Learnability for the Information Bottleneck

Abstract

The Information Bottleneck (IB) method provides an insightful and principled approach for balancing compression and prediction for representation learning. The IB objective $I(X;Z)-\beta I(Y;Z)$ employs a Lagrange multiplier $\beta$ to tune this trade-off. However, in practice, not only is $\beta$ chosen empirically without theoretical guidance, there is also a lack of theoretical understanding between $\beta$, learnability, the intrinsic nature of the dataset and model capacity. In this paper, we show that if $\beta$ is improperly chosen, learning cannot happen—the trivial representation $P\left(Z\right|X)=P(Z)$ becomes the global minimum of the IB objective. We show how this can be avoided, by identifying a sharp phase transition between the unlearnable and the learnable which arises as $\beta$ is varied. This phase transition defines the concept of IB-Learnability. We prove several sufficient conditions for IB-Learnability, which provides theoretical guidance for choosing a good $\beta$. We further show that IB-learnability is determined by the largest confident, typical and imbalanced subset of the examples (the conspicuous subset), and discuss its relation with model capacity. We give practical algorithms to estimate the minimum $\beta$ for a given dataset. We also empirically demonstrate our theoretical conditions with analyses of synthetic datasets, MNIST and CIFAR10.

1. Introduction

Tishby et al. [1] introduced the Information Bottleneck (IB) objective function which learns a representation Z of observed variables $(X,Y)$ that retains as little information about X as possible but simultaneously captures as much information about Y as possible:

$$min{\mathrm{IB}}_{\beta }(X,Y;Z)=min[I(X;Z)-\beta I(Y;Z)]$$

(1)

$I(·)$ is the mutual information. The hyperparameter $\beta$ controls the trade-off between compression and prediction, in the same spirit as Rate-Distortion Theory [2] but with a learned representation function $P\left(Z\right|X)$ that automatically captures some part of the “semantically meaningful” information, where the semantics are determined by the observed relationship between X and Y. The IB framework has been extended to and extensively studied in a variety of scenarios, including Gaussian variables [3], meta-Gaussians [4], continuous variables via variational methods [5,6,7], deterministic scenarios [8,9], geometric clustering [10] and is used for learning invariant and disentangled representations in deep neural nets [11,12].

From the IB objective (Equation (1)) we see that when $\beta \to 0$ it will encourage $I(X;Z)=0$ which leads to a trivial representation Z that is independent of X, while when $\beta \to +\infty$, it reduces to a maximum likelihood objective (e.g., in classification, it reduces to cross-entropy loss). Therefore, as we vary $\beta$ from 0 to $+\infty$, there must exist a point ${\beta }_0$ at which IB starts to learn a nontrivial representation where Z contains information about X.

As an example, we train multiple variational information bottleneck (VIB) models on binary classification of MNIST [13] digits 0 and 1 with 20% label noise at different $\beta$. The accuracy vs. $\beta$ is shown in Figure 1. We see that when $\beta <3.25$, no learning happens and the accuracy is the same as random guessing. Beginning with $\beta >3.25$, there is a clear phase transition where the accuracy sharply increases, indicating the objective is able to learn a nontrivial representation. In general, we observe that different datasets and model capacity will result in different ${\beta }_0$ at which IB starts to learn a nontrivial representation. How does ${\beta }_0$ depend on the aspects of the dataset and model capacity and how can we estimate it? What does an IB model learn at the onset of learning? Answering these questions may provide a deeper understanding of IB in particular and learning on two observed variables in general.

In this work, we begin to answer the above questions. Specifically:

• We introduce the concept of IB-Learnability and show that when we vary $\beta$, the IB objective will undergo a phase transition from the inability to learn to the ability to learn (Section 3).
• Using the second-order variation, we derive sufficient conditions for IB-Learnability, which provide upper bounds for the learnability threshold ${\beta }_0$ (Section 4).
• We show that IB-Learnability is determined by the largest confident, typical and imbalanced subset of the examples (the conspicuous subset), reveal its relationship with the slope of the Pareto frontier at the origin on the information plane $I(X;Z)$ vs. $I(Y;Z)$ and discuss its relation to model capacity (Section 5).
• We prove a deep relationship between IB-Learnability, our upper bounds on ${\beta }_0$, the hypercontractivity coefficient, the contraction coefficient and the maximum correlation (Section 5).

We also present an algorithm for estimating the onset of IB-Learnability and the conspicuous subset, which provide us with a tool for understanding a key aspect of the learning problem $(X,Y)$ (Section 6). Finally, we use our main results to demonstrate on synthetic datasets, MNIST [13] and CIFAR10 [14] that the theoretical prediction for IB-Learnability closely matches experiment, and show the conspicuous subset our algorithm discovers (Section 7).

2. Related Work

The seminal IB work [1] provides a tabular method for exactly computing the optimal encoder distribution $P\left(Z\right|X)$ for a given $\beta$ and cardinality of the discrete representation, $\left|Z\right|$. They did not consider the IB learnability problem as addressed in this work. Chechik et al. [3] presents the Gaussian Information Bottleneck (GIB) for learning a multivariate Gaussian representation Z of $(X,Y)$, assuming that both X and Y are also multivariate Gaussians. Under GIB, they derive analytic formula for the optimal representation as a noisy linear projection to eigenvectors of the normalized regression matrix ${\Sigma }_{x|y}{\Sigma }_x^{-1}$ and the learnability threshold ${\beta }_0$ is then given by ${\beta }_0=\frac{1}{1-{\lambda }_1}$ where ${\lambda }_1$ is the largest eigenvalue of the matrix ${\Sigma }_{x|y}{\Sigma }_x^{-1}$. This work provides deep insights about relations between the dataset, ${\beta }_0$ and optimal representations in the Gaussian scenario but the restriction to multivariate Gaussian datasets limits the generality of the analysis Another analytic treatment of IB is given in [4], which reformulates the objective in terms of the copula functions. As with the GIB approach, this formulation restricts the form of the data distributions—the copula functions for the joint distribution $(X,Y)$ are assumed to be known, which is unlikely in practice.

Strouse and Schwab [8] present the Deterministic Information Bottleneck (DIB), which minimizes the coding cost of the representation, $H\left(Z\right)$, rather than the transmission cost, $I(X;Z)$ as in IB. This approach learns hard clusterings with different code entropies that vary with $\beta$. In this case, it is clear that a hard clustering with minimal $H\left(Z\right)$ will result in a single cluster for all of the data, which is the DIB trivial solution. No analysis is given beyond this fact to predict the actual onset of learnability, however.

