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arxiv: 2606.31449 · v1 · pith:VSMK4E4M · submitted 2026-06-30 · cs.LG · stat.ML

Contextual Slate GLM Bandits with Limited Adaptivity

Tanmay Goyal , Sukruta Prakash Midigeshi , Gaurav Sinha This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-01 06:47 UTCgrok-4.3pith:VSMK4E4Mrecord.jsonopen to challenge →

classification cs.LG stat.ML
keywords contextual slate banditsgeneralized linear modelslimited adaptivityregret boundsbatched learningrarely switchingGLM bandits
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The pith

Two limited-adaptivity algorithms for contextual slate GLM bandits achieve regret bounds independent of the non-linearity parameter kappa.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines contextual slate bandits where each round presents N sets of d-dimensional items, the learner picks one item per set to form a slate, and the scalar reward follows a generalized linear model. It introduces B-SlateGLinCB, which partitions time into O(log log T) batches and updates only on prior-batch data, and RS-SlateGLinCB, which performs only O(Nd log T) parameter updates. Under a diversity assumption on the item sequences, the algorithms attain O(N d^{3/2} √T) and O(N d √T) regret respectively, with no dependence on the GLM non-linearity constant kappa. Both run in poly(N) time per round despite 2^Ω(N) possible slates. Simulations indicate they surpass other limited-adaptivity methods and approach fully adaptive performance, including on language-model in-context selection.

Core claim

Under a diversity assumption on the item sequences, B-SlateGLinCB and RS-SlateGLinCB achieve regret bounds of O(Nd^{3/2}√T) and O(Nd√T) respectively. Both bounds are independent of the non-linearity parameter kappa that typically scales GLM bandit regret. The algorithms remain computationally efficient, requiring only poly(N) time per round.

What carries the argument

B-SlateGLinCB and RS-SlateGLinCB, which enforce limited policy updates (O(log log T) batches or O(Nd log T) switches) using only data from prior periods.

Load-bearing premise

The sequences of presented items must satisfy a diversity assumption for the stated regret bounds to hold without scaling by kappa.

What would settle it

Empirical observation that regret scales linearly with kappa or exceeds O(Nd^{3/2}√T) when the presented item sequences lack the required diversity.

Figures

Figures reproduced from arXiv: 2606.31449 by Gaurav Sinha, Sukruta Prakash Midigeshi, Tanmay Goyal.

Figure 1
Figure 1. Figure 1: Comparison with limited adaptivity algorithms, [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison with the sequentially adaptive slate bandit algorithm [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Prompt Tuning on SST-2 6 Conclusions We present a batched algorithm B-SlateGLinCB and a rarely switching algorithm RS-SlateGLinCB for slate GLM bandits with bandit feedback. Under Assumption 2.1, we prove that B-SlateGLinCB and RS-SlateGLinCB incur O(Nd3/2√ T) and O(Nd√ T) regret respectively, while having poly(N) per round time complexity. Empirically, we show that our algorithms outperform all baseline l… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison with limited adaptivity algorithms, [PITH_FULL_IMAGE:figures/full_fig_p048_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison with fully sequential slate bandit algorithm [PITH_FULL_IMAGE:figures/full_fig_p049_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: B-SlateGLinCB 0 200 400 600 800 1000 1200 1400 Time rounds 20 21 22 23 24 25 Minimum Eigenvalue (a) Slot 1 0 200 400 600 800 1000 1200 1400 Time rounds 20 21 22 23 24 25 Minimum Eigenvalue (b) Slot 2 0 200 400 600 800 1000 1200 1400 Time rounds 20 21 22 23 24 25 Minimum Eigenvalue (c) Slot 3 0 200 400 600 800 1000 1200 1400 Time rounds 20 21 22 23 24 25 Minimum Eigenvalue (d) Slot 4 0 2000 4000 6000 8000 1… view at source ↗
Figure 7
Figure 7. Figure 7: RS-SlateGLinCB 0 2000 4000 6000 8000 10000 Time rounds 240 250 260 270 280 Minimum Eigenvalue (a) Slot 1 0 2000 4000 6000 8000 10000 Time rounds 240 250 260 270 280 Minimum Eigenvalue (b) Slot 2 0 2000 4000 6000 8000 10000 Time rounds 240 250 260 270 280 Minimum Eigenvalue (c) Slot 3 0 2000 4000 6000 8000 10000 Time rounds 240 250 260 270 280 Minimum Eigenvalue (d) Slot 4 [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 8
Figure 8. Figure 8: B-SlateGLinCB+ 51 [PITH_FULL_IMAGE:figures/full_fig_p051_8.png] view at source ↗
read the original abstract

