Suppose we have a two-party communication protocol for f which allows the parties to make queries to an oracle computing g; for example, they may query an Equality oracle. To translate this protocol into a randomized protocol, we must replace the oracle with a randomized subroutine for solving g. If q queries are made, the standard technique requires that we boost the error of each subroutine down to O(1/q), leading to communication complexity which grows as q log q. For which oracles g can this naïve boosting technique be improved? We focus on the oracles which can be computed by constant-cost randomized protocols, and show that the naïve boosting strategy can be improved for the Equality oracle but not the 1-Hamming Distance oracle. Two surprising consequences are (1) a new example of a problem where the cost of computing k independent copies grows superlinear in k, drastically simplifying the only previous example due to Blais & Brody (CCC 2019); and (2) a new proof that Equality is not complete for the class of constant-cost randomized communication (Harms, Wild, & Zamaraev, STOC 2022; Hambardzumyan, Hatami, & Hatami, Israel Journal of Mathematics 2022).
Suppose we have a two-party communication protocol for f which allows the parties to make queries to an oracle computing g; for example, they may query an Equality oracle. To translate this protocol into a randomized protocol, we must replace the oracle with a randomized subroutine for solving g. If q queries are made, the standard technique requires that we boost the error of each subroutine down to O(1/q), leading to communication complexity which grows as q log q. For which oracles g can this naïve boosting technique be improved? We focus on the oracles which can be computed by constant-cost randomized protocols, and show that the naïve boosting strategy can be improved for the Equality oracle but not the 1-Hamming Distance oracle. Two surprising consequences are (1) a new example of a problem where the cost of computing k independent copies grows superlinear in k, drastically simplifying the only previous example due to Blais & Brody (CCC 2019); and (2) a new proof that Equality is not complete for the class of constant-cost randomized communication (Harms, Wild, & Zamaraev, STOC 2022; Hambardzumyan, Hatami, & Hatami, Israel Journal of Mathematics 2022).
Suppose we have a two-party communication protocol for f which allows the parties to make queries to an oracle computing g; for example, they may query an Equality oracle. To translate this protocol into a randomized protocol, we must replace the oracle with a randomized subroutine for solving g. If q queries are made, the standard technique requires that we boost the error of each subroutine down to O(1/q), leading to communication complexity which grows as qlogq. For which oracles g can this naive boosting technique be improved? We focus on the oracles which can be computed by constant-cost randomized protocols, and show that the naive boosting strategy can be improved for the Equality oracle but not the 1-Hamming Distance oracle. Two surprising consequences are (1) a new example of a problem where the cost of computing k independent copies grows superlinear in k, drastically simplifying the only previous example due to Blais & Brody (CCC 2019); and (2) a new proof that Equality is not complete for the class of constant-cost randomized communication (Harms, Wild, & Zamaraev, STOC 2022; Hambardzumyan, Hatami, & Hatami, Israel Journal of Mathematics 2022).
