What are the fundamental limits of a given data science problem? The investigation of such questions typically follows a two-sided analysis. In theachievability part, one shows the existence of a procedure achieving a certain performance, e.g., error probability. In the converse part, one shows that no procedure can accomplish better performance than a certain lower bound. This basic structure is common to a wide range of problems that span information theory, statistics, and computer science. While the study of achievability generally depends strongly on the special features of the problem at hand, many converse bounds that arise in different areas rely essentially on information-theoretic methods.
Frequently, the upper and lower bounds on optimal performance provided by the achievability and converse analyses do not coincide. Starting with Shannon (1948) , the emphasis in information theory has been on the analysis of the fundamental limits in the asymptotic regime of long blocklengths. In that regime, the gap between many existing achievability and converse bounds does vanish and sharp answers can be found to questions such as the maximal transmission rate over noisy channels and the minimal data compression rate subject to a fidelity constraint. The last decade has witnessed a number of information-theoretic results (e.g., [40, 21, 22]), which are applicable in the non-asymptotic regime. These results are strongly motivated by the need to understand the relevance of asymptotic limits to practical systems that may be subject to severe delay constraints, or scenarios where the alphabet size or the number of users is large compared to the number of channels or sources [40, 37, 20, 39].
The development of a non-asymptotic information theory has required new and improved methods for investigating fundamental limits. In principle, it is not clear that the quantities that determine the fundamental limits in the asymptotic regime can accurately describe the performance at blocklengths of interest in realistic applications (such as, say, 1000). This has led to the development of nonasymptotic bounds that capture more sophisticated distributional information on the relevant information quantities. Unfortunately, however, such bounds are often difficult to compute. A less accurate but more tractable approach to understanding performance at smaller blocklengths is to focus attention on so-called second-order analysis, originally pioneered by Wolfowitz  and Strassen  in the 1960s and significantly refined in recent years. In such bounds, the fundamental limits in the first-order (linear in the blocklength) asymptotics are sharpened by investigating the deviation from this asymptotic behavior to second order (square-root of the blocklength). In particular, in those situations where the first- and second-order asymptotics can be established precisely, the resulting bounds have often proven to be quite accurate except for very short blocklengths.
While precise second-order results are available in various basic information-theoretic problems, more complicated setups, particularly those arising in multiuser information theory, have so far eluded second-order analysis. One of the challenges that emerged from this line of work is the development of second-order converses. Several existing approaches to obtaining second-order converses are briefly reviewed in Section 1.2; however, to date, a variety of information-theoretic problems have remained out of reach of such methods. On the other hand, the powerful general methodology introduced in 1976 by Ahlswede, Gács, and Körner  and exploited extensively in the classical book  (see also the recent survey ) has proven instrumental for proving converses in network information theory. Although this widely used technique yields converse results in a broad range of problems almost as a black box, it is fundamentally unable to yield second-order converses and is restricted to finite-alphabet settings.
Inspired by a result by Margulis , the method of Ahlswede, Gács, and Körner is based on a remarkable application of the concentration of measure phenomenon on the Hamming cube [24, 6], which is known in information theory as the “blowing-up lemma”. Historically, this is probably the very first application of modern measure concentration to a data science problem. One of the main messages of this paper is that, surprisingly, measure concentration turns out not to be the right approach after all in this original application. Instead, we will revisit the theory of Ahlswede, Gács, and Körner based not on the violent “blowing-up” operation, but on a new and more pacifist “smoothing out” principle that exploits reverse hypercontractivity of Markov semigroups and functional inequalities. With this gentler touch, we are able to eliminate the inefficiencies of the blowing-up method and obtain second-order converses, essentially for free, in many information-theoretic problems that were out of reach of previous methods.