The first amortized IB objective is in the Variational Information Bottleneck (VIB) of Alemi et al. [5]. VIB replaces the exact, tabular approach of IB with variational approximations of the classifier distribution ($P\left(Y\right|Z)$) and marginal distribution ($P\left(Z\right)$). This approach cleanly permits learning a stochastic encoder, $P\left(Z\right|X)$, that is applicable to any $x\in \mathcal{X}$, rather than just the particular X seen at training time. The cost of this flexibility is the use of variational approximations that may be less expressive than the tabular method. Nevertheless, in practice, VIB learns easily and is simple to implement, so we rely on VIB models for our experimental confirmation.

Closely related to IB is the recently proposed Conditional Entropy Bottleneck (CEB) [7]. CEB attempts to explicitly learn the Minimum Necessary Information (MNI), defined as the point in the information plane where $I(X;Y)=I(X;Z)=I(Y;Z)$. The MNI point may not be achievable even in principle for a particular dataset. However, the CEB objective provides an explicit estimate of how closely the model is approaching the MNI point by observing that a necessary condition for reaching the MNI point occurs when $I(X;Z|Y)=0$. The CEB objective $I(X;Z|Y)-\gamma I(Y;Z)$ is equivalent to IB at $\gamma =\beta +1$, so our analysis of IB-Learnability applies equally to CEB.

Kolchinsky et al. [9] show that when Y is a deterministic function of X, the “corner point” of the IB curve (where $I(X;Y)=I(X;Z)=I(Y;Z)$) is the unique optimizer of the IB objective for all $0<{\beta }^{\prime }<1$ (with the parameterization of Kolchinsky et al. [9], ${\beta }^{\prime }=1/\beta$), which they consider to be a “trivial solution”. However, their use of the term “trivial solution” is distinct from ours. They are referring to the observation that all points on the IB curve contain uninteresting interpolations between two different but valid solutions on the optimal frontier, rather than demonstrating a non-trivial trade-off between compression and prediction as expected when varying the IB Lagrangian. Our use of “trivial” refers to whether IB is capable of learning at all given a certain dataset and value of $\beta$.

Achille and Soatto [12] apply the IB Lagrangian to the weights of a neural network, yielding InfoDropout. In Achille and Soatto [11], the authors give a deep and compelling analysis of how the IB Lagrangian can yield invariant and disentangled representations. They do not, however, consider the question of the onset of learning, although they are aware that not all models will learn a non-trivial representation. More recently, Achille et al. [15] repurpose the InfoDropout IB Lagrangian as a Kolmogorov Structure Function to analyze the ease with which a previously-trained network can be fine-tuned for a new task. While that work is tangentially related to learnability, the question it addresses is substantially different from our investigation of the onset of learning.

Our work is also closely related to the hypercontractivity coefficient [16,17], defined as ${sup}_{Z-X-Y}\frac{I(Y;Z)}{I(X;Z)}$, which by definition equals the inverse of ${\beta }_0$, our IB-learnability threshold. In [16], the authors prove that the hypercontractivity cofficient equals the contraction coefficient ${\eta }_{KL}(P_{Y|X},P_X)$ and Kim et al. [18] propose a practical algorithm to estimate ${\eta }_{KL}(P_{Y|X},P_X)$, which provides a measure for potential influence in the data. Although our goal is different, the sufficient conditions we provide for IB-Learnability are also lower bounds for the hypercontractivity coefficient.

3. IB-Learnability

We are given instances of $(x,y)$ drawn from a distribution with probability (density) $P(X,Y)$ with support of $\mathcal{X}\times \mathcal{Y}$, where unless otherwise stated, both X and Y can be discrete or continuous variables. We use capital letters $X,Y,Z$ for random variables and lowercase $x,y,z$ to denote the instance of variables, with $P(·)$ and $p(·)$ denoting their probability or probability density, respectively. $(X,Y)$ is our training data and may be characterized by different types of noise. The nature of this training data and the choice of $\beta$ will be sufficient to predict the transition from unlearnable to learnable.

We can learn a representation Z of X with conditional probability $p\left(z\right|x)$, such that $X,Y,Z$ obey the Markov chain $Z\leftarrow X\leftrightarrow Y$. Equation (1) above gives the IB objective with Lagrange multiplier $\beta$, ${\mathrm{IB}}_{\beta }(X,Y;Z)$, which is a functional of $p\left(z\right|x)$: ${\mathrm{IB}}_{\beta }(X,Y;Z)={\mathrm{IB}}_{\beta }\left[p\left(z\right|x)\right]$. The IB learning task is to find a conditional probability $p\left(z\right|x)$ that minimizes ${\mathrm{IB}}_{\beta }(X,Y;Z)$. The larger $\beta$, the more the objective favors making a good prediction for Y. Conversely, the smaller $\beta$, the more the objective favors learning a concise representation.

How can we select $\beta$ such that the IB objective learns a useful representation? In practice, the selection of $\beta$ is done empirically. Indeed, Tishby et al. [1] recommends “sweeping $\beta$”. In this paper, we provide theoretical guidance for choosing $\beta$ by introducing the concept of IB-Learnability and providing a series of IB-learnable conditions.

Definition 1.

$(X,Y)$is${\mathit{IB}}_{\beta }$-learnable if there exists a Z given by some$p_1\left(z\right|x)$, such that${\mathit{IB}}_{\beta }(X,Y;Z){{|}_{p_1\left(z\right|x)}<{\mathit{IB}}_{\beta }(X,Y;Z)|}_{p\left(z\right|x)=p(z)}$, where$p\left(z\right|x)=p(z)$characterizes the trivial representation where$Z=Z_{\mathit{trivial}}$is independent of X.

If $(X;Y)$ is ${\mathrm{IB}}_{\beta }$-learnable, then when ${\mathrm{IB}}_{\beta }(X,Y;Z)$ is globally minimized, it will not learn a trivial representation. On the other hand, if $(X;Y)$ is not ${\mathrm{IB}}_{\beta }$-learnable, then when ${\mathrm{IB}}_{\beta }(X,Y;Z)$ is globally minimized, it may learn a trivial representation.

3.1. Trivial Solutions

Definition 1 defines trivial solutions in terms of representations where $I(X;Z)=I(Y;Z)=0$. Another type of trivial solution occurs when $I(X;Z)>0$ but $I(Y;Z)=0$. This type of trivial solution is not directly achievable by the IB objective, as $I(X;Z)$ is minimized but it can be achieved by construction or by chance. It is possible that starting learning from $I(X;Z)>0,I(Y;Z)=0$ could result in access to non-trivial solutions not available from $I(X;Z)=0$. We do not attempt to investigate this type of trivial solution in this work.

3.2. Necessary Condition for IB-Learnability

From Definition 1, we can see that ${\mathrm{IB}}_{\beta }$-Learnability for any dataset $(X;Y)$ requires $\beta >1$. In fact, from the Markov chain $Z\leftarrow X\leftrightarrow Y$, we have $I(Y;Z)\le I(X;Z)$ via the data-processing inequality. If $\beta \le 1$, then since $I(X;Z)\ge 0$ and $I(Y;Z)\ge 0$, we have that $min(I(X;Z)-\beta I(Y;Z))=0={\mathrm{IB}}_{\beta }(X,Y;Z_{trivial})$. Hence $(X,Y)$ is not ${\mathrm{IB}}_{\beta }$-learnable for $\beta \le 1$.

