Agreement-Based Batch-Size Policy

• Agreement-based batch-size policy is a framework that adapts batch sizes in sequential learning using statistical confidence to decide when to commit to actions.
• It employs strategies like explore-then-commit, successive elimination, and adaptive scheduling to optimize performance while managing operational and computational trade-offs.
• This approach is crucial in applications such as bandits, reinforcement learning, and online services, where resource constraints and latency necessitate robust, agreement-driven updates.

An agreement-based batch-size policy refers to a class of decision-making rules, primarily arising in sequential learning and stochastic control, that determine when and how to adapt batch sizes or commit to actions based on statistically principled agreement (statistical confidence, empirical consensus, or performance thresholds) among competing decisions. This approach is central to modern batched bandit algorithms, robust batch policy learning, and dynamic batching for efficient computation, where the overarching goal is to balance sample efficiency, regret, and operational or computational cost by making agreement-driven policy updates.

1. Foundational Concepts and Motivation

Agreement-based batch-size policies originate from the challenge of constraining learning agents to operate under batch-wise feedback or action commitment, as opposed to fully sequential adaptation. Such constraints are natural in domains including clinical trials, industrial process control, large-scale online systems, and cloud inference services, where continuous (per-round) updates are infeasible due to regulatory, computational, or latency considerations.

The canonical structure involves partitioning a fixed decision horizon $T$ into $M$ batches or windows (possibly determined adaptively), observing the consequences (feedback or rewards) within each, and updating the decision rule only at batch boundaries. An “agreement” is typically formalized as:

• Sufficient statistical evidence (e.g., confidence intervals, empirical means, cycle counts) that one arm, action, or policy outperforms others, or
• Achievement of a consensus according to a pre-specified test statistic or adaptive threshold. These policies are often contrasted with “fully adaptive” methods and are theoretically motivated by the objective of near-optimal regret, minimizing switching cost, and optimal resource/latency trade-offs.

2. Policy Structures: Explore-Then-Commit, Successive Elimination, and Adaptive Scheduling

A diverse suite of agreement-based batch-size policies has been advanced, especially in bandit and reinforcement learning contexts.

2.1 Explore-Then-Commit (ETC) under Predetermined Batches

In the two-armed stochastic bandit scenario (Perchet et al., 2015), the ETC policy partitions time into $M$ batches at preselected grid points $(t_1, t_2, ..., t_{M-1})$. In each exploration batch, arms are pulled equally and a statistical test

$$\varphi_t = \begin{cases} i, & \bar{\mu}_{t/2}(i) - B_{t/2}(i) > \bar{\mu}_{t/2}(j) + B_{t/2}(j),\quad j \neq i\ \bot, & \text{otherwise} \end{cases}$$

is computed, where $\bar{\mu}_{s}(i)$ is the empirical mean and $B_s(i)$ a confidence bound. Upon a decisive agreement, commitment to the superior arm occurs for all future pulls, ensuring high probability of committing to the optimal arm with regret

$$R_T(\Delta, T) \leq 9\Delta t_{m(\Delta, T)} + T\Delta \exp\left(-\frac{t_{M-1}\Delta^2}{16}\right)\mathbb{1}\{m(\Delta,T)=M-1\},$$

with $m(\Delta,T)$ defined via a minimax grid or geometric grid, depending on worst-case or problem-dependent preferences.

2.2 Batched Successive Elimination (BaSE) for Multi-Armed Bandits

For $K$-armed settings (Gao et al., 2019), BaSE generalizes ETC by maintaining an active arm set, at the end of each batch eliminating arms whose empirical means are sufficiently far from the leader beyond a batch-dependent confidence threshold. With carefully designed batch grids—minimax for worst-case, geometric for problem-dependent regret—BaSE secures

$$E[R_T(\text{minimax BaSE})] \leq C \log(K)\sqrt{K \log(KT)}T^{1/(2 - 2^{1-M})}$$

with only $O(\log \log T)$ batches for minimax optimality.

2.3 Adaptive and Agreement-Driven Batch Scheduling

Batched Thompson sampling (Kalkanli et al., 2021) introduces an anytime, agreement-based batch termination, using a per-arm “cycle count” and ending the batch when

$$U_{i,j} = \max\left\{1, \left\lceil \alpha \cdot M_i(T_{j-1}) \right\rceil\right\}$$

is exceeded for any arm, where $M_i(T_{j-1})$ is the number of cycles for arm $i$ up to batch $j-1$, and $\alpha>1$ is a tuning parameter. The method achieves (problem-dependent) $O(\log T)$ and (minimax) $O(\sqrt{T\log T})$ regret using $O(\log T)$ or even instance-dependent $O(\log\log T)$ batches, while needing no prior knowledge of the time horizon.

3. Regret Analysis and Fundamental Trade-offs

The regret of batch-centric policies is fundamentally influenced by the choice of batch-size policy, agreement criteria, and batch grid structure.

3.1 Regret Scaling with Batch Size

Both theoretical and empirical studies (Provodin et al., 2021, Provodin et al., 2022) establish that batching imposes an unavoidable penalty relative to online decision making, with regret scaling as

$R_n(\pi^b) \leq b \cdot R_M(\pi)$

where $b$ is the batch size, $n$ the total number of rounds, $M = n/b$ the number of batches, and $R_M(\pi)$ the regret of an online policy on a reduced horizon. This scaling reveals a direct trade-off: increasing batch size raises regret in proportion, motivating dynamic or agreement-adaptive batching to control performance loss.