1.2 Weak, strong, and second-order converses
As a concrete basis for discussion, let us consider the basic setup of single-user data transmission through noisy channels, in which there is a three-way tradeoff between code size, blocklength, and error probability. Suppose we wish to transmit an equiprobable message through a noisy channel with given blocklength .333A channel is a sequence of random transformations indexed by blocklength . We encode each possible message using a codebook . What is the largest possible size of the codebook that can be decoded with error probability (averaged over equiprobable codewords and channel randomness) at most ? For memoryless channels and in various more general situations, the maximum code size satisfies 
where the capacity and dispersion determine the precise first- and second-order asymptotics, and is the inverse Gaussian tail probability function. For memoryless channels (), channel capacity and dispersion are given, respectively, by the quantities [45, 40, 19]
To prove a result such as (1.1), we must address two separate questions. The achievability part (that is, the inequality ) requires us to show existence of a codebook that attains the prescribed error probability. This is usually accomplished using the probabilistic method due to Shannon  which analyzes the error probability not of a particular code, but rather its average when the codebooks are randomly drawn from an auxiliary distribution. For many problems in information theory with known first-order asymptotics, an achievability bound with second-order term can be derived using random coding in conjunction with other techniques [50, 57, 52].
In contrast, the converse part (that is, the inequality ) claims that no code can exceed the size given in (1.1) for the given error probability . The simplest and most widely used tool in converse analyses is Fano’s inequality 
, which yields, in the memoryless case, the following estimate:
Such a bound is called a weak converse: it yields the correct first-order asymptotics in the limit of vanishing error probability, namely,
However, (1.4) does not rule out that transmission rates exceeding may be feasible for any given nonzero error probability . To that end we need a more powerful result known as the strong converse, namely,
which not only implies a strong converse (1.6) but yields the behavior in (1.1). We caution that second-order converses do not always yield the sharpest possible constant in the -term; however, they aim to capture at least qualitatively various features of the sharp second-order asymptotics illustrated by (1.1).
Of course, the basic data transmission problem that we have discussed here for sake of illustration is particularly simple, as the exact second-order asymptotics (1.1) are known. This is not the case in more complicated information-theoretic problems. In particular, there are many problems in multiuser information theory for which either only weak converses are known, or, at most, strong converses have been obtained using the blowing-up method. It is precisely in such situations that new powerful and robust methods for obtaining second-order converses are needed. Next, we briefly discuss the three main approaches that have been developed in the literature for addressing such problems.
The goal of the single-shot method (e.g., [46, 51, 40, 21, 22, 42, 49]) is to obtain non-asymptotic achievability and converse bounds without imposing any probabilistic structure on either sources or channels. Therefore, only the essential features of the problem come into play. Those non-asymptotic bounds are expressed not in terms of average quantities such as entropy or mutual information, but in terms of information spectra, namely the distribution function of information densitiessuch as
. When coupled with the law of large of numbers or the ergodic theorem and with central-limit theorem tools, the bounds become second-order tight. The non-asymptotic converse bounds for single-user data transmission boil down to the derivation of lower bounds on the error probability of Bayesian-ary hypothesis testing followed by anonymation of the actual codebook. Often, those lower bounds are obtained by recourse to the analysis of an associated auxiliary binary hypothesis testing problem. This converse approach has been successfully applied to some problems of multiuser information theory such as Slepian-Wolf coding and multiple access channels , and broadcast channels . Its application to other network setups is however a work in progress.
Type class analysis has been used extensively since  and was popularized by  mainly in the context of error exponents; however, it applies also to second-order analysis (see, e.g., ). The idea behind this method is that to obtain lower bounds, we may consider a situation where the decoder is artificially given access to the type (empirical distribution) of the source or channel sequences. Conditioned on each type, the distribution is equiprobable on the type class, so the evaluation of the conditional error probability is reduced to a combinatorial problem (this has been referred to as the “skeleton” or “combinatorial kernel” of the information-theoretic problem ). However, this combinatorial problem is not easily solved in side information problems (without additional ideas such as the blowing-up lemma). Moreover, by its nature, the method of types is restricted to finite alphabets and memoryless channels.
The method using the blowing-up lemma (BUL) of Ahlswede-Gács-Körner [1, 10, 43] uses a completely different idea to attain converse bounds: rather than try to reduce the given converse problem to a simpler one (e.g., to a binary hypothesis testing problem or a codebook that uses a single type), the BUL method is in essence a general technique for bootstrapping a strong converse from a weak converse. Even when the error probability is fixed, the concentration of measure phenomenon implies that all sequences except those in a set of vanishing probability differ in at most a fraction of coordinates from a correctly decoded sequence. One can therefore effectively reduce the regime of fixed error probability to one of vanishing error probability, where a weak converse suffices, with negligible cost. The advantage of this method is that it is very broadly applicable. However, as will be discussed below, quantitative bounds obtained from this method are always suboptimal and second-order converses are fundamentally outside its reach. Moreover, the perturbation argument used in this approach is restricted to finite alphabets.
