Original: Successive Refinement and Coarsening of the Information Bottleneck
Revised: Successive Refinement and Coarsening of the Information Bottleneck

Original: Abstract
Revised: Abstract

Original: We discuss two models for two central issues appearing in conjunction with cognitive processing.
Revised: We discuss two models for two central issues appearing in conjunction with cognitive processing.

Original: One is the ability to incorporate fresh information to already learnt models; the other one is moving to a better understanding of what happens when information “trickles” through many layers of a cognitive processing pipeline.
Revised: One is the ability to incorporate fresh information to already learnt models; the other one is moving to a better understanding of what happens when information “trickles” through many layers of a cognitive processing pipeline.

Original: We do so by investigating formal properties of the Information Bottleneck method: namely how it relates to successive refinement, and successive coarsening of information.
Revised: We do so by investigating formal properties of the Information Bottleneck method: namely how it relates to successive refinement, and successive coarsening of information

Original:  Introduction
Revised:  Introduction

Original: We discuss two models for two central issues appearing in conjunction with cognitive processing. One is the ability to incorporate fresh information to already learnt models; the other one is moving to a better understanding of what happens when information “trickles” through many layers of  a  cognitive processing pipeline such as a deep network, or any cognitive system with a sensory interface and a rich inner information processing.
Revised: One of the most crucial tensions characterising the activity of biological and artificial embodied agents, is that between pursuing one’s own interests, thus behaving in a purposeful and relevant manner, and doing so given the constraints on cognitive processing specific to each agent and situation. To analyse how this fundamental tension impacts agents’ behaviour and cognitive dynamics, formal frameworks can be of great help and have indeed been proposed.

Original: Both can be treated using two — in a way — “dual” perspectives of the successive refinement and coarsening of the Information Bottleneck (IB) method [5, 2].
Revised: One of them is that of bounded rationality, and in particular its information-theoretic formulations [12, 5, 16].

Original: This method describes fundamental limits on the optimal relevancy/complexity trade-offs that one can achieve while extracting (in a single stage of information processing) relevant features within a given system. Similarly, one can use an analogous framework to understand optimal limits onseveral stages information processing, and compare them to the single-stage case.
Revised: From this perspective, the cognitive processing constraints are operationalised with limitations on the amount of information that can be extracted from the environment: i.e., with bounds on the mutual information

Original: Formally, the IB method considers a given joint probability p (X, Y ) , where X is the source of information (e.g., environmental variables), and Y the variable of interest (e.g., to the agent).
Revised: I ( X ; T ) between an environmental variable X and an agent’s representation T . On the other hand, behavioral relevancy is most often quantified with an explicit utility function, but it can also be measured implicitly through the mutual information

Original: The IB method produces a “bottleneck” T (e.g., the agent’s representation), which maximises the information about the relevancy Y under constraints on the information extracted  from  X : i.e., T  solves 
Revised: I ( Y ; T ) with a “relevancy” variable Y , extracted within the representation T from the environmental variable X . The representation T then solves the so-called Information Bottleneck (IB) problem [18, 6]

Original: [Equation]
Revised: [Equation]

Original: where λ ≥ 0 controls the trade-off between relevancy and complexity.
Revised: where λ > 0 controls the complexity-relevancy trade-off. The solutions to (1) for varying λ thus draw the optimal limits on the feasible complexity-relevancy trade-offs.

Revised: Note that for the actual computations of solutions to (1), due to the non-linearity of the constraint, one usually rather maximises the Lagrangian I ( Y ; T ) − βI ( X ; T ) .

Revised: However, a richer description of real-world biological and artificial agents demands taking into the picture another key characteristic of their information-processing dynamics: namely, that information- processing operates along hierarchies [9, 8, 3, 7], producing sequences of representations, instead of isolated ones.

Revised: Hierarchical and sequential generalisations of the IB problem (1), or related problems, have been proposed [5, 11, 17, 5, 20, 19].

Original: In the next sections, we study first refinement, and then coarsening of information 
Revised: In this paper, we are interested in the following question: how are the information-processing limits drawn by solutions to (1) affected when one takes into account these hierarchical sequential structures? We will take a look at two cases of sequential processing. The first one investigates to which extent the optimal information bounds on (1) can be implemented through incremental incorporation of fresh information into already learned models [10] — or in a developmental language, can the optimal bounds on the information trade-off (1) be achieved by a system which first passes through a given stage of development, and then builds on it to acquire further information about the environment [14, 15]? The second one considers the loss of optimality, in terms of discrepancy from the bound (1), induced by a “trickling down” of information through many layers of a cognitive processing pipeline.