3.2 Optimality and Lower Bounds

Carefully constructed policies using minimax or geometric grids—where batch sizes are chosen recursively or exponentially—can guarantee minimax optimal regret (order $\sqrt{T}$) with only $O(\log\log T)$ batches (Perchet et al., 2015, Gao et al., 2019). Lower bounds show that any $M$-batch policy must incur regret at least proportional to $T/M$ in certain regimes, demonstrating the near-optimality of agreement-based batch-size policies when batch count is constrained.

3.3 Policy-Agnostic Benchmarks

Recent analyses advocate evaluating any batch-size policy relative to its online “short” analog, formalizing agreement in terms of maintaining regret not much worse than a rescaled online baseline (Provodin et al., 2021, Provodin et al., 2022).

4. Practical Considerations: Switching Costs, Efficiency and Robustness

Agreement-based batch-size policies are engineered not only for statistical efficiency but also operational practicality.

4.1 Low Switching Strategies

By restricting policy updates to batch boundaries, the number of switches is explicitly bounded by the total number of batches, which, by design, can be logarithmic in $T$ (Perchet et al., 2015). Certain randomization strategies within batches reduce switching cost further to $O(\log\log T)$, a property crucial in high-cost environments (e.g., clinical trials, manufacturing).

4.2 Computational and Inference Efficiency

In modern machine learning inference with parallel hardware, batch size directly affects both responsiveness (latency) and energy efficiency. Dynamic batching (Xu et al., 4 Jan 2025) employs a semi-Markov decision process (SMDP) to select batch sizes at each decision epoch, minimizing

$$\text{Cost} = w_1 \cdot (\text{average response time}) + w_2 \cdot (\text{average power consumption}),$$

where $w_1, w_2$ are configurable weights. The SMDP-based agreement implicitly balances agreement between latency and efficiency goals, delivering near-optimal trade-offs.

4.3 Flexibility and Adaptation

SMDP-derived policies dynamically adapt batch thresholds in response to changing arrival rates, workload fluctuations, and system constraints, offering Pareto-optimal tradeoffs between responsiveness and resource utilization (Xu et al., 4 Jan 2025).

5. Applications in Bandits, Reinforcement Learning, and Online Services

The application of agreement-based batch-size policies extends across distinct areas:

• Stochastic Bandits: Clinical trials, A/B testing, online recommendation—where batch size is dictated by logistical or regulatory cycles, and agreement corresponds to statistical confidence or stopping rules (Perchet et al., 2015, Gao et al., 2019, Provodin et al., 2021, Provodin et al., 2022).
• Neural Bandits: Deep function approximation in bandit settings with batched contextual information, where policy updates are synchronized at batch boundaries to control computational cost (Gu et al., 2021).
• Dynamic Online Services: GPU-based ML inference and cloud services—agreement-based dynamic batching minimizes expected latency and power draw by adaptively matching the batch size to system state and workload, formalized via SMDPs (Xu et al., 4 Jan 2025).
• Multi-Agent Reinforcement Learning: Partitioning joint policy updates into “batches” of weakly dependent agents using agreement metrics on inter-agent dependencies to optimize both data efficiency and wall-clock time (Zhang et al., 21 Jul 2024).

6. Extensions and Implications for Robust and Adaptive Batch Policies

Robust batch policy optimization in Markov decision processes (Qi et al., 2020) draws on agreement-based ideas by optimizing for performance that is “in agreement” (min-max optimal) across neighborhoods of the stationary distribution, formalized via total variation balls: $\min_{u: TV(u, d^{(\pi)}) \leq c} E_u[R(S)].$ Doubly robust estimators and semi-parametric methods ensure rate-optimal regret bounds, quantifying the price of agreement under model uncertainty and finite batch data.

Further, batch size-invariance in policy optimization (e.g., PPO variants (Hilton et al., 2021)) can be regarded as a specific agreement principle—ensuring that learning dynamics remain consistent when batch sizes change, through careful decoupling of proximal and behavior policies and adaptive hyperparameter scaling.

7. Summary Table of Key Frameworks and Properties

Approach	Agreement Principle	Regret/Performance Guarantees
ETC (Batched Bandits)	Statistical test at batch boundary	Minimax $\sqrt{T}$ (with $O(\log\log T)$ batches)
BaSE	Successive empirical elimination via thresholds	Minimax and problem-dependent, rate-optimal with $O(\log\log T)$/$O(\log T)$ batches
Batched Thompson Sampling	Cycle count agreement triggers adaptive batch size	$O(\log T)$ instance, $O(\sqrt{T\log T})$ minimax regret, $O(\log\log T)$ batches
SMDP Dynamic Batching	Weighted latency/efficiency objective, cost agreement	Pareto-optimal latency/energy tradeoff; control-limit structure
MARL (B2MAPO)	Inter-agent attention-based dependency agreement	Monotonic improvement with tight bounds, improved efficiency

These methods collectively support stringent regret or efficiency guarantees while controlling for batch size, update frequency, and operational cost—embodying agreement-based principles in both statistical and resource-centric settings.