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Gated information bottleneck for generalization in sequential environments

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Deep neural networks suffer from poor generalization to unseen environments when the underlying data distribution is different from that in the training set. By learning minimum sufficient representations from training data, the information bottleneck (IB) approach has demonstrated its effectiveness to improve generalization in different AI applications. In this work, we propose a new neural network-based IB approach, termed gated information bottleneck (GIB), that dynamically drops spurious correlations and progressively selects the most task-relevant features across different environments by a trainable soft mask (on raw features). Using the recently proposed matrix-based Rényi’s \(\alpha \)-order mutual information estimator, GIB enjoys a simple and tractable objective, without any variational approximation or distributional assumption. We empirically demonstrate the superiority of GIB over other popular neural network-based IB approaches in adversarial robustness and out-of-distribution detection. Meanwhile, we also establish the connection between IB theory and invariant causal representation learning and observed that GIB demonstrates appealing performance when different environments arrive sequentially, a more practical scenario where invariant risk minimization fails.

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Correspondence to Francesco Alesiani or Shujian Yu.

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This paper was presented in part at the 2021 IEEE International Conference on Data Mining (ICDM) [1].

Appendix A Proofs to Lemma 1 and Lemma 2

Appendix A Proofs to Lemma 1 and Lemma 2

Before providing proofs to Lemma 1 and Lemma 2, let us first introduce the following three lemmas.

Lemma 3

Given two random variables XZ, whose joint distribution is given by p(zx) and conditional distribution \(p(z \vert x)\), the KL divergence between \(p(z \vert x)\) and p(z) is upper bounded by:

$$\begin{aligned} {\text {KL}}( p(z|x)||p(z))&\le {{\,\mathrm{{\mathbb {E}}}\,}}_{x'} {\text {KL}}( p(z|x)||p(z|x') ) \end{aligned}$$

Proof of Lemma 3

Indeed, we have:

$$\begin{aligned} {\text {KL}}( p(z|x)||p(z))&= \sum _{z} p(z|x) \ln \frac{p(z|x)}{p(z)} \\&= \sum _{z} p(z|x) \ln p(z|x) - \sum _{z} p(z|x) \ln {p(z)} \\&= H(z|x) - \sum _{z} p(z|x) \ln {\sum _{x'} p(z|x') p(x')} \\&\le H(z|x) - \sum _{x'} p(x') \sum _{z} p(z|x) \ln { p(z|x') } \\&= \sum _{z} p(z|x) \ln p(z|x) - \sum _{x'} p(x') \sum _{z} p(z|x) \ln { p(z|x') } \\&= \sum _{x'} p(x') \left[ \sum _{z} p(z|x) \ln p(z|x) \right. \left. - \sum _{z} p(z|x) \ln { p(z|x') } \right] \\&= \sum _{x'} p(x') \left[ \sum _{z} p(z|x) \frac{\ln p(z|x)}{p(z|x') } \right] \\&= {{\,\mathrm{{\mathbb {E}}}\,}}_{x'} {\text {KL}}( p(z|x)||p(z|x')), \end{aligned}$$

where we used \(p(z) = \sum _{x'} p(z|x') p(x') \) and \(\ln {\sum _{x'} p(z|x') p(x')} \ge \sum _{x'} p(x') \ln { p(z|x') }\). \(\square \)

Lemma 4

Given two random variables XZ, with two joint distributions given by \(p^1(z,x),p^2(z,x)\) and conditional distributions \(p^1(z|x),p^2(z|x)\), the KL divergence between \(p^1(z|x)\) and \(p^2(z)\) is upper bounded by:

$$\begin{aligned} {\text {KL}}( p^1(z|x)||p^2(z))&\le {{\,\mathrm{{\mathbb {E}}}\,}}_{x'} {\text {KL}}( p^1(z|x)||p^2(z|x')) \end{aligned}$$

Proof of Lemma 4

The derivation is similar to Lemma.3, where we used \(p^2(z) = \sum _{x'} p^2(z|x') p^2(x') \). \(\square \)

Given Lemma 3 and Lemma 4, we can now introduce the main inequality in the following Lemma.