Due to the reparameterization invariance of mutual information, we have the following theorem for ${\mathrm{IB}}_{\beta }$-Learnability:

Lemma 1.

Let$X^{\prime }=g\left(X\right)$be an invertible map (if X is a continuous variable, g is additionally required to be continuous). Then$(X,Y)$and$(X^{\prime },Y)$have the same${\mathit{IB}}_{\beta }$-Learnability.

The proof for Lemma 1 is in Appendix A.2. Lemma 1 implies a favorable property for any condition for ${\mathrm{IB}}_{\beta }$-Learnability: the condition should be invariant to invertible mappings of X. We will inspect this invariance in the conditions we derive in the following sections.

4. Sufficient Conditions for IB-Learnability

Given $(X,Y)$, how can we determine whether it is ${\mathrm{IB}}_{\beta }$-learnable? To answer this question, we derive a series of sufficient conditions for ${\mathrm{IB}}_{\beta }$-Learnability, starting from its definition. The conditions are in increasing order of practicality, while sacrificing as little generality as possible.

Firstly, Theorem 1 characterizes the ${\mathrm{IB}}_{\beta }$-Learnability range for $\beta$, with proof in Appendix A.3:

Theorem 1.

If$(X,Y)$is${\mathit{IB}}_{{\beta }_1}$-learnable, then for any${\beta }_2>{\beta }_1$, it is${\mathit{IB}}_{{\beta }_2}$-learnable.

Based on Theorem 1, the range of $\beta$ such that $(X,Y)$ is ${\mathrm{IB}}_{\beta }$-learnable has the form $\beta \in ({\beta }_0,+\infty )$. Thus, ${\beta }_0$ is the threshold of IB-Learnability.

Lemma 2.

$p\left(z\right|x)=p(z)$is a stationary solution for${\mathit{IB}}_{\beta }(X,Y;Z)$.

The proof in Appendix A.6 shows that both first-order variations $\delta I(X;Z)=0$ and $\delta I(Y;Z)=0$ vanish at the trivial representation $p\left(z\right|x)=p(z)$, so $\delta {\mathrm{IB}}_{\beta }\left[p\left(z\right|x)\right]=0$ at the trivial representation.

Lemma 2 yields our strategy for finding sufficient conditions for learnability: find conditions such that $p\left(z\right|x)=p(z)$ is not a local minimum for the functional ${\mathrm{IB}}_{\beta }\left[p\left(z\right|x)\right]$. Based on the necessary condition for the minimum (Appendix A.4), we have the following theorem (The theorems in this paper deal with learnability w.r.t. true mutual information. If parameterized models are used to approximate the mutual information, the limitation of the model capacity will translate into more uncertainty of Y given X, viewed through the lens of the model.):

Theorem 2

(Suff. Cond. 1). A sufficient condition for $(X,Y)$ to be ${\mathit{IB}}_{\beta }$-learnable is that there exists a perturbation function $h\left(z\right|x)$ (so that the perturbed probability (density) is $p^{\prime }\left(z\right|x)=p\left(z\right|x)+ϵ·h\left(z\right|x)$) with $\int h\left(z\right|x)dz=0$, such that the second-order variation ${\delta }^2{\mathit{IB}}_{\beta }\left[p\left(z\right|x)\right]<0$ at the trivial representation $p\left(z\right|x)=p(z)$.

The proof for Theorem 2 is given in Appendix A.4. Intuitively, if ${\delta }^2{\mathrm{IB}}_{\beta }\left[p\left(z\right|x)\right]{|}_{p\left(z\right|x)=p(z)}<0$, we can always find a $p^{\prime }\left(z\right|x)=p\left(z\right|x)+ϵ·h\left(z\right|x)$ in the neighborhood of the trivial representation $p\left(z\right|x)=p(z)$, such that ${\mathrm{IB}}_{\beta }\left[p^{\prime }\left(z\right|x)\right]<{\mathrm{IB}}_{\beta }\left[p\left(z\right|x)\right]$, thus satisfying the definition for ${\mathrm{IB}}_{\beta }$-Learnability.

To make Theorem 2 more practical, we perturb $p\left(z\right|x)$ around the trivial solution $p^{\prime }\left(z\right|x)=p\left(z\right|x)+ϵ·h\left(z\right|x)$ and expand ${\mathrm{IB}}_{\beta }[p\left(z\right|x)+ϵ·h\left(z\right|x)]-{\mathrm{IB}}_{\beta }\left[p\left(z\right|x)\right]$ to the second order of $ϵ$. We can then prove Theorem 3:

Theorem 3

(Suff. Cond. 2). A sufficient condition for $(X,Y)$ to be ${\mathit{IB}}_{\beta }$-learnable is X and Y are not independent and

$$\begin{array}{c}\hfill \beta >{\displaystyle \underset{h\left(x\right)}{inf}}{\beta }_0\left[h\left(x\right)\right]\end{array}$$

(2)

where the functional ${\beta }_0\left[h\left(x\right)\right]$ is given by

$${\beta }_0\left[h\left(x\right)\right]=\frac{{\mathbb{E}}_{x\sim p\left(x\right)}\left[h{\left(x\right)}^2\right]-{\left({\mathbb{E}}_{x\sim p\left(x\right)}\left[h\left(x\right)\right]\right)}^2}{{\mathbb{E}}_{y\sim p\left(y\right)}\left[{\left({\mathbb{E}}_{x\sim p\left(x\right|y)}\left[h\left(x\right)\right]\right)}^2\right]-{\left({\mathbb{E}}_{x\sim p\left(x\right)}\left[h\left(x\right)\right]\right)}^2}$$

Moreover, we have that${\left({inf}_{h\left(x\right)}\beta \left[h\left(x\right)\right]\right)}^{-1}$is a lower bound of the slope of the Pareto frontier in the information plane$I(Y;Z)$vs.$I(X;Z)$at the origin.

