IPS-weighted BPR in Recommender Systems

• IPS-weighted BPR is a pairwise collaborative filtering method that reweights user-item interactions using inverse propensity scores to counteract exposure bias.
• The approach uses stochastic gradient descent on reweighted losses, enhancing model robustness against popularity bias in implicit feedback data.
• Extensions such as propensity regularization, SNIPS evaluation, and adversarial training improve optimization stability and bias correction in recommendation systems.

IPS-weighted Bayesian Personalized Ranking (BPR) is a class of pairwise collaborative filtering algorithms aimed at debiased learning and evaluation from implicit feedback, notably accounting for selection and exposure bias through inverse propensity scoring (IPS) techniques. Rather than treating all observed interactions equally, as in classic BPR approaches, IPS-weighted variants reweight observed pairs by the inverse probability that they were exposed, mitigating the feedback-loop amplification of popularity bias prevalent in recommender systems. This paradigm is essential to counterfactual risk minimization in environments where logging policies strongly bias which items users see.

1. Foundations of Bayesian Personalized Ranking and IPS Weighting

Bayesian Personalized Ranking (BPR), as formalized by Rendle et al. (Rendle et al., 2012), is a pairwise ranking approach for implicit feedback settings. It targets the maximum a posteriori (MAP) solution for personalized order prediction. The canonical objective is

$$\mathcal{L}_{\text{BPR}} = -\sum_{(u,i,j)\in\mathcal{D}_S} \ln \sigma(\hat{x}_{uij})$$

where $(u,i,j)$ are user–item–negatives triples, $\hat{x}_{uij}$ is the model’s predicted score difference, and $\sigma(\cdot)$ is the logistic sigmoid.

IPS-weighted BPR modifies the above by introducing a weighting factor proportional to the inverse propensity, $w(u,i) = 1 / b(u,i)$, where $b(u,i)$ is the probability that item $i$ was exposed to user $u$ under the logging policy. The loss thus becomes: $$\mathcal{L}_{\text{IPS-BPR}} = -\sum_{(u,i,j)\in\mathcal{D}_S} w(u,i) \ln \sigma(\hat{x}_{uij})$$ This reweighting is essential for unbiased estimation in recommendation tasks—especially when item exposure is non-uniform and MNAR.

2. Algorithmic Structure and Numerical Optimization

IPS-weighted BPR is typically trained using stochastic gradient descent on reweighted pairwise losses. Each training triple, sampled according to user–item interaction logs, is weighted by its inverse propensity score. The generic learning algorithm (LEARNBPR (Rendle et al., 2012)) therefore consists of bootstrap sampling of $(u,i,j)$ tuples and per-step parameter updates as: $$\theta \leftarrow \theta + \alpha \cdot \left[w(u,i) \cdot \sigma(-\hat{x}_{uij}) \frac{\partial \hat{x}_{uij}}{\partial \theta} - \lambda \theta \right]$$ where $\alpha$ is the learning rate and $\lambda$ regularizes the model parameters. Crucially, excessive IPS weights can amplify gradient variance, destabilizing optimization. Several extension strategies have been developed:

• Propensity Regularization: Penalizing high IPS weights via regularizers, e.g.,

$$\mathcal{L}_{\text{IPS-BPR+PR}} = \mathcal{L}_{\text{IPS-BPR}} + \alpha \cdot \mathcal{R}(w(u,i))$$

with $\mathcal{R}(\cdot)$ a penalty function and $\alpha$ a trade-off hyperparameter (Raja et al., 30 Aug 2025).

• Self-Normalized IPS (SNIPS): For evaluation, SNIPS normalizes the sum of weights

$$\hat{V}_{\text{SNIPS}} = \frac{\sum_{\text{obs}} w(u,i) r_{ui}}{\sum_{\text{obs}} w(u,i)}$$

yielding lower variance, though introducing bias (Raja et al., 30 Aug 2025).

• Accelerated SGD: Sampling triples proportional to inverse propensities, scaling gradients by the mean (rather than maximum) IPS weight, reduces the convergence penalty from $O(M)$ to $O(\bar{M})$ where $M = \max_i 1/p_i$ and $\bar{M}$ is the mean (Jagerman et al., 2020).

