Slate-GLM-TS: Efficient Logistic Slate Bandits

• Slate-GLM-TS is an efficient Thompson Sampling algorithm designed for logistic contextual slate bandits, integrating local slot-wise maximization with joint parameter estimation.
• It minimizes cumulative regret by leveraging adaptive optimism and scalable local planning to handle exponentially large action spaces in polynomial time.
• Empirical results demonstrate low regret performance and competitive runtime, particularly in applications like in-context example selection for language models.

Slate-GLM-TS is an efficient Thompson Sampling algorithm for the logistic contextual slate bandit problem, where sequential slate selection is performed from exponentially large combinatorial action spaces, and only a single binary reward is observed per round. It operates under a global logistic reward model, leveraging local planning with independent slot-wise maximization and global learning via joint parameter estimation, and achieves both low regret and polynomial per-round computational complexity under mild diversity assumptions (Goyal et al., 16 Jun 2025).

1. Problem Formulation: Logistic Contextual Slate Bandits

The logistic contextual slate bandit framework models a sequential decision process over $T$ rounds. At each time step $t$, for each of $N$ slots, the agent faces a finite, potentially different item set $\mathcal X^i_t \subset \mathbb R^d$ and selects one item $x_t^i \in \mathcal X^i_t$. The chosen slate $x_t = (x^1_t, ..., x^N_t)$ lies in the Cartesian product $\mathcal X_t^1 \times \cdots \times \mathcal X_t^N$. A single binary reward $y_t \in \{0, 1\}$ is observed, drawn according to the probability $\mathbb P[y_t=1\mid x_t] = \mu(x_t^\top \theta^*)$ where $\mu(u) = 1/(1+e^{-u})$, and the true parameter $t$0 with $t$1 is unknown.

The objective is to minimize cumulative regret over $t$2 rounds: $t$3 The analysis assumes a slot-wise diversity condition: conditioned on previous history, $t$4 and $t$5 for each slot, with $t$6 dependent on the logistic curvature.

2. Algorithmic Structure of Slate-GLM-TS

Slate-GLM-TS adopts the Thompson Sampling (TS) paradigm, combining a Gaussian-style random perturbation of the regularized logistic-GLM estimator with local slot-wise greedy action selection. Parameter estimation relies on cumulative slot-wise statistics and is updated via an adaptive optimism-in-the-face-of-uncertainty (OFU) subroutine.

Algorithmic workflow at each round $t$7:

1. Context Observation: Observe item sets $t$8 for all slots.
2. Posterior Sampling: Draw $t$9, where $N$0, $N$1, and $N$2 satisfies TS distributional guarantees.
3. Local Planning: For each slot $N$3, extract the $N$4th block $N$5 and select $N$6. The slate $N$7 is the tuple of these choices.
4. Action and Update: Execute slate $N$8, observe reward $N$9, and update posterior parameters via the adaptive-OFU subroutine (recomputing $\mathcal X^i_t \subset \mathbb R^d$0, $\mathcal X^i_t \subset \mathbb R^d$1, $\mathcal X^i_t \subset \mathbb R^d$2, and $\mathcal X^i_t \subset \mathbb R^d$3).

The algorithm enforces that each sampled $\mathcal X^i_t \subset \mathbb R^d$4 falls within the admissible confidence set $\mathcal X^i_t \subset \mathbb R^d$5. The following table summarizes the action selection and update steps:

Step	Computation	Purpose
Posterior sample	$\mathcal X^i_t \subset \mathbb R^d$6 as above	TS for exploration
Slot selection	$\mathcal X^i_t \subset \mathbb R^d$7	Local maximization
Posterior update	Update via adaptive-OFU	Keep $\mathcal X^i_t \subset \mathbb R^d$8 in $\mathcal X^i_t \subset \mathbb R^d$9

3. Regret Bound and Theoretical Guarantees

In the fixed-arm scenario, Slate-GLM-TS-Fixed achieves (under the stated diversity assumption) $x_t^i \in \mathcal X^i_t$0 expected regret. Specifically, if the minimal eigenvalue of each slot-wise design matrix $x_t^i \in \mathcal X^i_t$1 grows at least linearly in $x_t^i \in \mathcal X^i_t$2 ($x_t^i \in \mathcal X^i_t$3), then employing:

• Local-global design equivalence ($x_t^i \in \mathcal X^i_t$4),
• TS posterior concentration ($x_t^i \in \mathcal X^i_t$5),
• Elliptical potential arguments ($x_t^i \in \mathcal X^i_t$6),

one can establish

$x_t^i \in \mathcal X^i_t$7

for fixed-arm settings, with the contextual bandit regret analysis relying on analogous design equivalence and confidence set invariance. The contextual case is conjectured to match this bound when design diversity holds.

4. Computational Complexity

Slate-GLM-TS ensures each round requires only polynomial time in the number of slots $x_t^i \in \mathcal X^i_t$8 and feature dimension $x_t^i \in \mathcal X^i_t$9. Action selection in round $x_t = (x^1_t, ..., x^N_t)$0 entails $x_t = (x^1_t, ..., x^N_t)$1 independent maximizations over $x_t = (x^1_t, ..., x^N_t)$2, with total time $x_t = (x^1_t, ..., x^N_t)$3. Posterior updates, including recalculation of parameter estimates and confidence sets, only require $x_t = (x^1_t, ..., x^N_t)$4 time. The exponential cost associated with evaluating all possible slates is thus replaced by scalable local maximization.

5. Comparison to Slate-GLM-OFU

Slate-GLM-TS and Slate-GLM-OFU both exploit the multiplicative-equivalence property of slot-wise and global slate design. However, the two algorithms differ in their approach:

• Slate-GLM-OFU: Maintains an explicit optimistic confidence set $x_t = (x^1_t, ..., x^N_t)$5 and computes the maximizer of an upper confidence bound (UCB) over the slate, relying on projection into $x_t = (x^1_t, ..., x^N_t)$6. The optimization decomposes into slot-level maximizations due to design equivalence.
• Slate-GLM-TS: Samples a perturbed parameter within the confidence set and performs greedy slot-wise action selection. It requires no UCB maximization, only TS sampling.
• Regret: OFU provides a clean high-probability $x_t = (x^1_t, ..., x^N_t)$7 guarantee for the contextual problem, while Slate-GLM-TS matches $x_t = (x^1_t, ..., x^N_t)$8 in the fixed-arm case with conjectured analogous guarantees under diversity for the contextual scenario.
• Implementation: TS is simpler when a valid posterior sampling distribution $x_t = (x^1_t, ..., x^N_t)$9 is available and does not require explicit confidence set optimization as in the OFU approach.
• Empirical Performance: Both methods demonstrate low regret and competitive runtime in experiments; TS sometimes explores more naturally and is simpler to code.

6. Practical Applications and Experimental Evidence

Slate-GLM-TS has been empirically validated across synthetic experiments, consistently outperforming state-of-the-art baselines in both cumulative regret and runtime. One key application area is the automated selection of in-context examples for LLM prompts in binary classification tasks, such as sentiment analysis, where it achieves competitive test accuracy for the end task (Goyal et al., 16 Jun 2025). The algorithm’s ability to efficiently handle exponentially large action sets via slot-wise decomposition makes it viable in both research and practical deployments.