8 : Learning Fully Observed Undirected Graphical Models

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1 10-708: Probabilisti Graphial Models , Spring : Learning Fully Observed Undireted Graphial Models Leturer: Kayhan Batmanghelih Sribes: Chenghui Zhou, Cheng Ran (Harvey) Zhang When learning fully observed Undireted Graphial Models (UGM), there are two settings in whih we onsider. The first setting only onsiders learning parameters when the random variables are disrete and the seond setting involving learning parameters when random variables an be both disrete and ontinuous. Note that throughout this leture we assume that 1) the graphial model struture is given and 2) every variable appears in the training examples. 1 Setting I: MLE of UGM with Disrete RV In this setion we mostly onsider the ase in whih φ C (x ) = θ C,x, e.x. eah variables are disrete, where θ k is the probability of a disrete value being equal to a ertain value. Consider ategorial distribution: p(x = t) k θ k I(x = t) Tabular lique potentials would look like: φ (X n ) = Y φ (Y ) I[Y=X n ] The log likelihood under this distribution is: where Z(φ) = Y φ (Y ) L(φ) = n Derivative of the log-likelihood is: Y I[Y = X n ] log φ (Y ) N log Z(φ) (1) φ (Y ) L(θ) = n I[Y = X n 1 ] φ (Y ) N p(y ) φ (Y ) (2) Condition to sastisfy to get maximum likelihood parameter: p(x ) = ɛ(x ) 1 N N I[X = X n ] n=1 The model marignals must be equal to the observed marginals. 1

2 2 8 : Learning Fully Observed Undireted Graphial Models 1.1 MLE by Inspetion (Deomposable Model) Definiteion: Graph is deomposable if it an be subdivded into sets A, B, and S suh S separates A and B. Reipe for MLE by guessing Conditions: 1. Graphial model is deomposable 2. Potentials defined on maximal liques 3. Potentials are parameterized as φ (x ) = θ C,x 1. Set eah lique potential to its empirial marginal 2. Divide out every non-empty intersetion between liques exatly one. NOTE: If the graph is not deomposable or the potentials are not defined on the maximal liques, we annot equate empirial marginals to MLE of the lique potentials. 1.2 Iterative Proportional Fitting (IPF) FIxed point iteration is a general tool for solving a system of equations. It an also iteratively optimize an objetive funtion until onvergene. General proedure: Given Objetive funtion: J(θ) Compute derivative, set to zero: dj(θ) dθ i = f(θ) Rearrange the equation s.t. one of parameters appears on the LHS: 0 = f(θ) θ i = g(θ) Initialize the parameters (only appliable in the first step) For parameters of i from (1,..., K), update eah parameters and inrement t: θ t+1 i = g(θ t ) Repeat until onvergene Properties of IPF updates: Applies only when potentials are parameterized as φ C (x ) = θ C,x IPF iterates a set of fixed point equations: φ (t+1) (Y ) φ (t) (Y ) ɛ(y) p (t) (Y ) It s proven that IPF is a oordiante asent algorithm. At eah step the log likelihood inreases and eventually onverge to a global maximum.

