Nearly sufficient statistics and information bottlenecks

1.1 “Nearly sufficient” IB

There are at least three ways to say “nearly sufficient” depending on what we want \(Y\) to do.

1. Information sufficiency (task‑conditional independence). Define \(T\) to be \(Y\)-sufficient if \(I(Y;X\mid T)=0\). Then “\(\varepsilon\)-sufficient” means \(I(Y;X\mid T)\le \varepsilon\). IB is the convex relaxation that searches for \(T\) with small \(I(T;X)\) subject to small \(I(Y;X\mid T)\), via a Lagrange multiplier \(\beta\).

2. Decision sufficiency. For a loss \(\ell(a,y)\), call \(T\) \(\varepsilon\)-sufficient if the Bayes risk gap using \(T\) instead of \(X\) is \(\le \varepsilon\). For log‑loss this is equivalent to #1; for other losses it’s stricter or looser. (This is where rate–distortion with a bespoke distortion can be a better fit than vanilla IB if the task is not log‑loss‑like.)

Pick the notion that matches our downstream risk. If we’re doing Bayesian filtering or SMC, #1 with \(Y\!=\!X_{t+1}\) or \(Y\!=\!\) future rewards is the right primitive; if we’re doing parameter estimation, #1 with \(Y\!=\!\theta\) (or a function like \(g(\theta)\)) is usually the cleanest.

2.1 Filtering and state compression

In Bayes filtering we want a recursively updated \(T_t\) that is small but “task‑sufficient”. In exponential families (e.g. linear-Gaussian) filters exist, e.g. the Kalman filter.

IB gives us a target for awkward models: minimize \(I(T_t; X_{1:t})\) while maximizing \(I(T_t; Y_t)\), where \(Y_t\) is whatever we care about next (the next observation, the latent state, the next reward). We can then:

• Parameter‑tracking: We set \(Y_t=\theta\) (or \(g(\theta)\)), turning \(T_t\) into a rolling compressed posterior proxy.
• Predictive control: We can set \(Y_t=X_{t+1}\) or a function of \(X_{t+1}\) and action \(A_t\).

2.2 In neural nets

Each layer in a trained DNN behaves approximately like a nearly sufficient statistic for its task variable, which is a helpful mental picture even if we don’t care about DNNs. The classical IB principle was defined as an optimization problem over encoders \(p(t|x)\): compress \(X\) into \(T\) while keeping as much information about \(Y\) as possible,

\[ \min_{p(t|x)} I(T;X) - \beta I(T;Y). \]

That’s clean, but impractical in raw form because neural networks are large and complicated. Deep Information Bottleneck parameterizes the encoder and decoder with neural nets and trains them using gradient methods.

• Encoder: \(p_\theta(t|x)\) (often Gaussian; mean and variance predicted by a neural net).
• Decoder: \(q_\phi(y|t)\), a variational approximation to \(p(y|t)\).
• Prior: \(r(t)\), usually standard Gaussian, to make the KL term computable.

So the training objective becomes:

\[ \max_{\theta,\phi}\; \mathbb{E}_{p_\theta(t|x)}[\log q_\phi(y|t)] - \beta\, \text{KL}(p_\theta(t|x)\|r(t)). \]

That’s the Deep Variational IB (DVIB) objective. If we set \(Y=X\), we get the VAE ELBO; if we set \(Y\) to be labels, we get a bottleneck tuned for classification.

The KL penalty is the mechanism of “compression”. It forces the encoder not to carry every pixel of \(X\) but only the degrees of freedom that actually help predict \(Y\). In practice, it behaves like information-aware dropout: we add noise to our hidden representation until only task-relevant features survive.

This gives the encoder a concrete way to approximate minimal sufficient statistics for \(Y\), even in messy, high-dimensional settings.

Shwartz-Ziv and Tishby (2017) famously plotted layer-wise mutual information during training. In their experiments each hidden layer \(T_i\) follows a path in the “information plane” \((I(T;X), I(T;Y))\):

• Phase 1 (fitting): mutual information with \(Y\) shoots up as the layers learn to predict.
• Phase 2 (compression): \(I(T;X)\) falls slowly — the representation forgets irrelevant details while keeping \(I(T;Y)\) high.

The two phases correspond to what we see if we measure gradient statistics: a drift phase (large, directed updates) followed by a diffusion phase (random fluctuations dominate). The diffusion adds noise that automatically compresses representations.

So in the Deep IB view, SGD itself is a bottleneck machine. Layers end up very close to the IB theoretical bound, which characterizes optimal trade-offs between memory \(I(T;X)\) and relevance \(I(T;Y)\).

2.3 Non-convexity

There are two competing terms in the functional

\[ \min_{p(t|x)} I(T;X) - \beta I(T;Y). \]

\(I(T;X)\) is convex in \(p(t|x)\), while \(I(T;Y)\) is concave in \(p(t|x)\). Their weighted difference is not convex. This means we don’t get the nice guarantees we’d have in rate–distortion theory with convex distortion measures. Instead, the IB Lagrangian can have multiple local minima.

The iterative IB algorithm (the original Blahut–Arimoto-style update rules from Tishby et al. 1999) only converges to a fixed point of the self-consistent equations, not necessarily the global optimum. This is why different runs with different initialisations can produce different clusterings or partitions of the data.

• Classical IB (discrete case): iterative IB converges to locally optimal solutions; sometimes multiple solutions lie on the same “information curve,” but we don’t know if we’ve found the best.
• Deep IB (DVIB): we’ve compounded the problem: now the encoder/decoder are parameterized by neural nets, so we inherit both the non-convexity of the IB objective and the non-convexity of deep learning. Optimizers (SGD, Adam) will tend to settle in some basin that’s “good enough” but not provably optimal.
• Information-plane diagnostics (Schwartz-Ziv & Tishby): One of the reasons they use mutual information plots is to check whether the layers land near the theoretical IB bound. But the fact that different training runs converge to slightly different places reflects the non-convexity.

Researchers have tried to tame this in several ways:

• Convex relaxations / surrogates: Variants like the Hilbert–Schmidt independence criterion bottleneck (HSIC-IB) replace mutual information with a kernel-based dependence measure, which yields a convex or easier objective for optimization.

• Variational bounds: DVIB itself is a relaxation — we replace intractable mutual information terms with variational lower bounds. These are optimizable with SGD, but they don’t restore convexity, they just make the problem numerically tractable.

• Deficiency and f-divergence bottlenecks: Alternatives like the variational deficiency bottleneck (VDB) swap in divergences with better optimization landscapes in some cases.

• Annealing β / continuation: A common heuristic is to slowly increase \(\beta\), following the IB curve and avoiding bad local minima.