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<time class="post-full-meta-date" datetime=" 5 March 2022"> 5 March 2022</time>
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<a href='/tag/cs224n/'>CS224N</a>
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<h1 class="post-full-title">cs224n - Lecture 2. Neural Classifiers</h1>
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<h3 id="review-main-idea-of-word2vec">Review: Main idea of word2vec</h3>
<ul>
<li>Start with random word vectors</li>
<li>Iterate through each word in the whole corpus</li>
<li>Try to predict surrounding words using word vectors: $P(o\mid c) = \frac{\exp(u_o^T v_c)}{\sum_{w \in V}\exp(u_w^T v_c)}$</li>
<li><strong>Learning</strong>: Update vectors so they can predict actual surrounding words better</li>
<li>Doing no more than this, this algorithm learns word vectors that capture well word similarity and meaningful directions in a wordspace.<br />
<img src="/assets/images/cs224n/lec2_0.png" alt="png" width="60%", height="80%" /></li>
<li>A “bag of words” model; doesn’t actually pay any attention to word order or position. The model makes the same predictions at each position; the probability estimate would be the same if it is next to the center word or a bit further away.</li>
<li>
<p>We want a model that gives a reasonably high probability estimate to <em>all</em> words that occur in the context(at all often)</p>
</li>
<li>Word2vec maximizes objective function by putting similar words nearby in high dimensional vector space</li>
</ul>
<h3 id="optimization-gradient-descent">Optimization: Gradient Descent</h3>
<ul>
<li>To learn good word vectors: minimize a cost function $J(\theta)$</li>
<li><strong>Gradient Descent</strong> is an algorithm to minimize $J(\theta)$ by changing $\theta$</li>
<li>idea: from current value of $\theta$, calculate gradient of $J(\theta)$, then take small step in the direction of <em>negative</em> gradient. Repreat.<br />
<img src="/assets/images/cs224n/lec2_1.png" alt="png" width="60%", height="80%" /></li>
<li>Algorithm:
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
<span class="n">theta_grad</span> <span class="o">=</span> <span class="n">evaluate_gradient</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">corpus</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">theta</span> <span class="o">-</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">theta_grad</span>
</code></pre></div> </div>
</li>
</ul>
<h3 id="stochastic-gradient-descent">Stochastic Gradient Descent</h3>
<ul>
<li><strong>Problem</strong>: $J(\theta)$ is a function of <strong>all</strong> windows in the corpus (often, billions!); so $\nabla_\theta J(\theta)$ is very expensive to compute</li>
<li>Solution: Stochastic gradient descent(SGD)
<ul>
<li>Repeatedly sample windows, and update after each one, or each small batch</li>
</ul>
</li>
<li>Algorithm:
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
<span class="n">window</span> <span class="o">=</span> <span class="n">sample_window</span><span class="p">(</span><span class="n">corpus</span><span class="p">)</span>
<span class="n">theta_grad</span> <span class="o">=</span> <span class="n">evaluate_gradient</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">window</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<span class="n">theta</span> <span class="o">=</span> <span class="n">theta</span> <span class="o">-</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">theta_grad</span>
</code></pre></div> </div>
</li>
<li>Stochastic gradients with word vectors (Aside)
<ul>
<li>iteratively take gradients at each such window for SGD</li>
<li>But in each window, we only have at most <em>2m + 1</em> words,<br />
so $\nabla_\theta J(\theta)$ is very sparse:<br />
\(\nabla_\theta J_t(\theta) = \begin{bmatrix} 0 \\ \vdots \\ \nabla_{v_{\text{like}}} \\ \vdots 0 \\ \nabla_{u_I} \\ \vdots \\ \nabla_{u_{\text{learning}}} \\ \vdots \end{bmatrix} \in \mathbb{R}^{2dV}\)</li>