Lemma 5

Given two random variables XZ, whose joint distribution is given by p(zx) and conditional distribution p(z|x), the Mutual Information I(xz) is upper bounded by:

$$\begin{aligned} I(x,z)&\le {{\,\mathrm{{\mathbb {E}}}\,}}_x {{\,\mathrm{{\mathbb {E}}}\,}}_{x'} {\text {KL}}( p(z|x)||p(z|x')) \end{aligned}$$

Proof of Lemma 5

Mutual information is the KL divergence between the join and product of marginal distributions

$$\begin{aligned} I(x,z)&= {\text {KL}}(p(x,z)||p(x)p(z)) \end{aligned}$$
$$\begin{aligned}&= {{\,\mathrm{{\mathbb {E}}}\,}}_x {\text {KL}}(p(z|x)||p(z)). \end{aligned}$$

We thus build an upper bound to the mutual information:

$$\begin{aligned} I(x,z)&\le {{\,\mathrm{{\mathbb {E}}}\,}}_x {{\,\mathrm{{\mathbb {E}}}\,}}_{x'} {\text {KL}}( p(z|x)||p(z|x')), \end{aligned}$$

where we use Lemma 3. \(\square \)

Now we provide proofs to Lemma 1 and Lemma 2, respectively.

1.1 Proof to Lemma 1

This inequality can be obtained by expanding the two distributions and using the convexity property of the logarithm:

$$\begin{aligned} {\text {KL}}( p^1(z) || p^2(z))&= {{\,\mathrm{{\mathbb {E}}}\,}}_{p^1(z)} \ln \frac{p^1(z)}{p^2(z)} \\&=\sum _{z} p^1(z) \ln p^1(z) - \sum _{z} p^1(z) \ln p^2(z) \\&=\sum _{x} p^1(x) \sum _{z} p^1(z|x) \ln \sum _{x'} p^1(x') p^1(z|x') - \sum _{z} p^1(z) \ln p^2(z) \\&\ge \sum _{x} p^1(x) \sum _{x'} p^1(x') \sum _{z} p^1(z|x) \ln p^1(z|x') - \sum _{z} p^1(z) \ln p^2(z) \\&= \sum _{x} p^1(x) \sum _{x'} p^1(x') \sum _{z} p^1(z|x) \ln p^1(z|x') \\&\quad - \sum _{x} p^1(x) \sum _{z} p^1(z|x) \ln p^2(z) \\&= \sum _{x} p^1(x) \sum _{x'} p^1(x') \sum _{z} p^1(z|x) \ln p^1(z|x') \\&\quad - \sum _{x} p^1(x) \sum _{x'} p^1(x') \sum _{z} p^1(z|x) \ln p^2(z) \\&= \sum _{x} p^1(x) \sum _{x'} p^1(x') ( \sum _{z} p^1(z|x) \ln p^1(z|x') - \sum _{z} p^1(z|x) \ln p^2(z) ) \\&= \sum _{x} p^1(x) \sum _{x'} p^1(x') ( \sum _{z} p^1(z|x) \ln p^1(z|x') \\&\quad - \sum _{z} p^1(z|x) \ln \sum _{x''} p^2(x'') p^2(z|x'') ) \\&\approx \sum _{x} p^1(x) \sum _{x'} p^1(x') ( \sum _{z} p^1(z|x) \ln p^1(z|x') \\&\quad - \sum _{x''} p^2(x'') \sum _{z} p^1(z|x) \ln p^2(z|x'') ) \\&= \sum _{x} p^1(x) \sum _{x'} p^1(x') \sum _{x''} p^2(x'') \left[ \sum _{z} p^1(z|x) \ln p^1(z|x') \right. \\&\quad \left. - \sum _{z} p^1(z|x) \ln p^2(z|x'') \right] \\&= \sum _{x} p^1(x) \sum _{x'} p^1(x') \sum _{x''} p^2(x'') \left[ \sum _{z} p^1(z|x) \ln \frac{p^1(z|x')}{ p^2(z|x'')} \right] \\&= \sum _{x} p^1(x) \sum _{x'} p^1(x') \sum _{x''} p^2(x'') {\text {KL}}( p^1(z|x') || p^2(z|x'')) \\&= {{\,\mathrm{{\mathbb {E}}}\,}}_{x\sim p^1(x),x'\sim p^1(x),x'' \sim p^2(x)} {\text {KL}}( p^1(z|x') || p^2(z|x'')) \\&\approx \sum _{x} p^1(x) \sum _{x''} p^2(x'') {\text {KL}}( p^1(z|x) || p^2(z|x'')) \\&= {{\,\mathrm{{\mathbb {E}}}\,}}_{x,x'} {\text {KL}}( p^1(z|x) || p^2(z|x')) \end{aligned}$$

where we used \(\ln \sum _{x'} p^1(x') p^1(z|x') \ge \sum _{x'} p^1(x') \ln p^1(z|x') \), \(\sum _{x'} p^1(x') \sum _{z} p^1(z|x') = p^1(z)\) and \(\sum _{x'} p^1(x') = 1\). In the last approximation we substitute \(p^1(z|x') \rightarrow p^1(z|x)\).