The single-shot and type class analysis methods yield second-order converses, but there are various problems in network information theory that have remained so far outside their reach. In contrast, the BUL method has been successful in establishing strong converses for a wide range of problems, including all settings in  with known single-letter rate region; see [10, Ch. 16]. For some problems in network information theory, such as source coding with compressed side information , BUL remained hitherto the only method for establishing a strong converse [49, Section 9.2]. However, the generality of the method comes at the cost of an inherent inefficiency, which prevents it from attaining second-order converses and prevents its application beyond the finite alphabet setting.
In this paper, we will show that one can have essentially the best of both worlds: the inefficiency of the blowing-up method can be almost entirely overcome by revisiting the foundation on which it is based. The resulting theory provides a canonical approach for proving second-order asymptotic converses and is applicable to a wide range of information-theoretic problems (including problems with general alphabets) for which no such results were known.
1.3 “Blowing up” vs “smoothing out”
In order to describe the core ingredients of our approach, let us begin by delving into the main elements of the blowing-up method of Ahlswede, Gács and Körner (a detailed treatment in a toy example will be given in Section 2).
The concentration of measure phenomenon is one of the most important ideas in modern probability [24, 6]. It states that for many high-dimensional probability measures, almost all points in the space are within a small distance of any set of fixed probability. This basic principle may be developed in different settings and has numerous important consequences; for example, it implies that Lipschitz functions on high-dimensional spaces are sharply concentrated around their median, a fact that will not be used in the sequel (but is crucial in many other contexts). The following modern incarnation of the concentration property used in the work of Ahlswede-Gács-Körner is due to Marton ; see also [43, Lemma 3.6.2].
Lemma 1.1 (Blowing-up lemma).
Denote the -blowup of by
where is the Hamming distance on . Then
for any and any probability measure on .
For example, if and , we can achieve by letting . Asymptotically, if does not vanish, then as long as . In other words, the rather remarkable fact is that we can drastically increase the probability of a set by perturbing only a very small () fraction of coordinates of each of its elements.
Ahlswede, Gács and Körner realized how to leverage the BUL to prove strong converses. Suppose one is in a situation where a weak converse, such as (1.4), can be proved through Fano’s inequality or any other approach. This can be done in all information-theoretic problems with known first-order asymptotics. However, a weak converse only yields the correct first-order constant when the error probability is allowed to vanish, while we are interested in the regime of constant . Let be the set of correctly decoded sequences, whose probability is . By the blowing-up lemma, a very slight blow-up of this set will already have probability . We now apply the weak converse argument using instead of . On the one hand, this provides the desired first-order term in (1.4), as is replaced by . On the other hand, we must pay a price in the argument for replacing the true decoding set by its blowup . If , the latter turns out to contribute only to lower order and thus a strong converse is obtained.
The beauty of this approach is that is provides a very general recipe for upgrading a weak converse to a strong converse, and is therefore widely applicable. However, the method has (at least) two significant drawbacks:
It is designed to yield a strong converse, not the stronger second-order asymptotic converse. Therefore, it is not surprising that it fails to yield second-order behavior that we expect from (1.1): when optimized, the BUL method appears unable to give a bound better than (e.g., [43, Thm. 3.6.7]). This is already suggested by Lemma 1.1 itself: to obtain a second-order term from (1.4), we would need to have probability at least . That would require perturbing at least coordinates, which already gives rise to additional logarithmic factors. Thus, the blowing-up operation is too crude to recover the correct second-order behavior. In Appendix A, we will show that this is not an inefficiency in the blowing-up lemma itself, but is in fact an insurmountable problem of any method that is based on set enlargement.
The argument relies essentially on the finite-alphabet setting. This is not because of the blowing-up lemma, which works for any alphabet , but because we must control the price paid for replacing by . While Lemma 1.1 gives a lower bound on as a function of , we can also upper bound as a function of by the following simple argument, which relies crucially on the finiteness of the alphabet.
