arXiv subjects

Pooya Hatami

Publications and source records attributed to Pooya Hatami.

At least 19 recordsLinked to original sources

arXiv2026

Sign-Rank, Index, and List Replicability: Connections and Separations

In learning theory, the sign-rank of a binary concept class captures the smallest dimension in which it can be represented by points and halfspaces. Despite tremendous interest, lower bounds on sign-rank are notoriously difficult to come by. Two recent approaches to the problem establish lower bounds on sign-rank by measures that are easier to analyze: the $\mathbb{Z}_2$-index and the list replicability number. We order these measures, showing that the $\mathbb{Z}_2$-index is upper-bounded by a linear function of the list replicability number. As a main consequence, we obtain a strong separation between sign-rank and $\mathbb{Z}_2$-index, thereby resolving a question of Frick, Hosseini, and Vasileuski. This motivates a thorough study of list replicability, the stronger of the two lower-bounding measures. We establish upper bounds on the list replicability number by two combinatorial measures: height and minimum star number. We also prove a fundamental composition result, showing that the product of two concept classes has list replicability number bounded by the sum of the list replicability numbers of the two classes.

cs.LG↗

arXiv2026

Tight list replicability bounds via a novel sphere covering theorem

In recent years, list replicability has emerged as a framework for formalizing reproducibility in learning theory. A central question is how the required list size relates to the accuracy parameter and natural complexity measures of the hypothesis class. To achieve sharp bounds on list replicability, we prove a novel topological sphere covering theorem, derived from the Borsuk-Ulam theorem. Specifically, if the $d$-sphere is covered by open sets, each of which lies in an open hemisphere, then $d+1$ of these sets must have a common intersection. Using this result, we obtain a sharp bound on the relationship between list size and accuracy for VC classes. We also show that for large-margin half-spaces, provided the margin is not too large, the optimal list size equals the ambient dimension. However, when the margin is taken to be very large, we devise a replicable algorithm achieving the minimal list size of $\lceil d/2 \rceil + 1$.

cs.LG↗

arXiv2025

Simplicial covering dimension of extremal concept classes

Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue covering) to binary concept classes. The topological space naturally associated with a concept class is its space of realizable distributions. The loss function and the class itself induce a simplicial structure on this space, with respect to which we define a simplicial covering dimension. We prove that for finite concept classes, this simplicial covering dimension exactly characterizes the list replicability number (equivalently, global stability) in PAC learning. This connection allows us to apply tools from classical dimension theory to compute the exact list replicability number of the broad family of extremal concept classes.

cs.LG↗

arXiv2025

Borsuk-Ulam and Replicable Learning of Large-Margin Halfspaces

We prove that the list replicability number of $d$-dimensional $\gamma$-margin half-spaces satisfies \[ \frac{d}{2}+1 \le \mathrm{LR}(H^d_\gamma) \le d, \] which grows with dimension. This resolves several open problems: $\bullet$ Every disambiguation of infinite-dimensional large-margin half-spaces to a total concept class has unbounded Littlestone dimension, answering an open question of Alon, Hanneke, Holzman, and Moran (FOCS '21). $\bullet$ Every disambiguation of the Gap Hamming Distance problem in the large gap regime has unbounded public-coin randomized communication complexity. This answers an open question of Fang, G\"o\"os, Harms, and Hatami (STOC '25). $\bullet$ There is a separation of $O(1)$ vs $\omega(1)$ between randomized and pseudo-deterministic communication complexity. $\bullet$ The maximum list-replicability number of any finite set of points and homogeneous half-spaces in $d$-dimensional Euclidean space is $d$, resolving a problem of Chase, Moran, and Yehudayoff (FOCS '23). $\bullet$ There exists a partial concept class with Littlestone dimension $1$ such that all its disambiguations have infinite Littlestone dimension. This resolves a problem of Cheung, H. Hatami, P. Hatami, and Hosseini (ICALP '23). Our lower bound follows from a topological argument based on a local Borsuk-Ulam theorem. For the upper bound, we construct a list-replicable learning rule using the generalization properties of SVMs.

