In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to obtain risk lower bounds involves the use of Fano's inequality. In an information-theoretic setting, it is known that Fano's inequality typically does not give a sharp converse result (error lower bound) for channel coding problems. Moreover, recent work has shown that an argument based on binary hypothesis testing gives tighter results. We adapt this technique to the statistical setting, and argue that Fano's inequality can always be replaced by this approach to obtain tighter lower bounds that can be easily computed and are asymptotically sharp. We illustrate our technique in three applications: density estimation, active learning of a binary classifier, and compressed sensing, obtaining tighter risk lower bounds in each case.
Birg ' e @cite_20 gives stronger, but less transparent, bounds than Fano's inequality using Fano-type arguments; again, @math is bounded in terms of an average of Kullback--Leibler divergences, but these are used as the argument for a function, rather than directly substituted. Sason and Verd 'u [Section 3] sasonV18arimoto recently derived a generalized Fano's inequality in terms of the Arimoto-R 'enyi conditional entropy. They also obtained upper bounds on @math in terms of the pairwise R 'enyi divergences [Section 4] sasonV18arimoto .
@cite_20
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Fano's inequality lower bounds the probability of transmission error through a communication channel. Applied to classification problems, it provides a lower bound on the Bayes error rate and motivates the widely used Infomax principle. In modern machine learning, we are often interested in more than just the error rate. In medical diagnosis, different errors incur different cost; hence, the overall risk is cost-sensitive. Two other popular criteria are balanced error rate (BER) and F-score. In this work, we focus on the two-class problem and use a general definition of conditional entropy (including Shannon's as a special case) to derive upper lower bounds on the optimal F-score, BER and cost-sensitive risk, extending Fano's result. As a consequence, we show that Infomax is not suitable for optimizing F-score or cost-sensitive risk, in that it can potentially lead to low F-score and high risk. For cost-sensitive risk, we propose a new conditional entropy formulation which avoids this inconsistency. In addition, we consider the common practice of using a threshold on the posterior probability to tune performance of a classifier. As is widely known, a threshold of 0.5, where the posteriors cross, minimizes error rate--we derive similar optimal thresholds for F-score and BER. In this technical note, we give two extensions of the classical Fano inequality in information theory. The first extends Fano's inequality to the setting of estimation, providing lower bounds on the probability that an estimator of a discrete quantity is within some distance @math of the quantity. The second inequality extends our bound to a continuum setting and provides a volume-based bound. We illustrate how these inequalities lead to direct and simple proofs of several statistical minimax lower bounds. Lower bounds involving f-divergences between the underlying probability measures are proved for the minimax risk in estimation problems. Our proofs just use simple convexity facts. Special cases and straightforward corollaries of our bounds include well known inequalities for establishing minimax lower bounds such as Fano's inequality, Pinsker's inequality and inequalities based on global entropy conditions. Two applications are provided: a new minimax lower bound for the reconstruction of convex bodies from noisy support function measurements and a different proof of a recent minimax lower bound for the estimation of a covariance matrix. where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 where A = P1 + P2 + * .. + Pn Naturally, this inequality contains the classical Poisson limit law (Just set pi = A n and note that the right side simplifies to 2A2 n), but it also achieves a great deal more. In particular, Le Cam's inequality identifies the sum of the squares of the pi as a quantity governing the quality of the Poisson approximation. Le Cam's inequality also seems to be one of those facts that repeatedly calls to be proved-and improved. Almost before the ink was dry on Le Cam's 1960 paper, an elementary proof was given by Hodges and Le Cam [18]. This proof was followed by numerous generalizations and refinements including contributions by Kerstan [19], Franken [15], Vervatt [30], Galambos [17], Freedman [16], Serfling [24], and Chen [11, 12]. In fact, for raw simplicity it is hard to find a better proof of Le Cam's inequality than that given in the survey of Serfling [25]. One purpose of this note is to provide a proof of Le Cam's inequality using some basic facts from matrix analysis. This proof is simple, but simplicity is not its raison d'etre. It also serves as a concrete introduction to the semi-group method for approximation of probability distributions. This method was used in Le Cam [20], and it has been used again most recently by Deheuvels and Pfeifer [13] to provide impressively precise results. The semi-group method is elegant and powerful, but it faces tough competition, especially from the coupling method and the Chen-Stein method. The literature of these methods is reviewed, and it is shown how they also lead to proofs of Le Cam's inequality. Let D and V denote respectively Information Divergence and Total Variation Distance. Pinsker's