Constant-Liar Batch Selector in FolDE

• The paper introduces a constant-liar approach that conditions the surrogate model on pessimistic outcomes to mitigate within-batch redundancy.
• It leverages deep ensemble surrogates and an adjustable noise parameter to balance exploration and exploitation in mutant selection.
• Empirical results demonstrate that lower alpha values improve batch diversity, leading to more informative data for subsequent optimization rounds.

The constant-liar batch selector is a batch Bayesian optimization heuristic for choosing, within a single experimental round, a set of mutants whose measurements will be performed in parallel. In FolDE, it is instantiated on top of a deep-ensemble surrogate for low-$N$ protein engineering and is designed to mitigate a specific failure mode of naïve batch construction: selecting the highest-predicted mutants in a batch tends to produce homogeneous training data concentrated at similar loci or local sequence neighborhoods. The method builds a batch sequentially, and after each selected mutant it conditions the surrogate on an imagined pessimistic outcome—a “lie”—so that correlated candidates become less attractive. In FolDE, this mechanism is explicitly presented as an adaptation of the classical “constant liar” or “Kriging believer” strategy from Gaussian-process Bayesian optimization, but with ensemble-derived means and covariance and an additional noise parameter $\alpha$ controlling the exploration–exploitation trade-off (Roberts et al., 28 Oct 2025).

1. Definition and problem setting

In FolDE, the constant-liar batch selector is used to choose a set of mutants for parallel experimental testing within an ALDE campaign. The underlying motivation is that protein optimization under low experimental budget is conducted in batches—for example, 16 mutants per round—and that a straightforward top-$N$ policy, whether based on predicted score or on a pointwise acquisition such as UCB, often yields homogeneous batches (Roberts et al., 28 Oct 2025).

The homogeneity problem is described concretely for multi-mutation search spaces. Top-$N$ selection tends to return many mutants that share mutations at the same loci or differ only slightly from already favored variants. The stated consequences are poor coverage of sequence space, highly redundant training data for subsequent rounds, difficulty stacking beneficial mutations from different loci, and increased risk of becoming trapped in local optima. In one quantified example, for a batch of 16 double mutants in round 2, only about 4 out of 32 total mutations target new loci under top-$N$ selection (Roberts et al., 28 Oct 2025).

Within this setting, batch diversity is treated as instrumentally important rather than as an end in itself. With only 16 mutants per round, each assay is expected to contribute as much information as possible; diverse batches improve mutational coverage, probe multiple loci in multi-mutation campaigns, and provide more informative data for downstream model fitting. The selector is therefore intended to preserve emphasis on promising mutants while explicitly discouraging redundancy inside the batch (Roberts et al., 28 Oct 2025).

A common misconception is to treat constant liar as merely another pointwise exploration bonus. In the FolDE formulation, its distinguishing property is batch awareness: later selections are conditioned on earlier ones in the same batch, so the procedure penalizes within-batch redundancy rather than only changing the score of each point independently (Roberts et al., 28 Oct 2025).

2. Mathematical formulation in FolDE

FolDE uses an ensemble of 5 neural networks trained with a Bradley–Terry ranking loss on ESMC-300M embeddings of dimension 960. For a candidate set $M=\{m_1,\dots,m_N\}$, the ensemble provides predicted activity means $y_i$ and a covariance matrix $\mathbf{\Sigma}\in\mathbb{R}^{N\times N}$ across candidates (Roberts et al., 28 Oct 2025).

The batching logic is described as a constant-liar approximation to UCB-like or more broadly Bayesian selection. The paper does not specify a closed-form acquisition such as expected improvement; operationally, candidates are ranked by the current posterior mean, with diversification emerging from the lie-based posterior updates rather than from a separate exploration term (Roberts et al., 28 Oct 2025).

