BLCE-G: Batched-Feedback Linear Contextual Bandit

• BLCE-G is a novel algorithm for linear contextual bandits that uses rare parameter updates and a static schedule to achieve near minimax-optimal regret.
• It combines a near G-optimal design phase, arm elimination, and phase-wise action selection to effectively balance exploration and exploitation.
• The algorithm strategically employs reward-free adaptivity within intervals and infrequent ridge-regression updates to match minimax lower bounds up to polylogarithmic factors.

Searching arXiv for BLCE-G and the cited paper to ground the article in current records. BLCE-G denotes Batched-Feedback Linear Contextual Bandit with Elimination and G-optimal design, a rare-update algorithm for linear contextual bandits introduced in “Practical and Optimal Algorithm for Linear Contextual Bandits with Rare Parameter Updates” (Yu et al., 31 May 2026). It is designed for settings in which reward feedback can be incorporated into the parameter estimate only at a small number of pre-specified update times, while contexts are still observed online and actions are selected sequentially. The algorithm combines a static update schedule with within-interval, reward-free adaptivity, using near G-optimal design, arm elimination, and greedy action selection. Its central result is minimax-optimal regret, up to polylogarithmic factors, in both the small-$K$ and large-$K$ regimes with only $\mathcal{O}(\log\log T)$ parameter updates (Yu et al., 31 May 2026).

1. Problem setting and rare-update model

BLCE-G is formulated for the linear contextual bandit model with i.i.d. context sets per round. At round $t \in [T]$, the learner observes

$$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$

and selects $x_{t,a_t} \in \mathcal{A}_t$, receiving reward

$$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$

where $\theta^* \in \mathbb{R}^d$ is unknown and $\eta_t$ is independent 1-subgaussian noise (Yu et al., 31 May 2026). The standing assumptions are $\|x\|_2 \leq 1$ for all $K$0 and $K$1.

The performance criterion is cumulative expected regret,

$K$2

where $K$3 (Yu et al., 31 May 2026). The analysis distinguishes a small-$K$4 regime, $K$5, from a large-$K$6 regime, $K$7.

A defining feature of BLCE-G is the distinction between rare parameter updates and strictly batched operation. Batched feedback partitions the horizon into intervals and reveals rewards only at interval endpoints. Strictly batched methods additionally prohibit within-interval context adaptivity: the action rule inside an interval cannot depend on the realized sequence of contexts and actions in that interval, beyond the current round’s context. BLCE-G does not impose that restriction. It permits reward-free, within-interval updates such as Gram-matrix maintenance and elimination-set updates, while restricting reward-dependent parameter recomputation to interval boundaries (Yu et al., 31 May 2026). This separation is operationally important because it preserves online responsiveness without frequent regression updates.

2. Static schedule and interval structure

BLCE-G uses a static grid with a minimal number of parameter updates, explicitly designed to be $K$8 (Yu et al., 31 May 2026). The first interval length is

$K$9

For intervals $\mathcal{O}(\log\log T)$0, rounds are split into three phases with counts proportional to terms involving $\mathcal{O}(\log\log T)$1, $\mathcal{O}(\log\log T)$2, and $\mathcal{O}(\log\log T)$3, and the interval lengths grow rapidly, yielding few updates overall. The paper proves that the total number of intervals $\mathcal{O}(\log\log T)$4 satisfies

$\mathcal{O}(\log\log T)$5

which implies interval complexity $\mathcal{O}(\log\log T)$6 (Yu et al., 31 May 2026).

Within each interval, BLCE-G divides action selection into exploration and exploitation sub-phases. The algorithm first uses a near G-optimal design phase, then an informative-direction phase, and finally a greedy phase. This schedule couples fast growth of interval lengths with progressively lower uncertainty, so that later greedy actions can exploit accumulated information without requiring frequent reward-driven re-estimation.

This suggests that BLCE-G is designed around a computational asymmetry: regression updates are treated as scarce, but matrix and set operations based only on observed contexts remain admissible online. That viewpoint is explicit in the paper’s rare-update formulation and is a practical distinction from strictly batched methods (Yu et al., 31 May 2026).

3. Online state, ridge updates, and elimination

BLCE-G maintains an inverse design matrix online within each interval using the Sherman–Morrison update

$\mathcal{O}(\log\log T)$7

The matrix is initialized as $\mathcal{O}(\log\log T)$8, and for BLCE-G the regularization is set to $\mathcal{O}(\log\log T)$9 (Yu et al., 31 May 2026).

At the end of interval $t \in [T]$0, BLCE-G performs a ridge-regression update: $t \in [T]$1 Thus the parameter estimate is recomputed only at interval boundaries, while the inverse Gram matrix is updated continuously inside intervals (Yu et al., 31 May 2026).

Elimination uses the past estimates $t \in [T]$2. For stage $t \in [T]$3,

$t \in [T]$4

and

$t \in [T]$5

The elimination mechanism prunes arms that are demonstrably suboptimal under confidence-controlled comparisons while preserving the optimal arm on the good event established in the analysis (Yu et al., 31 May 2026).

The confidence radii are

$t \in [T]$6

$t \in [T]$7

and

$t \in [T]$8

These bounds derive from self-normalized concentration and furnish uniform control over post-elimination arm sets (Yu et al., 31 May 2026).