Revised: We will investigate these questions from a formal point of view, with mathematical characterisations and numerical experiments on low-dimensional examples.

Original:  Successive refinement
Revised:  Successive refinement

Original: The IB framework, like classical rate-distortion problems before it[1, 4], has recently been adapted to the case of several-stages information processing [3].
Revised: Consider an incremental learning process where each step incorporates new information so as to refine the previously acquired representations. In a similar spirit as for the classic IB, one can describe the optimal informational bounds on this several-stages processing: this is precisely what the notion of successive refinement does. This notion has initially been formulated for classic rate-distortion problems [4, 13], but inspired in recent work [11, 5], we can adapt it to the IB framework.

Original: In learning-theoretic terms 1 , a two-stage bottleneck with parameters ( λ 1 , λ 2 ) , where λ 2 > λ 1  , can be described as the successive computation of a solution T 1 to the classic IB [Equation]
Revised: Consider the following two-stage bottleneck problem 1 : for λ 2 > λ 1 ≥ 0 , we first design a bottleneck T =

Original:  and then  a solution S 2 to
Revised: Tthat solves (1) with λ = λ 1 ; and then design a solution S 2 to

Original: [Equation]
Revised: [Equation]

Original: The second, conditional bottleneck S 2 optimally “supplements” the information already optimally acquired in T 1 , yielding a final representation T =
Revised: The second, conditional bottleneck S 2 optimally “supplements” the information already optimally acquired in T 1 , yielding a final “two-stage bottleneck” T 2 :=

Original: ( T 1 , S 2 ) .
Revised: ( T 1 , S 2 ) .

Original: The notion of successive refinement [1, 4, 3] describes if and when this two-stage IB (2)-(3) differs from the one-stage IB (1):

Original: Definition 1.
Revised: Definition 1.

Original: There is successive refinement from parameters λ 1 to λ 2 > λ 1 , if there is a two-stage bottleneck T =
Revised: There is successive refinement from parameters λ 1 to λ 2 > λ 1 , if there is a two-stage bottleneck T 2 := ( T 1 , S 2 ) with parameters ( λ 1 , λ 2 ) , such that T 2 is also a solution to the one-stage

Original: ( T 1 , S 2 ) with  parameters( λ 1 , λ 2) , such thatT is also a solution to the one-stage IB with parameter λ = λ 2 .
Revised: IB problem (1) with parameter λ = λ 2 . One can easily prove that there is successive refinement from λ 1 to λ 2 if and only if there exists one-stage bottlenecks T 1 and T 2 , with respective parameters λ 1 and λ 2 , such that the Markov chain T 1 − T 2 − X holds.

Original: We then have simple characterisations:
Revised: To investigate whether or not these equivalent properties hold true, we propose a geometric approach:

Original: Proposition 2.
Revised: Proposition 2.

Original: There is successive refinement from λ 1 to λ 2 > λ 1 if
Revised: Denote Hull ( E ) the convex hull of a set E , and consider the condition [Equation]

Original: and only if there exists distinct one-stage bottlenecks T 1 and T 2 , corresponding respectively to λ 1 and λ 2 , such that, equivalently
Revised: where t 1 , t 2 , resp. T 1 , T 2 , denote the symbols, resp. alphabets, of distinct one-stage bottlenecks Tand T 2 , with respective parameters λ 1 , λ 2 , where λ 1 < λ 2 .

Original: (i)
Revised: Then:

Original: We have the Markov chain T 1 − T 2 − X − Y ; (ii) Or for all t 1 ∈ T 1 , we have p ( X | t 1 ) 
Revised: (i) Successive refinement from λ 1 to λ 2 implies (3) , and (3) implies successive refinement if the transition matrix defined by q ( X | T 2 ) is injective.

Original: ∈
Revised: (ii)

Original: Hull { (cid:8) p ( X | t 2 ) , t 2 ∈ T2 } , where Hull ( E ) denotes the convex hull of a set E .
Revised: If the bottleneck problem has strictly concave information curve 3 , then for any bottleneck T , the transition matrix defined by q ( X | T ) can always be chosen injective. In short, Proposition 2 says that there is successive refinement if the convex hull of the q ( X | t 2 ) still contains the q ( X | t 1 ) .

Original: To investigate successive refinement, we computed bottlenecks on synthetic examples, with randomly chosen p ( X, Y ) .
Revised: Equipped with this new characterisation, we computed bottlenecks on synthetic examples, with example distributions p ( X, Y ) sampled uniformly on the simplex. We called “parallel information-processing hierarchy” in [5].

Revised: Point ( i ) is straightforward, and the proof of point ( ii ) will be presented in a publication under preparation.

