Derandomization in Game-theoretic Probability

We give a general method for constructing a deterministic strategy of Reality from a randomized strategy in game-theoretic probability. The construction can be seen as derandomization in game-theoretic probability.Comment: 19 page

Pseudorandomness for Approximate Counting and Sampling

We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions. Our main technical contribution allows one to “boost” a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize Arthur-Merlin games (i.e., show AM = NP) are in fact all equivalent. We also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the “boosting” theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM. We observe that Cai's proof that S_2^P ⊆ PP⊆(NP) and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice

Quantified Derandomization of Linear Threshold Circuits

One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for $TC^0$, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for $TC^0$. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of $TC^0$ circuits of depth $d>2$. Our first main result is a quantified derandomization algorithm for $TC^0$ circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a $TC^0$ circuit $C$ over $n$ input bits with depth $d$ and $n^{1+\exp(-d)}$ wires, runs in almost-polynomial-time, and distinguishes between the case that $C$ rejects at most $2^{n^{1-1/5d}}$ inputs and the case that $C$ accepts at most $2^{n^{1-1/5d}}$ inputs. In fact, our algorithm works even when the circuit $C$ is a linear threshold circuit, rather than just a $TC^0$ circuit (i.e., $C$ is a circuit with linear threshold gates, which are stronger than majority gates). Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of $TC^0$, and would consequently imply that $NEXP\not\subseteq TC^0$. Specifically, if there exists a quantified derandomization algorithm that gets as input a $TC^0$ circuit with depth $d$ and $n^{1+O(1/d)}$ wires (rather than $n^{1+\exp(-d)}$ wires), runs in time at most $2^{n^{\exp(-d)}}$, and distinguishes between the case that $C$ rejects at most $2^{n^{1-1/5d}}$ inputs and the case that $C$ accepts at most $2^{n^{1-1/5d}}$ inputs, then there exists an algorithm with running time $2^{n^{1-\Omega(1)}}$ for standard derandomization of $TC^0$.Comment: Changes in this revision: An additional result (a PRG for quantified derandomization of depth-2 LTF circuits); rewrite of some of the exposition; minor correction

Fine-Grained Derandomization: From Problem-Centric to Resource-Centric Complexity

We show that popular hardness conjectures about problems from the field of fine-grained complexity theory imply structural results for resource-based complexity classes. Namely, we show that if either k-Orthogonal Vectors or k-CLIQUE requires n^{epsilon k} time, for some constant epsilon>1/2, to count (note that these conjectures are significantly weaker than the usual ones made on these problems) on randomized machines for all but finitely many input lengths, then we have the following derandomizations: - BPP can be decided in polynomial time using only n^alpha random bits on average over any efficient input distribution, for any constant alpha>0 - BPP can be decided in polynomial time with no randomness on average over the uniform distribution This answers an open question of Ball et al. (STOC \u2717) in the positive of whether derandomization can be achieved from conjectures from fine-grained complexity theory. More strongly, these derandomizations improve over all previous ones achieved from worst-case uniform assumptions by succeeding on all but finitely many input lengths. Previously, derandomizations from worst-case uniform assumptions were only know to succeed on infinitely many input lengths. It is specifically the structure and moderate hardness of the k-Orthogonal Vectors and k-CLIQUE problems that makes removing this restriction possible. Via this uniform derandomization, we connect the problem-centric and resource-centric views of complexity theory by showing that exact hardness assumptions about specific problems like k-CLIQUE imply quantitative and qualitative relationships between randomized and deterministic time. This can be either viewed as a barrier to proving some of the main conjectures of fine-grained complexity theory lest we achieve a major breakthrough in unconditional derandomization or, optimistically, as route to attain such derandomizations by working on very concrete and weak conjectures about specific problems

Pseudorandomness and Average-Case Complexity via Uniform Reductions

Impagliazzo and Wigderson (36th FOCS, 1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP [not equal to] BPP). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worst-case hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. In this paper: 1. We obtain an optimal worst-case to average-case connection for EXP: if EXP is not a subset of BPTIME(t(n)), EXP has problems that cannot be solved on a fraction 1/2 +1/t'(n) of the inputs by BPTIME(t'(n)) algorithms, for t'=t^{\Omega(1)}. 2. We exhibit a PSPACE-complete self-correctible and downward self-reducible problem. This slightly simplifies and strengthens the proof of Impaglaizzo and Wigderson, which used a a #P-complete problem with these properties. 3. We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via "black-box" uniform reductions.Engineering and Applied Science

