arXiv subjects

Amir Yehudayoff

Publications and source records attributed to Amir Yehudayoff.

At least 55 records · Page 3Linked to original sources

arXiv2019

Anti-concentration in most directions

We prove anti-concentration bounds for the inner product of two independent random vectors. For example, we show that if $A,B$ are subsets of the cube $\{\pm 1\}^n$ with $|A| \cdot |B| \geq 2^{1.01 n}$, and $X \in A$ and $Y \in B$ are sampled independently and uniformly, then the inner product $\langle X, Y \rangle$ takes on any fixed value with probability at most $O(\tfrac{1}{\sqrt{n}})$. Extending Halász work, we prove stronger bounds when the choices for $x$ are unstructured. We also describe applications to communication complexity, randomness extraction and additive combinatorics.

math.PR↗

arXiv2019

On weak $ε$-nets and the Radon number

We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly's property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove a boosting-type result for weak $ε$-nets.

math.CO↗

arXiv2018

Average-Case Information Complexity of Learning

How many bits of information are revealed by a learning algorithm for a concept class of VC-dimension $d$? Previous works have shown that even for $d=1$ the amount of information may be unbounded (tend to $\infty$ with the universe size). Can it be that all concepts in the class require leaking a large amount of information? We show that typically concepts do not require leakage. There exists a proper learning algorithm that reveals $O(d)$ bits of information for most concepts in the class. This result is a special case of a more general phenomenon we explore. If there is a low information learner when the algorithm {\em knows} the underlying distribution on inputs, then there is a learner that reveals little information on an average concept {\em without knowing} the distribution on inputs.

cs.LG↗

arXiv2018

On the Perceptron's Compression

We study and provide exposition to several phenomena that are related to the perceptron's compression. One theme concerns modifications of the perceptron algorithm that yield better guarantees on the margin of the hyperplane it outputs. These modifications can be useful in training neural networks as well, and we demonstrate them with some experimental data. In a second theme, we deduce conclusions from the perceptron's compression in various contexts.

cs.LG↗

arXiv2018

On Communication Complexity of Classification Problems

This work studies distributed learning in the spirit of Yao's model of communication complexity: consider a two-party setting, where each of the players gets a list of labelled examples and they communicate in order to jointly perform some learning task. To naturally fit into the framework of learning theory, the players can send each other examples (as well as bits) where each example/bit costs one unit of communication. This enables a uniform treatment of infinite classes such as half-spaces in $\mathbb{R}^d$, which are ubiquitous in machine learning. We study several fundamental questions in this model. For example, we provide combinatorial characterizations of the classes that can be learned with efficient communication in the proper-case as well as in the improper-case. These findings imply unconditional separations between various learning contexts, e.g.\ realizable versus agnostic learning, proper versus improper learning, etc. The derivation of these results hinges on a type of decision problems we term "{\it realizability problems}" where the goal is deciding whether a distributed input sample is consistent with an hypothesis from a pre-specified class. From a technical perspective, the protocols we use are based on ideas from machine learning theory and the impossibility results are based on ideas from communication complexity theory.

cs.LG↗

arXiv2018

A Direct Sum Result for the Information Complexity of Learning

How many bits of information are required to PAC learn a class of hypotheses of VC dimension $d$? The mathematical setting we follow is that of Bassily et al. (2018), where the value of interest is the mutual information $\mathrm{I}(S;A(S))$ between the input sample $S$ and the hypothesis outputted by the learning algorithm $A$. We introduce a class of functions of VC dimension $d$ over the domain $\mathcal{X}$ with information complexity at least $Ω\left(d\log \log \frac{|\mathcal{X}|}{d}\right)$ bits for any consistent and proper algorithm (deterministic or random). Bassily et al. proved a similar (but quantitatively weaker) result for the case $d=1$. The above result is in fact a special case of a more general phenomenon we explore. We define the notion of information complexity of a given class of functions $\mathcal{H}$. Intuitively, it is the minimum amount of information that an algorithm for $\mathcal{H}$ must retain about its input to ensure consistency and properness. We prove a direct sum result for information complexity in this context; roughly speaking, the information complexity sums when combining several classes.