Original: We used the Lagrangian formulation of the IB [5] — and thus conventionally denote, from now on, β  the trade-off parameter — and  the corresponding Blahut-Arimoto algorithm [5], combined with reverse deterministic annealing (as in [6]), starting at β = ∞ from the (generically) optimal distribution T= X .
Revised: is curve by ( ( Y ; T )) every always ( Y | not deterministic used the Lagrangian formulation of the IB (see Section 1) — where β is the trade-off parameter — and optimised it with the corresponding Blahut-Arimoto algorithm [18] combined with reverse deterministic annealing [21], starting at β = ∞ from the generic optimum T

Revised: = X .

Original: We test our geometric characterisation by focusing on very lowdimensional distributions: |X| = |Y| = 3 . We can then plot the trajectories of q β ( X | T ), on the simplex ∆ X . 
Revised: We choose |X| = |Y| = 3 which in particular allows to plot the trajectories of q β ( X | T ) on the simplex ∆ X . We only focused on examples where condition ( ii ) of Proposition 2 was satisfied: strictly concave information curve and injective q ( X | T 2 ) — so that condition (3) characterises successive refinement.

Original: Figure 1 provides two representative examples.
Revised: Figure 1 shows two representative examples 4 .

Original: On these plots, we can visualise the bifurcations described in [6], whichcorrespond to points where some q β ( Y | t ) , and equivalently some q β ( X | t ), splits into two symbols.
Revised: On these plots, we can visualise the bifurcations at which some q β ( X | t ) splits into two distinct distributions as β grows, yielding an increase in effective bottleneck cardinality [21].

Original: Moreover, between each bifurcation, the bottleneck “trajectories” seem to target specific points, which are the p ([ x ]) for some merges [ x ] of source symbol probabilities.

Original: More importantly for us here, we observe that the conditional probabilities q β ( X | t ) do have a general tendency to contain the q  ( X | tβ ′ ) for β ′ > β .
Revised: More importantly for us here, we observe that Hull { q β ( X | t ) , t ∈ T } seems indeed to have a general tendency to contain the q β ′ ( X | t ) for β ′ < β .

Original: For instance, this isusually the case after the split into 3 distinct symbols.
Revised: For instance, this is the case after the split into 3 distinct symbols.

Original: But the condition can fail: for instance if the condition held, the first curve segments where we have only two symbols (bluer families of points in Figure 1) should be linear segments.
Revised: But the condition can fail: for instance if the condition held, the first curve segments where we have only two symbols (bluer families of points in Figure 1) should be linear segments.

Original: But it is clearly not the case for the right plot: on the latter, the first trajectory segment is non-negligibly curved - a relatively typical behavior.
Revised: But it is clearly not the case for the right plot: on the latter, the first trajectory segment is non-negligibly curved - a relatively typical behavior.

Original: Thus, even though there are many pairs of β and β ′ <
Revised: Thus, even though there are many pairs of β and β ′ <

Original: β for whichthe successive refinement condition  holds, these numerical results suggest that this is not generally the case.
Revised: β for which condition (3) holds, these numerical results suggest that the q β ′ ( X | t ) , for β ′ < β , are not always encompassed by Hull { q β ( X | t ) , t ∈ T } . We propose to quantify this “spreading out” from the convex hull with the measure of unique information proposed in [2], using the algorithm 5 designed in [1] [Equation]

Revised: where ∆ q is the set of r ( X, T β ′ , T β ) consistent with the bottleneck probabilities 6 , i.e., such that r ( X, T β ′ )

Revised: =

Original: Dense information coarsening
Revised: q ( X, T β ′ ) and r ( X, T β ) = q ( X, T β ) . This quantity vanishes if and only if there is successive refinement from T β to T β ′ . In the case of the left trajectory in Figure 1, where the convex hull condition (3) seems visually always satisfied, the maximum of UI ( X : T β ′ \ T β ) over all β ′ < β is approx.

Revised: 6 . 5 · 10 − 7 , whereas for the right trajectory, where (3) is clearly broken, it is approx.

Original: In the case of sequential information processing, if  X  is  the original source, and T 1 , T 2 are two representations atsuccessive stages, we must have the Markov chain X − T 1 − T 2 - and the question is rather: how much optimality do we lose as compared to a one stage processing?
Revised: Importantly, even if UI ( X : T β ′ \ T β ) is nonzero in the second case, it is still noticeably small. Among all the examples we studied, it never exceeded 5 . 18% of I ( X ; T β , T β ′ ) . Overall, these numerical results suggest that imposing two-stage processing, with incorporation of new information at the second stage, does not essentially degrade the optimal informational limits drawn by the classic IB — and the same statement can be made for n -stage processing, by direct iterations of the definitions, arguments and computations in this section.

