Sequential Support Network Learning (SSNL)

• Sequential Support Network Learning (SSNL) is a framework that identifies optimal support structures by leveraging overlapping evaluations among a set of entities.
• It employs the semi-overlapping multi-armed bandit (SOMMAB) model and a generalized GapE algorithm to improve sample complexity and error bounds.
• SSNL has practical applications in multi-task learning, federated learning, and multi-agent systems where joint evaluations accelerate decision-making.

Sequential Support Network Learning (SSNL) is a principled framework for identifying optimal contribution structures among a collection of entities, such as tasks, clients, or agents, by selecting the most beneficial candidate sets of partners through a sequence of computational trials. SSNL unifies a range of modern AI and ML problems where shared, asymmetric, and computationally intensive evaluations determine which participants should be selected to maximize collective or individual benefit. The central technical concept underpinning SSNL is the semi-overlapping multi-(multi-armed) bandit (SOMMAB) model, which generalizes the standard multi-bandit best-arm identification problem to settings where each evaluation can return feedback for multiple bandits, due to structural overlap among their arms. This intrinsic overlap enables significant gains in sample complexity and computational efficiency via shared evaluations. A generalized GapE algorithm for best-arm identification in SOMMABs establishes new theoretical guarantees with improved exponential error bounds, substantiating the effectiveness of SSNL in applied domains including multi-task learning (MTL), auxiliary task learning (ATL), federated learning (FL), and multi-agent systems (MAS) (Antos et al., 31 Dec 2025).

1. Formalization of the SOMMAB and SSNL Frameworks

Multi-Bandit Best-Arm Identification (MMAB)

In the MMAB paradigm, $M$ bandits are indexed by $m=1,\ldots,M$, with each bandit $m$ possessing $K_m$ arms. Each arm $k$ of bandit $m$ has an associated reward distribution $\nu_{mk}$ supported on $[0, b]$ with mean $\mu_{mk}$. The best-arm objective is defined via the gap $\Delta_{mk} := \mu_m^* - \mu_{mk}$ for each suboptimal arm, where $\mu_m^* = \max_j \mu_{mj}$. Arms are evaluated over a finite budget of $n$ trials, with each round allocating a pull to a specific arm, yielding a sample reward, updating empirical means $\hat{\mu}_{mk}(t)$ and gaps $\hat{\Delta}_{mk}(t)$. At termination, one recommends, for each bandit $m$, the arm $J_m(n) = \arg\max_k \hat{\mu}_{mk}(n)$. Performance is quantified by the worst-case error probability $L(n) = \max_m P[J_m(n) \neq k_m^*]$, with average error $e(n)$ and simple regret $r(n)$ being alternative analyses.

Semi-Overlapping Evaluation Groups

The SOMMAB extends MMAB to allow overlap: a single trial evaluates an "evaluation group" $G = \{(m_1, k_1), \ldots, (m_r, k_r)\}$, each $(m_i, k_i)$ from a distinct bandit, using one unit of budget but returning distinct rewards $X_{m_i k_i}$. If every group is a singleton ($r=1$), the model reduces to standard MMAB. If every group spans all bandits ($r=M$), the setting is maximally overlapping and analytic reductions to vector-reward bandit settings are possible.

Sequential Support Network Learning (SSNL)

In SSNL, the focus is on $M$ entities (e.g., tasks, clients, agents), each seeking the optimal "support set" $S \subseteq \{1,\ldots,M\} \setminus \{m\}$ from a candidate list $C_m$. For each $m$, the solution is encoded as a directed graph $G=(V,E)$ with edges $(i \to m)$ for $i \in S_m^*$. Under a strong duality constraint, if $S \in C_m$ and $b \in S$, then mutatis mutandis, $S' = S \setminus \{b\} \cup \{m\} \in C_b$, so that trialing $S$ for $m$ simultaneously evaluates the "role-swapped" configuration for $b$. In the SOMMAB formalism, these pairings induce semi-overlapping arms.