The proof is given in Appendix A.7, which also shows that if $\beta >{inf}_{h\left(x\right)}{\beta }_0\left[h\left(x\right)\right]$ in Theorem 3 is satisfied, we can construct a perturbation function $h\left(z\right|x)=h^{\ast }\left(x\right)h_2\left(z\right)$ with $h^{\ast }\left(x\right)={arg\; min}_{h\left(x\right)}{\beta }_0\left[h\left(x\right)\right]$, $\int h_2\left(z\right)dz=0,\int \frac{h_2^2\left(z\right)}{p\left(z\right)}dz>0$ for some $h_2\left(z\right)$, such that $h\left(z\right|x)$ satisfies Theorem 2. It also shows that the converse is true: if there exists $h\left(z\right|x)$ such that the condition in Theorem 2 is true, then Theorem 3 is satisfied, that is, $\beta >{inf}_{h\left(x\right)}{\beta }_0\left[h\left(x\right)\right]$. (We do not claim that any $h\left(z\right|x)$ satisfying Theorem 2 can be decomposed to $h^{\ast }\left(x\right)h_2\left(z\right)$ at the onset of learning. But from the equivalence of Theorems 2 and 3 as explained above, when there exists an $h\left(z\right|x)$ such that Theorem 2 is satisfied, we can always construct an $h^{\prime }\left(z\right|x)=h^{\ast }\left(x\right)h_2\left(z\right)$ that also satisfies Theorem 2.) Moreover, letting the perturbation function $h\left(z\right|x)=h^{\ast }\left(x\right)h_2\left(z\right)$ at the trivial solution, we have

$$\begin{array}{c}\hfill p_{\beta }\left(y\right|x)=p\left(y\right)+{ϵ}^2{C}_z(h^{\ast }\left(x\right)-{\overline{h}}_x^{\ast })\int p(x,y)(h^{\ast }\left(x\right)-{\overline{h}}_x^{\ast })dx\end{array}$$

(3)

where $p_{\beta }\left(y\right|x)$ is the estimated $p\left(y\right|x)$ by IB for a certain $\beta$, ${\overline{h}}_x^{\ast }=\int h^{\ast }\left(x\right)p\left(x\right)dx$ and $C_z=\int \frac{h_2^2\left(z\right)}{p\left(z\right)}dz>0$ is a constant. This shows how the $p_{\beta }\left(y\right|x)$ by IB explicitly depends on $h^{\ast }\left(x\right)$ at the onset of learning. The proof is provided in Appendix A.8.

Theorem 3 suggests a method to estimate ${\beta }_0$: we can parameterize $h\left(x\right)$ for example, by a neural network, with the objective of minimizing ${\beta }_0\left[h\left(x\right)\right]$. At its minimization, ${\beta }_0\left[h\left(x\right)\right]$ provides an upper bound for ${\beta }_0$, and $h\left(x\right)$ provides a soft clustering of the examples corresponding to a nontrivial perturbation of $p\left(z\right|x)$ at $p\left(z\right|x)=p(z)$ that minimizes ${\mathrm{IB}}_{\beta }\left[p\left(z\right|x)\right]$.

Alternatively, based on the property of ${\beta }_0\left[h\left(x\right)\right]$, we can also use a specific functional form for $h\left(x\right)$ in Equation (2) and obtain a stronger sufficient condition for ${\mathrm{IB}}_{\beta }$-Learnability. But we want to choose $h\left(x\right)$ as near to the infimum as possible. To do this, we note the following characteristics for the R.H.S of Equation (2):

• We can set $h\left(x\right)$ to be nonzero if $x\in {\Omega }_x$ for some region ${\Omega }_x\subset \mathcal{X}$ and 0 otherwise. Then we obtain the following sufficient condition:

  $$\begin{array}{c}\hfill \beta >{\displaystyle \underset{h\left(x\right),{\Omega }_x\subset \mathcal{X}}{inf}}\frac{\frac{{\mathbb{E}}_{x\sim p\left(x\right),x\in {\Omega }_x}\left[h{\left(x\right)}^2\right]}{{\left({\mathbb{E}}_{x\sim p\left(x\right),x\in {\Omega }_x}\left[h\left(x\right)\right]\right)}^2}-1}{\int \frac{dy}{p\left(y\right)}{\left(\frac{{\mathbb{E}}_{x\sim p\left(x\right),x\in {\Omega }_x}\left[p\left(y\right|x)h\left(x\right)\right]}{{\mathbb{E}}_{x\sim p\left(x\right),x\in {\Omega }_x}\left[h\left(x\right)\right]}\right)}^2-1}\end{array}$$

  (4)
• The numerator of the R.H.S. of Equation (4) attains its minimum when $h\left(x\right)$ is a constant within ${\Omega }_x$. This can be proved using the Cauchy-Schwarz inequality: $\langle u,u\rangle \langle v,v\rangle \ge {\langle u,v\rangle }^2$, setting $u\left(x\right)=h\left(x\right)\sqrt{p\left(x\right)}$, $v\left(x\right)=\sqrt{p\left(x\right)}$ and defining the inner product as $\langle u,v\rangle =\int u\left(x\right)v\left(x\right)dx$. Therefore, the numerator of the R.H.S. of Equation (4) $\ge \frac{1}{{\int }_{x\in {\Omega }_x}p\left(x\right)}-1$ and attains equality when $\frac{u\left(x\right)}{v\left(x\right)}=h\left(x\right)$ is constant.

Based on these observations, we can let $h\left(x\right)$ be a nonzero constant inside some region ${\Omega }_x\subset \mathcal{X}$ and 0 otherwise and the infimum over an arbitrary function $h\left(x\right)$ is simplified to infimum over ${\Omega }_x\subset \mathcal{X}$ and we obtain a sufficient condition for ${\mathrm{IB}}_{\beta }$-Learnability, which is a key result of this paper:

Theorem 4

(Conspicuous Subset Suff. Cond.). A sufficient condition for $(X,Y)$ to be ${\mathit{IB}}_{\beta }$-learnable is X and Y are not independent and

$$\begin{array}{c}\hfill \beta >{\displaystyle \underset{{\Omega }_x\subset \mathcal{X}}{inf}}{\beta }_0\left({\Omega }_x\right)\end{array}$$

(5)

where

$${\beta }_0\left({\Omega }_x\right)=\frac{\frac{1}{p\left({\Omega }_x\right)}-1}{{\mathbb{E}}_{y\sim p\left(y\right|{\Omega }_x)}\left[\frac{p\left(y\right|{\Omega }_x)}{p\left(y\right)}-1\right]}$$

${\Omega }_x$ denotes the event that $x\in {\Omega }_x$, with probability $p\left({\Omega }_x\right)$.

${\left({inf}_{{\Omega }_x\subset \mathcal{X}}{\beta }_0\left({\Omega }_x\right)\right)}^{-1}$gives a lower bound of the slope of the Pareto frontier in the information plane$I(Y;Z)$vs.$I(X;Z)$at the origin.

The proof is given in Appendix A.9. In the proof we also show that this condition is invariant to invertible mappings of X.

5. Discussion

5.1. The Conspicuous Subset Determines ${\beta }_0$

From Equation (5), we see that three characteristics of the subset ${\Omega }_x\subset \mathcal{X}$ lead to low ${\beta }_0$: (1) confidence: $p\left(y\right|{\Omega }_x)$ is large; (2) typicality and size: the number of elements in ${\Omega }_x$ is large or the elements in ${\Omega }_x$ are typical, leading to a large probability of $p\left({\Omega }_x\right)$; (3) imbalance: $p\left(y\right)$ is small for the subset ${\Omega }_x$ but large for its complement. In summary, ${\beta }_0$ will be determined by the largest confident, typical and imbalanced subset of examples or an equilibrium of those characteristics. We term ${\Omega }_x$ at the minimization of ${\beta }_0\left({\Omega }_x\right)$ the conspicuous subset.