If candidate $m_i$ is selected into the batch, the covariance is partitioned as

$$\mathbf{\Sigma} = \begin{pmatrix} \mathbf{\Sigma}_{M \setminus i} & \mathbf{v} \ \mathbf{v}^T & \sigma_i^2 \end{pmatrix},$$

where $\alpha$0 is the covariance among remaining candidates, $\alpha$1 is the covariance vector between $\alpha$2 and the remaining candidates, and $\alpha$3 is the variance of $\alpha$4 (Roberts et al., 28 Oct 2025).

Assuming an imagined observation $\alpha$5 at the selected candidate, the posterior covariance over the remaining candidates becomes

$\alpha$6

The posterior mean becomes

$\alpha$7

The covariance update is independent of the numerical value of the lie, but the mean update is not. If the lie is pessimistic, $\alpha$8, then highly correlated candidates have their means reduced, which directly induces diversity in the remainder of the batch (Roberts et al., 28 Oct 2025).

FolDE chooses a pessimistic lie equal to the minimum predicted mean among candidates,

$\alpha$9

Every selected candidate is therefore treated, for the purpose of batch construction, as if it had performed as badly as the worst-predicted remaining option. In the paper’s interpretation, this makes the algorithm explicitly exploration-oriented by suppressing the neighborhood of already selected high-mean candidates (Roberts et al., 28 Oct 2025).

A distinctive implementation feature is the introduction of additional observation noise governed by $N$0: $N$1 This term is stated to stabilize iterations of the constant-liar method when the covariance matrix has small or vanishing eigenvalues. It also provides a practical exploration–exploitation control. Small $N$2 makes the lie highly influential and thus strengthens diversification; large $N$3 weakens the update and causes the selector to approach greedy top-$N$4 exploitation. In the benchmarked configuration, $N$5 in round 2, while round 3 is effectively run with $N$6, corresponding to near-pure exploitation (Roberts et al., 28 Oct 2025).

This formulation differs from standard GP-based constant liar in three respects emphasized by FolDE: the surrogate is a deep ensemble rather than a GP, the covariance is constructed empirically from ensemble predictions, and the diagonal noise term scaled by $N$7 gives more direct control over lie strength and numerical stability (Roberts et al., 28 Oct 2025).

3. Placement within the FolDE workflow

The constant-liar selector is not a universal controller for the entire FolDE campaign; it is a mid-campaign batch-selection module. FolDE’s main benchmark consists of 3 rounds with batch size 16, for a total of 48 measured mutants per campaign (Roberts et al., 28 Oct 2025).

The first round is purely zero-shot. FolDE computes PLM naturalness for all single mutants and selects the top 16 mutants by naturalness for experimental testing. No constant liar is used in this round (Roberts et al., 28 Oct 2025).

For later rounds, sequences are embedded using ESMC-300M with mean pooling, and an ensemble of 5 MLPs with architecture $N$8, ReLU activations, batch norm, dropout $N$9, and no final bias is trained. Each MLP is warm-started for approximately 50 epochs to predict naturalness over all single mutants using ranking loss on naturalness scores, then fine-tuned on accumulated measured activities with Bradley–Terry pairwise ranking loss. Optimization uses Adam with learning rate $N$0 and weight decay $N$1 (Roberts et al., 28 Oct 2025).

Candidate generation for round $N$2 is performed over all single mutants of the wild type and hits from previous rounds. The ensemble then produces the predictive mean vector $N$3 and covariance matrix $N$4, which are the sole direct inputs to the constant-liar selector (Roberts et al., 28 Oct 2025).

In the main benchmark, the selector is used in round 2 with $N$5 to choose the next 16 mutants. In round 3, FolDE reverts to pure exploitation, described as top-$N$6 selection or constant liar with very high $N$7 and effectively no diversification (Roberts et al., 28 Oct 2025). Extended 6-round simulations apply constant liar in round 3 and beyond to examine longer-horizon diversity and model-quality effects (Roberts et al., 28 Oct 2025).