and Vajda's inequalities are respectively D ≥ [ 1 2] V2 and D ≥ log[( 2+V) ( 2-V)] - [( 2V) ( 2+V)]. In this paper, several generalizations and improvements of these inequalities are established for wide classes of f -divergences. First, conditions on f are determined under which an f-divergence Df will satisfy Df ≥ cf V2 or Df ≥ c2,f V2 + c4,f V4, where the constants cf, c2,f and c4,f are best possible. As a consequence, lower bounds in terms of V are obtained for many well known distance and divergence measures, including the χ2 and Hellinger's discrimination and the families of Tsallis' and Renyi's divergences. For instance, if D(α) (P||Q) = [α(α-1)]-1 [∫pαq1-αdμ-1] and ℑα (P||Q) = (α-1)-1 log[∫pαq1-αdμ] are respectively the relative information of type α and the Renyi's information gain of order α, it is shown that D(α) ≥ [ 1 2] V2 + [ 1 72] (α+1)(2-α) V4 whenever -1 ≤ α ≤ 2, α ≠ 0,1 and that ℑα ≥ [( α) 2] V2 + [ 1 36] α(1 + 5 α- 5 α2 ) V4 for 0 <; α <; 1. In a somewhat different direction, and motivated by the fact that these Pinsker's type lower bounds are accurate only for small variation (V close to zero), lower bounds for Df which are accurate for both small and large variation (V close to two) are also obtained. In the special case of the information divergence they imply that D ≥ log[ 2 ( 2-V)] - [( 2-V) 2] log[( 2+V) 2], which uniformly improves Vajda's inequality. In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–Łojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing. Since the very beginning the notion of complexity of finite objects was considered as an algorithmic counterpart to the notion of Shannon entropy. Kolmogorov's paper (1965) was called "Three approaches to the quantitative definition of information"; Shannon entropy and algorithmic complexity were among these approaches. It was mentioned by Kolmogorov later (1968) that the properties of algorithmic complexity and Shannon entropy are similar. We investigate one aspect of this similarity. Namely, we are interested in linear inequalities that are valid for Shannon entropies and for Kolmogorov complexities. It turns out that (1) all inequalities that are valid for Kolmogorov complexities, are also valid for Shannon entropies and vice versa; (2) all inequalities that are valid for Shannon entropies, are valid for ranks of finite subsets of linear spaces; (3) the opposite statement is not true: Ingleton's inequality (1971) is valid for ranks but not for Shannon entropies; (4) for some special cases all three classes of inequalities coincide and have simple description. We present an inequality for Kolmogorov complexities that implies Ingleton's inequality for ranks; another application of this inequality is a new simple proof of one of Gacs-Korner's results on common information. The paper investigates connections between linear inequalities that are valid for Shannon entropies and for Kolmogorov complexities. We derive an extension of McDiarmid’s inequality for functions f with bounded differences on a high probability set Y (instead of almost surely). The behavior of f outside Y may be arbitrary. The proof is short and elementary, and relies on an extension argument similar to Kirszbraun’s theorem [4]. We prove n (1) lower bounds on the multiparty communication complexity of AC 0 functions in the number-on-forehead (NOF) model for up to �(logn) players. These are the first lower bounds for any AC 0 function for !(loglogn) players. In particular we show that there are families of depth 3 read-once AC 0 formulas having k-player randomized multiparty NOF communication complexity n (1) 2 O(k) . We show similar lower bounds for depth 4 read-once AC 0 formulas that have nondeterministic communication complexity O(log 2 n), yielding exponential separations between k-party nondeterministic and randomized communication complexity for AC 0 functions. As a consequence of the latter bound, we obtain an n (1 k) 2 O(k) lower bound on the k-party NOF communication complexity of set disjointness. This is non-trivial for up to �( p logn) players which is significantly larger than the up to �(loglogn) players allowed in the best previous lower bounds for multiparty set disjointness given by Lee and Shraibman [LS08] and Chattopadhyay and Ada [CA08] (though our complexity bounds themselves are not as strong as those in [LS08, CA08] for o(loglogn) players). We derive these results by extending the k-party generalization in [CA08, LS08] of the pattern matrix method of Sherstov [She07, She08]. Using this technique, we derive a new sufficient criterion for strong communication complexity lower bounds based on functions having many diverse subfunctions that do not have good low-degree polynomial approximations. This criterion guarantees that such functions have orthogonalizing distributions that are “max-smooth” as opposed to the “min-smooth” orthogonalizing distributions used by Razborov and Sherstov [RS08] to analyze the sign-rank of AC 0 . Given an n × m nonnegative matrix A = (a ij ) and positive integral vectors r e ℜ n and c e Âm having a common one-norm h, the (r,c)-scaling problem is to obtain positive diagonal matrices X and Y, if they exist, such that XAY has row and column sums equal to r and c, respectively. The entropy minimization problem corresponding to A is to find an n × m matrix z = (z ij ) having the same zero pattern as A, the sum of whose entries is a given number h, its row and column sums are within given integral vectors of lower and upper bounds, and such that the entropy function consisting of the sum of the