Partial Batch Label Sampling (ParBaLS)

• The paper introduces ParBaLS as a batch acquisition strategy derived from myopic Bayesian decision theory that uses Monte Carlo pseudo-label sampling to approximate EPIG objectives.
• It incrementally constructs a partial batch by conditioning on already selected points, reducing redundancy and scaling linearly with both batch size and label samples.
• Empirical evaluations show that ParBaLS improves label efficiency across diverse datasets, outperforming traditional top‑B selection and BatchBALD in many scenarios.

Searching arXiv for the cited papers to ground the article in the current literature. Partial Batch Label Sampling (ParBaLS) is a batch acquisition strategy for active learning that is derived from myopic Bayesian decision theory for Expected Predictive Information Gain (EPIG) and Expected Error Reduction (EER). It addresses pool-based batch selection by incrementally constructing a batch, maintaining a partial batch of already chosen but as-yet-unlabeled points, sampling pseudo-labels for that partial batch, and averaging single-point EPIG scores across multiple pseudo-labeled “universes” (Hu et al., 10 Oct 2025). In a related but distinct usage, the same phrase is applied to the Global Optimal Transport Planner (GOP) component of "Holistic Optimal Label Selection for Robust Prompt Learning under Partial Labels," where batch-level optimal transport over partial candidate label sets functions as a form of partial batch label assignment under weak supervision (Zhao et al., 8 Apr 2026).

1. Decision-theoretic foundation

ParBaLS is formulated in the setting of pool-based batch active learning for classification. The basic objects are an unlabeled pool $D \subset \mathcal{X}$, a labeled set $L \subset \mathcal{X} \times \mathcal{Y}$, a validation set $V \subset \mathcal{X}$ sampled from the test distribution, a batch size $B$, and a Bayesian predictive model $M_L$ induced by a posterior $p(w \mid L)$ (Hu et al., 10 Oct 2025). At each active learning round, an algorithm selects a batch $S_t \subset D$, queries their true labels, adds the labeled points to $L$, and retrains or updates the model.

The derivation begins from myopic Bayesian decision theory. Under the myopic assumption, only one additional label is imagined to be acquired before making predictions on the evaluation distribution. With negative log-likelihood loss, the optimal predictions after acquiring a label are the posterior predictive distributions, and the resulting acquisition objective reduces to maximizing

$$\argmax_{\hat{x} \in D} \sum_{x \in V} I(Y_x; Y_{\hat{x}} \mid L).$$

This is exactly the EPIG objective, and in this formulation EPIG and EER coincide (Hu et al., 10 Oct 2025).

The significance of this derivation is that ParBaLS is not introduced as an ad hoc heuristic. It is obtained by extending a myopic BDT objective to the case where batch construction must account for dependencies among unlabeled points. This directly targets the known failure mode of naive top-$B$ selection, namely redundancy among individually high-scoring points.

2. Partial-batch conditioning and pseudo-label universes

The defining idea of ParBaLS is the introduction of a partial batch $L \subset \mathcal{X} \times \mathcal{Y}$0: a set of points already committed to the current batch, but whose true labels are still unknown (Hu et al., 10 Oct 2025). When selecting the next point $L \subset \mathcal{X} \times \mathcal{Y}$1, the relevant objective is no longer the unconditional EPIG score, but the batch-aware criterion conditioned on the unknown labels of $L \subset \mathcal{X} \times \mathcal{Y}$2: $L \subset \mathcal{X} \times \mathcal{Y}$3 The difficulty is that the expectation is over $L \subset \mathcal{X} \times \mathcal{Y}$4 label configurations.

ParBaLS approximates this expectation by Monte Carlo label sampling. It samples $L \subset \mathcal{X} \times \mathcal{Y}$5 label configurations $L \subset \mathcal{X} \times \mathcal{Y}$6, restricts each sample to the partial batch $L \subset \mathcal{X} \times \mathcal{Y}$7, treats $L \subset \mathcal{X} \times \mathcal{Y}$8 as if they were true labels in universe $L \subset \mathcal{X} \times \mathcal{Y}$9, updates a model $V \subset \mathcal{X}$0 on $V \subset \mathcal{X}$1, computes single-point EPIG scores in each universe, and averages them (Hu et al., 10 Oct 2025). The resulting approximation is

$V \subset \mathcal{X}$2

This construction is “partial” in a precise sense: it samples labels only for the already committed subset $V \subset \mathcal{X}$3, not for the full future batch. That is the central distinction between ParBaLS and exact joint batch objectives. It is also the main reason its cost scales linearly rather than exponentially with batch size.