<li>We might only update the word vectors that actually appear.</li>
<li>Solution: either you need sparse matrix update operations to only update certain <strong>rows</strong>(in most DL packages) of full embedding matrices <em>U</em> and <em>V</em>, or you need to keep around a hash for word vectors.<br />
<img src="/assets/images/cs224n/lec2_2.png" alt="png" width="40%", height="80%" /></li>
<li>If you have millions of word vectors and do distributed computing, it is important to not have to send gigantic updates around.</li>
</ul>
</li>
</ul>
<h3 id="2b-word2vec-algorithm-family-more-details">2b. Word2vec algorithm family: More details</h3>
<ul>
<li>Why two vectors? $\rightarrow$ Easier optimization. Average both at the end
<ul>
<li>But can implement the algorithm with just one vector per word, and it works slightly better, but it makes the algorithm much more complicated.</li>
</ul>
</li>
<li>Two model variants:
<ol>
<li>Skip-grams(SG)<br />
Predict context(“outside”) words (position independent) given center word</li>
<li>Continuous Bag of Words(CBOW)<br />
Predict center word from (bag of) context words<br />
<em>We presented: Skip-gram model</em></li>
</ol>
</li>
<li>Additional efficiency in training:
<ul>
<li>Negative sampling<br />
<em>So far: Focus on naive softmax(simpler, but expensive training method)</em></li>
</ul>
</li>
</ul>
<h3 id="the-skip-gram-model-with-negative-samplingsgns">The skip-gram model with negative sampling(SGNS)</h3>
<ul>
<li>The normalization term is computationally expensive, especially on the denominator of $P(o\mid c)$.</li>
<li>Main idea: train binary logistic regressions for a true pair (center word and a word in its context window) versus several noise pairs (the center word paired with a random word)</li>
<li>From paper: <em>“Distributed Representations of Words and Phrases and their Compositionality” (Mikolov et al. 2013)</em></li>
<li>Overall objective function(to maximize):<br />
\(J(\theta) = \frac{1}{T}\sum_{t=1}^T J_t(\theta)\)<br />
\(J_t(\theta) = \log\sigma(u_o^T u_c) + \sum_{i=1}^k \mathbb{E}_{j~P(w)}\left[ \log\sigma(-u_j^T v_c) \right]\)<br />
where the logistic/sigmoid function: $\sigma(x) = \frac{1}{1+ e^{-x}}$</li>
<li>We maximize the probability of two words co-occuring in first log and minimize probability of noise words:<br />
$J_{\text{neg-sample}}(u_o, v_c, U) = -\log \sigma(u_o^T v_c) - \sum_{k\in { \text{K sampled indicies} }} \log \sigma(-u_k^T v_c)$</li>
<li>We take <em>k</em> negative samples (using word probabilities)</li>
<li>Maximize probability that real outside word appears, minimize probability that random words appear around center word</li>
<li>Another trick: sample with $P(w) = U(w)^{3/4} / Z$, the unigram distribution $U(w)$ raised to the $3/4$ power (We provide this function in the starter code)</li>
<li>The power makes less frequent words be sampled more often</li>
</ul>
<h3 id="why-not-capture-co-occurrence-counts-directly">Why not capture co-occurrence counts directly?</h3>
<ul>
<li>Building a co-occurrence matrix <em>X</em>
<ul>
<li>2 options: windows vs. full document</li>
<li>Window: Similar to word2vec, use window around each word and captures some syntactic and semantic information<br />
<img src="/assets/images/cs224n/lec2_3.png" alt="png" width="80%", height="80%" /></li>
<li>Word-document co-occurrence matrix will give general topics (all sports terms will have similar entries) learning to “Latent Semantic Analysis”; in tasks like information retrieval</li>
</ul>
</li>
</ul>
<h3 id="co-occurrence-vectors">Co-occurrence vectors</h3>
<ul>
<li>Simple count co-occurrence vectors
<ul>
<li>Vectors increase in size with vocabulary</li>
<li>Very high dimensional: require a lot of storage (though sparse)</li>
<li>Subsequent classification models have sparsity issues $\rightarrow$ Models are less robust</li>