On Closeness to k-Wise Uniformity

A probability distribution over {-1, 1}^n is (epsilon, k)-wise uniform if, roughly, it is epsilon-close to the uniform distribution when restricted to any k coordinates. We consider the problem of how far an (epsilon, k)-wise uniform distribution can be from any globally k-wise uniform distribution. We show that every (epsilon, k)-wise uniform distribution is O(n^{k/2}epsilon)-close to a k-wise uniform distribution in total variation distance. In addition, we show that this bound is optimal for all even k: we find an (epsilon, k)-wise uniform distribution that is Omega(n^{k/2}epsilon)-far from any k-wise uniform distribution in total variation distance. For k=1, we get a better upper bound of O(epsilon), which is also optimal. One application of our closeness result is to the sample complexity of testing whether a distribution is k-wise uniform or delta-far from k-wise uniform. We give an upper bound of O(n^{k}/delta^2) (or O(log n/delta^2) when k = 1) on the required samples. We show an improved upper bound of O~(n^{k/2}/delta^2) for the special case of testing fully uniform vs. delta-far from k-wise uniform. Finally, we complement this with a matching lower bound of Omega(n/delta^2) when k = 2. Our results improve upon the best known bounds from [Alon et al., 2007], and have simpler proofs

Satisfiability and Derandomization for Small Polynomial Threshold Circuits

A polynomial threshold function (PTF) is defined as the sign of a polynomial p : {0,1}^n ->R. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth. - Satisfiability (#SAT). We give the first zero-error randomized algorithm faster than exhaustive search that counts the number of satisfying assignments of a given constant-depth circuit with a super-linear number of wires whose gates are s-sparse PTFs, for s almost quadratic in the input size of the circuit; here a PTF is called s-sparse if its underlying polynomial has at most s monomials. More specifically, we show that, for any large enough constant c, given a depth-d circuit with (n^{2-1/c})-sparse PTF gates that has at most n^{1+epsilon_d} wires, where epsilon_d depends only on c and d, the number of satisfying assignments of the circuit can be computed in randomized time 2^{n-n^{epsilon_d}} with zero error. This generalizes the result by Chen, Santhanam and Srinivasan (CCC, 2016) who gave a SAT algorithm for constant-depth circuits of super-linear wire complexity with linear threshold function (LTF) gates only. - Quantified derandomization. The quantified derandomization problem, introduced by Goldreich and Wigderson (STOC, 2014), asks to compute the majority value of a given Boolean circuit, under the promise that the minority-value inputs to the circuit are very few. We give a quantified derandomization algorithm for constant-depth PTF circuits with a super-linear number of wires that runs in quasi-polynomial time. More specifically, we show that for any sufficiently large constant c, there is an algorithm that, given a degree-Delta PTF circuit C of depth d with n^{1+1/c^d} wires such that C has at most 2^{n^{1-1/c}} minority-value inputs, runs in quasi-polynomial time exp ((log n)^{O (Delta^2)}) and determines the majority value of C. (We obtain a similar quantified derandomization result for PTF circuits with n^{Delta}-sparse PTF gates.) This extends the recent result of Tell (STOC, 2018) for constant-depth LTF circuits of super-linear wire complexity. - Pseudorandom generators. We show how the classical Nisan-Wigderson (NW) generator (JCSS, 1994) yields a nontrivial pseudorandom generator for PTF circuits (of unrestricted depth) with sub-linearly many gates. As a corollary, we get a PRG for degree-Delta PTFs with the seed length exp (sqrt{Delta * log n})* log^2(1/epsilon)

Sparse multivariate polynomial interpolation in the basis of Schubert polynomials

Schubert polynomials were discovered by A. Lascoux and M. Sch\"utzenberger in the study of cohomology rings of flag manifolds in 1980's. These polynomials generalize Schur polynomials, and form a linear basis of multivariate polynomials. In 2003, Lenart and Sottile introduced skew Schubert polynomials, which generalize skew Schur polynomials, and expand in the Schubert basis with the generalized Littlewood-Richardson coefficients. In this paper we initiate the study of these two families of polynomials from the perspective of computational complexity theory. We first observe that skew Schubert polynomials, and therefore Schubert polynomials, are in \CountP (when evaluating on non-negative integral inputs) and \VNP. Our main result is a deterministic algorithm that computes the expansion of a polynomial $f$ of degree $d$ in $\Z[x_1, \dots, x_n]$ in the basis of Schubert polynomials, assuming an oracle computing Schubert polynomials. This algorithm runs in time polynomial in $n$, $d$, and the bit size of the expansion. This generalizes, and derandomizes, the sparse interpolation algorithm of symmetric polynomials in the Schur basis by Barvinok and Fomin (Advances in Applied Mathematics, 18(3):271--285). In fact, our interpolation algorithm is general enough to accommodate any linear basis satisfying certain natural properties. Applications of the above results include a new algorithm that computes the generalized Littlewood-Richardson coefficients.Comment: 20 pages; some typos correcte