5.2. Multiple Phase Transitions

Based on this characterization of ${\Omega }_x$, we can hypothesize datasets with multiple learnability phase transitions. Specifically, consider a region ${\Omega }_{x0}$ that is small but “typical”, consists of all elements confidently predicted as $y_0$ by $p\left(y\right|x)$ and where $y_0$ is the least common class. By construction, this ${\Omega }_{x0}$ will dominate the infimum in Equation (5), resulting in a small value of ${\beta }_0$. However, the remaining $\mathcal{X}-{\Omega }_{x0}$ effectively form a new dataset, ${\mathcal{X}}_1$. At exactly ${\beta }_0$, we may have that the current encoder, $p_0\left(z\right|x)$, has no mutual information with the remaining classes in ${\mathcal{X}}_1$; that is, $I(Y_1;Z_0)=0$. In this case, Definition 1 applies to $p_0\left(z\right|x)$ with respect to $I(X_1;Z_1)$. We might expect to see that, at ${\beta }_0$, learning will plateau until we get to some ${\beta }_1>{\beta }_0$ that defines the phase transition for ${\mathcal{X}}_1$. Clearly this process could repeat many times, with each new dataset ${\mathcal{X}}_i$ being distinctly more difficult to learn than ${\mathcal{X}}_{i-1}$.

5.3. Similarity to Information Measures

The denominator of ${\beta }_0\left({\Omega }_x\right)$ in Equation (5) is closely related to mutual information. Using the inequality $x-1\ge \log \left(x\right)$ for $x>0$, it becomes:

$$\begin{array}{cc}\hfill {\mathbb{E}}_{y\sim p\left(y\right|{\Omega }_x)}\left[\frac{p\left(y\right|{\Omega }_x)}{p\left(y\right)}-1\right]& \ge {\mathbb{E}}_{y\sim p\left(y\right|{\Omega }_x)}\left[\log \frac{p\left(y\right|{\Omega }_x)}{p\left(y\right)}\right]=\tilde{I}({\Omega }_x;Y)\hfill \end{array}$$

where $\tilde{I}({\Omega }_x;Y)$ is the mutual information “density” at ${\Omega }_x\subset \mathcal{X}$. Of course, this quantity is also ${\mathbb{D}}_{KL}[p\left(y\right|{\Omega }_x)\left|\right|p\left(y\right)]$, so we know that the denominator of Equation (5) is non-negative. Incidentally, ${\mathbb{E}}_{y\sim p\left(y\right|{\Omega }_x)}\left[\frac{p\left(y\right|{\Omega }_x)}{p\left(y\right)}-1\right]$ is the density of “rational mutual information” [19] at ${\Omega }_x$.

Similarly, the numerator of ${\beta }_0\left({\Omega }_x\right)$ is related to the self-information of ${\Omega }_x$:

$$\frac{1}{p\left({\Omega }_x\right)}-1\ge \log \frac{1}{p\left({\Omega }_x\right)}=-\log p\left({\Omega }_x\right)=h\left({\Omega }_x\right)$$

so we can estimate ${\beta }_0$ as:

$${\beta }_0\simeq {\displaystyle \underset{{\Omega }_x\subset \mathcal{X}}{inf}}\frac{h\left({\Omega }_x\right)}{\tilde{I}({\Omega }_x;Y)}$$

(6)

Since Equation (6) uses upper bounds on both the numerator and the denominator, it does not give us a bound on ${\beta }_0$, only an estimate.

5.4. Estimating Model Capacity

The observation that a model cannot distinguish between cluster overlap in the data and its own lack of capacity gives an interesting way to use IB-Learnability to measure the capacity of a set of models relative to the task they are being used to solve. For example, for a classification task, we can use different model classes to estimate $p\left(y\right|x)$. For each such trained model, we can estimate the corresponding IB-learnability threshold ${\beta }_0$. A model with smaller capacity than the task needs will translate to more uncertainty in $p\left(y\right|{\Omega }_x)$, resulting in a larger ${\beta }_0$. On the other hand, models that give the same ${\beta }_0$ as each other all have the same capacity relative to the task, even if we would otherwise expect them to have very different capacities. For example, if two deep models have the same core architecture but one has twice the number of parameters at each layer and they both yield the same ${\beta }_0$, their capacities are equivalent with respect to the task. Thus, ${\beta }_0$ provides a way to measure model capacity in a task-specific manner.

5.5. Learnability and the Information Plane

Many of our results can be interpreted in terms of the geometry of the Pareto frontier illustrated in Figure 2, which describes the trade-off between increasing $I(Y;Z)$ and decreasing $I(X;Z)$. At any point on this frontier that minimizes ${\mathrm{IB}}_{\beta }^{min}\equiv minI(X;Z)-\beta I(Y;Z)$, the frontier will have slope ${\beta }^{-1}$ if it is differentiable. If the frontier is also concave (has negative second derivative), then this slope ${\beta }^{-1}$ will take its maximum ${\beta }_0^{-1}$ at the origin, which implies ${\mathrm{IB}}_{\beta }$-Learnability for $\beta >{\beta }_0$, so that the threshold for ${\mathrm{IB}}_{\beta }$-Learnability is simply the inverse slope of the frontier at the origin. More generally, as long as the Pareto frontier is differentiable, the threshold for ${\mathrm{IB}}_{\beta }$-learnability is the inverse of its maximum slope. Indeed, Theorem 3 and Theorem 4 give lower bounds of the slope of the Pareto frontier at the origin.

5.6. IB-Learnability, Hypercontractivity and Maximum Correlation

IB-Learnability and its sufficient conditions we provide harbor a deep connection with hypercontractivity and maximum correlation:

$$\begin{array}{cc}\hfill \frac{1}{{\beta }_0}& =\xi (X;Y)={\eta }_{\mathrm{KL}}\ge {\displaystyle \underset{h\left(x\right)}{sup}}\frac{1}{{\beta }_0\left[h\left(x\right)\right]}={\rho }_m^2(X;Y)\hfill \end{array}$$

(7)

which we prove in Appendix A.11. Here ${\rho }_m(X;Y)\equiv {max}_{f,g}\mathbb{E}\left[f\left(X\right)g\left(Y\right)\right]$ s.t. $\mathbb{E}\left[f\right(X\left)\right]=\mathbb{E}\left[g\right(Y\left)\right]=0$ and $\mathbb{E}\left[f^2\left(X\right)\right]=\mathbb{E}\left[g^2\left(Y\right)\right]=1$ is the maximum correlation [20,21], $\xi (X;Y)\equiv {sup}_{Z-X-Y}\frac{I(Y;Z)}{I(X;Z)}$ is the hypercontractivity coefficient and ${\eta }_{\mathrm{KL}}(p\left(y\right|x),p\left(x\right))\equiv {sup}_{r\left(x\right)\ne p\left(x\right)}\frac{{\mathbb{D}}_{\mathrm{KL}}(r\left(y\right)\Vert p\left(y\right))}{{\mathbb{D}}_{\mathrm{KL}}(r\left(x\right)\Vert p\left(x\right))}$ is the contraction coefficient. Our proof relies on Anantharam et al. [16]’s proof $\xi (X;Y)={\eta }_{\mathrm{KL}}$. Our work reveals the deep relationship between IB-Learnability and these earlier concepts and provides additional insights about what aspects of a dataset give rise to high maximum correlation and hypercontractivity: the most confident, typical, imbalanced subset of $(X,Y)$.