cs.LG↗

arXiv2025

Stability and List-Replicability for Agnostic Learners

Two seminal papers--Alon, Livni, Malliaris, Moran (STOC 2019) and Bun, Livni, and Moran (FOCS 2020)--established the equivalence between online learnability and globally stable PAC learnability in binary classification. However, Chase, Chornomaz, Moran, and Yehudayoff (STOC 2024) recently showed that this equivalence does not hold in the agnostic setting. Specifically, they proved that in the agnostic setting, only finite hypothesis classes are globally stable learnable. Therefore, agnostic global stability is too restrictive to capture interesting hypothesis classes. To address this limitation, Chase et al. introduced two relaxations of agnostic global stability. In this paper, we characterize the classes that are learnable under their proposed relaxed conditions, resolving the two open problems raised in their work. First, we prove that in the setting where the stability parameter can depend on the excess error (the gap between the learner's error and the best achievable error by the hypothesis class), agnostic stability is fully characterized by the Littlestone dimension. Consequently, as in the realizable case, this form of learnability is equivalent to online learnability. As part of the proof of this theorem, we strengthen the celebrated result of Bun et al. by showing that classes with infinite Littlestone dimension are not stably PAC learnable, even if we allow the stability parameter to depend on the excess error. For the second relaxation proposed by Chase et al., we prove that only finite hypothesis classes are globally stable learnable, even if we restrict the agnostic setting to distributions with small population loss.

cs.LG↗

arXiv2024

Constant-Cost Communication is not Reducible to k-Hamming Distance

Every known communication problem whose randomized communication cost is constant (independent of the input size) can be reduced to $k$-Hamming Distance, that is, solved with a constant number of deterministic queries to some $k$-Hamming Distance oracle. We exhibit the first examples of constant-cost problems which cannot be reduced to $k$-Hamming Distance. To prove this separation, we relate it to a natural coding-theoretic question. For $f : \{2, 4, 6\} \to \mathbb{N}$, we say an encoding function $E : \{0, 1\}^n \to \{0, 1\}^m$ is an $f$-code if it transforms Hamming distances according to $\mathrm{dist}(E(x), E(y)) = f(\mathrm{dist}(x, y))$ whenever $f$ is defined. We prove that, if there exist $f$-codes for infinitely many $n$, then $f$ must be affine: $f(4) = (f(2) + f(6))/2$.

cs.CC↗

arXiv2024

Hilbert Functions and Low-Degree Randomness Extractors

For $S\subseteq \mathbb{F}^n$, consider the linear space of restrictions of degree-$d$ polynomials to $S$. The Hilbert function of $S$, denoted $\mathrm{h}_S(d,\mathbb{F})$, is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets $S$ of arbitrary finite grids in $\mathbb{F}^n$ with a fixed size $|S|$. We achieve this by proving that this value coincides with a combinatorial quantity, namely the smallest number of low Hamming weight points in a down-closed set of size $|S|$. Understanding the smallest values of Hilbert functions is closely related to the study of degree-$d$ closure of sets, a notion introduced by Nie and Wang (Journal of Combinatorial Theory, Series A, 2015). We use bounds on the Hilbert function to obtain a tight bound on the size of degree-$d$ closures of subsets of $\mathbb{F}_q^n$, which answers a question posed by Doron, Ta-Shma, and Tell (Computational Complexity, 2022). We use the bounds on the Hilbert function and degree-$d$ closure of sets to prove that a random low-degree polynomial is an extractor for samplable randomness sources. Most notably, we prove the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits. Until recently, even the existence of arbitrary deterministic extractors for such sources was not known.

cs.CC↗

arXiv2024

No Complete Problem for Constant-Cost Randomized Communication

We prove that the class of communication problems with public-coin randomized constant-cost protocols, called $BPP^0$, does not contain a complete problem. In other words, there is no randomized constant-cost problem $Q \in BPP^0$, such that all other problems $P \in BPP^0$ can be computed by a constant-cost deterministic protocol with access to an oracle for $Q$. We also show that the $k$-Hamming Distance problems form an infinite hierarchy within $BPP^0$. Previously, it was known only that Equality is not complete for $BPP^0$. We introduce a new technique, using Ramsey theory, that can prove lower bounds against arbitrary oracles in $BPP^0$, and more generally, we show that $k$-Hamming Distance matrices cannot be expressed as a Boolean combination of any constant number of matrices which forbid large Greater-Than subproblems.

cs.CC↗

arXiv2024

Structure in Communication Complexity and Constant-Cost Complexity Classes

Several theorems and conjectures in communication complexity state or speculate that the complexity of a matrix in a given communication model is controlled by a related analytic or algebraic matrix parameter, e.g., rank, sign-rank, discrepancy, etc. The forward direction is typically easy as the structural implications of small complexity often imply a bound on some matrix parameter. The challenge lies in establishing the reverse direction, which requires understanding the structure of Boolean matrices for which a given matrix parameter is small or large. We will discuss several research directions that align with this overarching theme.