2. Reward, Feedback, and Evaluation Dynamics

The reward model for SOMMAB-based SSNL is characterized by trial-level evaluation: when group $G = \{(m_i, k_i)\}$ is pulled, one unit of budget generates independent samples $X_{m_i k_i} \sim \nu_{m_i k_i}$ for each. Each arm’s pull count $T_{m_i k_i}(t)$ increments, empirical mean $\hat{\mu}_{m_i k_i}(t)$ is recalculated, and the process iterates. For non-overlapping cases, this reduces to standard one-arm pulls.

The effect of overlap is central: in practical SSNL deployments (e.g., testing a coalition of agents, distributed learning across clients), a single system call or joint evaluation returns feedback for all entities involved, modeling structural duality between entities and their candidate sets.

3. The Generalized GapE Algorithm for Semi-Overlapping Bandits

The classical GapE algorithm of Gabillon et al. for best-arm identification is generalized for overlap. The algorithm proceeds as follows:

1. For each arm $(m, k)$, obtain $l$ initialization samples (ideally pulled jointly within overlap groups), setting $T_{mk} = l$ and computing empirical means $\hat{\mu}_{mk}(l)$ and gaps $\hat{\Delta}_{mk}(l)$.
2. For $t = l \sum_m K_m + 1, \ldots, n$:
   • Compute, for every arm, the index $$B_{mk}(t) = -\hat{\Delta}_{mk}(t-1) + b \sqrt{a / T_{mk}(t-1)}$$.
   • Select $(m^*, k^*) = \arg\max_{m,k} B_{mk}(t)$, and pull the entire group $G$ containing $(m^*, k^*)$.
   • For all $(m, k) \in G$, observe $X_{mk}$, increment $T_{mk}$, update $\hat{\mu}_{mk}$ and recompute $\hat{\Delta}_{mk}$ for relevant arms.
3. At $t=n$, recommend $J_m(n) = \arg\max_k \hat{\mu}_{mk}(n)$ for each bandit.

Key properties:

• For no overlap, the algorithm coincides with the original GapE.
• All overlapping arms are updated from a single pull, which accelerates learning if overlap is present.
• The index $B_{mk}(t)$ ensures a balance between exploiting confident arms and exploring uncertain ones, analogous to confidence-based approaches but adapted for overlapping structures (Antos et al., 31 Dec 2025).

4. Theoretical Guarantees and Sample Complexity

Complexity Parameter and Error Bounds

Define the global complexity as $$H = \sum_{m=1}^M \sum_{k=1}^{K_m} (b^2 / \Delta_{mk}^2)$$. For classic (non-overlapping) MMAB, running GapE with $l=1$ and $a \leq (4/9)(n-MK)/H$ yields $L(n) \leq 2 M K n \exp(-a/64)$. This leads to a bound $L(n) \leq 2 M K n \exp(- (n-MK)/(144 H))$.

In the semi-overlapping scenario, improved bounds are established. Key quantities:

• $\rho = \sqrt{\min(l/(l-1), 2)}$
• $c = 1 / [2\sqrt{3\rho+\rho^2} + 2\rho + 1]$
• $Q_c = 3(1+5c)(1+c)/4$

Running GapE with $a \leq (n - MK + 1) / ((1 + 2c)^2 H - Q_c)$ (with $b$ normalized) gives $L(n) \leq 2 M K n \exp(-2 a c^2)$, or equivalently,

$$L(n) \leq 2 M K n \exp\left( - \frac{n-MK+1}{(1/c+2)^2 H - Q_c / c^2} \right).$$

For $l=152$ ($c \approx 1/41$), this specializes to

$$L(n) < 2 M K n \exp\left( - \frac{n-MK+1}{41H - 36} \right).$$

In an $r$-order SOMMAB (each pull evaluates $r$ arms), the effective sample count is $rn$ so

$$L(n) < 2 M K n \exp\left( - \frac{r n - MK + 1}{41H - 36} \right).$$

Proof Strategy

• Concentration: Using Hoeffding's inequality and a union bound, the probability of empirical means deviating more than $c\sqrt{a/T_{mk}(t)}$ is exponentially small for all arms and rounds.
• Gap-Index Invariant: Inductive argument establishes the index gap maintains separation of best and suboptimal arms under the good event.
• Stopping Condition: Any arm that remains significantly under-sampled leads to insufficient sampling overall, contradicting the trial budget; thus all arms are sufficiently explored.
• The final result ensures the best arm is empirically identified with high probability, yielding the stated exponential error bounds.