A frequent misconception is to equate ParBaLS with simple stochastic diversification. The formalism is narrower and more structured than that. Pseudo-labels are not used merely to perturb scores; they are used to approximate the conditional expectation appearing in the batch-aware myopic BDT objective (Hu et al., 10 Oct 2025).

3. Algorithmic form and computational properties

The operational form of "ParBaLS EPIG" is an iterative batch-construction procedure (Hu et al., 10 Oct 2025). It starts from $V \subset \mathcal{X}$4, trains a Bayesian model $V \subset \mathcal{X}$5, samples $V \subset \mathcal{X}$6 pseudo-label universes over the unlabeled pool, and then repeats $V \subset \mathcal{X}$7 times: compute a ParBaLS score for each candidate point, select the maximizer, add it to $V \subset \mathcal{X}$8, remove it from $V \subset \mathcal{X}$9, and update each universe-specific model with the corresponding pseudo-labeled point.

The score used at each step is

$B$0

Within each universe, earlier members of the partial batch are treated as labeled according to the sampled pseudo-labels. Later selections are therefore conditioned on what earlier selections would imply about the test distribution under that universe.

The paper states a round-level complexity of $B$1 in the number of EPIG score evaluations, ignoring training cost (Hu et al., 10 Oct 2025). The crucial scaling statement is that complexity is linear in batch size $B$2 and in the number of label samples $B$3, and does not incur the exponential growth in $B$4 that affects exact joint-label objectives such as BatchBALD-style formulations.

A formal Monte Carlo error guarantee is also given. If $B$5 is the optimizer of the exact expectation and $B$6 is the Monte Carlo choice, then with probability $B$7,

$B$8

which is $B$9 and does not depend on batch size $M_L$0 (Hu et al., 10 Oct 2025). This bound is one of the distinguishing theoretical features of ParBaLS.

The paper also defines a cheaper variant, ParBaLS-MAP, in which $M_L$1 and the pseudo-label for each unlabeled point is taken to be the MAP class under the current model (Hu et al., 10 Oct 2025). This replaces Monte Carlo label sampling with a deterministic pseudo-label universe.

4. Empirical behavior in batch active learning

The empirical study evaluates ParBaLS on two tabular datasets from Kaggle—Airline Passenger Satisfaction and Credit Card Fraud—and four image datasets: CIFAR-10, CIFAR-100, iWildCam, and fMoW from WILDS (Hu et al., 10 Oct 2025). For image data, two derived scenarios are used: one-vs-all binary classification and subpopulation-shift with three classes in the pool but only the first two in the test set.

The predictive model is Bayesian Logistic Regression on fixed embeddings. For WILDS image datasets, embeddings come from CLIP-ViT-B/32; for CIFAR-10/100, from DINOv2-ViT-S/14; PCA retains $M_L$2 variance; and posterior inference uses NUTS in PyMC with $M_L$3 posterior samples (Hu et al., 10 Oct 2025). Two main regimes are reported: an initial labeled set of 500 with $M_L$4 rounds and $M_L$5, and an initial labeled set of 100 with $M_L$6 and $M_L$7.

The main empirical findings reported are that ParBaLS EPIG has the highest mean performance in 4 of 10 datasets/settings and is “among the top” in 9, while ParBaLS-MAP EPIG has the highest mean in 2 and is among top in 7 (Hu et al., 10 Oct 2025). The paper also reports specific examples in which ParBaLS EPIG or ParBaLS-MAP improves over top-$M_L$8 EPIG, including Credit Fraud, subpopulation-shift CIFAR-10, and subpopulation-shift iWildCam (Hu et al., 10 Oct 2025).

The comparisons emphasize three patterns. First, top-$M_L$9 EPIG is generally strong but is consistently slightly worse than ParBaLS EPIG. Second, BatchBALD is competitive on some image tasks but becomes too slow in the larger-batch regime. Third, stochastic batch heuristics such as PowerBALD, SoftmaxBALD, SoftRankBALD, and their EPIG variants show mixed performance and require tuning hyperparameters such as temperature $p(w \mid L)$0, which the authors describe as impractical in active learning (Hu et al., 10 Oct 2025).

These results support the interpretation that the partial-batch conditioning mechanism improves label-efficiency by reducing redundancy while retaining the predictive-space rationale of EPIG.

5. Relation to top-$p(w \mid L)$1, BatchBALD, and BAIT

ParBaLS occupies an intermediate position between cheap but myopically independent top-$p(w \mid L)$2 selection and expensive joint batch objectives. Top-$p(w \mid L)$3 EPIG simply ranks points by single-point EPIG and selects the $p(w \mid L)$4 largest scores. This ignores dependencies among the labels of batch points and can yield highly redundant batches (Hu et al., 10 Oct 2025). ParBaLS instead recomputes scores conditioned on pseudo-labeled versions of the already selected subset, so later points are chosen with awareness of earlier ones.

