Ranking Bandits for Carousel Interfaces with Observable Browsing Depth — Deep dive

In carousels users stop at a browsing depth; slots below that depth are neither clicked nor exposed, yet classical PBM still treats them as seen-but-not-clicked. Under observable depth $V(t)$, this paper proposes OD-UCB / OD-TS / OD-DMED-L that update stats only from truly exposed positions. OD-UCB has an instance-dependent log regret bound; OD-DMED-L’s asymptotic upper bound matches the lower-bound leading term. At $T = 500, 000$ on shallow/deep synthetic and RecGaze-calibrated environments, OD-TS matches informed PBM-TS on final mean regret (1121.8 vs 1159.9 shallow) while all OD methods beat PBM-UCB. When browse depth is observed, do not treat unseen slots as negative feedback.

1. Problem

Each round shows an $L$ -slot ranking; the user stops after depth $V(t)$; clicks occur only on observed slots with unknown attractiveness $θ_{i}$ . PBM folds unobserved slots into the likelihood via examination probabilities $P_{j}$ . Here $V(t)$ is observed and independent of display order; the goal is expected-click maximization / regret minimization.

Setup page: carousel and observable depth $V(t)$.

2. Method

Exposure and click counts $n_{i} (t), s_{i} (t)$ accumulate only for positions $j \leq V (s)$ . OD-UCB ranks by UCB of observed means; OD-TS samples Beta posteriors; OD-DMED-L follows DMED-style exploration. Baselines PBM-UCB/TS still get true $P_{j}$ but not realized depth.

Bounds page: finite-time OD-UCB and asymptotic OD-DMED-L.

3. Core claim

Observable depth replaces a position-bias table: OD-UCB’s leading term avoids a $P_{L}^{- 2}$ factor vs PBM-UCB; OD-DMED-L is asymptotically tight; OD-TS needs neither true $P_{j}$ nor non-conjugate rejection sampling.

4. Experiments

$K=50$, $L=5$, $T = 5 \times 1 0^{5}$ ; shallow/deep synthetic and RecGaze-driven ( $P$ and CTR from eye-tracking carousels). Final mean regrets:

Env	OD-TS	PBM-TS	OD-DMED-L	OD-UCB	PBM-UCB
shallow	1121.8	1159.9	2137.2	6280.6	19568.2
deep	1326.9	1356.8	4006.6	8761.5	12226.3
RecGaze	856.1	876.9	1910.5	higher	higher

OD-TS≈PBM-TS with overlapping CIs; long-horizon $T = 1 0^{7}$ diagnostics pull OD-DMED-L toward its asymptotic coefficient.

5. Limits

Independent clicks and ranking-independent depth are assumptions; RecGaze is log-calibrated simulation, not an online deployment.

6. Practice

If real browse depth is logged, mask unexposed slots in bandit updates. Existing PBM-TS stacks can A/B OD-TS to drop dependence on $P_{j}$ while keeping similar regret.