cs.CC↗

arXiv2023

Online Learning and Disambiguations of Partial Concept Classes

In a recent article, Alon, Hanneke, Holzman, and Moran (FOCS '21) introduced a unifying framework to study the learnability of classes of partial concepts. One of the central questions studied in their work is whether the learnability of a partial concept class is always inherited from the learnability of some ``extension'' of it to a total concept class. They showed this is not the case for PAC learning but left the problem open for the stronger notion of online learnability. We resolve this problem by constructing a class of partial concepts that is online learnable, but no extension of it to a class of total concepts is online learnable (or even PAC learnable).

cs.LG↗

arXiv2021

The Implicit Graph Conjecture is False

An efficient implicit representation of an $n$-vertex graph $G$ in a family $\mathcal{F}$ of graphs assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency between every pair of vertices can be determined only as a function of their codes. This function can depend on the family but not on the individual graph. Every family of graphs admitting such a representation contains at most $2^{O(n\log(n))}$ graphs on $n$ vertices, and thus has at most factorial speed of growth. The Implicit Graph Conjecture states that, conversely, every hereditary graph family with at most factorial speed of growth admits an efficient implicit representation. We refute this conjecture by establishing the existence of hereditary graph families with factorial speed of growth that require codes of length $n^{\Omega(1)}$.

math.CO↗

arXiv2021

A counter-example to the probabilistic universal graph conjecture via randomized communication complexity

We refute the Probabilistic Universal Graph Conjecture of Harms, Wild, and Zamaraev, which states that a hereditary graph property admits a constant-size probabilistic universal graph if and only if it is stable and has at most factorial speed. Our counter-example follows from the existence of a sequence of $n \times n$ Boolean matrices $M_n$, such that their public-coin randomized communication complexity tends to infinity, while the randomized communication complexity of every $\sqrt{n}\times \sqrt{n}$ submatrix of $M_n$ is bounded by a universal constant.

cs.CC↗

arXiv2019

Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions

The seminal result of Kahn, Kalai and Linial shows that a coalition of $O(\frac{n}{\log n})$ players can bias the outcome of any Boolean function $\{0,1\}^n \to \{0,1\}$ with respect to the uniform measure. We extend their result to arbitrary product measures on $\{0,1\}^n$, by combining their argument with a completely different argument that handles very biased coordinates. We view this result as a step towards proving a conjecture of Friedgut, which states that Boolean functions on the continuous cube $[0,1]^n$ (or, equivalently, on $\{1,\dots,n\}^n$) can be biased using coalitions of $o(n)$ players. This is the first step taken in this direction since Friedgut proposed the conjecture in 2004. Russell, Saks and Zuckerman extended the result of Kahn, Kalai and Linial to multi-round protocols, showing that when the number of rounds is $o(\log^* n)$, a coalition of $o(n)$ players can bias the outcome with respect to the uniform measure. We extend this result as well to arbitrary product measures on $\{0,1\}^n$. The argument of Russell et al. relies on the fact that a coalition of $o(n)$ players can boost the expectation of any Boolean function from $\epsilon$ to $1-\epsilon$ with respect to the uniform measure. This fails for general product distributions, as the example of the AND function with respect to $\mu_{1-1/n}$ shows. Instead, we use a novel boosting argument alongside a generalization of our first result to arbitrary finite ranges.

cs.DM↗

arXiv2018

On Multilinear Forms: Bias, Correlation, and Tensor Rank

In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for multilinear forms and tensors. 1. Correlation bounds : We show that a random $d$-linear form has exponentially low correlation with low-degree polynomials. More precisely, for $d \ll 2^{o(k)}$, we show that a random $d$-linear form $f(X_1,X_2, \dots, X_d) : \left(GF(2)^{k}\right)^d \rightarrow GF(2)$ has correlation $2^{-k(1-o(1))}$ with any polynomial of degree at most $d/10$. This result is proved by giving near-optimal bounds on the bias of random $d$-linear form, which is in turn proved by giving near-optimal bounds on the probability that a random rank-$t$ $d$-linear form is identically zero. 2. Tensor-rank vs Bias : We show that if a $d$-dimensional tensor has small rank, then the bias of the associated $d$-linear form is large. More precisely, given any $d$-dimensional tensor $$T :\underbrace{[k]\times \ldots [k]}_{\text{$d$ times}}\to GF(2)$$ of rank at most $t$, the bias of the associated $d$-linear form $$f_T(X_1,\ldots,X_d) := \sum_{(i_1,\dots,i_d) \in [k]^d} T(i_1,i_2,\ldots, i_d) X_{1,i_1}\cdot X_{1,i_2}\cdots X_{d,i_d}$$ is at least $\left(1-\frac1{2^{d-1}}\right)^t$. The above bias vs tensor-rank connection suggests a natural approach to proving nontrivial tensor-rank lower bounds for $d=3$. In particular, we use this approach to prove that the finite field multiplication tensor has tensor rank at least $3.52 k$ matching the best known lower bound for any explicit tensor in three dimensions over $GF(2)$.

