CPE: Confidence Optimization in Probabilistic Encoding

• CPE is a probabilistic encoding method that represents inputs as Gaussian latent distributions to capture uncertainty more effectively.
• It introduces a confidence-aware mechanism and L2 variance regularization to stabilize class discrimination in stochastic latent sampling.
• Empirical results on the TweetEval benchmark with BERT and RoBERTa show approximate 5% improvements over standard cross-entropy classifiers.

Searching arXiv for the specified paper and closely related work on probabilistic encoding for classification. I’m retrieving the arXiv record for (Xia et al., 22 Jul 2025) and nearby related papers to ground the article. Confidence Optimization Probabilistic Encoding (CPE) is a classification-oriented refinement of probabilistic encoding in which each input is represented as a Gaussian latent distribution rather than a single deterministic point. In the formulation introduced in "Confidence Optimization for Probabilistic Encoding" (Xia et al., 22 Jul 2025), the method is motivated by a specific failure mode of Gaussian latent sampling: while probabilistic encoding can improve uncertainty modeling and generalization, point-based classification distances become unstable when prediction is performed on stochastic samples drawn from the latent distribution. CPE addresses this instability through two coupled modifications: a confidence-aware mechanism that adjusts the way class-relative reliability is modeled, and a variance regularization strategy that replaces KL-divergence prior matching with a direct $L_2$ constraint on variance. The method is presented as model-agnostic and is evaluated on natural language classification with BERT and RoBERTa on TweetEval (Xia et al., 22 Jul 2025).

1. Probabilistic encoding and the classification problem

In the underlying probabilistic encoding setup, an input $x_i$ is mapped to a Gaussian latent distribution

$$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$

and a stochastic latent representation is obtained by reparameterization:

$$s_i = \mu_i + \epsilon \sigma_i, \qquad \epsilon \sim \mathcal{N}(0, I).$$

Classification is then performed on the sampled representation $s_i$ rather than directly on the mean $\mu_i$ (Xia et al., 22 Jul 2025).

The central issue identified by the paper is geometric rather than merely statistical. In classification, decision boundaries and class scores still depend on point-based latent comparisons or linear scores. Because $s_i$ is stochastic, samples drawn from the same latent Gaussian may occupy substantially different positions relative to class centers. A sample near the edge of the Gaussian can appear artificially close to or far from a class prototype, even when the latent mean suggests a different relation. The paper argues that this weakens intra-class compactness and inter-class separability, because the classification distance becomes inconsistent with the confidence structure implied by the probabilistic representation (Xia et al., 22 Jul 2025).

This framing distinguishes CPE from deterministic representation learning. The objective is not to discard uncertainty, but to preserve uncertainty modeling while making downstream class discrimination less sensitive to incidental sample placement. A common simplification is to describe CPE as probabilistic encoding augmented with auxiliary losses; the paper instead treats it as an attempt to make class-relative confidence explicit within the latent geometry.

2. Confidence-aware mechanism

CPE operationalizes confidence through the relation between latent statistics and class centers. The classifier weight matrix $W \in \mathbb{R}^{D \times C}$ is interpreted as a set of class centers, where $w_c$ denotes the center for class $c$, $x_i$0 is the embedding dimension, and $x_i$1 is the number of classes (Xia et al., 22 Jul 2025).

The paper first introduces a Gaussian density-based quantity and then defines a class confidence score using the error function:

$x_i$2

Conceptually, confidence is intended to be high when the class center lies near the Gaussian mean and low when it lies farther away. This makes confidence a function of both latent location and latent dispersion, rather than only of a sampled feature vector (Xia et al., 22 Jul 2025).

Raw confidence is then replaced by a normalized version, termed N-confidence in the paper. The normalization is softmax-like over Gaussian density values across candidate classes, using terms of the form $x_i$3. The stated purpose is to prevent multiple classes from simultaneously receiving high confidence and thereby weakening discrimination (Xia et al., 22 Jul 2025).