The unifying topic of this thesis is the relation of randomness to computation. We study two different facets of this: the hardness of sampling from probability distributions and the extent of usefulness of randomness in communication complexity. Hardness of Sampling. A circuit samples the distribution of its output on a uniformly random input. What distributions are samplable with small circuits? What about local circuits (NC0)? We make two contributions in this direction. The first is to show that sampling uniformly from the set of Hamming-weight-k n-bit string is hard for NC0-circuits (and even constant-width DNFs) as long as k=o(n). The second is to show that sampling a uniformly random permutation of [n] elements requires polylog(n) adaptive queries to a sequence of uniform input symbols from [n]---adaptive cell probes. These results refine our understanding of sampling by constant-depth de Morgan circuits (AC0). Randomness in Communication. The classical example separating deterministic and randomized communication is the Equality problem: Alice and Bob are to decide whether their n-bit inputs are equal. It requires transmitting n+1 in the deterministic model, but is solvable with a constant-cost protocol using public randomness. An active line of research found many non-trivial problems that are solvable with just constant-cost randomized communication [Harms-Wild-Zamaraev, STOC 2022; Hambardzumyan-Hatami-Hatami, Isr. J. Math.; Fang-Hambardzumyan-Harms-Hatami, STOC 2024; Fang-Göös-Harms-Hatami, STOC 2025]. All of those problems can be solved with polylog(n) queries to Equality oracle. Our first contribution is to exhibit a problem that is solvable with constant-cost randomized communication, but requires polynomially many calls to Equality oracle. This resolves an open problem in the survey [Hatami-Hatami, ACM SIGACT News]. Our second contribution is to investigate when the randomized communication complexity can be localized or condensed: for a matrix of randomized communication cost k, can we find its 2^k-by-2^k submatrix that retains communication cost at least linear in k$? In general this is impossible [Hambardzumyan-Hatami-Hatami, Discrete Appl. Math.]. We show, however, that in an important case it can be done: we condense randomized communication cost of sink-of-xor function---a recent counterexample to log-approximate-rank conjecture by [Chattopadhyay-Mande-Sherif, J. ACM], and thereby quantitatively improve the counterexample.
The canonical problem that gives an exponential separation between deterministic and randomized communication complexity in the classical two-party communication model is "Equality". In this work we show that even allowing access to an "Equality" oracle, deterministic protocols remain exponentially weaker than randomized ones. More precisely, we exhibit a total function on n bits with randomized one-sided communication complexity O(log n), but such that every deterministic protocol with access to "Equality" oracle needs Omega(n) cost to compute it. Additionally we exhibit a natural and strict infinite hierarchy within BPP, starting with the class P^{EQ} at its bottom.
In an influential paper, Linial and Shraibman (STOC '07) introduced the factorization norm as a powerful tool for proving lower bounds against randomized and quantum communication complexities. They showed that the logarithm of the approximate γ₂-factorization norm is a lower bound for these parameters and asked whether a stronger lower bound that replaces approximate γ₂ norm with the γ₂ norm holds. We answer the question of Linial and Shraibman in the negative by exhibiting a 2ⁿ×2ⁿ Boolean matrix with γ₂ norm 2^Ω(n) and randomized communication complexity O(log n). As a corollary, we recover the recent result of Chattopadhyay, Lovett, and Vinyals (CCC '19) that deterministic protocols with access to an Equality oracle are exponentially weaker than (one-sided error) randomized protocols. In fact, as a stronger consequence, our result implies an exponential separation between the power of unambiguous nondeterministic protocols with access to Equality oracle and (one-sided error) randomized protocols, which answers a question of Pitassi, Shirley, and Shraibman (ITSC '23). Our result also implies a conjecture of Sherif (Ph.D. thesis) that the γ₂ norm of the Integer Inner Product function (IIP) in dimension 3 or higher is exponential in its input size.
The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models. To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations: Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most õ(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021]. Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].
We study an extension of the standard two-party communication model in which Alice and Bob hold probability distributions p and q over domains X and Y, respectively. Their goal is to estimate Ex∼p,y∼q[f(x,y)] to within additive error ε for a bounded function f, known to both parties. We refer to this as the distributed estimation problem. Special cases of this problem arise in a variety of areas including sketching, databases and learning. Our goal is to understand how the required communication scales with the communication complexity of f and the error parameter ε. The random sampling approach -- estimating the mean by averaging f over O(1/ε2) random samples -- requires O(R(f)/ε2) total communication, where R(f) is the randomized communication complexity of f. We design a new debiasing protocol which improves the dependence on 1/ε to be linear instead of quadratic. Additionally we show better upper bounds for several special classes of functions, including the Equality and Greater-than functions. We introduce lower bound techniques based on spectral methods and discrepancy, and show the optimality of many of our protocols: the debiasing protocol is tight for general functions, and that our protocols for the equality and greater-than functions are also optimal. Furthermore, we show that among full-rank Boolean functions, Equality is essentially the easiest.