Naturalness is therefore related to the constant-liar selector only indirectly. The selector itself operates on surrogate predictions and covariance, not on naturalness scores. However, the naturalness-based warm-start improves global surrogate quality and thus plausibly improves the meaningfulness of the covariance structure on which constant liar relies (Roberts et al., 28 Oct 2025). This suggests that FolDE treats batch diversification and surrogate regularization as complementary mechanisms rather than interchangeable ones.

4. Empirical behavior and observed effects

FolDE evaluates the selector’s effects on batch diversity using two measures: the number of unique loci mutated in a batch and the number of newly explored loci, defined as loci mutated in that batch but not previously mutated in earlier rounds (Roberts et al., 28 Oct 2025).

In extended 6-round simulations, lowering $N$8 increases diversity. In the single-mutation benchmark at round 2, $N$9 yields approximately 9.5 new loci per batch, whereas $N$0 yields approximately 12.2 new loci. In the multi-mutation benchmark at round 2, $N$1 yields approximately 3.8 new loci, whereas $N$2 yields approximately 7.3 new loci. The effect is therefore especially pronounced in more combinatorial multi-mutation landscapes (Roberts et al., 28 Oct 2025).

The paper also states that UCB alone, without constant liar, does not improve batch diversity. Top-$N$3 and top-$N$4+UCB show nearly identical declining exploration patterns, which is presented as evidence that pointwise uncertainty bonuses do not solve the within-batch redundancy problem (Roberts et al., 28 Oct 2025).

More aggressive constant liar also modestly improves held-out predictive quality in later rounds: lower $N$5 is associated with higher Spearman correlation between predicted and true activities in both single- and multi-mutation benchmarks (Roberts et al., 28 Oct 2025). The stated interpretation is that more diverse batches produce more informative training data and therefore improve the surrogate.

By contrast, the effect on end-of-campaign optimization metrics is limited. The primary campaign metrics are the number of top-10% mutants discovered and the probability of discovering at least one top-1% mutant (Roberts et al., 28 Oct 2025). In sweeps over $N$6, aggressive constant liar decreases the number of top-10% hits, while some intermediate values around 3–6 slightly improve the probability of finding a top-1% mutant. In the main 3-round benchmark, medium constant liar slightly improves top-1% probability for the single-mutation setting and has near no effect for the multi-mutation setting (Roberts et al., 28 Oct 2025).

FolDE’s ablation results further indicate that constant liar contributes little to the overall improvement of FolDE over baseline ALDE methods; most gains come from zero-shot naturalness in round 1, naturalness warm-start, and the ranking-loss neural network (Roberts et al., 28 Oct 2025). A second common misconception is therefore that constant liar is the principal reason FolDE outperforms baselines. The reported evidence does not support that interpretation.

5. Relation to alternative batch-selection frameworks

Within FolDE, the constant-liar selector is positioned against three main alternatives: greedy top-$N$7 selection, UCB-based top-$N$8, and other Bayesian optimization methods such as TuRBO and Thompson sampling as used in prior ALDE work (Roberts et al., 28 Oct 2025).

The contrast with greedy top-$N$9 is straightforward. Top-$M=\{m_1,\dots,m_N\}$0 is purely exploitative and ignores redundancy inside the batch. Constant liar modifies the posterior after each selection and thereby down-weights correlated candidates, explicitly introducing a within-batch dissimilarity mechanism (Roberts et al., 28 Oct 2025).

The contrast with UCB and Thompson sampling is more specific. These methods encourage exploration at the point level, but in the settings discussed they still select top-$M=\{m_1,\dots,m_N\}$1 points individually and do not condition on what has already been placed in the same batch. FolDE’s empirical finding that UCB alone does not improve diversity is used to argue that pointwise uncertainty is not equivalent to batch-aware exploration (Roberts et al., 28 Oct 2025).