The resulting confidence optimization term is

$x_i$4

This loss trains the model not only to predict the correct label, but also to place the representation in a region where the correct class has high normalized confidence. In effect, CPE ties classification reliability to the latent distribution itself rather than treating uncertainty as a post hoc statistic.

A further component is introduced to prevent what the paper calls variance collapse. If confidence becomes too high in a way that drives variance toward zero, the stochastic encoder degenerates toward a deterministic one, undermining the intended benefits of probabilistic encoding. To counter this, the method introduces an Overly Mask, a binary mask aligned with the class-center matrix $x_i$5. Dimensions whose confidence exceeds thresholds $x_i$6 and $x_i$7 are masked before classifier scoring, with the aim of preventing overly confident dimensions from dominating the classifier update (Xia et al., 22 Jul 2025).

3. Objective function and variance regularization

The full CPE training objective is

$x_i$8

where $x_i$9 is the cross-entropy loss, $$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$0 is the confidence optimization term, and $$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$1 is a variance regularizer instantiated either as KL divergence or, in the proposed method, as a direct $$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$2 term (Xia et al., 22 Jul 2025).

The regularization choice is one of the defining features of the method. Standard probabilistic encoders frequently use KL divergence to push the latent posterior toward a standard Gaussian prior:

$$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$3

with closed form

$$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$4

The paper argues that this is ill-suited to the classification setting considered, because the latent distribution induced by confidence constraints need not resemble $$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$5 (Xia et al., 22 Jul 2025).

CPE therefore replaces KL-based prior matching with

$$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$6

The stated rationale is that this term directly constrains variance without imposing a potentially inaccurate prior assumption. In the paper’s comparison, KL regularization shapes the full latent distribution toward a standard normal prior, whereas the proposed $$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$7 term acts directly on variance and is therefore more task-focused (Xia et al., 22 Jul 2025).

This separation between confidence optimization and prior matching is central to the method’s identity. CPE does not reject probabilistic encoding; rather, it retains stochastic latent representations while revising the way confidence and variance are optimized for classification.

4. Model-agnostic design and experimental setting

The method is described as model-agnostic and is instantiated in the paper with BERT-base and RoBERTa-base encoders, using 768-dimensional text embeddings and a softmax classifier with weight matrix $$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$8 (Xia et al., 22 Jul 2025). This suggests that the method is intended to be inserted into existing probabilistic encoding pipelines without requiring a specialized encoder architecture.

The empirical evaluation is conducted on the TweetEval benchmark, comprising seven tasks: Emotion, Emoji, Hate, Irony, Offensive language, Sentiment, and Stance. The main evaluation metric is macro-F1, with task-specific metrics also used where appropriate: StanceEval uses the average F1 of against and favor, IronyEval uses F1 for the ironic class, and Sentiment uses recall (Xia et al., 22 Jul 2025).

The reported implementation details are specific. Training uses an NVIDIA RTX 2080Ti, batch size 128, the AdamX optimizer, learning rate $$P(z_i \mid x_i) = \mathcal{N}(z_i \mid \mu_i, \sigma_i^2 I),$$9, 20 epochs, and maximum sequence length 128. The hyperparameters $$s_i = \mu_i + \epsilon \sigma_i, \qquad \epsilon \sim \mathcal{N}(0, I).$$0 and $$s_i = \mu_i + \epsilon \sigma_i, \qquad \epsilon \sim \mathcal{N}(0, I).$$1 are tuned by grid search over values in the range 0.1 to 1 (Xia et al., 22 Jul 2025).

The experimental comparison includes SVM, FastText, BiLSTM, a CE baseline, MEIB, and SPC. Within this setup, CPE is evaluated not as a stand-alone classifier but as a probabilistic encoding strategy layered onto standard Transformer backbones.