cs.CC↗

arXiv2018

An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean Function

We prove that there is a constant $C\leq 6.614$ such that every Boolean function of degree at most $d$ (as a polynomial over $\mathbb{R}$) is a $C\cdot 2^d$-junta, i.e. it depends on at most $C\cdot 2^d$ variables. This improves the $d\cdot 2^{d-1}$ upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)]. Our proof uses a new weighting scheme where we assign weights to variables based on the highest degree monomial they appear on. The bound of $C\cdot 2^d$ is tight up to the constant $C$ as a lower bound of $2^d-1$ is achieved by a read-once decision tree of depth $d$. We slightly improve the lower bound by constructing, for each positive integer $d$, a function of degree $d$ with $3\cdot 2^{d-1}-2$ relevant variables. A similar construction was independently observed by Shinkar and Tal.

math.CO↗

arXiv2016

Low-Sensitivity Functions from Unambiguous Certificates

We provide new query complexity separations against sensitivity for total Boolean functions: a power $3$ separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power $2.22$ separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a seemingly unrelated measure called one-sided unambiguous certificate complexity ($UC_{min}$). We also show that $UC_{min}$ is lower-bounded by fractional block sensitivity, which means we cannot use these techniques to get a super-quadratic separation between $bs(f)$ and $s(f)$. We also provide a quadratic separation between the tree-sensitivity and decision tree complexity of Boolean functions, disproving a conjecture of Gopalan, Servedio, Tal, and Wigderson (CCC 2016). Along the way, we give a power $1.22$ separation between certificate complexity and one-sided unambiguous certificate complexity, improving the power $1.128$ separation due to G\"o\"os (FOCS 2015). As a consequence, we obtain an improved $\Omega(\log^{1.22} n)$ lower-bound on the co-nondeterministic communication complexity of the Clique vs. Independent Set problem.

cs.CC↗

arXiv2015

On the Structure of Quintic Polynomials

We study the structure of bounded degree polynomials over finite fields. Haramaty and Shpilka [STOC 2010] showed that biased degree three or four polynomials admit a strong structural property. We confirm that this is the case for degree five polynomials also. Let $\mathbb{F}=\mathbb{F}_q$ be a prime field. [1.] Suppose $f:\mathbb{F}^n\rightarrow \mathbb{F}$ is a degree five polynomial with bias(f)=\delta. Then f can be written in the form $f= \sum_{i=1}^{c} G_i H_i + Q$, where $G_i$ and $H_i$s are nonconstant polynomials satisfying $deg(G_i)+deg(H_i)\leq 5$ and $Q$ is a degree $\leq 4$ polynomial. Moreover, $c=c(\delta)$ does not depend on $n$ and $q$. [2.] Suppose $f:\mathbb{F}^n\rightarrow \mathbb{F}$ is a degree five polynomial with $bias(f)=\delta$. Then there exists an $\Omega_\delta(n)$ dimensional affine subspace $V$ of $\mathbb{F}^n$ such that $f$ restricted to $V$ is a constant. Cohen and Tal [Random 2015] proved that biased polynomials of degree at most four are constant on a subspace of dimension $\Omega(n)$. Item [2.] extends this to degree five polynomials. A corollary to Item [2.] is that any degree five affine disperser for dimension $k$ is also an affine extractor for dimension $O(k)$. We note that Item [2.] cannot hold for degrees six or higher. We obtain our results for degree five polynomials as a special case of structure theorems that we prove for biased degree d polynomials when $d<|\mathbb{F}|+4$. While the $d<|\mathbb{F}|+4$ assumption seems very restrictive, we note that prior to our work such structure theorems were only known for $d<|\mathbb{F}|$ by Green and Tao [Contrib. Discrete Math. 2009] and Bhowmick and Lovett [arXiv:1506.02047]. Using algorithmic regularity lemmas for polynomials developed by Bhattacharyya, et. al. [SODA 2015], we show that whenever such a strong structure exists, it can be found algorithmically in time polynomial in n.

math.CO↗

arXiv2015

A characterization of functions with vanishing averages over products of disjoint sets

Given $α_1,...,α_m \in (0,1)$, we characterize all integrable functions $f:[0,1]^m \to \mathbb{C}$ satisfying $\int_{A_1 \times ...\times A_m} f =0$ for any collection of disjoint sets $A_1,...,A_m \subseteq [0,1]$ of respective measures $α_1,...,α_m$. We use this characterization to settle some of the conjectures in [S. Janson and V. Sós, More on quasi-random graphs, subgraph counts and graph limits, arXiv:1405.6808].

math.CO↗