Suppose that and that for all . Then
where . Therefore, if , then
The main contribution of this paper is to show that the shortcomings of the blowing-up method can be essentially eliminated while retaining its wide applicability. This is enabled by two key ideas that play a central role in our theory.
Functional inequalities. To prove a weak converse such as (1.4), one must relate the relevant information-theoretic quantity (e.g., mutual information) to the error probability (i.e., the probability of the decoding sets). This connection is generally made using a data processing inequality. However, in BUL-type methods, we no longer work directly with the original decoding sets, but rather with a perturbation of these that has better properties. Thus, there is also no reason to restrict attention to sets: we can replace the decoding set by an arbitrary function, and then control the relevant information functionals through their variational characterization (that is, by convex duality). Unlike the data processing inequality, such variational characterizations are in principle sharp and provide a lot more freedom in how to approximate the decoding set.
“Smoothing out” vs. “blowing up”. Once one makes the psychological step of working with functional inequalities, it becomes readily apparent that the idea of “blowing up” the decoding sets is much more aggressive than necessary to obtain a strong converse. What turns out to matter is not the overall size of the decoding set, but only the presence of very small values of the function used in the variational principle. Modifying the indicator function of a set to eliminate its small values can be accomplished by a much smaller perturbation than is needed to drastically increase its probability: this basic insight explains the fundamental inefficiency of the classical BUL approach.
To implement this idea, we must identify an efficient method to improve the positivity of an indicator function. To this end, rather than “blowing up” the set by adding all points within Hamming distance , we will “smooth out” the indicator function by averaging it locally over points at distance . More precisely, this averaging will be performed by means of a suitable Markov semigroup, which enables us to apply the reverse hypercontractivity phenomenon [5, 35] to establish strong positivity-improving properties. Such hypercontractivity phenomenon replaces, in our approach, the much better known concentration of measure phenomenon that was exploited in the BUL method. We will show, moreover, that Markov semigroup perturbations are much easier to control than their blowing-up counterparts, so that our method extends readily to general alphabets, Gaussian channels, and channels with memory.
When combined, the above ideas provide a powerful machinery for developing second-order converses. For example, in the basic data transmission problem that we discussed at the beginning of Section 1.2, our method yields
(cf. Theorem 3.2), where is a certain quantity closely related to the dispersion. When compared to the exact second-order asymptotics (1.1), we see that the second-order term has the correct scaling not only in the blocklength , but also in the error probability (as as ). However, our method does not recover the exact dispersion in (1.1); nor does it capture the fact that in the small error regime , the second-order term in (1.1) is in fact negative (the second-order term in (1.14) is always positive). While the latter features are not directly achievable by the methods of this paper, they can be addressed by combining our methods with type class analysis . The details of such a refined analysis are beyond the scope of this paper.
Let us remark that there are various connections between the notions of (reverse) hypercontractivity and concentration of measure; see, e.g.,  and [23, p. 116] in the continuous case and [34, 35] in the discrete case. However, our present application of reverse hypercontractivity is different in spirit: we are not using it to achieve concentration but only a much weaker effect, which is the key to the efficiency of our method. The sharpness and broad applicability of our method suggests that this may be the “right” incarnation of the pioneering ideas of Ahlswede-Gács-Körner: one might argue that the blowing-up method succeeded in its aims, in essence, because it approximates the natural smoothing-out operation.
We have sketched the main ideas behind our approach in broad terms. To describe the method in detail, it is essential to get our hands dirty. To this end, we develop both the blowing-up and smoothing-out approaches in Section 2 in the simplest possible toy example: that of binary hypothesis testing. While this problem is amenable to a (completely classical) direct analysis, we view it as the ideal pedagogical setting in which to understand the main ideas behind our general theory.
The remainder of the paper is devoted to implementing these ideas in increasingly nontrivial situations.
In Section 3, we use our approach to strengthen Fano’s inequality with an optimal second-order term. We develop both the discrete and the Gaussian cases, and illustrate their utility in applications to broadcast channels and the output distribution of good channel codes.