6. Estimating the IB-Learnability Condition

Theorem 4 not only reveals the relationship between the learnability threshold for $\beta$ and the least noisy region of $P\left(Y\right|X)$ but also provides a way to practically estimate ${\beta }_0$, both in the general classification case and in more structured settings.

6.1. Estimation Algorithm

Based on Theorem 4, for general classification tasks we suggest Algorithm 1 to empirically estimate an upper-bound ${\tilde{\beta }}_0\ge {\beta }_0$, as well as discovering the conspicuous subset that determines ${\beta }_0$.

We approximate the probability of each example $p\left(x_i\right)$ by its empirical probability, $\widehat{p}\left(x_i\right)$. For example, for MNIST, $p\left(x_i\right)=\frac{1}{N}$, where N is the number of examples in the dataset. The algorithm starts by first learning a maximum likelihood model of $p_{\theta }\left(y\right|x)$, using for example, feed-forward neural networks. It then constructs a matrix $P_{y|x}$ and a vector $p_y$ to store the estimated $p\left(y\right|x)$ and $p\left(y\right)$ for all the examples in the dataset. To find the subset $\Omega$ such that the ${\tilde{\beta }}_0$ is as small as possible, by previous analysis we want to find a conspicuous subset such that its $p\left(y\right|x)$ is large for a certain class j (to make the denominator of Equation (5) large) and containing as many elements as possible (to make the numerator small).

We suggest the following heuristics to discover such a conspicuous subset. For each class j, we sort the rows of $\left(P_{y|x}\right)$ according to its probability for the pivot class j by decreasing order and then perform a search over $i_{\mathrm{left}},i_{\mathrm{right}}$ for $\Omega =\{i_{\mathrm{left}},i_{\mathrm{left}}+1,\dots ,i_{\mathrm{right}}\}$. Since ${\tilde{\beta }}_0$ is large when $\Omega$ contains too few or too many elements, the minimum of ${\tilde{\beta }}_0^{\left(j\right)}$ for class j will typically be reached with some intermediate-sized subset and we can use binary search or other discrete search algorithm for the optimization. The algorithm stops when ${\tilde{\beta }}_0^{\left(j\right)}$ does not improve by tolerance $\epsilon$. The algorithm then returns the ${\tilde{\beta }}_0$ as the minimum over all the classes ${\tilde{\beta }}_0^{\left(1\right)},\dots {\tilde{\beta }}_0^{\left(N\right)}$, as well as the conspicuous subset that determines this ${\tilde{\beta }}_0$.

After estimating ${\tilde{\beta }}_0$, we can then use it for learning with IB, either directly or as an anchor for a region where we can perform a much smaller sweep than we otherwise would have. This may be particularly important for very noisy datasets, where ${\beta }_0$ can be very large.

Algorithm 1 Estimating the upper bound for ${\beta }_0$ and identifying the conspicuous subset

Require: Dataset $\mathcal{D}=\left\{(x_i,y_i)\right\},i=1,2,\dots N$. The number of classes is C. Require$\epsilon$: tolerance for estimating ${\beta }_0$ 1:  Learn a maximum likelihood model $p_{\theta }\left(y\right|x)$ using the dataset $\mathcal{D}$. 2:  Construct matrix $\left(P_{y|x}\right)$ such that ${\left(P_{y|x}\right)}_{ij}=p_{\theta }(y=y_j|x=x_i)$. 3:  Construct vector $p_y=(p_{y1},..,p_{yC})$ such that $p_{yj}=\frac{1}{N}{\sum }_{i=1}^N{\left(P_{y|x}\right)}_{ij}$. 4:  forjin$\{1,2,\dots C\}$: 5:   $P_{y|x}^{\left(\mathrm{sort}j\right)}\leftarrow$Sort the rows of $P_{y|x}$ in decreasing values of ${\left(P_{y|x}\right)}_{ij}$. 6:   ${\tilde{\beta }}_0^{\left(j\right)},{\Omega }^{\left(j\right)}\leftarrow$Search $i_{\mathrm{left}}$, $i_{\mathrm{right}}$ until ${\tilde{\beta }}_0^{\left(j\right)}=\mathbf{Get}\mathit{\beta }(P_{y|x},p_y,\Omega )$ is minimal with tolerance $\epsilon$, where $\Omega =\{i_{\mathrm{left}},i_{\mathrm{left}}+1,\dots i_{\mathrm{right}}\}$. 7:  end for 8:  $j^{\ast }\leftarrow {arg\; min}_j\{{\tilde{\beta }}_0^{\left(j\right)}\},j=1,2,\dots N$.    9:  ${\tilde{\beta }}_0\leftarrow {\tilde{\beta }}_0^{\left(j^{\ast }\right)}$. 10:  $P_{y|x}^{\left({\tilde{\beta }}_0\right)}\leftarrow$ the rows of $P_{y|x}^{\left(\mathrm{sort}{j}^{\ast }\right)}$ indexed by ${\Omega }^{\left(j^{\ast }\right)}$. 11:  return${\tilde{\beta }}_0,P_{y|x}^{\left({\tilde{\beta }}_0\right)}$

subroutine Get$\mathit{\beta }$($P_{y|x},p_y,\Omega$): s1:  $N\leftarrow$ number of rows of $P_{y|x}$. s2:  $C\leftarrow$ number of columns of $P_{y|x}$. s3:  $n\leftarrow$ number of elements of $\Omega$. s4:  ${\left(p_{y|\Omega }\right)}_j\leftarrow \frac{1}{n}{\sum }_{i\in \Omega }{\left(P_{y|x}\right)}_{ij}$, $j=1,2,\dots ,C$. s5:  ${\tilde{\beta }}_0\leftarrow \frac{\frac{N}{n}-1}{{\sum }_j\left[\frac{{\left(p_{y|{\Omega }_x}\right)}_j^2}{p_{yj}}-1\right]}$ s6:  return${\tilde{\beta }}_0$

6.2. Special Cases for Estimating ${\beta }_0$

Theorem 4 may still be challenging to estimate, due to the difficulty of making accurate estimates of $p\left({\Omega }_x\right)$ and searching over ${\Omega }_x\subset \mathcal{X}$. However, if the learning problem is more structured, we may be able to obtain a simpler formula for the sufficient condition.