</ul>
</li>
<li>Low-dimensional vectors
<ul>
<li>idea: store “most” of the important information in a fixed, small number of directions: a dense vector</li>
<li>Usually 25-1000 directions, similar to word2vec</li>
<li>How to reduce the dimensionality?</li>
</ul>
</li>
</ul>
<h3 id="classic-method-dimensionality-reduction-on-x">Classic Method: Dimensionality Reduction on X</h3>
<ul>
<li>Singular Value Decomposition of co-occurrence matrix <em>X</em><br />
Factorizes <em>X</em> into $U\Sigma V^T$, where <em>U</em> and <em>V</em> are orthonormal<br />
<img src="/assets/images/cs224n/lec2_4.png" alt="png" width="60%", height="80%" /><br />
Corresponding to the columns without singular values in $\Sigma$, bottom rows in $V^T$ are ignored. The singular values inside the diagonal matrix $\Sigma$ are ordered from largest down to smallest. Retaining only <em>k</em> singular values, in order to generalize, the lower dimensional representation $\hat{X}$ is the best rank <em>k</em> approximation to <em>X</em>, in terms of least squares.</li>
</ul>
<h3 id="hacks-to-x-several-used-in-rohde-et-al-2005-in-coals">Hacks to X (several used in Rohde et al. 2005 in COALS)</h3>
<ul>
<li>
<p>Running an SVD on a raw count co-occurrence matrix works poorly; In the mathematical assumptions of SVD, we are expecting to have normally distributed errors. But there are exceedingly common words like “a”, “the”, and “and”, and there is a very large number of rare words.</p>
</li>
<li>Scaling the counts in the cells can help <strong>a lot</strong>
<ul>
<li>Problem: function words(<em>the, he, has</em>) are too frequent $\rightarrow$ syntax has too much impact. Some fixes:
<ul>
<li>log the frequencies</li>
<li>$min(X,t)$, with $t\approx 100$</li>
<li>ignore the function words</li>
</ul>
</li>
</ul>
</li>
<li>Ramped windows that count closer words more than further away words</li>
<li>Use Pearson correlations instead of counts, then set negative values to <em>0</em></li>
<li>Etc.</li>
<li>Result:<br />
Interesting semantic patterns emerge in the scaled vectors; something like a word vector analogies.<br />
<img src="/assets/images/cs224n/lec2_5.png" alt="png" width="60%", height="80%" /></li>
</ul>
<h3 id="towards-glove-count-based-vs-direct-prediction">Towards GloVe: Count based vs. direct prediction</h3>
<p><img src="/assets/images/cs224n/lec2_6.png" alt="png" width="80%", height="80%" /></p>
<h3 id="encoding-meaning-components-in-vector-differences">Encoding meaning components in vector differences</h3>
<ul>
<li><em>Pennington, Socher, and Manning, EMNLP 2014</em></li>
<li>What properties needed to make vector analogies work?</li>
<li>
<p>Crucial insight: Ratios of co-occurrence probabilities can encode meaning components<br />
<img src="/assets/images/cs224n/lec2_7.png" alt="png" width="80%", height="80%" /><br />
<img src="/assets/images/cs224n/lec2_8.png" alt="png" width="80%", height="80%" /></p>
</li>
<li>Q: How can we capture ratios of co-occurrence probabilities as linear meaning components in a word vector space?</li>
<li>A: Log-bilinear model: the dot product between two word vectors attempts to approximate the log of the probability of co-occurrence; \(w_i \cdot w_j = \log P(i|j)\)<br />
$\rightarrow$ with vector differences \(w_x \cdot (w_a - w_b) = \log \frac{P(x\mid a)}{P(x \mid b)}\)</li>
</ul>
<h3 id="combining-the-best-of-both-worlds-glove">Combining the best of both worlds: GloVe</h3>
<ul>
<li><em>Pennington, Socher, and Manning, EMNLP 2014</em></li>
<li>With \(w_i \cdot w_j = \log P(i|j)\),<br />
explicit loss function \(J = \sum_{i,j=1}^V f(X_{ij})(w_i^T \tilde{w}_j + b_i + \tilde{b}_j - \log X_{ij})^2\)<br />
to make the dot product to be similar to the log of the co-occurrence. To not have very common words dominate, capped the effect of high word counts using $f$ function. Optimize <em>J</em> directly on the co-occurrence count matrix.
