Uniform laws of large numbers form a cornerstone of Vapnik–Chervonenkis theory, where they are characterized by the finiteness of the VC dimension. In this work, we study uniform convergence phenomena in \emph{cartesian product spaces}, under assumptions on the underlying distribution that are compatible with the product structure. Specifically, we assume that the distribution is absolutely continuous with respect to the product of its marginals, a condition that captures many natural settings, including product distributions, sparse mixtures of product distributions, distributions with low mutual information, and more. We show that, under this assumption, a uniform law of large numbers holds for a family of events if and only if the \emph{linear VC dimension} of the family is finite. The linear VC dimension is defined as the maximum size of a shattered set that lies on an \emph{axis-parallel line}, namely, a set of vectors that agree on all but at most one coordinate. This dimension is always at most the classical VC dimension, yet it can be arbitrarily smaller. For instance, the family of convex sets in $\mathbb{R}^d$ has linear VC dimension $2$, while its VC dimension is infinite already for $d \ge 2$. Our proofs rely on an estimator that departs substantially from the standard empirical mean estimator and exhibits a more intricate structure. We show that such deviations from the standard empirical mean estimator are unavoidable in this setting. Throughout the paper, we propose several open questions, with a particular focus on quantitative sample complexity bounds.

We consider the problem of fair division, where a set of indivisible goods should be distributed fairly among a set of agents with combinatorial valuations. To capture fairness, we adopt the notion of shares, where each agent is entitled to a fair share, based on some fairness criterion, and an allocation is considered fair if the value of every agent (weakly) exceeds her fair share. A share-based notion is considered universally feasible if it admits a fair allocation for every profile of monotone valuations. A major question arises: is there a non-trivial share-based notion that is universally feasible? The most well-known share-based notions, namely the proportional share and the maximin share, are not universally feasible, nor are any constant approximations of them.

We extend the classical theory of PAC learning in a way which allows to model a rich variety of practical learning tasks where the data satisfy special properties that ease the learning process. For example, tasks where the distance of the data from the decision boundary is bounded away from zero, or tasks where the data lie on a lower dimensional surface. The basic and simple idea is to consider partial concepts: these are functions that can be undefined on certain parts of the space. When learning a partial concept, we assume that the source distribution is supported only on points where the partial concept is defined. This way, one can naturally express assumptions on the data such as lying on a lower dimensional surface, or that it satisfies margin conditions. In contrast, it is not at all clear that such assumptions can be expressed by the traditional PAC theory using learnable total concept classes, and in fact we exhibit easy-to-learn partial concept classes which provably cannot be captured by the traditional PAC theory. This also resolves, in a strong negative sense, a question posed by Attias, Kontorovich, and Mansour (2019). We characterize PAC learnability of partial concept classes and reveal an algorithmic landscape which is fundamentally different than the classical one. For example, in the classical PAC model, learning boils down to Empirical Risk Minimization (ERM). This basic principle follows from Uniform Convergence and the Fundamental Theorem of PAC Learning (Vapnik and Chervonenkis, 1971, 1974b; Blumer, Ehrenfeucht, Haussler, and Warmuth, 1989; Hodges, 1993). In stark contrast, we show that the ERM principle fails spectacularly in explaining learnability of partial concept classes. In fact, we demonstrate classes that are incredibly easy to learn, but such that any algorithm that learns them must use an hypothesis space with unbounded VC dimension. We also find that the sample compression conjecture of Littlestone and Warmuth fails in this setting. Our impossibility results hinge on the recent breakthroughs in communication complexity and graph theory by Göös (2015); Ben-David, Hatami, and Tal (2017); Balodis, Ben-David, Göös, Jain, and Kothari (2021). Thus, this theory features problems that cannot be represented in the traditional way and cannot be solved in the traditional way. We view this as evidence that it might provide insights on the nature of learnability in realistic scenarios which the classical theory fails to explain. We include in the paper suggestions for future research and open problems in several contexts, including combinatorics, geometry, and learning theory.

We prove that every family of (not necessarily distinct) odd cycles $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$'s, forming an odd cycle). As part of the proof, we characterize those families of $n$ odd cycles in $K_{n+1}$ that do not have any rainbow odd cycle. We also characterize those families of $n$ cycles in $K_{n+1}$, as well as those of $n$ edge-disjoint nonempty subgraphs of $K_{n+1}$, without any rainbow cycle.

Let P be a set of n points in the plane and let C be a family of simple closed curves in the plane each of which avoids the points of P. For every curve C ∈ C we denote by disc(C) the region in the plane bounded by C. Fix an integer s > 0 and assume that every two curves in C intersect at most s times and that for every two curves C,C' ∈ C the intersection disc(C) ∩ disc(C') is a connected set. We consider the family F = {P ∩ disc(C) | C ∈ C}. When s is even, we provide sharp bounds, in terms of n, s, and k, for the number of sets in F of cardinality k, assuming that ∩C ∈Cdisc(C) is nonempty. In particular, we provide sharp bounds for the number of halving pseudo-parabolas for a set of n points in the plane. Finally, we consider the VC-dimension of F and show that F has VC-dimension at most s+1.

This article contains a construction for independent sets in the powers of the complements of odd cycles. In particular, we show that /spl alpha/(C~/sub 2n+3/(2/sup n/))/spl ges/2(2/sup n/)+1. It follows that for n/spl ges/0 we have /spl Theta/(C~/sub 2n+3/)>2, where /spl Theta/(G) denotes the Shannon (1956) capacity of graph G.

This paper introduces and studies the question of balancing the load on processors participating in a given quorum system. Our proposed measure for the degree of balancing is the ratio between the load on the least frequently referenced element and on the most frequently used one. We give some simple sufficient and necessary conditions for perfect balancing. We then look at the balancing properties of the common class of voting systems and prove that every voting system with odd total weight is perfectly balanced. (This holds, in fact, for the more general class of ordered systems.) We also give some characterizations for the balancing ratio in the worst case. It is shown that for any quorum system with a universe of size n, the balancing ratio is no smaller than $1/(n-1)$, and this bound is the best possible. When restricting attention to nondominated coteries (NDCs), the bound becomes $2/\bigl(n-\log_2 n+o(\log n)\bigr)$, and there exists an NDC with ratio $2/\bigl(n-\log_2 n-o(\log n)\bigr)$. Next, we study the interrelations between the two basic parameters of load balancing and quorum size. It turns out that the two size parameters suitable for our investigation are the size of the largest quorum and the optimally weighted average quorum size(OWAQS) of the system. For the class of ordered NDCs (for which perfect balancing is guaranteed), it is shown that over a universe of size n, some quorums of size $\lceil(n+1)/2\rceil$ or more must exist (and this bound is the best possible). A similar lower bound holds for the OWAQS measure if we restrict attention to voting systems. For nonordered systems, perfect balancing can sometimes be achieved with much smaller quorums. A lower bound of $\Omega(\sqrt{n})$ is established for the maximal quorum size and the OWAQS of any perfectly balanced quorum system over n elements, and this bound is the best possible. Finally, we turn to quorum systems that cannot be perfectly balanced, but have some balancing ratio $0 < \rho < 1$. For such systems we study the trade-offs between the required balancing ratio $\rho$ and the quorum size it admits in the best case. It is easy to get an analogue of the result for perfect balancing, yielding a lower bound of $\sqrt{n\rho}$. We actually get a better estimate by a refinement of the argument.