cs.LG↗

arXiv2018

Learners that Use Little Information

We study learning algorithms that are restricted to using a small amount of information from their input sample. We introduce a category of learning algorithms we term $d$-bit information learners, which are algorithms whose output conveys at most $d$ bits of information of their input. A central theme in this work is that such algorithms generalize. We focus on the learning capacity of these algorithms, and prove sample complexity bounds with tight dependencies on the confidence and error parameters. We also observe connections with well studied notions such as sample compression schemes, Occam's razor, PAC-Bayes and differential privacy. We discuss an approach that allows us to prove upper bounds on the amount of information that algorithms reveal about their inputs, and also provide a lower bound by showing a simple concept class for which every (possibly randomized) empirical risk minimizer must reveal a lot of information. On the other hand, we show that in the distribution-dependent setting every VC class has empirical risk minimizers that do not reveal a lot of information.

cs.LG↗

arXiv2017

A learning problem that is independent of the set theory ZFC axioms

We consider the following statistical estimation problem: given a family F of real valued functions over some domain X and an i.i.d. sample drawn from an unknown distribution P over X, find h in F such that the expectation of h w.r.t. P is probably approximately equal to the supremum over expectations on members of F. This Expectation Maximization (EMX) problem captures many well studied learning problems; in fact, it is equivalent to Vapnik's general setting of learning. Surprisingly, we show that the EMX learnability, as well as the learning rates of some basic class F, depend on the cardinality of the continuum and is therefore independent of the set theory ZFC axioms (that are widely accepted as a formalization of the notion of a mathematical proof). We focus on the case where the functions in F are Boolean, which generalizes classification problems. We study the interaction between the statistical sample complexity of F and its combinatorial structure. We introduce a new version of sample compression schemes and show that it characterizes EMX learnability for a wide family of classes. However, we show that for the class of finite subsets of the real line, the existence of such compression schemes is independent of set theory. We conclude that the learnability of that class with respect to the family of probability distributions of countable support is independent of the set theory ZFC axioms. We also explore the existence of a "VC-dimension-like" parameter that captures learnability in this setting. Our results imply that that there exist no "finitary" combinatorial parameter that characterizes EMX learnability in a way similar to the VC-dimension based characterization of binary valued classification problems.

cs.LG↗

arXiv2017

Submultiplicative Glivenko-Cantelli and Uniform Convergence of Revenues

In this work we derive a variant of the classic Glivenko-Cantelli Theorem, which asserts uniform convergence of the empirical Cumulative Distribution Function (CDF) to the CDF of the underlying distribution. Our variant allows for tighter convergence bounds for extreme values of the CDF. We apply our bound in the context of revenue learning, which is a well-studied problem in economics and algorithmic game theory. We derive sample-complexity bounds on the uniform convergence rate of the empirical revenues to the true revenues, assuming a bound on the $k$th moment of the valuations, for any (possibly fractional) $k>1$. For uniform convergence in the limit, we give a complete characterization and a zero-one law: if the first moment of the valuations is finite, then uniform convergence almost surely occurs; conversely, if the first moment is infinite, then uniform convergence almost never occurs.

cs.LG↗

arXiv2017

Geometric stability via information theory

The Loomis-Whitney inequality, and the more general Uniform Cover inequality, bound the volume of a body in terms of a product of the volumes of lower-dimensional projections of the body. In this paper, we prove stability versions of these inequalities, showing that when they are close to being tight, the body in question is close in symmetric difference to a 'box'. Our results are best possible up to a constant factor depending upon the dimension alone. Our approach is information theoretic. We use our stability result for the Loomis-Whitney inequality to obtain a stability result for the edge-isoperimetric inequality in the infinite $d$-dimensional lattice. Namely, we prove that a subset of $\mathbb{Z}^d$ with small edge-boundary must be close in symmetric difference to a $d$-dimensional cube. Our bound is, again, best possible up to a constant factor depending upon $d$ alone.