<ul>
<li>Fast training</li>
<li>Scalable to hugh corpora</li>
<li>Good performance even with small corpus and small vectors</li>
</ul>
</li>
</ul>
<h3 id="how-to-evaluate-word-vectors">How to evaluate word vectors?</h3>
<ul>
<li>Related to general evaluation in NLP: intrinsic vs. extrinsic</li>
<li>Intrinsic:
<ul>
<li>Evaluation on a specific/intermediate subtask</li>
<li>Fast to compute</li>
<li>Helps to understand that system</li>
<li>Not clear if really helpful unless correlation to real task is established</li>
</ul>
</li>
<li>Extrinsic:
<ul>
<li>Evaluation on a real task</li>
<li>Can take a long time to compute accuracy</li>
<li>Unclear if the subsystem is the problem or its interaction or other subsystems</li>
<li>If replacing exactly one subsystem with another improves accuracy $\rightarrow$ Winning!</li>
</ul>
</li>
</ul>
<h3 id="intrinsic-word-vector-evaluation">Intrinsic word vector evaluation</h3>
<ul>
<li>Word Vector Analogies<br />
|a:b :: c:?| $\rightarrow$ $d = \text{argmax}_i \frac{(x_b -x_a +x_c)^T x_i}{\lVert x_b -x_a +x_c\rVert}$<br />
(e.g. man:woman :: king:?)</li>
<li>Evalute word vectors by how well their cosine distance after addition captures intuitive semantic and syntactic analogy questions</li>
<li>Discarding the input words from the search!</li>
<li>Problem: What if the information is there but not linear?</li>
</ul>
<h3 id="glove-visualizations">GloVe Visualizations</h3>
<p><img src="/assets/images/cs224n/lec2_9.png" alt="png" width="70%", height="80%" /></p>
<h3 id="analogy-evaluation-and-hyperparameters">Analogy evaluation and hyperparameters</h3>
<p><img src="/assets/images/cs224n/lec2_10.png" alt="png" width="70%", height="80%" /></p>
<p><img src="/assets/images/cs224n/lec2_11.png" alt="png" width="100%", height="100%" /></p>
<h3 id="another-intrinsic-word-vector-evaluation">Another intrinsic word vector evaluation</h3>
<ul>
<li>Word vector distances and their correlation with human judgements</li>
<li>Example dataset: WordSim353 <a href="http://www.cs.technion.ac.il/~gabr/resources/data/wordsim353/">http://www.cs.technion.ac.il/~gabr/resources/data/wordsim353/</a></li>
</ul>
<p><img src="/assets/images/cs224n/lec2_12.png" alt="png" width="50%", height="80%" /><br />
<img src="/assets/images/cs224n/lec2_13.png" alt="png" width="70%", height="80%" /></p>
<ul>
<li>Some ideas from Glove paper have been shown to improve skip-gram(SG) model also (e.g., average both vectors)</li>
</ul>
<h3 id="extrinsic-word-vector-evaluation">Extrinsic word vector evaluation</h3>
<ul>
<li>All subsequent NLP tasks</li>
<li>One example where good word vectors should help directly: <strong>named entity recognition</strong>: identifying references to a person, organization or location</li>
</ul>
<p><img src="/assets/images/cs224n/lec2_14.png" alt="png" width="60%", height="80%" /></p>
<h3 id="word-senses">Word senses</h3>
<ul>
<li>Most words have lots of meanings
<ul>
<li>Especially common words</li>
<li>Especially words that have existed for a long time</li>
</ul>
</li>
<li>Does one vector caputre all these meanings or do we have a mess?</li>
</ul>
<h4 id="linear-algebric-structure-of-word-senses-with-applications-to-polysemy-arora--ma--tacl-2018"><em>“Linear Algebric Structure of Word Senses, with Applications to Polysemy”, Arora, …, Ma, …, TACL 2018</em></h4>
<ul>
<li>Different senses of a word reside in a linear superposition(weighted sum) in standard word embeddings like word2vec</li>
<li>\(v_{\text{pike}} = \alpha_1 v_{\text{pike}_2} + \alpha_2 v_{\text{pike}_2} + \alpha_3 v_{\text{pike}_3}\)<br />
where $\alpha_1 = \frac{f_1}{f_1+f_2+f_3}$, etc., for frequency <em>f</em></li>
<li>Surprising result:<br />
Commonly, it is impossible to reconstruct the original components from their sum, but, because of ideas from <em>sparse coding</em> you can actually separate out the senses(providing they are relatively common)!<br />
<img src="/assets/images/cs224n/lec2_15.png" alt="png" width="80%", height="80%" /></li>
</ul>
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