In Section 4, we use our approach to strengthen a basic image size characterization bound of , obtaining a second-order term. The theory is developed once again both in the discrete and the Gaussian cases. We illustrate the utility of our results in applications to hypothesis testing under communication constraints, and to source coding with compressed side information.
We end this section by collecting common notation that will be used throughout the paper. First, we record two conventions that will always be in force:
All information-theoretic quantities (such as entropy, relative entropy, mutual information, etc.) will be defined in base .
In the sequel, we make use of standard information-theoretic notations for relative entropy and conditional relative entropy , mutual information , entropy , and differential entropy .
We denote by the set of nonnegative Borel measurable functions on , and by the subset of with range in . For a measure and , we write and . We will frequently use for a probability measure . The measure of a set is denoted as , and the restriction of a measure to a set is denoted . A random transformation , mapping measures on to measures on , is viewed as an operator mapping to according to where . The notation
denotes that the random variables
form a Markov chain. The cardinality ofis denoted , and
denotes the Euclidean norm of a vector. Finally, denotes the th element of a sequence or vector, while denotes the components up to the th one .
2 Prelude: Binary Hypothesis Testing
The most elementary setting in which the ideas of this paper can be developed is the classical problem of binary hypothesis testing. It should be emphasized that our theory does not prove anything new in this setting: due to the Neyman-Pearson lemma, the exact form of the optimal tests is known and thus the analysis is amenable to explicit computation (we will revisit this point in Section 2.4). Nonetheless, the simplicity of this setting makes it the ideal toy example in which to introduce and discuss the main ideas of this paper.
In the binary hypothesis testing problem, we consider two competing hypotheses: data is drawn from a probability distribution on, which we know is either or . Our aim is to test, on the basis of a data sample, whether it was drawn from or . More precisely, a (possibly randomized) test is defined by a function : when a data sample is observed, we decide hypothesis with probability , and decide hypothesis otherwise. Thus, we must consider two error probabilities:
We aim to investigate the fundamental tradeoff between and in the case of product measures , ; in other words, what is the smallest error probability that may be achieved by a test that satisfies ? In this setting, the exact first-order and second-order asymptotics are due to Chernoff  and Strassen , respectively, resulting in
The achievability () part of (2.1) is straightforward (see Section 2.4), so the main interest in the proof is to obtain the converse (). In this section, we will illustrate both the blowing-up method of Ahlswede, Gács and Körner and the new approach of this paper in the context of this simple problem, and compare the resulting bounds to the exact second-order asymptotics (2.1).
2.2 The blowing-up method
For simplicity, to illustrate the blowing-up method in the context of the binary hypothesis testing problem, we restrict attention to deterministic tests for some (that is, we decide hypothesis if and hypothesis otherwise). This is not essential, but it simplifies the analysis.
As we mentioned, the blowing-up method is a general technique for upgrading weak converses to strong converses. In the present setting, a weak converse (for any , not necessarily product measures) follows in a completely elementary manner from the data processing property of relative entropy.
Lemma 2.1 (Weak converse bound for binary hypothesis testing).
Let be probability measures on and define the set of observations for which the deterministic test decides . If , then satisfies
By the data processing property of relative entropy, we have
where is the binary entropy function. ∎
Specializing to product measures , , Lemma 2.1 yields
for any test that satisfies . While this is sufficient to conclude the weak converse [if , then cannot vanish faster than ] it falls short of recovering the correct first-order asymptotics in the regime of fixed , as we saw in (2.1).
but to its blow-up . Then
We must now control both the gain and the loss caused by the blowup. On the one hand, by the blowing-up Lemma 1.1, as long as , which eliminates the factor in the weak converse (2.5). On the other hand, assuming finite alphabets we can invoke Lemma 1.2 with and (we may assume without loss of generality that is positive on ) to obtain the strong converse
With a little more effort, we can optimize the argument over and quantify the magnitude of the lower-order term.
Assume . Any deterministic test between and on such that satisfies
Re-examining the proof of Proposition 2.2, we can easily verify that no other choice for the growth of with may accelerate the decay of the slack term in (2.9). While the blowing-up method almost effortlessly turns a weak converse into a strong one, it evidently fails to result in a second-order converse. The rather crude bounds provided by the blowing-up Lemma 1.1 and Lemma 1.2 may be expected to be the obvious culprits. It will shortly become evident, however, that the inefficiency of the method lies much deeper than expected: the major loss occurs already in the very first step (2.3) where we apply the data processing inequality. We will in fact show in Appendix A that any method based on the data processing inequality necessarily yields a slack term at least of order . To surmount this obstacle, we have no choice but to go back to the drawing board.
