFs under the uniform distribution. To obtain our bound on the Boolean average sensitivity of PTFs, we generalize the “critical-index” machinery of [R. Servedio, Comput. Complexity, 16 (2007), pp. 180--209] (which in that work applies to halfspaces, i.e., degree-1 PTFs) to general...