6.2.1. Class-Conditional Label Noise

Classification with noisy labels is a common practical scenario. An important noise model is that the labels are randomly flipped with some hidden class-conditional probabilities and we only observe the corrupted labels. This problem has been studied extensively [22,23,24,25,26]. If IB is applied to this scenario, how large $\beta$ do we need? The following corollary provides a simple formula.

Corollary 1.

Suppose that the true class labels are$y^{\ast }$and the input space belonging to each$y^{\ast }$has no overlap. We only observe the corrupted labels y with class-conditional noise$p\left(y\right|x,y^{\ast })=p\left(y\right|y^{\ast })$and Y is not independent of X. We have that a sufficient condition for${\mathit{IB}}_{\beta }$-Learnability is:

$$\begin{array}{c}\hfill \beta >{\displaystyle \underset{y^{\ast }}{inf}}\frac{\frac{1}{p\left(y^{\ast }\right)}-1}{{\sum }_y\frac{p{\left(y\right|y^{\ast })}^2}{p\left(y\right)}-1}\end{array}$$

(8)

We see that under class-conditional noise, the sufficient condition reduces to a discrete formula which only depends on the noise rates $p\left(y\right|y^{\ast })$ and the true class probability $p\left(y^{\ast }\right)$, which can be accurately estimated via, for example, Northcutt et al. [26]. Additionally, if we know that the noise is class-conditional but the observed ${\beta }_0$ is greater than the R.H.S. of Equation (8), we can deduce that there is overlap between the true classes. The proof of Corollary 1 is provided in Appendix A.10.

6.2.2. Deterministic Relationships

Theorem 4 also reveals that ${\beta }_0$ relates closely to whether Y is a deterministic function of X, as shown by Corollary 2:

Corollary 2.

Assume that Y contains at least one value y such that its probability$p\left(y\right)>0$. If Y is a deterministic function of X and not independent of X, then a sufficient condition for${\mathit{IB}}_{\beta }$-Learnability is$\beta >1$.

The assumption in the Corollary 2 is satisfied by classification and certain regression problems. (The following scenario does not satisfy this assumption: for certain regression problems where Y is a continuous random variable and the probability density function $p_Y\left(y\right)$ is bounded, then for any y, the probability $P(Y=y)$ has measure 0.)This corollary generalizes the result in Reference [9] which only proves it for classification problems. Combined with the necessary condition $\beta >1$ for any dataset $(X,Y)$ to be ${\mathrm{IB}}_{\beta }$-learnable (Section 3), we have that under the assumption, if Y is a deterministic function of X, then a necessary and sufficient condition for ${\mathrm{IB}}_{\beta }$-learnability is $\beta >1$; that is, its ${\beta }_0$ is 1. The proof of Corollary 2 is provided in Appendix A.10.

Therefore, in practice, if we find that ${\beta }_0>1$, we may infer that Y is not a deterministic function of X. For a classification task, we may infer that either some classes have overlap or the labels are noisy. However, recall that finite models may add effective class overlap if they have insufficient capacity for the learning task, as mentioned in Section 4. This may translate into a higher observed ${\beta }_0$, even when learning deterministic functions.

7. Experiments

To test how the theoretical conditions for ${\mathrm{IB}}_{\beta }$-learnability match with experiment, we apply them to synthetic data with varying noise rates and class overlap, MNIST binary classification with varying noise rates and CIFAR10 classification, comparing with the ${\beta }_0$ found experimentally. We also compare with the algorithm in Kim et al. [18] for estimating the hypercontractivity coefficient (=$1/{\beta }_0$) via the contraction coefficient ${\eta }_{\mathrm{KL}}$. Experiment details are in Appendix A.12.

7.1. Synthetic Dataset Experiments

We construct a set of datasets from 2D mixtures of 2 Gaussians as X and the identity of the mixture component as Y. We simulate two practical scenarios with these datasets: (1) noisy labels with class-conditional noise and (2) class overlap. For (1), we vary the class-conditional noise rates. For (2), we vary class overlap by tuning the distance between the Gaussians. For each experiment, we sweep $\beta$ with exponential steps and observe $I(X;Z)$ and $I(Y;Z)$. We then compare the empirical ${\beta }_0$ indicated by the onset of above-zero $I(X;Z)$ with predicted values for ${\beta }_0$.

7.1.1. Classification with Class-Conditional Noise

In this experiment, we have a mixture of Gaussian distribution with 2 components, each of which is a 2D Gaussian with diagonal covariance matrix $\Sigma =\mathrm{diag}(0.25,0.25)$. The two components have distance 16 (hence virtually no overlap) and equal mixture weight. For each x, the label $y\in \{0,1\}$ is the identity of which component it belongs to. We create multiple datasets by randomly flipping the labels y with a certain noise rate $\rho =P(y=0|y^{\ast }=1)=P(y=1|y^{\ast }=0)$. For each dataset, we train VIB models across a range of $\beta$ and observe the onset of learning via random $I(X;Z)$ (Observed). To test how different methods perform in estimating ${\beta }_0$, we apply the following methods: (1) Corollary 1, since this is classification with class-conditional noise and the two true classes have virtually no overlap; (2) Algorithm 1 with true $p\left(y\right|x)$; (3) The algorithm in Kim et al. [18] that estimates ${\widehat{\eta }}_{KL}$, provided with true $p\left(y\right|x)$; (4) ${\beta }_0\left[h\left(x\right)\right]$ in Equation (2); (2′) Algorithm 1 with $p\left(y\right|x)$ estimated by a neural net; (3′) ${\widehat{\eta }}_{KL}$ with the same $p\left(y\right|x)$ as in (2′). The results are shown in Figure 3 and in

Table 1. Full table of values used to generate Figure 3.

