「word2vecは実際に何を学習するのか？」

What exactly does word2vec

Learning dynamics of word2vec. When trained from small initialization, word2vec learns in discrete, sequential steps. Left: rank-incrementing learning steps in the weight matrix, each decreasing the loss. Right: three time slices of the latent embedding space showing how embedding vectors expand into subspaces of increasing dimension at each learning step, continuing until model capacity is saturated.

Before elaborating on this result, let’s motivate the problem. word2vec

Maybe this shouldn’t be too surprising: after all, the word2vec

With this motivation in mind, let’s describe the main result. Concretely, suppose we initialize all the embedding vectors randomly and very close to the origin, so that they’re effectively zero-dimensional. Then (under some mild approximations) the embeddings collectively learn one “concept” (i.e., orthogonal linear subspace) at a time in a sequence of discrete learning steps.

It’s like when diving head-first into learning a new branch of math. At first, all the jargon is muddled — what’s the difference between a function and a functional? What about a linear operator vs. a matrix? Slowly, through exposure to new settings of interest, the words separate from each other in the mind and their true meanings become clearer.

As a consequence, each new realized linear concept effectively increments the rank of the embedding matrix, giving each word embedding more space to better express itself and its meaning. Since these linear subspaces do not rotate once they’re learned, these are effectively the model’s learned features. Our theory allows us to compute each of these features a priori in closed form – they are simply the eigenvectors of a particular target matrix which is defined solely in terms of measurable corpus statistics and algorithmic hyperparameters.

What are the features?

The answer is remarkably straightforward: the latent features are simply the top eigenvectors of the following matrix:

where $i$ and $j$ index the words in the vocabulary, $P(i,j)$ is the co-occurrence probability for words $i$ and $j$, and $P(i)$ is the unigram probability for word $i$ (i.e., the marginal of $P(i,j)$).

Constructing and diagonalizing this matrix from the Wikipedia statistics, one finds that the top eigenvector selects words associated with celebrity biographies, the second eigenvector selects words associated with government and municipal administration, the third is associated with geographical and cartographical descriptors, and so on.

The takeaway is this: during training, word2vec

The following plots illustrate this behavior.

Learning dynamics comparison showing discrete, sequential learning steps.

On the left, the key empirical observation is that word2vec

What are the mild approximations? They are: 1) quartic approximation of the objective function around the origin; 2) a particular constraint on the algorithmic hyperparameters; 3) sufficiently small initial embedding weights; and 4) vanishingly small gradient descent steps. Thankfully, these conditions are not too strong, and in fact they’re quite similar to the setting described in the original word2vec

Importantly, none of the approximations involve the data distribution! Indeed, a huge strength of the theory is that it makes no distributional assumptions. As a result, the theory predicts exactly what features are learned in terms of the corpus statistics and the algorithmic hyperparameters. This is particularly useful, since fine-grained descriptions of learning dynamics in the distribution-agnostic setting are rare and hard to obtain; to our knowledge, this is the first one for a practical natural language task.

As for the approximations we do make, we empirically show that our theoretical result still provides a faithful description of the original word2vec

To demonstrate the usefulness of the result, we apply our theory to study the emergence of abstract linear representations (corresponding to binary concepts such as masculine/feminine or past/future). We find that over the course of learning, word2vec

All in all, this result gives one of the first complete closed-form theories of feature learning in a minimal yet relevant natural language task. In this sense, we believe our work is an important step forward in the broader project of obtaining realistic analytical solutions describing the performance of practical machine learning algorithms.

Learn more about our work: Link to full paper

This post originally appeared on Dhruva Karkada’s blog.