2.3 The smoothing-out method
The aim of this section is to introduce the key ingredients of the new method proposed in this paper. As will be illustrated throughout this paper, this method yields second-order converses while retaining the broad range of applicability of the blowing-up method. In the following, there will be no reason to restrict attention to deterministic tests as in the previous section, so we will consider arbitrary randomized tests from now on.
2.3.1 Functional inequalities
To prove a converse, we must relate the relevant information-theoretic quantity to the properties of any given test . This was accomplished above by means of the data processing inequality (2.3). However, as was indicated at the end of the previous section, this already precludes us from obtaining sharp quantitative bounds. The first idea behind our approach is to replace the data processing argument by a different lower bound: we will use throughout this paper functional inequalities associated to information-theoretic quantities by convex duality. In the present setting, the relevant inequality follows from the Donsker-Varadhan variational principle for relative entropy  (see, e.g., [43, (3.4.67)])
Unlike the data processing inequality, which can only attain equality in trivial situations, the variational principle (2.12) always attains its supremum by choosing . Therefore, unlike the data processing inequality, in principle an application of (2.12) need not entail any loss.
What we must now show is how to choose the function in (2.12) to capture the properties of a given test . Tempting as it is, the choice is dismal: for example, in the case of deterministic tests , generally and we do not even obtain a weak converse. Instead, inspired by the blowing-up method, we may apply (2.12) to a suitably chosen perturbation of . Let us first develop the argument abstractly so that we may gain insight into the requisite properties. Suppose we can design a mapping (which plays the role of the blowing-up operation in the present setting) that satisfies:
For any test on with , we have
For any test on , we have
Setting in (2.12) and using (2.13) and (2.14), we immediately deduce a strong converse: for any test such that , the error probability satisfies (2.8). Besides replacing the data processing inequality by the variational principle, the above logic parallels the blowing-up method: (2.13) plays the role of the blowing-up Lemma 1.1, while (2.14) plays the role of the counting estimate (1.11).
Nonetheless, this apparently minor change of perspective lies at the heart of our theory. To explain why it provides a crucial improvement, let us pinpoint the origin of the inefficiency of the blowing-up method. The purpose of the blowing-up operation is to increase the probability of a test: given , one designs a blow-up so that . However, when we use the the sharp functional inequality (2.12) rather than the data processing inequality, we do not need to control , but rather . The latter is dominated by the small values of , not by its overall magnitude. Therefore, an efficient perturbation of should not seek to blow it up but only to boost its small values, which may be accomplished at a much smaller cost than the blowing-up operation. It is precisely this insight that will allow us to eliminate the inefficiency of the blowing-up method and attain sharp second-order bounds.
In order to take full advantage of this insight, we must understand how to design efficient perturbations . The second key ingredient of our method is its main workhorse: a general mechanism to implement (2.13) and (2.14) so that their speed of decay will be such that the slack term in (2.8) is in fact
2.3.2 Simple semigroups
The essential intuition that arises from the above discussion is that in order to obtain efficient bounds in (2.13) and (2.14), we must design an operation that is positivity-improving: it boosts the small values of sufficiently to ensure that is not too small. In this subsection we design a suitable transformation , and in Section 2.3.3 we show that it achieves the desired goal.
Let be an arbitrary alphabet and let be any probability measure thereon. We say is a simple semigroup444 Readers who are unfamiliar with semigroups may ignore this terminology; while the semigroup property plays an important role in the proof of Theorem 2.3, it is not used directly in this paper. with stationary measure if
In the i.i.d. case ,
we consider their tensor product
We will use , for a suitable choice of , as a positivity-improving operation.