3. Theoretical Properties and Bias Correction

IPS-weighted BPR provides an unbiased estimator of the ranking risk under exposure bias: $$\mathbb{E}_{\text{logging policy}}\left[ w(u,i) \ell(\hat{x}_{uij}) \right] = \mathbb{E}_{\text{target policy}} \left[ \ell(\hat{x}_{uij}) \right]$$ where $\ell$ denotes the pairwise ranking loss. The unbiasedness critically depends on accurate estimation of propensities $b(u,i)$. Under misspecified propensities or unmodeled trust bias, IPS estimators can retain residual bias (Vardasbi et al., 2020). Affine corrections—subtracting an additive bias term and dividing by the multiplicative propensity—are necessary for bias removal when user trust systematically affects exposure probability: $$\text{Affine Estimate}: \quad \frac{c(d) - \beta_k}{\alpha_k}$$ where $\alpha_k$ and $\beta_k$ are rank-specific parameters reflecting both position and trust bias (Vardasbi et al., 2020).

4. Extensions, Variants, and Related Methodologies

Numerous extensions enhance robustness, performance, or interpretability:

• Propensity Regularizer: Explicit regularization on large IPS weights to mitigate variance (Raja et al., 30 Aug 2025).
• Adversarial Training: APR applies adversarial perturbations to IPS-weighted BPR, forming a minimax objective for further robustness against noise and overfitting (He et al., 2018).
• Sampler Design: Incorporating user view data, adaptive hard negative mining, or spectral clustering can make sampling more efficient and loss weighting more expressive (Ding et al., 2018, Shi et al., 28 Mar 2024, Yu et al., 2019).
• Explainable BPR: Combining IPS-weighted loss with item-level explanation embeddings and propensity-aware regularization yields more interpretable recommendations (Damak et al., 2021).
• Variational BPR: The ELBO-KL decomposition substitutes instance-level propensity weighting, introducing latent prototype aggregation, hard mining, and noise reduction (Liu et al., 14 Mar 2025).
• Cross Pairwise Ranking (CPR): CPR structurally cancels out propensity-induced biases via cross-coupling of multiple positive samples rather than explicit weighting, achieving unbiased learning without propensity scores (Wan et al., 2022).

The implementation of IPS-weighted BPR often depends on careful tuning of hyperparameters and correct estimation of exposure probabilities. Replicability studies indicate that differences in optimizer choice, regularization, and negative sampling can produce large performance variations, and the same sensitivities apply to IPS-weighted cases (Milogradskii et al., 21 Sep 2024).

Extension	Mechanism	Effect
Propensity Regularizer	$\mathcal{R}(w(u,i))$	Variance reduction
Affine Correction	$(c(d)-\beta_k)/\alpha_k$	Trust bias removal
Adversarial BPR	Minimax adversarial regularizer	Robustness
CPR	Loss structure	Implicit debiasing

5. Counterfactual Learning and Evaluation Techniques

IPS-weighted BPR fits naturally in counterfactual risk minimization frameworks. Policy learning with exposure bias entails training from logged data generated by a previous policy. Offline evaluation then uses IPS, SNIPS, or direct methods (DM):

• DM: Direct regression of rewards can have low variance but is misspecification sensitive.
• IPS: Unbiased but suffers high variance, especially with small propensities.
• SNIPS: Trades small bias for lower variance in policy value estimates, empirically shown to yield more stable evaluation (Raja et al., 30 Aug 2025).

The combined pipeline (IPS-weighted BPR + PR for training, SNIPS for evaluation) yields robust recommendation models generalizing better to unbiased exposure, reducing evaluation variance, and offering stable practical deployment (Raja et al., 30 Aug 2025).

6. Limitations and Open Challenges

While IPS-weighted BPR robustly addresses exposure bias, several limitations and unresolved challenges remain:

• IPS-weighting assumes accurate propensities—estimation can be difficult under MNAR data or unobserved confounders (Vardasbi et al., 2020).
• Variance explosion from extreme IPS weights can cause unstable training and poor generalization—hence necessity of regularizers or alternative sampling (Jagerman et al., 2020, Raja et al., 30 Aug 2025).
• IPS, as a linear correction, cannot resolve additive trust bias without affine adjustments (Vardasbi et al., 2020).
• Integrating hard negative mining, explainability, and spectral features with IPS weighting enhances performance but introduces additional optimization intricacies (Ding et al., 2018, Shi et al., 28 Mar 2024, Yu et al., 2019, Damak et al., 2021).

Collectively, IPS-weighted BPR serves as a fundamental family of unbiased collaborative filtering algorithms. Recent research focuses on mitigating variance, integrating robust debiasing, and building practical evaluation pipelines for counterfactual learning to rank in real-world recommender systems.