Michael Collins, Nigel Duffy: Convolution kernels for natural language

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1. What is a generative model, what is a discriminative model and what is their main difference?
2. What are the “fairly strong independence assumptions” in PCFG? Come up with an example tree that can't be modelled by a PCFG.
3. Derive and explain the formula for h(T1)*h(T2) on page 3 at the bottom.
4. What is a convolution? Why are “convolution” kernels called like this?
5. Find an error in one of the formulae in the paper.

1. • Generative models use a two-step setup. They learn class-conditional (likelihood) <latex>P(x|y)</latex>, prior <latex>P(y)</latex> and use the Bayes rule to obtain the posterior.
     • they learn the joint distributions: marginalize P(y), condition P(y|x) = P(x,y) / P(x)
     • They learn more than is actually needed, but are not prone to partially missing input data.
     • They are able to “generate” fake inputs, but this feat is not used very often.
     • Examples: Naive Bayes, Mixtures of Gaussians, HMM, Bayesian Networks, Markov Random Fields
   • Discriminative models do everything in one-step – they learn the posterior <latex>P(y|x)</latex> as a function of some features of <latex> x</latex>.
     • They are simpler and can use many more features, but are prone to missing inputs.
     • Examples: SVM, Logistic Regression, Neural network, k-NN, Conditional Random Fields
2. Each CFG rule generates just one level of the derivation tree. Therefore, using “standard” nonterminals, it is not possible to generate e.g. this sentence:
   • (S (NP (PRP He)) (VP (VBD saw)(NP (PRP himself))))
     • It could be modelled with an augmentation of the nonterminal labels.
   • CFGs can't generate non-projective sentences.
     • But they can be modelled using traces.
3. The derivation is actually quite simple:
   1. <latex>h(T_a)\cdot h(T_b) = \sum_i h_i(T_a) \cdot h_i(T_b)</latex> – (definition of the dot product)
   2. <latex>= \sum_i \left(\sum_{n_a \in N_a} I_i(n_a)\right) \left(\sum_{n_b \in N_b} I_i(n_b)\right)</latex> (from the definition of <latex>I</latex> in the paragraph above the formula)
   3. <latex>= \sum_i\sum_{n_a \in N_a}\sum_{n_b \in N_b} I_i(n_b)\cdot I_i(n_a)</latex> (since <latex>(a+b)(c+d) = ac+ad+bc+bd</latex>)
   4. <latex>= \sum_{n_a \in N_a}\sum_{n_b \in N_b}\sum_i I_i(n_b)\cdot I_i(n_a)</latex> (change summation order)
   5. <latex>= \sum_{n_a \in N_a}\sum_{n_b \in N_b}C(n_a, n_b)</latex> (definition of <latex> C </latex>)
4. Convolution is defined like this: <latex>(f*g)_k = \sum_i f_i g_{k-i}</latex>, so it measures the presence of structures that complement each other. Here, we have a measure of structures that are similar. So it is something different. But the main idea is the same – we can combine smaller structures (kernels) into more complex ones.
5. There is a (tiny) error in the last formula of Section 3. You cannot actually multiply tree parses, so it should read: <latex>\bar{w}^{*} \cdot h(\mathbf{x}) = \dots</latex>

We discussed the answers to the questions most of the time. Other issues raised in the discussion were:

• Usability – the approach is only usable for reranking the output of some other parser.
• Scalability – they only use 800 sentences and 20 candidates per sentence for training. We believe that for large data (milions of examples) this will become too complex.
• Evaluation – it looks as if they used a non-standard evaluation metric to get “better” results. The standard here would be F1-score.