Hierarchical Transformer Encoders

• Hierarchical Transformer Encoders are neural architectures that explicitly model multiscale dependencies through hierarchical structures and blockwise attention patterns.
• They approximate optimal inference by emulating belief propagation, using layer-wise aggregation to process local to global information in tree-structured data.
• Key design principles include matching transformer depth to data hierarchy, employing curriculum training via hierarchical filtering, and leveraging interpretable block-diagonal attention patterns.

Hierarchical Transformer Encoders are neural architectures that extend the standard Transformer model by explicitly modeling latent or observed hierarchical structure in input sequences, typically through multi-level representation, blockwise attention patterns, or explicit tree/binary partitioning. These hierarchical mechanisms enable the model to aggregate, process, and interpret information at multiple scales, from local to global, thereby improving sample efficiency, interpretability, and performance on data exhibiting multiscale or tree-like dependencies.

1. Formal Framework: Hierarchical Sequence Models and Filtering

The precise analysis of hierarchical Transformer encoders begins with their application to data generated by explicit hierarchical processes. A canonical example is the generative model over tree-structured sequences as detailed in (Garnier-Brun et al., 27 Aug 2024). Here, data $x = (x_1, \ldots, x_{2^\ell})$ is the set of leaves of a full binary tree of depth $\ell$. The root $x_0$ is sampled from

$P_0(x_0=a),\ a\in\{1,\ldots,q\}$

and each internal node at depth $d < \ell$ generates its left and right children via a fixed transition tensor,

$$M \in \mathbb{R}_+^{q \times q \times q},\qquad M_{a\,b\,c}=P(x_\ell=b,\;x_r=c\mid x_u=a)$$

with normalization $\sum_{b,c}M_{a\,b\,c}=1$.

A hierarchical filtering procedure enables fine-grained control over the range of correlations present in the observed data; for a cutoff scale $k\le \ell$, correlations above depth $k$ are removed by independently redrawing nodes above $k$ from their marginals: $$P_{F_k}(x_j\mid x_0)=P(x_j\mid x_0),\quad x_{j_1}\perp x_{j_2}\mid x_0 \text{ for } j_1 \neq j_2$$ with full branching structure restored in each remaining subtree under depth $k$. This constructs a family of data distributions with precise scale-localization of correlations.

2. Hierarchical Inference: Belief Propagation as Computational Target

For hierarchical data, the exact inference algorithm is upward–downward belief propagation (BP) on the tree factor graph. This consists of recursive message passing:

• Upward messages $\nu_{i \to \alpha}(x_i)$ propagate evidence from leaves toward the root, combining at internal nodes via: $$\hat\nu_{\alpha\to u}(x_u) \propto \sum_{x_\ell,x_r}M_{x_u\,x_\ell\,x_r}\,\nu_{\ell\to\alpha}(x_\ell)\,\nu_{r\to\alpha}(x_r)$$
• Above a certain scale $k$, conditional independence decouples nodes, and messages are replaced by appropriate marginal computations.

Marginals at each node are computed after one upward and one downward pass: $$\mu_i(x_i) \propto \prod_{\alpha\in\partial i}\hat\nu_{\alpha\to i}(x_i)$$ This BP algorithm is optimal—computing exact node posteriors in $O(2^\ell)$ time.

3. Transformer Encoders as Approximate Hierarchical Inference Machines

A standard encoder-only Transformer with $n_L=\ell$ layers can, when trained on masked language modeling and root classification tasks over tree-structured data, approximate the exact BP algorithm (Garnier-Brun et al., 27 Aug 2024). Each attention layer $m=1,\ldots,\ell$ implements blockwise aggregation: $$A^{(m)}_{ij} =\mathrm{softmax}_j\!\left(\frac{Q^{(m)}h_i^{(m-1)} \cdot K^{(m)}h_j^{(m-1)}}{\sqrt{d}}\right)$$ with weights learning to select all tokens $j$ that share the same ancestor as $i$ at tree depth $\ell-m$. Empirically, this manifests as

$$A^{(m)}_{ij}\approx \begin{cases} 1/|\mathcal B^{(m)}(i)| & \text{if }j\in\mathcal B^{(m)}(i) \ 0 & \text{otherwise} \end{cases}$$

where $\mathcal B^{(m)}(i)$ is the block (subtree) of size $2^m$.

Within each layer, the residual and feed-forward update

$$\tilde h_i^{(m)} = h_i^{(m-1)} + \sum_j A_{ij}^{(m)}\,V^{(m)}\,h_j^{(m-1)}$$

enables layer-wise propagation of information, allowing the network to recursively simulate BP's upward pass modulo learned parameterization in $V^{(m)}$ and the FFN.

4. Empirical Evidence: Staged Learning, Blockwise Attention, and Selectivity

Hierarchical Transformer encoders, when trained from scratch in this controlled setting, display staged learning dynamics that reflect underlying data scales:

• The decrease in Kullback-Leibler divergence between model and BP marginals,

$$D_{\rm KL}^{(k)}(t) =\mathbb{E}_{x\sim F_k}\left[ \mathrm{KL}\bigl(\mu^{\rm Tr}_t(x)\;\|\;\mu^{\rm BP}_k(x)\bigr) \right]$$

provably occurs first at shorter-range scales (large $k$) before extending to longer-range (small $k$), i.e. the model learns correlations from local to global.

• Visualization of attention matrices $A^{(m)}$, when averaged, reveals emergence of block-diagonal structure aligned with tree hierarchy. As the data's scale of correlation increases, so do the block sizes in the respective layers.
• Probing activations at each layer $m$ using small supervised classifiers shows that only information about ancestors up to depth $m$ is accessible, confirming that each SBP “up-pass” is realized at the matching layer.

5. Design Principles and Implementation Guidelines

Three key principles for deploying hierarchical Transformer encoders emerge:

Depth-Layer Matching: To mirror a depth-$\ell$ BP computation, the Transformer should possess at least $\ell$ layers; additional layers yield no measurable benefit in controlled settings.

Curriculum via Hierarchical Filtering: Progressive training on $F_k$ distributions with decreasing $k$—i.e., initially exposing the model only to short-range dependencies, then gradually reintroducing longer-range structure—facilitates more efficient learning of global correlations.

Interpretable Blockwise Attention: The presence of visible block patterns in self-attention matrices at each layer provides a mechanistic interpretability handle: each block corresponds to aggregation at a specific hierarchical level, offering a direct mapping from architecture to computation.

A summary table encapsulating these principles:

Principle	Implementation Action	Empirical Impact
Match depth to layers	$n_L \geq \ell$	Accurate BP approximation
Curriculum by filtering	Train on $F_k$, gradually lower $k$	Staged multiscale generalization
Blockwise attention	Visualize $A^{(m)}$ for block-diagonals	Mechanistic computation insight

6. Implications and Extensions

This analysis confirms that vanilla Transformer encoders can, when trained appropriately, discover multiscale computation and approximate optimal algorithms for structured data, despite having no explicit prior for hierarchy. For general applications:

• Matching the data's intrinsic hierarchical depth by layer count improves performance and efficiency.
• Where data exhibits unknown or latent multi-scale dependencies, curriculum strategies based on data filtering may accelerate learning and improve generalization.
• Inspection of intermediate attention patterns enables researchers to diagnose the scales at which a model has or has not learned the requisite structure.

These insights offer a rigorous, mechanistic interpretation of deep self-attention networks in a well-defined computational context, with broad ramifications for interpretable AI and design of multiscale sequence models (Garnier-Brun et al., 27 Aug 2024).