4. Near G-optimal design and phase-wise action selection

The most distinctive component of BLCE-G is its near G-optimal design phase. For any arm set $t \in [T]$9, the paper establishes the existence of a design distribution $$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$0 supported on $$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$1 such that

$$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$2

and such a design can be computed in time $$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$3 (Yu et al., 31 May 2026). In BLCE-G, this distribution is denoted $$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$4, and the algorithm samples

$$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$5

during the first fraction of each interval.

The phase-wise selection rules are compactly summarized below.

Phase	Action rule	Role
Near G-opt phase	sample $$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$6	covariance control
Informative-direction phase	pick $$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$7	uncertainty-guided exploration
Greedy phase	pick $$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$8	exploitation

The informative-direction phase selects the arm with maximal self-normalized uncertainty score $$\mathcal{A}_t := \{x_{t,1}, \dots, x_{t,K}\} \subseteq \mathbb{R}^d$$9, which is the quantity controlled by the elliptical potential argument in the regret proof (Yu et al., 31 May 2026). The greedy phase then exploits the latest available estimate.

The conceptual division is precise. The near G-optimal phase controls worst-case variance across directions in the surviving arm set; the informative-direction phase sharpens estimation in directions that remain uncertain under the current covariance; and the greedy phase converts that reduction in uncertainty into low instantaneous regret. In the large-$x_{t,a_t} \in \mathcal{A}_t$0 regime, the paper identifies the near G-optimal design step as crucial for achieving the $x_{t,a_t} \in \mathcal{A}_t$1 minimax scale (Yu et al., 31 May 2026).

5. Regret guarantees, optimality, and proof structure

The main guarantee for BLCE-G is Theorem 1 of the paper: $x_{t,a_t} \in \mathcal{A}_t$2 Accordingly, BLCE-G attains $x_{t,a_t} \in \mathcal{A}_t$3 in the small-$x_{t,a_t} \in \mathcal{A}_t$4 regime and $x_{t,a_t} \in \mathcal{A}_t$5 in the large-$x_{t,a_t} \in \mathcal{A}_t$6 regime (Yu et al., 31 May 2026).

The paper further states that known minimax lower bounds for fully sequential contextual bandits are $x_{t,a_t} \in \mathcal{A}_t$7 in small-$x_{t,a_t} \in \mathcal{A}_t$8 and $x_{t,a_t} \in \mathcal{A}_t$9 in large-$$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$0, so BLCE-G matches these lower bounds up to polylogarithmic factors in $$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$1 and $$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$2 while using only $$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$3 parameter updates (Yu et al., 31 May 2026). This is the core optimality claim.

The proof strategy combines several ingredients. Self-normalized concentration yields high-probability control of $$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$4 over post-elimination sets. Elimination is then shown to preserve the optimal arm. The elliptical potential lemma controls cumulative uncertainty terms through log-determinant growth of the design matrix. A design-transfer argument shows that the near G-optimal design phase induces expected determinant growth that is logarithmic in dimension and interval lengths. Finally, the rapidly growing static schedule ensures that earlier exploration phases regularize the covariance sufficiently that later greedy phases incur small regret (Yu et al., 31 May 2026).

A plausible implication is that BLCE-G is not merely a batched approximation to fully sequential algorithms. Its regret proof relies essentially on reward-free within-interval adaptivity, especially through covariance updates and dynamic elimination, rather than only on coarse interval-level planning.

6. Computational profile, comparison with BLCE, and nomenclature

BLCE-G uses $$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$5 ridge-regression recomputations under its static schedule (Yu et al., 31 May 2026). Its computational cost has several components: near G-optimal design calls cost $$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$6 each; arm elimination costs $$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$7 per round; and the overall time complexity is stated as

$$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$8

The memory footprint is described as typically $$r_t = \langle x_{t,a_t}, \theta^* \rangle + \eta_t,$$9, reflecting storage of $\theta^* \in \mathbb{R}^d$0 matrices, vector sums, and elimination sets (Yu et al., 31 May 2026).

The paper introduces a second algorithm, BLCE, which removes the near G-optimal design step entirely and relies on uncertainty-driven exploration, elimination, and greedy selection. BLCE retains minimax-optimal regret up to slightly larger polylogarithmic factors,

$\theta^* \in \mathbb{R}^d$1

while reducing runtime to

$\theta^* \in \mathbb{R}^d$2

The paper therefore recommends BLCE-G when one can afford the near G-optimal design step and seeks the tightest regret bound, especially in large-$\theta^* \in \mathbb{R}^d$3 regimes, and BLCE when runtime is the priority (Yu et al., 31 May 2026).

The same work also extends the rare-update perspective to generalized linear contextual bandits through BGLE, which uses weighted Gram matrices and elimination without relying on G-optimal design and achieves near-optimal regret independent of the worst-case curvature parameter $\theta^* \in \mathbb{R}^d$4 (Yu et al., 31 May 2026). This places BLCE-G within a broader program: rare update schedules, combined with within-interval context adaptivity, can retain optimal or near-optimal statistical efficiency without frequent reward-dependent estimation.

A common source of confusion is the acronym itself. In the arXiv record, “BLCE-G” is also used in an unrelated 2025 condensed-matter paper to denote bicircular-light-induced multi-state geometric current (Guo et al., 5 Mar 2025). The contextual-bandit BLCE-G discussed here is the algorithmic object introduced in (Yu et al., 31 May 2026). The coincidence of abbreviations is terminological rather than conceptual.