Hierarchical ABoB Structure

• Hierarchical ABoB structure is a multi-level framework that organizes atomic descriptors, evidential trees, and bandit clusters to efficiently capture local dependencies and reduce complexity.
• It applies recursive algorithms in evidence theory, neighbor-aware descriptor concatenation in molecular ML, and hierarchical clustering in adversarial bandits to achieve significant performance gains.
• The framework enhances computational efficiency, data fusion, and online optimization by leveraging natural data-driven hierarchies and local smoothness properties.

A Hierarchical ABoB Structure refers to any partitioned, multi-level architecture derived from the atomic Bag-of-Bonds (aBoB) descriptor, Adversarial Bandit over Bandits (ABoB) algorithm, or evidence theory constructs (“bodies of evidence”), where the fundamental building blocks are systematically organized in layers or clusters that capture local, structural, or set-theoretic dependencies. Hierarchical ABoB frameworks have been deployed in machine learning for molecules, nonstochastic multi-armed bandit optimization, and belief function computations, exploiting the compositional or inclusion structure to yield improved computational efficiency, data efficiency, or regret guarantees.

1. Hierarchical ABoB in Evidence Theory

The Hierarchical ABoB Structure in the context of evidence theory is formalized via hierarchical trees over bodies of evidence, as proposed in (Sandri, 2013). A body of evidence $(F,m)$ on a frame of discernment $\Omega$ consists of a set of focal elements $F \subseteq \mathcal{P}(\Omega)$ and a mass function $m:\mathcal{P}(\Omega)\to[0,1]$ with $m(A)>0$ iff $A\in F$ and $\sum_{A\subseteq\Omega} m(A)=1$. Traditionally, the computation of set functions such as belief $\mathrm{Bel}(A)$, plausibility $\mathrm{Pl}(A)$, and the commonality function $Q(A)$ requires operations with complexity $O(|F|^2)$ or worse due to the need to sum over subset or superset relationships.

The hierarchical structure is constructed by partitioning $F$ by cardinality and connecting each $A\in F$ to its minimal-cardinality superset $B\in F$. The resulting tree (or forest) has nodes representing focal elements and directed edges from child $A$ to parent $B$. This enables recursive, depth-first computations of $Q(A)$:

$Q(A) = m(A) + \sum_{B \supset A,\, B\in F} m(B)$

with reuse of partial sums across descendants. Resulting algorithms evaluate all desired set functions in $O(|F|)$ tree traversals and similarly reduce the cost of Dempster’s combination, avoiding explicit enumeration of $\mathcal{P}(\Omega)$.

Table 1: Complexity Comparison in Evidence Aggregation

Operation	Brute-Force	Hierarchical Tree
$\mathrm{Bel},\mathrm{Pl}, Q$ (all $A$)	$O(|F|^2)$	$O(|F|)$
Dempster’s Combination	$O(|F_1|\cdot|F_2|)$	$O(|F_1|+\sum\limits_f |S_2(f)|)$

The significance of this approach is evident in large-scale evidence aggregation and uncertainty quantification, where redundant subset and intersection computations dominate cost. The method directly prunes infeasible pairwise tasks and elegantly exploits set-inclusion lattices (Sandri, 2013).

2. Hierarchical aBoB Descriptors in Molecular Machine Learning

In the context of atom-in-molecule representations, the hierarchical structure arises in the multi-level aBoB-RBF($n$) family introduced in (Das et al., 7 Oct 2025). The starting point, the atomic Bag-of-Bonds (aBoB), generates for each atom $I$ a vectorized descriptor by gathering, bond-type by bond-type, all atomic-pair Coulomb-matrix features involving $I$, applying a distance-damping term:

$$M_{IJ}^{(A,B)} = \frac{Z_I Z_J}{R_{IJ} \cdot s(R_{IJ})} \quad s(R_{IJ}) = R_{IJ}^{-\beta},\: \beta\approx 3$$

with $J \ne I$ and types $(A,B)$. Continuity is injected via Gaussian radial basis functions (RBF):

$$g_{IJ}(r) = \frac{1}{\sigma_{\mathrm{rad}}\sqrt{2\pi}}\exp\left(-\frac{(r-R_{IJ})^2}{2\sigma_{\mathrm{rad}}^2}\right),$$

producing for each bond type $(A,B)$ a channel $$d^{(A,B)}(r) = \sum_{I \in A,\, J \in B,\, J \neq I} g_{IJ}(r) \cdot M_{IJ}^{(A,B)}$$.

The hierarchy is established by concatenating these per-atom, continuous aBoB-RBF descriptors for a query atom $I$ and its $n$ nearest neighbors ($J_1,\dots,J_n$), weighted by a smooth cosine cutoff $f_{\mathrm{cos}}(R;r_{\mathrm{cut}})$. The composite descriptor,

$$d_I^{\text{aBoB‐RBF}(n)}(r) = [d_I(r), f_{\mathrm{cos}}(R_{IJ_1})d_{J_1}(r),\,\dots,\, f_{\mathrm{cos}}(R_{IJ_n})d_{J_n}(r)]$$

defines a layered, neighbor-aware structure that systematically encodes not only first-shell two-body but also approximate multi-body environments. Empirical results indicate aBoB-RBF(4) achieves out-of-sample mean errors as low as 1.69 ppm for $^{13}$C NMR shielding on QM9NMR, outperforming descriptor families that are not structurally hierarchical (Das et al., 7 Oct 2025).