Revised: This is encouraging for the feasibility of the optimal IB trade-offs (1) in real-world processing systems, as the representations they design about their environment are most often the result of a sequential – and in particular, developmental — process.

Revised: However, as the modified BA algorithm is only guaranteed to find local optima, one should be wary that the observed patterns do not necessarily reflect the behavior of actual, global solutions to (1).

Original: Moreover it is natural to assume thateach stage of  the informationprocessing is a bottleneck - where the relevancy
Revised: The IB method provides us only with and q (  Dense information coarsening Here we consider the converse situation as that of Section 2, where along a feed-forward cognitive processing pipeline, the information is coarsened rather than refined, without additional external input between layers.

Original: Y keeps being  the same.
Revised: In other words, we consider a hierarchy of representations T 1 , . . . , T n , that, by assumption, satisfies the Markov chain X − T 1 − · · · − T n , where X is an environmental variable, and T k has only direct access to the previous layer T k − 1 — deep neural networks being an example of this setting (note that the indices are in reverse order as those of the previous section, because we stick to an indexing describing the order in which representations are produced). Thus if each layer extracts information about a fixed relevancy variable Y , the optimal information trade-off at each layer is drawn by bottlenecks T k with relevancy Y , but where the source X in (1) is replaced by T k − 1 .

Original: In thecase of cognitivesystems, the number of processing stages stemming from the sensory interfaces might be extremely large, and thus it is natural to consider the limit of an infinite number of processing stages.
Revised: The question is then: given these hard constraints on the cognitive processing pipeline, how much information optimality do we lose as compared to a one-stage processing as described by (1)? We focus on the limit of an infinite number of processing stages, upper-bounding the number of processing stages stemming from the sensory interfaces in cognitive systems.

Original: We chose to focus on the case where the processing is highly distributed among stages: in other words, each stage performs only a small amount of compression.
Revised: Moreover we consider the case where the processing is highly distributed among stages: in other words, each stage performs only a small amount of compression, resulting in a “trickling down” of information through dense information processing layers.

Original: We formalised this situation by considering an original source T 0 = X , a fixed relevancy variable Y , and a sequence of bottlenecks ( T k ) k ∈ N ∗ , where T k is  bottleneck with source T k − 1 and relevancy Y .
Revised: We thus consider a sequence of bottlenecks ( T k ) k ∈ N ∗ , where T k is a bottleneck with source T k − 1 and relevancy Y .

Original: We chose the sequence ( β k ) k ∈ N ∗ with small increments, such that I ( X ; T 1 )=
Revised: We choose the sequence ( β k ) k ∈ N ∗ with small increments, such that I ( X ; T 1 )

Original:  I ( X ; Y )and
Revised: ≈ I ( X ; Y )

Original:  I ( X ; T k ) → 0 when k → + ∞ .
Revised: and I ( X ; T k ) → 0 when k → + ∞ .

Original: We will refer  to the described method as the “dense information coarsening” (DIC for short).
Revised: We will refer here to this process as the “dense information coarsening” (DIC for short).

Original: We then plotted, in Figure 2, the q β k (  Y| T k ) trajectories, on the∆ Y simplex,for a representative example, comparing them to the trajectories of a regular IB  with  source X and relevancy Y  . 
Revised: We compared, for distributions p ( X, Y ) sampled uniformly on the simplex, the information curve of the regular IB (1) with fixed source X , with the DIC information curve — i.e., the curve drawn by points ( I ( X ; T k ) , I ( Y ; T k )) . Figure 2 (left) shows a representative example 7 with |X| = |Y| = 3 . Here, like other examples explored (not shown), the DIC curve tracks rather neatly the regular IB curve.

Original: We see that the  two trajectories present similarities: they seem to “target”, before reaching a bifurcation point, thesame.
Revised: However, the discrepancy between the curves is non-negligible and the DIC curve displays a distinctive structure: as opposed to the IB curve which is always concave, the DIC curve is only piece-wise concave, resulting in slight cusps at the junctions — in Figure 2 (left), there are two such pieces of concave curve and one cusp. These cusps correspond to symbol splitting points of the

Original: probabilities on the simplex, which are some convex combinations of the p ( Y | x ) withweights given by the corresponding p ( x ) — this corresponds to the similar phenomenon seen on the ∆ X simplex in Figure 1. The main difference here is that the DIC trajectories move strictly along linear segments. But more crucial are the differences exhibited by the information curves (Figure 3).
Revised: DIC trajectories on the simplex ∆ X (see Appendix A for further details). In Figure 2 (right), we superimposed these graphs with the trajectories of points ( I ( T k − 1 ; T k ) , I ( Y ; T k )) which suggests an hypothesis on the reason for the discrepancy between the DIC and IB curve.

