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.
Using Fano's inequality, Yang and Barron @cite_2 obtained order-optimal minimax risk lower bounds that depend only on global metric entropy features of the underlying function class, without explicitly constructing a packing set. The required metric entropy features (bounds on the packing number and covering number) are available from results in approximation theory for many function classes of interest. Guntuboyina @cite_1 obtained a lower bound on the average error probability in terms of general @math -divergences, and also generalized the metric entropy results of Yang and Barron @cite_2 to certain @math -divergences such as the @math -divergence. An interesting direction for future work would be to obtain a non-asymptotic result analogous to Theorem for the case where only the global metric entropy features are available.
@cite_1 @cite_2
2101210333 1524622012
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. Deep neural nets have caused a revolution in many classification tasks. A related ongoing revolution---also theoretically not understood---concerns their ability to serve as generative models for complicated types of data such as images and texts. These models are trained using ideas like variational autoencoders and Generative Adversarial Networks. We take a first cut at explaining the expressivity of multilayer nets by giving a sufficient criterion for a function to be approximable by a neural network with @math hidden layers. A key ingredient is Barron's Theorem Barron1993 , which gives a Fourier criterion for approximability of a function by a neural network with 1 hidden layer. We show that a composition of @math functions which satisfy certain Fourier conditions ("Barron functions") can be approximated by a @math -layer neural network. For probability distributions, this translates into a criterion for a probability distribution to be approximable in Wasserstein distance---a natural metric on probability distributions---by a neural network applied to a fixed base distribution (e.g., multivariate gaussian). Building up recent lower bound work, we also give an example function that shows that composition of Barron functions is more expressive than Barron functions alone. 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. 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.