It is instructive to examine the effect of the operator in (2.16) on indicator functions. For that purpose, we introduce the following ad-hoc notation: if and , then is defined by
the application of the operator in (2.16) to the indicator function becomes
where is a random subset of obtained by including each element independently with probability (in particular, ), and is independent of . In contrast, the blowing-up operation may be expressed in terms of indicator functions as
From this perspective, we see that the semigroup operation is a a smoothing out counterpart of the blowing-up operation: while the blowing-up operation maximizes the function over a local neighborhood of size , the semigroup operation averages the function over a random neighborhood of size . What we will gain from smoothing is that it increases the small values of (it is positivity-improving) without increasing the total mass , so that the mass under cannot grow too much. In contrast, blowing-up is designed to increase the mass ; but then the mass under becomes large as well, which yields the suboptimal rate achieved by the blowing-up method.
2.3.3 Reverse hypercontractivity
It is intuitively clear that is positivity improving: it maps any nonnegative function to a strictly positive function. But the goal of lower bounding
is more ambitious. This idea already appears in the probability theory literature in a very different context: it was realized long ago by Borell that Markov semigroups possess very strong positivity-improving properties, which are described quantitatively by a reverse form of the classical hypercontractivity phenomenon. While Borell was motivated by applications in quantum field theory, we will show in this paper that reverse hypercontractivity provides a powerful mechanism that appears almost tailor-made for our present purposes.
We will presently describe an important generalization of Borell’s ideas to general alphabets due to Mossel et al. , and show how it may be combined with the above ideas to obtain sharp non-asymptotic converses.
Theorem 2.3 (Reverse hypercontractivity).
An estimate of the form (2.13) is almost immediate from (2.22). now follow from a simple change of measure argument. The following result combines these ingredients to derive a second-order converse for binary hypothesis testing.
Any test between and such that satisfies
On the other hand, to establish (2.14), we argue as follows:
where , and (2.27) is just the definition of .
Beside resulting in a second-order converse, the smoothing-out method has an additional major advantage over the blowing-up method: the change-of-measure argument (2.30) is purely measure-theoretic in nature and sidesteps the counting argument of Lemma 1.2. No analogue of the latter can hold beyond the finite alphabet case: indeed, in general alphabets even the blowup of a set of measure zero will have positive measure, ruling out any estimate of the form (1.11). In contrast, the result of Theorem 2.4 holds for any alphabet and requires only a bounded density assumption. Even the latter assumption is not an essential restriction and can be eliminated in specific situations by working with semigroups other than (2.16). A particularly important example that will be developed in detail later on in this paper is the case of Gaussian measures (see, for example, Section 3.2).
The proof of Theorem 2.4 illustrates the approach of this paper in its simplest possible setting. However, the basic ideas that we have introduced here form the basis of a general recipe that will be applied repeatedly in the following sections to obtain second-order converses in a broad range of applications. The comparison between the key ingredients of the blowing-up method and the smoothing-out approach proposed in this paper is summarized in Table 1.
|Blowing-up method||Smoothing-out method|
|Connecting information measures and observables||Data processing property (2.3)||Convex duality (2.12)|
|Lower bound w.r.t. a given measure||Concentration of measure (Lemma 1.1)||Reverse hypercontractivity (Theorem 2.3)|
|Upper bound w.r.t. reference measure||Counting argument (Lemma 1.2)||Change of measure (2.29) [or scaling argument (3.21)]|
2.3.4 Beyond product measures
Throughout this paper we focus for simplicity on stationary memoryless systems, that is, those defined by product measures. However, our approach is by no means limited to this setting. For the benefit of the interested reader, let us briefly sketch in the context of Theorem 2.4 what modifications would be needed to adapt our approach to general dependent measures. For an entirely different application of our approach in a dependent setting, see .
Consider the problem of testing between two arbitrary (non-product) hypotheses on . To adapt the proof of Theorem 2.4, we need to introduce a hypercontractive semigroup with stationary measure . A natural candidate in this general setting is the so-called Gibbs sampler , where the Markov process is defined by replacing each coordinate with an independent draw from its conditional distribution (where
) at independent exponentially distributed intervals. It was shown in that any semigroup that satisfies a modified log-Sobolev inequality is reverse hypercontractive. In particular, such inequalities may be established for the Gibbs sampler under rather general weak dependence assumptions [8, 32].