math.MG↗

arXiv2016

On statistical learning via the lens of compression

This work continues the study of the relationship between sample compression schemes and statistical learning, which has been mostly investigated within the framework of binary classification. The central theme of this work is establishing equivalences between learnability and compressibility, and utilizing these equivalences in the study of statistical learning theory. We begin with the setting of multiclass categorization (zero/one loss). We prove that in this case learnability is equivalent to compression of logarithmic sample size, and that uniform convergence implies compression of constant size. We then consider Vapnik's general learning setting: we show that in order to extend the compressibility-learnability equivalence to this case, it is necessary to consider an approximate variant of compression. Finally, we provide some applications of the compressibility-learnability equivalences: (i) Agnostic-case learnability and realizable-case learnability are equivalent in multiclass categorization problems (in terms of sample complexity). (ii) This equivalence between agnostic-case learnability and realizable-case learnability does not hold for general learning problems: There exists a learning problem whose loss function takes just three values, under which agnostic-case and realizable-case learnability are not equivalent. (iii) Uniform convergence implies compression of constant size in multiclass categorization problems. Part of the argument includes an analysis of the uniform convergence rate in terms of the graph dimension, in which we improve upon previous bounds. (iv) A dichotomy for sample compression in multiclass categorization problems: If a non-trivial compression exists then a compression of logarithmic size exists. (v) A compactness theorem for multiclass categorization problems.

cs.LG↗

arXiv2016

Teaching and compressing for low VC-dimension

In this work we study the quantitative relation between VC-dimension and two other basic parameters related to learning and teaching. Namely, the quality of sample compression schemes and of teaching sets for classes of low VC-dimension. Let $C$ be a binary concept class of size $m$ and VC-dimension $d$. Prior to this work, the best known upper bounds for both parameters were $\log(m)$, while the best lower bounds are linear in $d$. We present significantly better upper bounds on both as follows. Set $k = O(d 2^d \log \log |C|)$. We show that there always exists a concept $c$ in $C$ with a teaching set (i.e. a list of $c$-labeled examples uniquely identifying $c$ in $C$) of size $k$. This problem was studied by Kuhlmann (1999). Our construction implies that the recursive teaching (RT) dimension of $C$ is at most $k$ as well. The RT-dimension was suggested by Zilles et al. and Doliwa et al. (2010). The same notion (under the name partial-ID width) was independently studied by Wigderson and Yehudayoff (2013). An upper bound on this parameter that depends only on $d$ is known just for the very simple case $d=1$, and is open even for $d=2$. We also make small progress towards this seemingly modest goal. We further construct sample compression schemes of size $k$ for $C$, with additional information of $k \log(k)$ bits. Roughly speaking, given any list of $C$-labelled examples of arbitrary length, we can retain only $k$ labeled examples in a way that allows to recover the labels of all others examples in the list, using additional $k\log (k)$ information bits. This problem was first suggested by Littlestone and Warmuth (1986).

cs.LG↗

arXiv2016

Distributed Construction of Purely Additive Spanners

This paper studies the complexity of distributed construction of purely additive spanners in the CONGEST model. We describe algorithms for building such spanners in several cases. Because of the need to simultaneously make decisions at far apart locations, the algorithms use additional mechanisms compared to their sequential counterparts. We complement our algorithms with a lower bound on the number of rounds required for computing pairwise spanners. The standard reductions from set-disjointness and equality seem unsuitable for this task because no specific edge needs to be removed from the graph. Instead, to obtain our lower bound, we define a new communication complexity problem that reduces to computing a sparse spanner, and prove a lower bound on its communication complexity using information theory. This technique significantly extends the current toolbox used for obtaining lower bounds for the CONGEST model, and we believe it may find additional applications.