Sample Complexity Comparison

Algorithm	Exponent in Error Bound	Remarks
GapE (no overlap)	$-(n-MK)/(144H)$	Baseline (Gabillon et al. 2011)
Uniform+UCB-E	$-(n-MK)/(18 M H_\text{max})$	Inefficient for large $M$
New GapE (SOMMAB)	$-(n-MK)/(41H)$ (improves with overlap)	Linear improvement in $r$-order overlap

For error tolerance $\delta$, the required sample count is $n = O(H [\ln(MK) + \ln(1/\delta)])$, and for $r$-order overlap, $n = O( (H/r) [\ln(MK) + \ln(1/\delta)] )$.

5. Practical Applications and Implications

Sequential Support Network Learning

Each of $M$ entities possesses a candidate list $C_m$ of $K_m$ donor sets. Each $S \in C_m$ forms an arm $(m,S)$. If $S \in C_m$ and $b \in S$, then the role-swapped $S' = S \setminus \{b\} \cup \{m\} \in C_b$ creates an overlap: a single joint evaluation provides feedback for both $(m, S)$ and $(b, S')$.

The SSNL objective is, for each $m$, to identify the best $S^*_m \in C_m$, thereby constructing a directed support network where edge $(i \to m)$ signifies $i$ in $S^*_m$.

Domains

• Multi-Task Learning (MTL) and Auxiliary Task Learning (ATL): Each task $m$ is a bandit, with arms as subsets of auxiliaries $S$. Joint evaluation (e.g., cross-validation) on $\{m\} \cup S$ yields performance measurements for each member. Duality leads to overlap where role-swapped candidate lists share configurations.
• Federated Learning (FL): Each client $m$ is a bandit, with arms as subsets $S$ of clients. A federated round over $S \cup \{m\}$ reports local metrics. Semi-overlap is realized when client partner selections reciprocally test each client as donor.
• Multi-Agent Systems (MAS): Agents form bandits; groups $S$ are coalitions. A simulation run provides returns to all. Coalition duality ensures overlap in arms across agents.

Implications

The SOMMAB abstraction supplies a well-founded pure-exploration protocol for SSNL problems, offering both theoretical and practical advancements:

• Shared evaluations yield linear sample efficiency gains.
• The generalized GapE algorithm is algorithmically straightforward, fitting within established MAB architectures, and features exponential error decay with tight constants.
• Architectural integration in FL or cloud-edge SSNL is straightforward: a central coordinator can deploy GapE while distributed nodes generate joint feedback per group.

6. Comparative and Conceptual Insights

The fundamental insight supporting SSNL is that structural overlap—when evaluations for one entity automatically inform others—drives a reduction in required trials for fixed error. For scenarios with strong duality and high overlap ($r \approx M$), the system approaches the efficiency of a single-bandit with $K$ arms and high-dimensional rewards. A plausible implication is that, in highly symmetric SSNL instances (e.g., fully connected support networks), architects should maximize overlap to achieve near-optimal performance in both sample complexity and runtime.

A potential misconception is that overlap complicates learning; the established results demonstrate the opposite: overlap, when exploited algorithmically, accelerates convergence and strengthens theoretical guarantees (Antos et al., 31 Dec 2025).

7. Outlook and Future Directions

While the current SOMMAB and generalized GapE framework provides strong guarantees for SSNL, several research vistas remain:

• Rigorous exploration of dynamic candidate lists and non-i.i.d. reward structures.
• Application to adaptive, decentralized, or heterogeneous-agent systems.
• Extension of the theory to partial or state-dependent overlap, where evaluation groups are not fixed in size nor symmetric.

A plausible implication is that further integration of overlap-aware algorithms into large-scale federated and distributed learning systems could yield both practical and theoretical benefits, especially as support networks become increasingly complex and computationally interdependent.