The selector is also situated within a broader literature on batch BO. FolDE explicitly links its approach to the classical constant liar and Kriging believer heuristics of Ginsbourger et al. and notes that the method is easy to implement relative to more theoretically optimal but more complex batch methods such as the parallel knowledge gradient of Wu and Frazier (Roberts et al., 28 Oct 2025).

A distinct line of work on adaptive batch sizing provides a useful foil. The paper "Adaptive Batch Sizes for Active Learning A Probabilistic Numerics Approach" formulates batch construction as a quadrature problem, measures batch quality using worst-case integration error or MMD, and chooses batch size adaptively through a linear program driven by a tolerance $M=\{m_1,\dots,m_N\}$2 rather than fixing batch size in advance (Adachi et al., 2023). That work does not mention constant liar by name, but it explicitly discusses the broader limitations of fixed batch-size heuristic methods, including hallucination-like procedures, and argues that batch size should respond to the geometry of the acquisition landscape and to changing cost–speed trade-offs over the run (Adachi et al., 2023).

From that perspective, the constant-liar selector in FolDE should be understood as a fixed-size, batch-aware diversification heuristic, not as an adaptive batch-sizing method. A plausible implication is that the limited effect observed in FolDE’s 3-round benchmark may partly reflect the fact that only point choice, and not batch cardinality, is being adapted (Adachi et al., 2023, Roberts et al., 28 Oct 2025). FolDE itself does not make that claim, but the juxtaposition clarifies a conceptual boundary between two families of batch methods.

6. Limitations, caveats, and likely directions

FolDE states three principal limitations of the constant-liar selector. First, under the benchmarked setting of 3 rounds and batch size 16, its effect on end-of-campaign performance is small relative to other components. Second, lower $M=\{m_1,\dots,m_N\}$3 produces a clear trade-off with exploitation by reducing the number of top-10% mutants found in early rounds. Third, the method depends on the quality of the surrogate covariance estimate: if ensemble covariance does not reflect useful similarity in sequence–function space, the resulting diversification may not align with productive exploration (Roberts et al., 28 Oct 2025).

The paper also offers several reasons why the measured effect may be small in its benchmarks. The campaigns are short; benefits from improved diversity and improved surrogate quality may not have sufficient time to accumulate. ProteinGym landscapes used in simulation, although large, are still shallower than real campaigns that may proceed 6–29 mutations away from wild type. In such shallower settings, strong zero-shot priors and naturalness warm-start may already supply much of the attainable benefit, leaving less room for a batch diversification heuristic to improve outcomes (Roberts et al., 28 Oct 2025).

These caveats matter because they constrain the interpretation of the empirical results. The selector successfully improves batch diversity and modestly improves model predictions, but in the reported 3-round benchmark it has limited effect on overall optimization performance (Roberts et al., 28 Oct 2025). That conclusion is narrower than a blanket endorsement or rejection of constant liar as a batch strategy.

Future work suggested in the source material includes evaluating more theoretically grounded batch BO methods such as parallel knowledge gradient, testing constant liar in longer and more combinatorial campaigns, and improving uncertainty modeling through true GP surrogates, better ensembles, or more calibrated uncertainty estimates (Roberts et al., 28 Oct 2025). In parallel, adaptive-batch frameworks based on probabilistic numerics suggest an alternative direction in which batch size itself becomes a dynamic quantity controlled by quadrature precision objectives or estimated violation risk (Adachi et al., 2023).

In contemporary protein engineering terms, the constant-liar batch selector is best understood as an explicit importation of batch Bayesian optimization ideas into ALDE under realistic parallel-assay constraints. In FolDE, it is neither the dominant source of performance gains nor a negligible embellishment: it is a targeted mechanism for reducing within-batch redundancy, most evidently beneficial for mutational coverage and surrogate quality, and potentially more consequential in longer, deeper, and more combinatorial search campaigns than in the 3-round benchmark where it was introduced (Roberts et al., 28 Oct 2025).