cs.DS↗

arXiv2016

Sign rank versus VC dimension

This work studies the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is {three}. For $d=2$, this maximum is $\tildeΘ(N^{1/2})$. For $d >2$, similar but slightly less accurate statements hold. {The lower bounds improve over previous ones by Ben-David et al., and the upper bounds are novel.} The lower bounds are obtained by probabilistic constructions, using a theorem of Warren in real algebraic topology. The upper bounds are obtained using a result of Welzl about spanning trees with low stabbing number, and using the moment curve. The upper bound technique is also used to: (i) provide estimates on the number of classes of a given VC dimension, and the number of maximum classes of a given VC dimension -- answering a question of Frankl from '89, and (ii) design an efficient algorithm that provides an $O(N/\log(N))$ multiplicative approximation for the sign rank. We also observe a general connection between sign rank and spectral gaps which is based on Forster's argument. Consider the $N \times N$ adjacency matrix of a $Δ$ regular graph with a second eigenvalue of absolute value $λ$ and $Δ\leq N/2$. We show that the sign rank of the signed version of this matrix is at least $Δ/λ$. We use this connection to prove the existence of a maximum class $C\subseteq\{\pm 1\}^N$ with VC dimension $2$ and sign rank $\tildeΘ(N^{1/2})$. This answers a question of Ben-David et al.~regarding the sign rank of large VC classes. We also describe limitations of this approach, in the spirit of the Alon-Boppana theorem. We further describe connections to communication complexity, geometry, learning theory, and combinatorics.

math.CO↗

arXiv2016

On the Theoretical Capacity of Evolution Strategies to Statistically Learn the Landscape Hessian

We study the theoretical capacity to statistically learn local landscape information by Evolution Strategies (ESs). Specifically, we investigate the covariance matrix when constructed by ESs operating with the selection operator alone. We model continuous generation of candidate solutions about quadratic basins of attraction, with deterministic selection of the decision vectors that minimize the objective function values. Our goal is to rigorously show that accumulation of winning individuals carries the potential to reveal valuable information about the search landscape, e.g., as already practically utilized by derandomized ES variants. We first show that the statistically-constructed covariance matrix over such winning decision vectors shares the same eigenvectors with the Hessian matrix about the optimum. We then provide an analytic approximation of this covariance matrix for a non-elitist multi-child $(1,λ)$-strategy, which holds for a large population size $λ$. Finally, we also numerically corroborate our results.

cs.NE↗

arXiv2015

Inequalities and tail bounds for elementary symmetric polynomial with applications

We study the extent of independence needed to approximate the product of bounded random variables in expectation, a natural question that has applications in pseudorandomness and min-wise independent hashing. For random variables whose absolute value is bounded by $1$, we give an error bound of the form $σ^{Ω(k)}$ where $k$ is the amount of independence and $σ^2$ is the total variance of the sum. Previously known bounds only applied in more restricted settings, and were quanitively weaker. We use this to give a simpler and more modular analysis of a construction of min-wise independent hash functions and pseudorandom generators for combinatorial rectangles due to Gopalan et al., which also slightly improves their seed-length. Our proof relies on a new analytic inequality for the elementary symmetric polynomials $S_k(x)$ for $x \in \mathbb{R}^n$ which we believe to be of independent interest. We show that if $|S_k(x)|,|S_{k+1}(x)|$ are small relative to $|S_{k-1}(x)|$ for some $k>0$ then $|S_\ell(x)|$ is also small for all $\ell > k$. From these, we derive tail bounds for the elementary symmetric polynomials when the inputs are only $k$-wise independent.

cs.CC↗

arXiv2015

An elementary exposition to topological overlap in the plane

The aim of this text is to provide an elementary and self-contained exposition of Gromov's argument on topological overlap (the presentation is based on Gromov's work, as well as two follow-up papers of Matousek and Wagner, and of Dotterrer, Kaufman and Wagner). We also discuss a simple generalization in which the vertices are weighted according to some probability distribution. This allows to use von Neumann's minimax theorem to deduce a dual statement.

math.GT↗

arXiv2015

Sample compression schemes for VC classes

Sample compression schemes were defined by Littlestone and Warmuth (1986) as an abstraction of the structure underlying many learning algorithms. Roughly speaking, a sample compression scheme of size $k$ means that given an arbitrary list of labeled examples, one can retain only $k$ of them in a way that allows to recover the labels of all other examples in the list. They showed that compression implies PAC learnability for binary-labeled classes, and asked whether the other direction holds. We answer their question and show that every concept class $C$ with VC dimension $d$ has a sample compression scheme of size exponential in $d$. The proof uses an approximate minimax phenomenon for binary matrices of low VC dimension, which may be of interest in the context of game theory.

cs.LG↗