5. Reported performance and ablation evidence

The paper reports average performance improvements for both BERT and RoBERTa backbones on TweetEval (Xia et al., 22 Jul 2025).

Backbone	CE average	CPE average
BERT	59.76	64.68
RoBERTa	61.21	65.94

For BERT, the reported averages are CE 59.76, MEIB 61.33, SPC 63.32, and CPE 64.68. For RoBERTa, the averages are CE 61.21, MEIB 62.70, SPC 65.65, and CPE 65.94. The paper summarizes these gains as improvements of about 4.92% over CE for BERT and about 4.73% over CE for RoBERTa (Xia et al., 22 Jul 2025).

At the task level, the paper highlights particularly strong performance on HateEval and IronyEval with the BERT backbone, while RoBERTa yields the best overall average and is described as robust across tasks. The additional computational burden is reported as small, about 0.9% training-time overhead (Xia et al., 22 Jul 2025).

The ablation study is used to isolate the contribution of each design choice. Replacing the proposed $$s_i = \mu_i + \epsilon \sigma_i, \qquad \epsilon \sim \mathcal{N}(0, I).$$2 variance term with KL divergence in a variant denoted $$s_i = \mu_i + \epsilon \sigma_i, \qquad \epsilon \sim \mathcal{N}(0, I).$$3 reduces average performance. Removing normalization also reduces performance, indicating that normalized confidence rather than raw confidence is important for class discrimination. Removing $$s_i = \mu_i + \epsilon \sigma_i, \qquad \epsilon \sim \mathcal{N}(0, I).$$4 lowers performance relative to full CPE, which the paper interprets as evidence that explicit confidence optimization is beneficial. Adding the confidence term to SPC improves SPC, and removing regularization altogether leads to a further drop. Taken together, these ablations support the claim that CPE’s effect depends on the interaction among normalized confidence, confidence loss, and explicit variance control (Xia et al., 22 Jul 2025).

6. Interpretation, scope, and acronym disambiguation

CPE occupies a specific position within uncertainty-aware representation learning. Its core claim is that uncertainty should influence classification through the geometry of the latent distribution rather than through stochastic sampling alone. This suggests a view of probabilistic encoding in which class confidence is an internal structural property of the representation, not merely an after-the-fact scalar attached to a prediction.

It is also important to distinguish this method from other confidence-related paradigms. "Confidence Before Answering: A Paradigm Shift for Efficient LLM Uncertainty Estimation" introduces CoCA, a confidence-first reinforcement learning framework for LLMs in which the model emits a confidence segment before the answer and optimizes calibration and answer accuracy jointly via segmented GRPO (Li et al., 6 Mar 2026). That work is adjacent in its concern with confidence optimization, but it is not a probabilistic encoding method and does not define CPE as Confidence Optimization Probabilistic Encoding.

The acronym itself is overloaded across fields. In differential geometry, CPE denotes critical point equation metrics, the critical points of the total scalar curvature functional under constant-scalar-curvature and unit-volume constraints (Andrade, 2020). In three-dimensional Riemannian geometry, that terminology appears again in work proving Besse’s conjecture for CPE metrics via minimal surface theory (Leandro, 2022). In cybersecurity NLP, "CPE-Identifier" refers to automated extraction of Common Platform Enumeration entities from CVE summaries (Hu et al., 2024). None of these uses are related to probabilistic latent encoding.

Within machine learning, therefore, Confidence Optimization Probabilistic Encoding designates the 2025 classification framework introduced in (Xia et al., 22 Jul 2025): a model-agnostic method that combines Gaussian latent representations, normalized class-relative confidence, explicit confidence loss, an Overly Mask against variance collapse, and direct $$s_i = \mu_i + \epsilon \sigma_i, \qquad \epsilon \sim \mathcal{N}(0, I).$$5 variance regularization in place of KL prior matching. Its reported contribution is to improve the reliability of distance-based classification under stochastic latent sampling while preserving the generalization advantages of probabilistic encoding.