Original: Differently from the IB information curve, the  DIC  informationcurve is not concave.
Revised: Indeed, the curves show that the increase in discrepancy between IB and DIC curves occur concomitantly with a brutal decrease of the information

Original: At the contrary, it clearly displays cusps — which correspond to the bifurcation one can observe on the simplex. These bifurcations are points at which the DIC information curve is  the closest to the  IB information curve.
Revised: I ( T k − 1 ; T k ) between consecutive layers, suggesting that this abrupt decrease in cross-layer information induces a divergence from the optimal informational trade-off. Whether this hypothesis is satisfied in real-world cognitive systems or not, these numerical results suggest a phenomenon of divergence, at critical bifurcation points, from the optimal trade-off drawn by the regular IB (1), for information-processing pipelines resembling very deep neural networks.

Original: The overall feature which is interesting to us here, and was present in all the synthetic examples we observed, is that along  the flow of  the DIC, there seems to be crucial points at which the non-optimality of the trade-off, resulting from the fact that the information has successively degraded along the various stages of processing, is minimal.
Revised: This phenomenon will be better clarified in future work. Appendices [I still have to include the figures in this Appendix, along with the real values of p ( X, Y ) . I am not sure I still have the data, so I might have to change also the previous Figures in the paper (I will choose plots which look qualitatively the same.)]

Revised: A IB and DIC trajectories on the simplex

Revised: One can compare the simplex trajectories of the IB and DIC methods.

Revised: Even though a deeper understanding of the observed patterns (in particular how it relates to the main content in Section 3) is left to future work, we still describe them here.

Revised: In Figure 3 (left), we plot the IB trajectory on the simplex ∆ X of the q β ( X | T ) distributions corresponding to the IB solutions drawing the optimal IB curve in Figure 2 (blue curve).

Revised: In Figure 3 (right), we compare this IB trajectory to the DIC trajectory, still on ∆ X , of the q ( X | T k ) distributions that draw the DIC curve in Figure 2 (red curve).

Original: In particular, it seems that at least in some cases, there is virtually no loss of optimality at these crucial, bifurcation points.
Revised: See resp. Sections 2 and 3 for notations. The two kinds of trajectories are to a large extent similar. Crucially, the DIC trajectory displays symbol merges as k increases, just like the IB trajectory displays symbol merges when β increases — note that the symbol merge in Figure ??

Revised: (right) corresponds exactly to the cusp which can be observed on the DIC information curve in Figure 2.

Revised: However we notice two key differences: first, contrarily to the IB trajectories which have in general non-zero curvatures, the DIC trajectories are always composed of straight lines, between two symbol merges.

Original: Thus these points are advantageous to sit onwithin the information processing of any cognitive system, and  it seems reasonable to hypothesise that living organisms have evolved towards organising their information-processing around those critical points.
Revised: Second, the symbol merge, which can occur in the interior of the simplex for the IB case, happens on the border of the simplex for the DIC trajectories — a feature which was always satisfied for DIC trajectories. These numerical results hint towards possible ways of understanding the discrepancy between the IB and DIC and information curve: it might be partly explained by the discrepancy between the distributions q β ( X | T ) , resp. q

Revised: ( X | T k ) , at which symbol merge occurs in resp.

Revised: the IB and DIC case; and it might also be related to a different nature of the trajectories on the simplex.

Revised: B Values of sample distributions

Revised: The distributions p ( X, Y ) used for our numerical experiments were sampled from the uniform distribution on the probability simplex.

Revised: In this paper, we presented only features which were qualitatively always satisfied among the examples we studied.

Revised: The sample p ( X, Y ) was defined fromp ( X ) = (cid:32) 0 .0 .0 .(cid:33) , p ( Y | X ) = (cid:32) 0 .

Revised: 389 0 .(cid:33)

Revised: for Figure 1 left p ( X ) =

Revised: (cid:32) 0 .0 .0 .(cid:33) ,

Revised: p ( Y | X ) =

Revised: (cid:32) 0 .

Revised: 158 0 .(cid:33) .

Revised: for Figure 1 right, from p ( X ) =

Revised: (cid:32) 0 .0 .0 .(cid:33) ,

Revised: p ( Y | X ) =

Revised: (cid:32) 0 .

Revised: 550 0 .(cid:33) .

Revised: The conditional distributions p ( Y | X ) are represented on the Y simplex in Figure 4.