terms z ij ln (z ij a ij ) is minimized. When the lower and upper bounds coincide, matrix scaling and entropy minimization are closely related. In this paper we present several complexity bounds for the e-approximate (r,c)-scaling problem, polynomial in n,m,h, 1 e, and ln V v, where V and v are the largest and the smallest positive entries of A, respectively. These bounds, although not polynomial in ln(1 1 e), not only provide alternative complexities for the polynomial time algorithms, but could result in better overall complexities. In particular, our theoretical analysis supports the practicality of the well-known RAS algorithm. In our analysis we obtain bounds on the norm of scaling vectors which will be used in deriving not only some of the above complexities, but also a complexity for square nonnegative matrices having positive permanent. In particular, our results extend, nontrivially, many bounds for the doubly stochastic scaling of square nonnegative matrices previously given by Kalantari and Khachiyan to the case of general (r,c)-scaling. Finally, we study a more general entropy minimization problem where row and column sums are constrained to lie in prescribed intervals, and the sum of all entries is also prescribed. Balinski and Demange described an RAS type algorithm for its continuous version, but did not analyze its complexity. We show that this algorithm produces an 1 e-approximate solution within complexity polynomial in n, m, h, V v and 1 1 e. We study the quantitative geometry of graphs in terms of their genus, using the structure of certain "cut graphs," i.e. subgraphs whose removal leaves a planar graph. In particular, we give optimal bounds for random partitioning schemes, as well as various types of embeddings. Using these geometric primitives, we present exponentially improved dependence on genus for a number of problems like approximate max-flow min-cut theorems, approximations for uniform and nonuniform Sparsest Cut, treewidth approximation, Laplacian eigenvalue bounds, and Lipschitz extension theorems and related metric labeling problems. We list here a sample of these improvements. All the following statements refer to graphs of genus g, unless otherwise noted. • We show that such graphs admit an O(log g)-approximate multi-commodity max-flow min-cut theorem for the case of uniform demands. This bound is optimal, and improves over the previous bound of O(g) [KPR93, FT03]. For general demands, we show that the worst possible gap is O(log g + CP), where CP is the gap for planar graphs. This dependence is optimal, and already yields a bound of O(log g + √log n), improving over the previous bound of O(√g log n) [KLMN04]. • We give an O(√log g)-approximation for the uniform Sparsest Cut, balanced vertex separator, and treewidth problems, improving over the previous bound of O(g) [FHL05]. • If a graph G has genus g and maximum degree D, we show that the kth Laplacian eigenvalue of G is (log g)2 · O(kgD n), improving over the previous bound of g2·O(kgD n) [KLPT09]. There is a lower bound of Ω(kgD n), making this result almost tight. • We show that if (X, d) is the shortest-path metric on a graph of genus g and S ⊆ X, then every L-Lipschitz map f: S → Z into a Banach space Z admits an O(L log g)-Lipschitz extension f: X → Z. This improves over the previous bound of O(Lg) [LN05], and compares to a lower bound of Ω(L√log g). In a related way, we show that there is an O(log g)-approximation for the 0-extension problem on such graphs, improving over the previous O(g) bound. • We show that every n-vertex shortest-path metric on a graph of genus g embeds into L2 with distortion O(log g + √log n), improving over the previous bound of O(√g log n). Our result is asymptotically optimal for every dependence g = g(n). In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30], several inequalities for tail probabilities of sums M n = X 1 + ... + X n of bounded independent random variables X j were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ. Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257-314] and refined (and extended) by me. In this article, I introduce a new type of inequality. Namely, I show that P M n ≥ x ≤ cP S n ≥ x , where c is an absolute constant and S n = e 1 + ... + e n is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those x ∈ Ρ where the survival function x → P S n ≥ x has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or log-linearly. If it is necessary, to estimate P S n ≥ x special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important. The inequalities have applications to measure concentration, leading to results of the type where, up to an absolute constant, the measure concentration is dominated by the concentration in a simplest appropriate model, such results will be considered elsewhere. The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem. We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures Ω∈P(P(M))Ω∈P(P(M)) on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier [1] to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschlager [12] for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where Ω is not finitely supported. As applications, we prove versions of Jensen's inequality on Wasserstein space and a generalized Brunn–Minkowski inequality for a random measurable set on a Riemannian manifold. We introduce a new approach to characterizing the unobserved portion of a distribution, which provides sublinear--sample estimators achieving arbitrarily small additive constant error for a class of properties that includes