3 8 : Learning Fully Observed Undireted Graphial Models 3 2 Setting II: MLE of UGM with Continuous RV 2.1 Feature-Based Models For the ase where potential funtions in an UGM may be ontinuous, we resort to a speifi type of representation where the liques an be represented as the inner produt of parameters θ with some feature feature funtion f(x ) whih we an design. This is the idea behind feature-based models and the potential funtions will take the following form: φ(x ) = exp(θ f(x )) It is important to note that feature-based models are useful even for graphs with disrete potential funtions. For large liques, general potentials are exponentially ostly for inferene and we must learn exponential number of parameters. In this ase, we an hoose to hange the graphial model to make the liques smaller; or we an keep the same graphial model but use features to ahieve a less general parametrization of the lique potentials with less parameters. An example of this is provided below. Example - Optial Charater Reognition (OCR) Given images of hand-writings, we want to reognize hand written words and make sure the written words is in the ditionary. Eah harater i of the word is a node in the graph and an be one of 26 letters. Assume we want to model all the dependenies between the nodes and have a fully onneted graph, a graph of just 3 nodes will have parameters. However, we often know that some partiular joint settings of the variables in a lique are quite likely or unlikely. For example, words ending with ed is likely a past tense while no words have jkx together. We an design features over the variables of the graph to redue the number of parameters in the joint distribution. Formally, a feature is a funtion whih is vauous over all joint settings exept a few partiular ones on whih it is high or low. For example, we an have f ing(1, 2, 3) whih is 1 if the string is ing and 0 otherwise. We an also design features over the input when the input is ontinuous. Eah feature funtion an be made into a miro-potential and we an multiply these miro-potentials together to get a lique potential. For example, a lique potential φ( 1, 2, 3 ) an be expressed as: K φ( 1, 2, 3 ) = e θingfing e θ edf ed... = exp( θ k f k ( 1, 2, 3 )) Note that instead of using parameters to express the potential, we are only using K parameters, where K is the number of features we use. Parameters θ i an be interpreted as the numerial strength of feature i. If we write out the Gibb s distribution using the lique potentials with features, we have: p(x) = 1 Z(θ) exp( i k=1 θ i f i (x i )) The joint distribution is in the form of exponential family, where h(x) = 1 and T (x) = f(x). 2.2 Generalized Iterative Saling We an modify the IPF algorithm to solve for lique potentials of feature-based models. The key ideas of Generalized Iterative Saling (GIS) is to:

4 4 8 : Learning Fully Observed Undireted Graphial Models 1. Define a funtion that is the lower-bound of the log-likelihood that we are trying to optimize 2. Observe that the bound is tight at urrent parameters 3. Compute MLE on the lower bound using fixed-point iteration in order to inrease log-likelihood. The algorithm is presented below: The lower bound is obtained by linearizing a log and applying Jensen-Shannon in the following form: Note that the idea of GIS is very similar to that of Expetation-Maximization (EM) Algorithm. We an ontrast IPF and GIS and notie that GIS looks like IPF in the log spae. If we take the log of the IPF update equation, we have the following: log(φ t+1 ) log(φ t ɛ ) + log( p t (y) ) Both IPF and GIS then have update equation in the form of: old parameter new parameter + log( emprial marginal ) 2.3 Gradient-Based Methods We know that the joint distribution of feature-based models is in the exponential family and we an also use gradient-based methods to optimize the likelihood funtion. Reipe for Gradient-Based Learning

5 8 : Learning Fully Observed Undireted Graphial Models 5 1. Write down the objetive funtion 2. Compute the partial derivative of the objetive 3. feed objetive funtion and derivative into blak box gradient-based optimizer 4. retrieve optimal parameters from blak box We an use any optimization algorithms suh as Newton s Methods, Conjugate Gradient, or Stohasti Gradient Desent. The pseudo-ode for Stohasti Gradient Desent is presented below: To derive the gradient of the likelihood funtion in a feature based model, we know that for one data point: log(p(x; θ)) = θ T f(x ) log(z(θ)) We know that θ log(p(x; θ)) = f(x ) Z((θ))/Z(θ) So for a single data point: Z(θ) = e A(θ) θ Z(θ) = A θ e A (θ) θ Z(θ) = E[f (x )]Z(θ) θ log(p(x; θ)) = f(x ) E[f (x )] and the gradient of the likelihood over all data points an be written as: θ L(D; θ) = 1 N f (x ) E[f (x )] N n=1 Computing E[f (x )] requires us to do inferene to ompute the marginals. But the above equation does provide us with a way to ompute the gradient and therefore we an use gradient-based optimizers to optimize the likelihood funtion. We an interpret optimizing a likelihood as optimizing a loss funtion. It s also a good idea to add regularizer to your loss funtion. In this ase, we would be omputing the regularized maximum likelihood. The regularization an be L1 or L2 regularization et. Note that SGD also has effets of regularization as well.