We investigate the contextual slate bandit problem with generalized linear rewards under limited adaptivity. At each round, the learner is presented with $N$ sets of items, where each item is represented by a $d$-dimensional feature vector. The learner then constructs a slate by selecting one item per set; the resulting slate yields a scalar reward sampled from a Generalized Linear Model (GLM). We propose algorithms under two limited-adaptivity settings: (a) Batched and (b) Rarely-Switching. For the batched setting, we introduce B-SlateGLinCB, which partitions the time horizon into $\mathcal{O}(\log\log T)$ batches such that each batch's policy relies only on data from previous batches. For the rarely-switching setting, we propose RS-SlateGLinCB, which adaptively performs only $\mathcal{O}(Nd\log T)$ parameter updates. Under a diversity assumption on the item sequences, we prove that B-SlateGLinCB and RS-SlateGLinCB achieve regret bounds of $\mathcal{O}(Nd^{3/2}\sqrt{T})$ and $\mathcal{O}(Nd\sqrt{T})$, respectively. Notably, both bounds are independent of the non-linearity parameter $\kappa$ that is typically found to scale the regret of GLM bandit algorithms. Our algorithms are computationally efficient, requiring only $\text{poly}(N)$ time per round despite $2^{\Omega(N)}$ possible slates. Simulations show our algorithms outperform existing baselines with limited adaptivity and remain competitive with Slate-GLM-OFU, a fully adaptive state-of-the-art algorithm. Notably, a slightly modified B-SlateGLinCB empirically matches this baseline. Finally, we demonstrate strong performance in a practical in-context example selection task for language models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript studies the contextual slate bandit problem with GLM rewards under limited adaptivity. It introduces B-SlateGLinCB, which uses O(log log T) batches where each batch's policy depends only on prior data, and RS-SlateGLinCB, which performs only O(Nd log T) parameter updates. Under a diversity assumption on the presented item sequences, the paper claims regret bounds of O(N d^{3/2} √T) for B-SlateGLinCB and O(N d √T) for RS-SlateGLinCB; both are independent of the GLM nonlinearity parameter κ. The algorithms run in poly(N) time per round despite an exponential number of possible slates and are evaluated in simulations and a language-model in-context example selection task.

Significance. If the regret analysis holds, the work would be a meaningful contribution by delivering limited-adaptivity algorithms for slate GLM bandits whose rates do not scale with κ, a factor that ordinarily appears through the link-function curvature. The poly(N) per-round complexity and the practical demonstration on language-model example selection are concrete strengths. The diversity assumption is presented as the mechanism that removes κ dependence, which would be a useful structural insight if rigorously established.

major comments (2)
  1. [Abstract] Abstract: the diversity assumption on item sequences is invoked to obtain κ-independent regret, yet its precise quantitative form (e.g., a lower bound on the minimum eigenvalue of the N feature matrices or a uniform spread condition across rounds) is not stated. This assumption is load-bearing for the central claim, because it is what purportedly supplies the eigenvalue lower bounds that absorb the GLM curvature factor and thereby eliminate κ from the final rates.
  2. [Theoretical analysis] Theoretical analysis (regret proofs): the stated bounds O(N d^{3/2} √T) and O(N d √T) are asserted to hold under the diversity assumption, but the provided description gives no derivation showing how the batching schedule (O(log log T) batches) or the O(Nd log T) update limit interacts with the GLM MLE estimation to preserve these rates without reintroducing a κ factor. Verification of the eigenvalue control step that absorbs the link-function derivative is required.
minor comments (1)
  1. [Abstract] Abstract: the per-round runtime is described only as 'poly(N)'; stating the explicit degree would help readers assess practicality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address the two major comments point by point below, indicating the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the diversity assumption on item sequences is invoked to obtain κ-independent regret, yet its precise quantitative form (e.g., a lower bound on the minimum eigenvalue of the N feature matrices or a uniform spread condition across rounds) is not stated. This assumption is load-bearing for the central claim, because it is what purportedly supplies the eigenvalue lower bounds that absorb the GLM curvature factor and thereby eliminate κ from the final rates.

    Authors: We agree that the abstract should state the diversity assumption more explicitly, as it is central to the κ-independent bounds. In the revised manuscript we will update the abstract to read: 'Under a diversity assumption ensuring that the minimum eigenvalue of each of the N feature matrices is bounded below by a positive constant λ independent of κ and T...' This makes the quantitative form and its role in absorbing the link-function curvature clear from the outset. revision: yes

  2. Referee: [Theoretical analysis] Theoretical analysis (regret proofs): the stated bounds O(N d^{3/2} √T) and O(N d √T) are asserted to hold under the diversity assumption, but the provided description gives no derivation showing how the batching schedule (O(log log T) batches) or the O(Nd log T) update limit interacts with the GLM MLE estimation to preserve these rates without reintroducing a κ factor. Verification of the eigenvalue control step that absorbs the link-function derivative is required.

    Authors: The full proofs appear in the appendix and establish the claimed rates. The diversity assumption supplies a uniform lower bound λ on the eigenvalues of the N per-set Gram matrices; this bound enters the GLM MLE concentration inequality and cancels the 1/κ factor that would otherwise arise from the link-function derivative. The O(log log T) batching schedule is constructed so that each batch collects enough samples to maintain the eigenvalue lower bound while using only prior-batch data, and the O(Nd log T) update limit for the rarely-switching algorithm is chosen to keep the estimation error controlled at the same rate. If the main-text presentation is insufficiently transparent, we will add a one-paragraph high-level sketch of these steps (eigenvalue control → MLE error → regret decomposition) to Section 4 in the revision. revision: partial

Circularity Check

0 steps flagged

No circularity; regret bounds derived from diversity assumption without reduction to inputs

full rationale

The paper states algorithms B-SlateGLinCB and RS-SlateGLinCB and proves regret bounds O(Nd^{3/2}√T) and O(Nd√T) under an explicit diversity assumption on item sequences. The claimed independence from the GLM nonlinearity parameter κ is presented as a structural consequence of that assumption absorbing curvature effects in the analysis, not as a fitted quantity or self-referential definition. No equations or steps in the abstract reduce by construction to prior outputs, self-citations, or renamed empirical patterns. The derivation chain is therefore self-contained against the stated assumption, consistent with standard proof structures in bandit literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on a diversity assumption on item sequences that is not derived in the abstract; standard GLM and bandit technical assumptions are also invoked but not enumerated here.

axioms (1)
  • domain assumption Diversity assumption on the item sequences
    Explicitly required for the regret bounds of both algorithms to hold.

pith-pipeline@v0.9.1-grok · 5857 in / 1283 out tokens · 22841 ms · 2026-07-01T06:47:59.471126+00:00 · methodology

discussion (0)

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