			(2) Algorithm 1	(3) ${\widehat{\mathit{\eta }}}_{KL}$
Noise Rate	Observed	(1) Corollary 1	True $\mathit{p}\left(\mathit{y}\right|\mathit{x})$	True $\mathit{p}\left(\mathit{y}\right|\mathit{x})$	(4) Equation (2)	(2′) Algorithm 1	(3′) ${\widehat{\mathit{\eta }}}_{KL}$
0.02	1.06	1.09	1.09	1.10	1.08	1.08	1.10
0.04	1.20	1.18	1.18	1.21	1.18	1.19	1.20
0.06	1.26	1.29	1.29	1.33	1.30	1.31	1.33
0.08	1.40	1.42	1.42	1.45	1.42	1.43	1.46
0.10	1.52	1.56	1.56	1.60	1.55	1.58	1.60
0.12	1.70	1.73	1.73	1.78	1.71	1.73	1.77
0.14	1.99	1.93	1.93	1.99	1.90	1.91	1.95
0.16	2.04	2.16	2.16	2.24	2.15	2.15	2.16
0.18	2.41	2.44	2.44	2.49	2.43	2.42	2.49
0.20	2.74	2.78	2.78	2.86	2.76	2.77	2.71
0.22	3.15	3.19	3.19	3.29	3.19	3.21	3.29
0.24	3.75	3.70	3.70	3.83	3.71	3.75	3.72
0.26	4.40	4.34	4.34	4.48	4.35	4.31	4.17
0.28	5.16	5.17	5.17	5.37	5.12	4.98	4.55
0.30	6.34	6.25	6.25	6.49	6.24	6.03	5.58
0.32	8.06	7.72	7.72	8.02	7.63	7.19	7.33
0.34	9.77	9.77	9.77	10.13	9.74	8.95	7.37
0.36	12.58	12.76	12.76	13.21	12.51	11.11	10.09
0.38	16.91	17.36	17.36	17.96	16.97	14.55	10.49
0.40	24.66	25.00	25.00	25.99	25.01	20.36	17.27
0.42	39.08	39.06	39.06	40.85	39.48	30.12	10.89
0.44	64.82	69.44	69.44	71.80	76.48	51.95	21.95
0.46	163.07	156.25	156.26	161.88	173.15	114.57	21.47
0.48	599.45	625.00	625.00	651.47	838.90	293.90	8.69

.

From Figure 3 and Table 1 we see the following. (A) When using the true $p\left(y\right|x)$, both Algorithm 1 and ${\widehat{\eta }}_{\mathrm{KL}}$ generally upper bound the empirical ${\beta }_0$ and Algorithm 1 is generally tighter. (B) When using the true $p\left(y\right|x)$, Algorithm 1 and Corollary 1 give the same result. (C) Comparing Algorithm 1 and ${\widehat{\eta }}_{KL}$ both of which use the same empirically estimated $p\left(y\right|x)$, both approaches provide good estimation in the low-noise region; however, in the high-noise region, Algorithm 1 gives more precise values than ${\widehat{\eta }}_{KL}$, indicating that Algorithm 1 is more robust to the estimation error of $p\left(y\right|x)$. (D) Equation (2) empirically upper bounds the experimentally observed ${\beta }_0$ and gives almost the same result as theoretical estimation in Corollary 1 and Algorithm 1 with the true $p\left(y\right|x)$. In the classification setting, this approach does not require any learned estimate of $p\left(y\right|x)$, as we can directly use the empirical $p\left(y\right)$ and $p\left(x\right|y)$ from SGD mini-batches.

This experiment also shows that for dataset where the signal-to-noise is small, ${\beta }_0$ can be very high. Instead of blindly sweeping $\beta$, our result can provide guidance for setting $\beta$ so learning can happen.

7.1.2. Classification with Class Overlap

In this experiment, we test how different amounts of overlap among classes influence ${\beta }_0$. We use the mixture of Gaussians with two components, each of which is a 2D Gaussian with diagonal covariance matrix $\Sigma =\mathrm{diag}(0.25,0.25)$. The two components have weights 0.6 and 0.4. We vary the distance between the Gaussians from 8.0 down to 0.8 and observe the ${\beta }_{0,exp}$. Since we do not add noise to the labels, if there were no overlap and a deterministic map from X to Y, we would have ${\beta }_0=1$ by Corollary 2. The more overlap between the two classes, the more uncertain Y is given X. By Equation (5) we expect ${\beta }_0$ to be larger, which is corroborated in Figure 4.

7.2. MNIST Experiments

We perform binary classification with digits 0 and 1 and as before, add class-conditional noise to the labels with varying noise rates $\rho$. To explore how the model capacity influences the onset of learning, for each dataset we train two sets of VIB models differing only by the number of neurons in their hidden layers of the encoder: one with $n=512$ neurons, the other with $n=128$ neurons. As we describe in Section 4, insufficient capacity will result in more uncertainty of Y given X from the point of view of the model, so we expect the observed ${\beta }_0$ for the $n=128$ model to be larger. This result is confirmed by the experiment (Figure 5). Also, in Figure 5 we plot ${\beta }_0$ given by different estimation methods. We see that the observations (A), (B), (C) and (D) in Section 7.1 still hold.

7.3. MNIST Experiments Using Equation (2)

To see what IB learns at its onset of learning for the full MNIST dataset, we optimize Equation (2) w.r.t. the full MNIST dataset and visualize the clustering of digits by $h\left(x\right)$. Equation (2) can be optimized using SGD using any differentiable parameterized mapping $h\left(x\right):\mathcal{X}\to \mathbb{R}$. In this case, we chose to parameterize $h\left(x\right)$ with a PixelCNN++ architecture [27,28], as PixelCNN++ is a powerful autoregressive model for images that gives a scalar output (normally interpreted as $\log p\left(x\right)$). Equation (2) should generally give two clusters in the output space, as discussed in Section 4. In this setup, smaller values of $h\left(x\right)$ correspond to the subset of the data that is easiest to learn. Figure 6 shows two strongly separated clusters, as well as the threshold we choose to divide them. Figure 7 shows the first 5776 MNIST training examples as sorted by our learned $h\left(x\right)$, with the examples above the threshold highlighted in red. We can clearly see that our learned $h\left(x\right)$ has separated the “easy” one (1) digits from the rest of the MNIST training set.

7.4. CIFAR10 Forgetting Experiments

For CIFAR10 [14], we study how forgetting varies with $\beta$. In other words, given a VIB model trained at some high ${\beta }_2$, if we anneal it down to some much lower ${\beta }_1$, what $I(Y;Z)$ does the model converge to? Using Algorithm 1, we estimated ${\beta }_0=1.0483$ on a version of CIFAR10 with 20% label noise, where the $P_{y|x}$ is estimated by maximum likelihood training with the same encoder and classifier architectures as used for VIB. For the VIB models, the lowest $\beta$ with performance above chance was $\beta =1.048$ (Figure 8), a very tight match with the estimate from Algorithm 1. See Appendix A.12 for details.

8. Conclusions

In this paper, we have presented theoretical results for predicting the onset of learning and have shown that it is determined by the conspicuous subset of the training examples. We gave a practical algorithm for predicting the transition as well as discovering this subset and showed that those predictions are accurate, even in cases of extreme label noise. We proved a deep connection between IB-learnability, our upper bounds on ${\beta }_0$, the hypercontractivity coefficient, the contraction coefficient and the maximum correlation. We believe that these results provide a deeper understanding of IB, as well as a tool for analyzing a dataset by discovering its conspicuous subset and a tool for measuring model capacity in a task-specific manner. Our work also raises other questions, such as whether there are other phase transitions in learnability that might be identified. We hope to address some of those questions in future work.