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 $\lceil d/2 \rceil + 1$.

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.

We prove that the list replicability number of d-dimensional γ-margin half-spaces satisfies d/2+1 ≤ LR(Hγd) ≤ d. In particular, it grows with the dimension. Our lower bound uses a topological argument based on a local Borsuk–Ulam theorem. Our upper bound is proved by constructing a list-replicable learning rule from the generalization properties of SVMs. These bounds yield several consequences in learning theory and communication complexity.

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 \emph{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 \emph{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 \emph{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.

We study the discrepancy of the following communication problem. Alice receives a halfplane, and Bob receives a point in the plane, and their goal is to determine whether Bob’s point belongs to Alice’s halfplane. This communication task corresponds to determining whether x₁y₁+y₂ ≥ x₂, where the first player knows (x₁,x₂) and the second player knows (y₁,y₂). Denoting n = m³, we show that when the inputs are chosen from [m] × [m²], the communication discrepancy of the above problem is O(n^{-1/6} log^{3/2} n). On the other hand, through the connections to the notion of hereditary discrepancy by Matoušek, Nikolov, and Tawler (IMRN 2020) and a classical result of Matoušek (Discrete Comput. Geom. 1995), we show that the communication discrepancy of every set of n points and n halfplanes is at least Ω(n^{-1/4} log^{-1} n).

In a recent breakthrough, Kelley and Meka (FOCS 2023) obtained a strong upper bound on the density of sets of integers without non-trivial three-term arithmetic progressions. In this work, we extend their result, establishing similar bounds for all linear patterns defined by binary systems of linear forms, where “binary” indicates that every linear form depends on exactly two variables. Prior to our work, no strong bounds were known for such systems even in the finite field model setting. A key ingredient in our proof is a graph counting lemma. The classical graph counting lemma, developed by Thomason (Random Graphs 1985) and Chung, Graham, and Wilson (Combinatorica 1989), is a fundamental tool in combinatorics. For a fixed graph$H$, it states that the number of copies of$H$in a pseudorandom graph$G$is similar to the number of copies of$H$in a purely random graph with the same edge density as$G$. However, this lemma is only non-trivial when$G$is a dense graph. In this work, we prove a graph counting lemma that is also effective when$G$is sparse. Moreover, our lemma is well-suited for density increment arguments in additive number theory. As an immediate application, we obtain a strong bound for the Turán problem in abelian Cayley sum graphs: let$\Gamma$be a finite abelian group with odd order. If a Cayley sum graph on$\Gamma$does not contain any r-elique as a sub graph, it must have at most$2^{-\Omega_r\left(\log ^{1 / 16}\vert \Gamma\vert \right)} \cdot\vert \Gamma\vert ^2$edges. These results hinge on the technology developed by Kelley and Meka and the follow-up work by Kelley, Lovett, and Meka (STOC 2024).

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang et al. (FOCS'13), and the second one is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

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.

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).

We introduce a new topological argument based on the Borsuk-Ulam theorem to prove a lower bound on sign-rank.

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 $n^{\Omega(1)}$.

Sign-rank and discrepancy are two central notions in communication complexity. The seminal work of Babai, Frankl, and Simon from 1986 initiated an active line of research that investigates the gap between these two notions. In this article, we establish the strongest possible separation by constructing a boolean matrix whose sign-rank is only 3, and yet its discrepancy is 2^{-Ω(n)}. We note that every matrix of sign-rank 2 has discrepancy n^{-O(1)}. Our result in particular implies that there are boolean functions with O(1) unbounded error randomized communication complexity while having Ω(n) weakly unbounded error randomized communication complexity.

The seminal result of Kahn, Kalai and Linial shows that a coalition of O(n/(log n)) players can bias the outcome of any Boolean function {0,1}^n -> {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 input bits. 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,...,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.

Let $f:\{0,1\}^n\to\{0,1\}$ be a boolean function. Its associated XOR function is the two-party function $f_{\oplus}(x,y)=f(x\oplus y)$. We show that, up to polynomial factors, the deterministic communication complexity of $f_{\oplus}$ is equal to the parity decision tree complexity of $f$. This relies on a novel technique of entropy reduction for protocols, combined with existing techniques in Fourier analysis and additive combinatorics.

The information complexity of a function $f$ is the minimum amount of information Alice and Bob need to exchange to compute the function $f$. In this paper we provide an algorithm for approximating the information complexity of an arbitrary function $f$ to within any additive error $α> 0$, thus resolving an open question as to whether information complexity is computable. In the process, we give the first explicit upper bound on the rate of convergence of the information complexity of $f$ when restricted to $b$-bit protocols to the (unrestricted) information complexity of $f$.

Let f be a boolean function on n variables. Its associated XOR function is the two-party function F(x, y) = f(x xor y). We show that, up to polynomial factors, the deterministic communication complexity of F is equal to the parity decision tree complexity of f. This relies on a novel technique of entropy reduction for protocols, combined with existing techniques in Fourier analysis and additive combinatorics.

Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function $f:\mathbb{F}_p^n\to\mathbb{F}_p$ with polynomials of degree at most $d\leq p$ is nonnegligible, while making only a constant number of queries to the function. This is an instance of correlation testing. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analogue of proximity oblivious testing, a notion coined by Goldreich and Ron, in the high error regime. In this work, we study general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by Gowers uniformity tests, and hence having correlation with the property is equivalent to having correlation with degree $d$ polynomials for some fixed $d$. We stress that our result holds also for nonlinear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable. The proof is based on higher-order Fourier analysis. Another ingredient is a nontrivial extension of a graph theoretical theorem of Erdös, Lovász, and Spencer to the context of additive number theory.

Let P be an affine invariant property of multivariate functions over a constant size finite field. We show that if P is locally testable with a constant number of queries, then one can estimate the distance of a function f from P with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over the binary field. Our test is simple: take a restriction of f to a constant dimensional affine subspace, and measure its distance from P. We show that by choosing the dimension large enough, this approximates with high probability the global distance of f from P. The analysis combines the approach of Fischer and Newman [SIAM J. Comp 2007] who established a similar result for graph properties, with recently developed tools in higher order Fourier analysis, in particular those developed in Bhattacharyya et al. [STOC 2013].

Set F = Fp for any fixed prime p ≥ 2. An affine-invariant property is a property of functions over Fn that is closed under taking affine transformations of the domain. We prove that all affine-invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property cP, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies cP, and otherwise reject with probability larger than a positive number that depends only on the distance between f and cP. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable.

Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function f:Fpn -> Fp with polynomials of degree at most d ≤ p is non-negligible, while making only a constant number of queries to the function. This is an instance of correlation testing. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analog of proximity oblivious testing, a notion coined by Goldreich and Ron, in the high error regime. We study in this work general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by the Gowers uniformity test, and hence having correlation with the property is equivalent to having correlation with degree d polynomials for some fixed d. We stress that our result holds also for non-linear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable.

Given a discrete random walk on a finite graph $G$, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step $t$. Let $\Gamma(t)$ be the subgraph of $G$ induced by the vacant set of the walk at step $t$. Similarly, let $\widehat \Gamma(t)$ be the subgraph of $G$ induced by the edges of the vacant net. For random $r$-regular graphs $G_r$, it was previously established that for a simple random walk the graph $\Gamma(t)$ of the vacant set undergoes a phase transition in the sense of the phase transition on Erdös--Renyi graphs $G_{n,p}$. Thus, for $r \ge 3$ there is an explicit value $t^*=t^*(r)$ of the walk such that for $t\leq (1-\epsilon)t^*$, $\Gamma(t)$ has a unique giant component, plus components of size $O(\log n)$, whereas for $t\geq (1+\epsilon)t^*$ all the components of $\Gamma(t)$ are of size $O(\log n)$. In this paper we establish the threshold value $\widehat t$ for a phase transition in the graph $\widehat \Gamma(t)$ of the vacant net of a simple random walk on a random $r$-regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a nonbacktracking random walk and, for $r$ even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random $r$-regular graphs. The main findings are the following: As $r$ increases, the threshold for the vacant set converges to $n \log r$ in all three walks. For the vacant net, the threshold converges to $rn/2 \; \log n$ for both the simple random walk and the nonbacktracking random walk. When $r\ge 4$ is even, the threshold for the vacant net of the unvisited edge process converges to $rn/2$, which is also the vertex cover time of the process.

This paper studies the fractional chromatic number of graphs with maximum degree at most 3. It is proved that if G is triangle free and has maximum degree at most 3, then $\chi_f(G)\leq3-\frac{3}{64}$. If G has girth at least k and maximum degree at most 3, then $\chi_f(G)\leq c_k$, where $c_k$ is a decreasing sequence converging to $8/3$, and $c_{15}\approx2.66681$.

We study various semidefinite programming (SDP) formulations for Vertex Cover by adding different constraints to the standard formulation. We show that Vertex Cover cannot be approximated better than $2-O(\sqrt{\log\log n/\log n})$ even when we add the so-called pentagonal inequality constraints to the standard SDP formulation, and thus almost meet the best upper bound known due to Karakostas [Proceedings of the 32nd International Colloquium on Automata, Languages and Programming, 2005], of $2-\Omega(\sqrt{1/\log n})$. We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution embeds into $\ell_1$ with no distortion, we get an exact relaxation (integrality gap is 1), and on the other hand, if the solution is arbitrarily close to being $\ell_1$ embeddable, the integrality gap is $2-o(1)$. Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar [SODA '02: Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2002, pp. 616–620] to provide a family of simple examples for negative type metrics that cannot be embedded into $\ell_1$ with distortion better than $8/7-\epsilon$. To this end we prove a new isoperimetric inequality for the hypercube.