Relative to BatchBALD, the difference is both conceptual and computational. BatchBALD maximizes $p(w \mid L)$5, a parameter-space mutual information objective, whereas ParBaLS remains in the predictive space and approximates a batch-aware EPIG/EER objective through label-space sampling (Hu et al., 10 Oct 2025). The paper states that BatchBALD requires many posterior weight samples as batch size grows and that its computation and memory become prohibitive for large batches. By contrast, ParBaLS has linear dependence on $p(w \mid L)$6 and $p(w \mid L)$7, and its Monte Carlo approximation error does not increase with $p(w \mid L)$8 (Hu et al., 10 Oct 2025).

The same work also shows that BAIT can be derived from the myopic BDT viewpoint via asymptotic approximations involving Taylor expansion, Laplace approximation, and a V-optimal design criterion

$p(w \mid L)$9

(Hu et al., 10 Oct 2025). ParBaLS differs in that it does not rely on those asymptotic parameter-space approximations; it addresses batching for EPIG/EER directly via pseudo-label conditioning.

A plausible implication is that ParBaLS is best understood not as a replacement for all batch acquisition methods, but as a scalable mechanism for importing batch awareness into predictive mutual-information objectives that would otherwise be applied pointwise.

6. Partial batch label sampling in prompt learning under partial labels

In "Holistic Optimal Label Selection for Robust Prompt Learning under Partial Labels," the phrase partial batch label sampling is used in a different learning problem: partial-label prompt learning with a frozen CLIP-like vision-LLM (Zhao et al., 8 Apr 2026). The training set is

$S_t \subset D$0

where each candidate set $S_t \subset D$1 contains exactly one true label and $S_t \subset D$2 confusing labels, with confusion rate $S_t \subset D$3 (Zhao et al., 8 Apr 2026). Predictions are produced by a frozen vision encoder $S_t \subset D$4, a frozen text encoder $S_t \subset D$5, and learnable prompt parameters $S_t \subset D$6.

The proposed method, HopS, has two components: a Local Density-based Filter (LDF) and a Global Optimal Transport Planner (GOP) (Zhao et al., 8 Apr 2026). The paper explicitly states that GOP is “the part that directly realizes partial batch label sampling.” For a mini-batch of size $S_t \subset D$7, GOP defines a source distribution $S_t \subset D$8 that is uniform over batch instances and a candidate-aware target distribution $S_t \subset D$9 over labels: $L$0 It then solves an entropic optimal transport problem over a transport plan $L$1 with costs

$L$2

subject to row-sum, column-sum, and candidate-set constraints (Zhao et al., 8 Apr 2026).

The resulting transport plan is interpreted as a batch-level probabilistic assignment from instances to labels, and the hard pseudo-label for each instance is obtained by row-wise argmax: $L$3 The paper characterizes this as “partial batch label sampling” because assignments are computed jointly over the batch, under partial candidate-label constraints, before being collapsed to discrete labels (Zhao et al., 8 Apr 2026).

This usage is distinct from ParBaLS EPIG in active learning. In the active learning setting, ParBaLS samples hypothetical labels for already selected but unlabeled points in order to choose future query points (Hu et al., 10 Oct 2025). In HopS, batch-level optimal transport assigns pseudo-label mass across instances and classes within a training batch in order to disambiguate partial labels during prompt learning (Zhao et al., 8 Apr 2026). The shared conceptual element is the joint treatment of ambiguous or unknown labels at batch level rather than independent per-instance selection.

The HopS paper reports that GOP, together with LDF, improves performance under partial supervision on eight benchmark datasets—Caltech-101, DTD, EuroSAT, FGVCAircraft, Food-101, Oxford Flowers-102, OxfordPets, and UCF101—and that combining local and global strategies yields higher accuracy and faster convergence than either alone (Zhao et al., 8 Apr 2026). It also notes that performance is sensitive when batch size $L$4 is close to the number of classes $L$5, with best performance typically occurring when $L$6 is near $L$7 (Zhao et al., 8 Apr 2026). This suggests that the batch-level allocation constraints central to GOP play a role analogous to the redundancy-reduction role of partial-batch conditioning in active learning, although the two mechanisms are mathematically different.

ParBaLS therefore denotes a family resemblance rather than a single invariant algorithmic form: in one case, Monte Carlo pseudo-label conditioning for batch active learning; in another, optimal-transport-based batch label assignment under partial labels. In both cases, the defining feature is that ambiguous label decisions are made jointly at batch level instead of independently per sample.