entropy and distribution support size. Additionally, we show new matching lower bounds. Together, this settles the longstanding question of the sample complexities of these estimation problems, up to constant factors. Our algorithm estimates these properties up to an arbitrarily small additive constant, using O(n log n) samples, where n is a bound on the support size, or in the case of estimating the support size, 1 n is a lower bound on the probability of any element of the domain. Previously, no explicit sublinear--sample algorithms for either of these problems were known. Our algorithm is also computationally extremely efficient, running in time linear in the number of samples used. In the second half of the paper, we provide a matching lower bound of Ω(n log n) samples for estimating entropy or distribution support size to within an additive constant. The previous lower-bounds on these sample complexities were n 2O(√log n). To show our lower bound, we prove two new and natural multivariate central limit theorems (CLTs); the first uses Stein's method to relate the sum of independent distributions to the multivariate Gaussian of corresponding mean and covariance, under the earthmover distance metric (also known as the Wasserstein metric). We leverage this central limit theorem to prove a stronger but more specific central limit theorem for "generalized multinomial" distributions---a large class of discrete distributions, parameterized by matrices, that represents sums of independent binomial or multinomial distributions, and describes many distributions encountered in computer science. Convergence here is in the strong sense of statistical distance, which immediately implies that any algorithm with input drawn from a generalized multinomial distribution behaves essentially as if the input were drawn from a discretized Gaussian with the same mean and covariance. Such tools in the multivariate setting are rare, and we hope this new tool will be of use to the community. We prove new upper and lower bounds on the sample complexity of (a#x03B5;, a#x03B4;) differentially private algorithms for releasing approximate answers to threshold functions. A threshold function cx over a totally ordered domain X evaluates to cx(y) = 1 if y a#x2264; x, and evaluates to 0 otherwise. We give the first nontrivial lower bound for releasing thresholds with (a#x03B5;, a#x03B4;) differential privacy, showing that the task is impossible over an infinite domain X, and moreover requires sample complexity n a#x2265; (log* |X|), which grows with the size of the domain. Inspired by the techniques used to prove this lower bound, we give an algorithm for releasing thresholds with n a#x2264; 2(1+o(1)) log* |X| samples. This improves the previous best upper bound of 8(1+o(1)) log* |X| (, RANDOM'13). Our sample complexity upper and lower bounds also apply to the tasks of learning distributions with respect to Kolmogorov distance and of properly PAC learning thresholds with differential privacy. The lower bound gives the first separation between the sample complexity of properly learning a concept class with (a#x03B5;, a#x03B4;) differential privacy and learning without privacy. For properly learning thresholds in 'dimensions, this lower bound extends to n a#x2265; (l a#x2219; log* |X|). To obtain our results, we give reductions in both directions from releasing and properly learning thresholds and the simpler interior point problem. Given a database D of elements from X, the interior point problem asks for an element between the smallest and largest elements in D. We introduce new recursive constructions for bounding the sample complexity of the interior point problem, as well as further reductions and techniques for proving impossibility results for other basic problems in differential privacy. We consider a non-Bayesian infinite horizon version of the multi-armed bandit problem with the objective of designing simple policies whose regret increases sldwly with time. In their seminal work on this problem, Lai and Robbins had obtained a O(logn) lower bound on the regret with a constant that depends on the KullbackLeibler number. They also constructed policies for some specific families of probability distributions (including exponential families) that achieved the lower bound. In this paper we construct index policies that depend on the rewards from each arm only through their sample mean. These policies are computationally much simpler and are also applicable much more generally. They achieve a O(logn) regret with a constant that is also based on the Kullback-Leibler number. This constant turns out to be optimal for one-parameter exponential families; however, in general it is derived from the optimal one via a 'contraction' principle. Our results rely entirely on a few key lemmas from the theory of large deviations. We present simple and computationally efficient nonparametric estimators of R 'enyi entropy and mutual information based on an i.i.d. sample drawn from an unknown, absolutely continuous distribution over @math . The estimators are calculated as the sum of @math -th powers of the Euclidean lengths of the edges of the generalized nearest-neighbor' graph of the sample and the empirical copula of the sample respectively. For the first time, we prove the almost sure consistency of these estimators and upper bounds on their rates of convergence, the latter of which under the assumption that the density underlying the sample is Lipschitz continuous. Experiments demonstrate their usefulness in independent subspace analysis.