# Bayesian Negative Sampling

## Required Packages

• numpy : Implement BNS for Matrix Factorization (MF) (run main_MF.py);
• pytorch: Implement BNS for light Graph Convolution Network (LightGCN) (run main_lightGCN.py).

We summarize the Bayesian negative sampling algorithm as: Randomly select a small set of candidates $\mathcal{M}_u$ from the negative instances set $\mathcal{I}_u^-$, for each negative instances $l$ in the samll candidate set $\mathcal{M}_u$:

### (i) Computing prior probability

By assuming $pop_l \sim B (N, P_{fn}(l))$, we compute prior probability as: $$P_{fn}(l) = \frac{pop_l}{N}$$ This step needs $\mathcal{O}(1)$.

### (ii) Computing cumulative distribution function (likelihood)

$$F_{n}(\hat{x} _ l) = \frac{1}{n} \sum I_ {|x_ \cdot \leq \hat{x}_l |}$$

This step needs $\mathcal{O}(|\mathcal{I}|)$.

The empirical distribution function $F_n (\cdot)$ converges to common cumulative distribution function $F(\cdot)$ almost surely by the strong law of large numbers. Glivenko Theorem (1933) strengthened this result by proving uniform convergence of $F_n(\cdot)$ to $F(\cdot)$. This property makes it possible for us to compute the $F(\cdot)$, even if we do not know its explicit expression. Given the observation $\hat{x}_l$, $F(\hat{x}_l)$ describes the joint probability of the observed instance $\hat{x}_l$ as a function of the parameters of the ranking model. For the specific parameter $l \in fn$, $F(\hat{x}_l)$ assigns a probabilistic prediction valued in $[0,1]$ of $l$ being false negative (positive).

### (iii) Computing negative signal (posterior probability)

$$\mathtt{unbias}(l) = \frac{[1 – F(\hat{x} _ l)] [1-P _ {fn} (l)] }{1 – F(\hat{x}_ l) -P_{fn}(l) + 2F(\hat{x}_ l)P_{fn}(l)}$$

Note $\mathtt{unbias}(j)$ is an unbiased estimator for $l$ being true negative (see Lemma 3.1). This step needs $\mathcal{O}(1)$.

### (iv) Performing Bayesian negative sampling

$$j = \mathop{\arg\min}\limits_{l \in\mathcal{M}_u}~ \mathtt{info}(l)\cdot [1-(1+\lambda)\mathtt{unbias}(l)]$$

If the sampler $h^*$ minimizes the conditional sampling risk $R(l|i)$, then the empirical sampling risk will be minimized (see Theorem 3.1). Thus the Bayesian optimal sampling rule is essentially sampling the instance with minimal conditional sampling risk.This step needs $\mathcal{O}(1)$.

More details about BNS(Bin Liu & Bang Wang, 2022) see our paper or poster at https://doi.org/10.48550/arXiv.2204.06520 .

## Distribution Analysis

The key is the class conditional density. On the basis of order relation analysis of negatives’ scores, we derive the class conditional density of true negatives and that of false negatives, and provide an affirmative answer from a Bayesian viewpoint to distinguish true negatives from false negatives.

Real distribution

Theoretical distribution

We exhibits the distribution morphology of false negatives and true negatives derived from the ordinal relation with different types of $f(x)$: Gaussian distribution $x \sim \mathcal{N}(\mu,\sigma)$ (symmetrical), student distribution $x \sim t(n)$ (symmetrical), and Gamma distribution $x\sim Ga(\alpha,\lambda)$ (asymmetrical) . As we can see, during the training process, the real distribution of true/false negatives gradually exhibit the same structure as theoretical distribution. Although the probability density function $f(x)$ changes during the training process, this separated structure is sufficient for us to classify the true negative instances from false negative instances.

## Dataset Description:

MovieLens100K; MovieLens1M; Yahoo!-R3.

Our Bayesian Negative Sampling algothrim does not depend on the datasets. Just split the dataset you are interested in into the training set and test set, replace the train.csv and test.csv files, you can run Bayesian negative sampling easily. You just need to make sure that you have chosen appropriate prior information for modeling prior probability $P_{tn}(\cdot)$ or $P_{fn}(\cdot)$.

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