On the other hand, we need to establish an upper bound on . This may be done as follows. The Gibbs sampler satisfies the differential equation 
We may therefore estimate for any
where we used the tower property of conditional expectations and we defined
Solving the differential inequality yields precisely the same estimate as was obtained in (2.30) in the product case.
Putting together these estimates, we obtain for any test with that
where is the modified log-Sobolev constant of . This extends our approach for the memoryless case to any dependent situation where a modified log-Sobolev inequality is available for . A deeper investigation of dependent processes is beyond the scope of this paper.
2.4 Achievability and optimality
Theorem 2.4 gives a non-asymptotic converse bound for binary hypothesis testing. To understand whether it is accurate, we also need an upper bound (achievability). Such a bound was already stated in (2.1). To motivate the following discussion, it is instructive to give a quick proof of the achievability part of (2.1).
There exist a sequence of binary hypothesis tests with such that
provided that .
For typographical convenience we will write and . By the central limit theorem
where we have defined the relative information
whose mean and variance withare and , respectively. We may therefore choose a deterministic sequence such that the deterministic test defined by
satisfies for all . But a simple Chernoff bound now yields
and the proof is completed by noting that . ∎
Comparing the achievability bound (2.36) with our converse (2.23), we see that the main features of the second-order term are captured faithfully by the smoothing-out method, although it fails to recover the precise constant in the second-order term: is replaced by its natural uniform bound in our converse.555To see this, use to show . Beside the optimal order scaling , we recall that the bound behaves correctly as a function of (up to universal constant) for large error probabilities ; cf. the disucussion following (1.14).
As is illustrated by Lemma 2.5, the achievability analysis is conceptually simple in the binary hypothesis testing case thanks to the Neyman-Pearson lemma which identifies the optimal test. However, in information theory optimal procedures are very seldom known explicitly. Thus, the methodology we have introduced says nothing new about binary hypothesis testing. The point of the present method, however, is that it applies broadly in situations where such a direct analysis is far out of reach. In particular, in the general setting of Section 4 the present approach is currently the only known method to achieve sharp second-order converses in a variety of multiuser information theory problems.
3 Second-Order Fano’s Inequality
The aim of this section is to develop the smoothing-out methodology for channel coding problems, of which a basic example was discussed in Section 1.2.
Weak converses for channel coding problems can be obtained in great generality (cf. ): this is the domain of Fano’s inequality , one of the most basic results in information theory, which gives an implicit upper bound on the error probability of an -ary hypothesis testing problem. For discrete memoryless channels, when combined with a list decoding argument, the blowing-up method strengthens Fano’s inequality to a strong converse with second-order term [1, 33, 41, 43]. In this section, we will show that the smoothing-out method results in a strong form of Fano’s inequality that not only attains the optimal second-order term, but is applicable to a much broader class of channels. The power of this machinery will be illustrated in two typical applications.
Before we turn to the main results of this section, it is instructive to give a short proof of a basic form of Fano’s inequality. Although we state it and prove it for deterministic decoding, it also holds for stochastic decoders.
Lemma 3.1 (Fano’s inequality).
Let be an equiprobable message to be transmitted over a noisy channel . Let be the codewords corresponding to , and let be the disjoint decoding sets. Suppose the average probability of correct decoding satisfies
where is the output of the channel with input .
Let be the decoded message, that is, when the output . The bound can be shown by reduction to an auxiliary binary hypothesis testing problem: , . Then the conclusion follows by applying Lemma 2.1 with since . ∎
The proof of Fano’s inequality highlights the connection between the channel coding problems investigated in this section and the simple hypothesis testing problem of Section 2. In particular, it suggests that the weak converse (3.2) may be strengthened to a strong converse of the form in the setting of memoryless channels . Unfortunately, the latter does not follow from the strong converse obtained in Section 2 for binary hypothesis testing. Indeed, the framework of Section 2 does not apply as the measures that appear in the proof of Lemma 3.1 are not product measures even when the channel is memoryless. To sidestep this hurdle we will apply the smoothing-out operation conditionally on : that is, we will introduce semigroups for which the channel is the stationary measure. The main additional challenge that arises is that the semigroup depends on the channel input and must therefore be controlled uniformly in .
3.1 Bounded probability density case
In this section, we consider a random transformation from to