1.2 Proof to Lemma 2

Similar to Lemma 5, this property follows from Lemma 4. Indeed, when we consider the cross-domain mutual information between XZ on the two distributions, we have:

$$\begin{aligned} I^{12}(X;Z)&= {\text {KL}}(p^1(z,x)||p^2(z)p^2(x)) \end{aligned}$$
$$\begin{aligned}&= {{\,\mathrm{{\mathbb {E}}}\,}}_x {\text {KL}}(p^1(z|x)||p^2(z)) \end{aligned}$$
$$\begin{aligned}&\le {{\,\mathrm{{\mathbb {E}}}\,}}_x {{\,\mathrm{{\mathbb {E}}}\,}}_{x'} {\text {KL}}( p^1(z|x)||p^2(z|x')), \end{aligned}$$

where \({\text {KL}}(p^1(z,x)||p^2(z)p^2(x)) = {{\,\mathrm{{\mathbb {E}}}\,}}_x {\text {KL}}(p^1(z|x)||p^2(z))\), when \(p^2(x)=p^1(x)\) and \({\text {KL}}( p^1(z|x)||p^2(z)) \le {{\,\mathrm{{\mathbb {E}}}\,}}_{x'} {\text {KL}}( p^1(z|x)||p^2(z|x')) \) from Lemma 4.

1.3 Simulation details

1.3.1 Synthetic experiment

In this experiment, the features have fix size of \(d = 20\). Hyper-parameter search was done using grid search method over the following values \(\lambda \in [0, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1]\), 5 repetitions, learning rate \(lr=1e-4\), 10’000 iterations, 1’000 samples. Experiments were performed on a server with 8 CPUs, 64Gb RAM, and one GPU with 8Gb RAM.

1.3.2 Colored dataset experiment

Experiments have been conducted using an MLP with 3 hidden linear layer with drop-out, each of 200 neurons and Relu activation function. Cross Entropy is used as minimization loss. Hyper-parameters search based on grid search. For IRM and IRMG the best parameters suggested in the original works have been used. Regularization terms have been selected based on heuristics. Each environment has \(5'000\) samples, while the test has \(10'0000\) samples, trained over 100 epochs. For MER, EWC, and GEM settings are those from the authors, when possible.

1.4 Analysis of the complexity (time, space, sample size)

The proposed GIB algorithm requires only marginal complexity in time and space. The space complexity is two times the size of features, marginal w.r.t the number of variables in a typical DNN.

Fig. 10
figure 10

(a) Total dependence value among each dimension of representation measured by \(T_\alpha ^*\) and IDD; and (b) PGD on MNIST

1.5 The independence assumption in drop-bottleneck

Although Drop-Bottleneck [11] shares similar ideas to ours, i.e., discretely drops redundant features that are irrelevant to tasks. We emphasize two significant differences here:

  • Drop-Bottleneck does not build the connection between IB and invariance representation learning. It targets reinforcement learning and implements the maximization of mutual information I(zy) by the mutual information neural estimator (MINE) as has been used in the famed deep infoMax. By contrast, same to other popular IB approaches, we simply implement this term by cross-entropy.

  • Drop-Bottleneck assumes the independence between each dimension of the representation Z for simplicity (i.e., ), and claims that the total dependence among \({Z_1,Z_2,...,Z_p}\) could decrease to zero as the optimization progresses.

Although it is hard to directly compare GIB with Drop-Bottleneck, we additionally provide two empirical justifications for why the independence assumption is harmful. Figure (10) shows the total dependence when we train an MLP on MNIST. As can be seen, the total dependence (measured by state-of-the-art \(T_\alpha ^*\) [14] and IDD [61]) is far away from zero, which indicates that the pairwise independence assumption does not hold across training. Figure (10) shows the result when we replace our entropy estimator (i.e., Eq. (12)) with a simpler Shannon entropy estimator that assumes fully independence (i.e., \(H(z)=\sum _i^p H(z_i)\)). As can be seen, such assumptions significantly degrade the performance.

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Alesiani, F., Yu, S. & Yu, X. Gated information bottleneck for generalization in sequential environments. Knowl Inf Syst 65, 683–705 (2023).

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