Notes on Noise Contrastive Estimation (NCE)
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1 Notes on Noise Contrastive Estimation NCE) David Meyer March 0, 207 Introduction In this note we follow the notation used in [2]. Suppose X x, x 2,, x Td ) is a sample of a random vector x R n, where each x i is drawn unknown probability density function p d. Now, one way to describe the properties of the observed data X is to describe its properties relative to the properties of some reference data Y. The basic idea behind Noise Contrastive Estimation NCE) is to draw the reference noise) sample Y y, y 2,, y Tn ) from a known pdf p n and then estimate the ratio p d /p n, which will in turn give us an estimate of p d. 2 Definitions Let U X Y {u, u 2,, u Td +T n }, where is the number of data samples and T n is the number of samples from a noise distribution. We associate with each datapoint u t a class label C t such that C t { if ut X 0 if u t Y Since p d is unknown, we model p. C ) p m.; θ). Note that we re making an assumption here that there exists a θ such that p d.) p m.; θ ). That is, we assume that the empirical distribution p d.) is a member of the parameterized family p m.; θ). Given these definitions, the likelihoods are Some authors characterize the problem as drawing sample from the empirical distribution ) and k samples from the reference noise) distribution T n k), where the total number of samples per turn is k +. See e.g.,
2 pu C ) p m u; θ) # data ) pu C 0) p n u) # noise 2) The priors are The probability of any given u, P u), is thus P C ) + T n 3) P C 0) T n + T n 4) Remembering Bayes Rule: P u) P C ) pu C )) + P C 0) pu C 0)) 5) T ) d T ) n p m u; θ) + p n u) 6) + T n + T n P Θ X ) P X Θ)P Θ) P X ) # posterior likelihood prior evidence we get the posterior probabilities P u C ; θ) P C ) P C u; θ) P u) p m u; θ) ) Td +T n p m u; θ) Td +T n p m u; θ) p m u; θ) p m u; θ) + Tn )p n u) p m u; θ) P C u; θ) p m u; θ) + vp n u) vp n u) P C 0 u; θ) p m u; θ) + vp n u) +T n + Tn +T n p n u) p m u; θ) ) + Tn +T n p n u) 7) ) 8) )) # v T n P C 0) P C ) +T n 9) 0) ) # same analysis 2) 2
3 Now we have p m u; θ) P C u; θ) 3) p m u; θ) + vp n u) p m u; θ) p mu;θ) 4) p m u; θ) + vp n u)) p mu;θ) + v pnu) 5) p mu;θ) + v p ) nu) 6) p m u; θ) This is the first time we see an estimate of ratio we are looking for, namely define Gu; θ) Gu; θ) ln p mu; θ) p n u) pnu) p mu;θ). Now, 7) ln p m u; θ) ln p n u) 8) Gu; θ) is called the log-ratio between p m u; θ) and p n u). If we let P C u; θ) hu; θ) then hu; θ) can be written as hu; θ) r v Gu; θ)) 9) where r v u) + v exp u) that is, r v.) is the sigmoid/logistic function parameterized by v. 20) Note that the class labels C t are assumed to be Bernoulli and iid. The conditional loglikelihood is given by lθ) +T n C t ln P C t u t ; θ) + C t ) ln P C 0 u t ; θ) 2) ln [ hx t ; θ) ] + T n ln [ hy t ; θ) ] 22) Of the many interesting things here: optimizing lθ) with respect to θ leads to an estimate G.; ˆθ) of the log-ratio lnp d /p n ). Said another way, an approximate description of X 3
4 relative to Y can be obtained by optimizing Equation 22. cross-entropy function. Note also that lθ) is the This result shows that density estimation, an unsupervised learning task, can be performed by logistic regression which is supervised learning). 3 The NCE Estimator Recall that the normalized pdf p m.; ˆθ) satisfies the following constraints: p m u; θ)du # normalization constraint 23) p m.; θ) 0 # positivity constraint 24) Note that if the constraints for the model p m.; θ) hold for all θ and not just ˆθ) then the model is said to be normalized and Maximum Likelihood can be used to estimate θ. If only the positivity constraint and not the normalization constraint) is satisfied then we say the model is unnormalized. The main assumption here is that there exists at least one set of parameters, θ, for which the unnormalized model integrations to one 2. That is, p d.) p m.; θ ). The unnormalized model is usually denoted by p 0 m.; α). 3 Now, define the partition function Zα) as Zα) p 0 mu; α)du 25) Zα) can be used to convert an unnormalized model p 0 m.; α) into a normalized one: p 0 m.; α)/zα) which integrates to one for every value of α. Examples of distributions that are typically specified by unnormalized models include Gibbs distributions and Markov networks. The problem, however, is that the mapping α Zα) is defined by an integral Equation 25), so unless p 0 m.; α) has some convenient form, the integral usually cannot be computed analytically and thus Zα) is not available in closed form). For low-dimensional problems numerical integration can provide good accuracy, but for high-dimensional problems computation of Zα) becomes intractable cf the curse of dimensionality). 2 Another way to think about this: as mentioned above, p d.) comes from the family of parameterized distributions, p mu; θ). 3 p 0 m.; α), the unnormalized model, is sometimes written φξ; θ) []; the usual lack of standardized notation in machine learning... 4
5 It is worth noting that unnormalized models are widely used, with examples including models of images Markov Random Fields), models of text neural probabilistic language models), various models in physics Ising models), and more. The advantages of using unnormalized models in that they are often easier to specify than normalized models. The disadvantage is that the likelihood function is generally intractable. Noise Contrastive Estimation NCE) is an estimation method for unnormalized models. The basic idea here is to consider Z, or equivalently c ln /Z as a parameter to the model rather than a partition function). In particular, NCE considers p 0 m.; α) to include a new normalizing parameter c and estimates ln p m ;.; θ) ln p 0 m.; α) + c 26) where the parameter θ α, c). In particular, the estimate ˆθ ˆα, ĉ) then causes the unnormalized model p 0 m.; α)) to match the shape of p d and ĉ provides the appropriate scaling so that the normalization constraint Equation 24) holds. Observe that after training, ĉ provides an estimate for ln /Zα). Of course, if the model is normalized in the first place c is unnecessary. Given these definitions, we can define the estimator ˆθ T as the value of θ which maximizes { J T θ) Td ln [ hx t ; θ) ] + T n ln [ hy t ; θ) ]} 27) where the nonlinearity h.; θ) is defined in Equation 9. Note that the objection function J T is the log-likelihood Equation 22) divided by. J T can be rewritten as J T θ) ln [ hx t ; θ) ] + v T n T n ln [ hy t ; θ) ] 28) Interestingly, note that h.; θ) 0, ) and h.; θ) { 0 lim G.; θ) lim G.; θ) 5
6 As a result, the optimal paratement ˆθ T makes Gu T ; ˆθ T ) as large as possible for u T X and as small as possible for u T Y. Said another way, we determine the parameters θ such that P C ; u; θ) is large for most x i and small for most y i. Amazingly, this means that in this case logistic regression has learned to discriminate classify) between the data and noise sets. That is, what we end up with is unsupervised learning by supervised learning, where successful classification is equivalent to learning the differences between the data and the noise. Here we used nonlinear logistic regression for classification, but other classifiers are possible. 6
7 References [] Noise-Contrastive Estimation and its Generalizations. [2] Aapo Hyvarinen Michael U. Gutmann. Noise-contrastive estimation of unnormalized statistical models, with applications to natural image statistics. Journal of Machine Learning Research, 2:307 36,
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