3. Adversarial Bandit over Bandits (ABoB) as a Hierarchical Structure

The ABoB algorithm (Avin et al., 25 May 2025) is a two-level hierarchical adversarial bandit framework for large action spaces. The set of $k$ arms $K = \{1,\dots,k\}$ is partitioned into $p$ clusters $P^1,\dots,P^p$ under a metric $D$, with each cluster representing a “virtual arm.” The top (parent) level runs a standard adversarial MAB algorithm such as EXP3 or Tsallis-INF across the clusters, while each cluster maintains an independent instance (“child bandit”) over its constituent arms.

The per-round protocol is hierarchical: the parent bandit samples a cluster according to $\pi^{\mathrm{parent}}_t$; the child bandit for that cluster then samples a concrete arm according to $\pi^{\mathrm{child},P_t}_t$. Rewards are fed back at both levels. The algorithm admits “flat” adversarial MABs as special cases (by taking $p = k$), but hierarchy enables clustering to exploit local smoothness and structure.

The regret $R(T)$ against the best fixed arm satisfies (in worst case):

$$R(T) \leq O\Big(\sqrt{pT\ln p}\Big) + O\Big(\sqrt{p^*T\ln p^*}\Big) + O\bigg(\sqrt{kT\ln \frac{k}{p}}\bigg)$$

where $p^* = |P^*|$ is the size of the best cluster $P^*$, and $T$ is the time horizon. Under local Lipschitz continuity within clusters,

$|c_t(a) - c_t(b)| \leq \ell$

for $a,b \in P^i$, the regret sharpens to $O\left(k^{1/4}\sqrt{T}\right)$ with optimal clustering and small $\ell$, representing a significant improvement over flat approaches (Avin et al., 25 May 2025).

4. Algorithmic Principles and Pseudocode Structures

The common algorithmic signature of hierarchical ABoB classes is the explicit construction and layered traversal or update of sub-blocks:

• Hierarchical trees in evidence theory are built by connecting each focal element to its minimal superset, followed by depth-first traversals for computing $Q,\,\mathrm{Bel},\,\mathrm{Pl}$ and recursive Dempster’s combination. Tree construction pseudocode partitions by cardinality, iteratively assigns unique parents, and prunes redundant subset search (Sandri, 2013).
• Hierarchical aBoB-RBF(n) descriptors concatenate a base aBoB-RBF block with damped blocks from neighbor atoms. Computational complexity in kernel regression or GPR tasks scales linearly in $n$ and total descriptor length, with diminishing returns once all first-shell neighbors are included (Das et al., 7 Oct 2025).
• ABoB bandit meta-routines instantiate a parent–child composition of any “flat” adversarial MABs (EXP3, Tsallis-INF), with per-round feedback and parameter setting shared by established regret-optimal policies. The top-level pseudocode initializes both parent and per-cluster children, loops through rounds by sequential sampling, reward relay, and joint updates (Avin et al., 25 May 2025).

5. Practical Impact and Typical Use-Cases

Hierarchical ABoB structures are prominent in three settings:

• Uncertainty fusion in belief function theory: dramatically improving the speed and tractability of evaluating belief, plausibility, and combining evidence, especially as focal sets grow large and structurally inhomogeneous (Sandri, 2013).
• Molecular property prediction: aBoB-RBF(n) and similar neighbor-aware descriptors yield lower mean absolute errors and steeper learning curves in ML models for NMR shielding, outperforming non-hierarchical descriptors and competitive with many-body potentials at far lower cost (Das et al., 7 Oct 2025).
• Online hyperparameter or configuration optimization: ABoB enables bandit-based control in large, structured action spaces, leveraging metric-based clusters for accelerated adaptation and reduced regret in both adversarial and stochastic regimes (Avin et al., 25 May 2025).

A plausible implication is that hierarchy, whether set-theoretic, metric, or spatial, provides principled opportunities to factor computation or learning along natural data-driven or structural axes, allowing for both theoretical and empirical gains without loss of generality.

6. Limitations and Theoretical Considerations

While hierarchical ABoB structures offer practical and asymptotic advantages, they also inherit specific limitations:

• For hierarchical evidence trees, the worst-case complexity remains exponential if the number of focal elements approaches $2^{|\Omega|}$, and benefits are strongest when inclusion relationships are rich rather than “flat” (Sandri, 2013).
• In neighbor-augmented molecular descriptors, the computational cost scales with the neighborhood size $n$, and improvements beyond the first shell of atoms rapidly saturate (Das et al., 7 Oct 2025).
• In ABoB bandit clustering, no regret improvement is available unless within-cluster reward functions satisfy some smoothness (Lipschitz) property; otherwise, the worst-case regret matches the flat approach and tree construction brings only minor overhead (Avin et al., 25 May 2025).

Implementation burden is higher than for naive or “flat” methods, especially regarding data structure design, recursive traversals, and in some cases cluster selection or neighborhood tuning.

7. Connections and Broader Context

The hierarchical ABoB paradigm unifies several research threads:

• It generalizes “bag-of-X” constructions by introducing explicit, layered locality, systematic parent–child structure, and local-smoothness assumptions.
• The approach aligns with many-body expansions in physics-informed ML (e.g., systematic addition of neighbors in descriptors is analogous to higher-order terms in SLATM or FCHL) (Das et al., 7 Oct 2025).
• In optimization, it bridges stable regret bounds of flat bandit algorithms with the adaptivity and computational exploitation of metric-induced clustering, and interfaces with bandit meta-algorithms applied in online learning, model selection, and automated control (Avin et al., 25 May 2025).
• The evidence-theoretic instantiation is harmonized with efficient graph-based reasoning in probabilistic graphical models, such as local propagation in join trees (Sandri, 2013).

This suggests that the hierarchical ABoB construct is a representative instance of a broader class of efficient, structure-exploiting algorithms in modern computational statistics, learning theory, and uncertainty quantification.