Technical Summary: PIT-SUN

Problem Statement

In value-driven recommender systems, predicting continuous business metrics such as dwell time, Gross Merchandise Value (GMV), and Lifetime Value (LTV) is central to ranking and allocation. These targets are characteristically sparse, heavy-tailed, multimodal, and zero-inflated.

The core dilemma identified in the paper is the trade-off between optimization stability and expectation consistency:

1. Original-Space MSE: While minimizing Mean Squared Error (MSE) in the original space theoretically targets the conditional expectation $m(x) = E[Y|X=x]$, finite-sample gradients are unstable due to extreme values and zero plateaus. This leads to "mean collapse" or "tail shrinkage."
2. Transformed Space: Applying monotonic transformations (e.g., log, sqrt, Box-Cox) stabilizes the learning coordinate but introduces retransformation bias. Directly inverting the transformed prediction, $T^{-1}(E[T(Y)|x])$, does not equal the original expectation $E[Y|x]$ unless $T^{-1}$ is affine. However, affine transforms cannot provide the necessary bounded tail compression for heavy-tailed data.

Existing recovery methods (e.g., TranSUN/GTS) restore expectation consistency via conditionally linear recovery but rely on manual selection of transforms and recovery bases. The paper argues that simply replacing a manual transform with an empirical Probability Integral Transform (PIT) is insufficient without a unified framework for the coordinate, inverse lookup, recovery base, and drift monitoring.

Methodology: PIT-SUN

The authors propose PIT-SUN (Probability-Integral-TranSformed Unbiased recovery), a deployable empirical marginal recovery framework. It treats the empirical marginal table not merely as a transform, but as a unified "closure" that defines the coordinate, inverse lookup, recovery base, and monitoring mechanism simultaneously.

1. Theoretical Foundation

The paper establishes two key impossibility results:

• Nonlinear Direct Inversion: A nonlinear, tail-compressing transform cannot be universally expectation-preserving under direct inversion.
• PIT-Only Bias: Even with a perfect PIT coordinate, direct inversion $C^{-1}(E[C(Y)|x])$ exhibits systematic Taylor bias due to the curvature of the inverse quantile function.

2. The PIT-SUN Closure

The framework consists of three coupled components derived from a single empirical Cumulative Distribution Function (CDF) table:

• Empirical PIT Coordinate (Scale-Bounded Learning):
  Instead of manual transforms, the model learns in a bounded normal-score space. The target is transformed via the empirical CDF $\hat{F}_Y$ clipped to $[\delta, 1-\delta]$ and mapped to a standard normal quantile:
  $$C(y) = \Phi^{-1}(\text{clip}(\hat{F}_Y(y), \delta, 1-\delta))$$
  This converts heavy-tailed amplitudes into bounded coordinates, stabilizing gradients.

• Inverse-Quantile Base (Stabilized Recovery):
  The predicted PIT coordinate is mapped back to the original scale to form a recovery base $b(x)$:
  $$b(x) = \max(C^{-1}(\bar{f}(x)), b_{min}) + \epsilon$$
  Here, $C^{-1}$ is the empirical inverse quantile lookup. Crucially, $b_{min}$ (a lower bound based on the quantile of positive targets) prevents denominator collapse in zero-inflated regions and controls the variance of the recovery label.

• SUN Recovery (Conditionally Linear):
  A second branch learns the proportional target $z(x) \approx E[Y/b(x) | x]$ using a stop-gradient operator on the base $b(x)$. The loss is:
  $$L_{SUN} = E\left[ \left( z_\psi(x) - \frac{y}{\text{sg}[b(x)]} \right)^2 \right]$$
  This ensures the recovery path is conditionally linear with respect to the fixed base, restoring expectation consistency without corrupting the PIT geometry with high-variance residuals.

• Final Prediction:
  $$\hat{y}(x) = z_\psi(x) \cdot b(x)$$

3. Handling Sparse and Zero-Inflated Targets

For targets with extreme zero-inflation (e.g., GMV), the paper introduces PIT-SUN-ZI, which factorizes the problem:

1. A hurdle branch estimates $P(Y > 0 | x)$.
2. PIT-SUN estimates $E[Y | x, Y > 0]$ on positive samples.
3. The final prediction is the product of the occurrence probability and the positive amount.

Key Contributions

1. Impossibility-Guided Formulation: The paper formally proves that nonlinear stable coordinates cannot be expectation-preserving under direct inversion, necessitating a separate recovery path rather than just a better scalar transform.
2. The PIT-SUN Closure: It introduces a unified framework where a single empirical CDF table defines the stable coordinate, the inverse lookup, the variance-controlled recovery base, and the drift monitor. This replaces manual heuristic choices with a data-driven, deployable closure.
3. Empirical Validation: The method is validated across synthetic distributions, public benchmarks (CIKM16, DTMart), large-scale industrial datasets (Dwell Time, GMV), and online A/B tests.

Experimental Results

• Synthetic & Public Benchmarks: PIT-SUN demonstrates robust calibration (measured by SRE, TRE, MRE) and ranking quality (xAUC, Gini) across various marginal shapes (heavy-tailed, zero-inflated, multimodal). It outperforms distributional models (ZILN, EGMN), discretized models (TPM, CREAD), and manual recovery baselines (TranSUN).
• Industrial Offline Results: On large-scale Dwell Time and Zero-inflated GMV datasets, PIT-SUN achieves the lowest point errors (NMAE, NRMSE) and best aggregate calibration. Diagnostic variants (e.g., PIT-SUN-w/o SUN, PIT-TranSUN) confirm that the full closure (bounded coordinate + stabilized base + gradient isolation) is necessary; simply plugging empirical PIT into existing frameworks yields inferior results.
• Online A/B Testing: Deployed in an incentive allocation system, PIT-SUN showed statistically significant improvements in dwell time, video view counts, and deeply engaged users. Daily Active Users (DAU) and ad revenue remained stable, indicating the model improves engagement without harming other business metrics.
• Efficiency: PIT-SUN reduces training time and inference latency (P50/P99) compared to two-stage baselines, making it suitable for high-QPS production environments.

Significance and Claims

The paper claims that PIT-SUN reframes target transformation from a heuristic preprocessing step into a deployable empirical-marginal closure. Its significance lies in solving the "stability vs. expectation" dilemma without relying on complex distributional assumptions or manual hyperparameter tuning for transforms.

The authors emphasize that the framework is model-agnostic and interpretable, serving as a practical scale-aligner for heterogeneous continuous targets (dwell time, GMV, LTV) in industrial recommendation systems. The work demonstrates that a complete, data-driven closure—integrating the coordinate, base, and recovery—is essential for achieving both point accuracy and expectation consistency in complex, real-world regression tasks.