What precisely does word2vec study, and the way? Answering this query quantities to understanding illustration studying in a minimal but fascinating language modeling process. Even supposing word2vec is a well known precursor to fashionable language fashions, for a few years, researchers lacked a quantitative and predictive idea describing its studying course of. In our new paper, we lastly present such a idea. We show that there are sensible, sensible regimes wherein the educational downside reduces to unweighted least-squares matrix factorization. We clear up the gradient move dynamics in closed type; the ultimate realized representations are merely given by PCA.

Earlier than elaborating on this outcome, let’s inspire the issue. word2vec is a well known algorithm for studying dense vector representations of phrases. These embedding vectors are educated utilizing a contrastive algorithm; on the finish of coaching, the semantic relation between any two phrases is captured by the angle between the corresponding embeddings. In actual fact, the realized embeddings empirically exhibit putting linear construction of their geometry: linear subspaces within the latent area usually encode interpretable ideas corresponding to gender, verb tense, or dialect. This so-called linear illustration speculation has not too long ago garnered numerous consideration since LLMs exhibit this conduct as properly, enabling semantic inspection of inner representations and offering for novel mannequin steering strategies. In word2vec, it’s exactly these linear instructions that allow the realized embeddings to finish analogies (e.g., “man : lady :: king : queen”) by way of embedding vector addition.

Possibly this shouldn’t be too stunning: in any case, the word2vec algorithm merely iterates by a textual content corpus and trains a two-layer linear community to mannequin statistical regularities in pure language utilizing self-supervised gradient descent. On this framing, it’s clear that word2vec is a minimal neural language mannequin. Understanding word2vec is thus a prerequisite to understanding function studying in additional refined language modeling duties.

The Outcome

With this motivation in thoughts, let’s describe the primary outcome. Concretely, suppose we initialize all of the embedding vectors randomly and really near the origin, in order that they’re successfully zero-dimensional. Then (underneath some delicate approximations) the embeddings collectively study one “idea” (i.e., orthogonal linear subspace) at a time in a sequence of discrete studying steps.

It’s like when diving head-first into studying a brand new department of math. At first, all of the jargon is muddled — what’s the distinction between a operate and a useful? What a couple of linear operator vs. a matrix? Slowly, by publicity to new settings of curiosity, the phrases separate from one another within the thoughts and their true meanings develop into clearer.

As a consequence, every new realized linear idea successfully increments the rank of the embedding matrix, giving every phrase embedding more room to raised specific itself and its that means. Since these linear subspaces don’t rotate as soon as they’re realized, these are successfully the mannequin’s realized options. Our idea permits us to compute every of those includes a priori in closed type – they’re merely the eigenvectors of a specific goal matrix which is outlined solely when it comes to measurable corpus statistics and algorithmic hyperparameters.

What are the options?

The reply is remarkably simple: the latent options are merely the highest eigenvectors of the next matrix:

[M^{star}_{ij} = frac{P(i,j) – P(i)P(j)}{frac{1}{2}(P(i,j) + P(i)P(j))}]

the place $i$ and $j$ index the phrases within the vocabulary, $P(i,j)$ is the co-occurrence chance for phrases $i$ and $j$, and $P(i)$ is the unigram chance for phrase $i$ (i.e., the marginal of $P(i,j)$).

Establishing and diagonalizing this matrix from the Wikipedia statistics, one finds that the highest eigenvector selects phrases related to movie star biographies, the second eigenvector selects phrases related to authorities and municipal administration, the third is related to geographical and cartographical descriptors, and so forth.

The takeaway is that this: throughout coaching, word2vec finds a sequence of optimum low-rank approximations of $M^{star}$. It’s successfully equal to operating PCA on $M^{star}$.

The next plots illustrate this conduct.

On the left, the important thing empirical statement is that word2vec (plus our delicate approximations) learns in a sequence of primarily discrete steps. Every step increments the efficient rank of the embeddings, leading to a stepwise lower within the loss. On the precise, we present three time slices of the latent embedding area, demonstrating how the embeddings increase alongside a brand new orthogonal path at every studying step. Moreover, by inspecting the phrases that almost all strongly align with these singular instructions, we observe that every discrete “piece of information” corresponds to an interpretable topic-level idea. These studying dynamics are solvable in closed type, and we see a wonderful match between the idea and numerical experiment.

What are the delicate approximations? They’re: 1) quartic approximation of the target operate across the origin; 2) a specific constraint on the algorithmic hyperparameters; 3) small enough preliminary embedding weights; and 4) vanishingly small gradient descent steps. Fortunately, these circumstances are usually not too sturdy, and actually they’re fairly much like the setting described within the unique word2vec paper.

Importantly, not one of the approximations contain the info distribution! Certainly, an enormous power of the idea is that it makes no distributional assumptions. Because of this, the idea predicts precisely what options are realized when it comes to the corpus statistics and the algorithmic hyperparameters. That is significantly helpful, since fine-grained descriptions of studying dynamics within the distribution-agnostic setting are uncommon and laborious to acquire; to our information, that is the primary one for a sensible pure language process.

As for the approximations we do make, we empirically present that our theoretical outcome nonetheless offers a devoted description of the unique word2vec. As a rough indicator of the settlement between our approximate setting and true word2vec, we are able to examine the empirical scores on the usual analogy completion benchmark: word2vec achieves 68% accuracy, the approximate mannequin we research achieves 66%, and the usual classical different (often called PPMI) solely will get 51%. Take a look at our paper to see plots with detailed comparisons.

To reveal the usefulness of the outcome, we apply our idea to review the emergence of summary linear representations (similar to binary ideas corresponding to masculine/female or previous/future). We discover that over the course of studying, word2vec builds these linear representations in a sequence of noisy studying steps, and their geometry is well-described by a spiked random matrix mannequin. Early in coaching, semantic sign dominates; nevertheless, later in coaching, noise could start to dominate, inflicting a degradation of the mannequin’s potential to resolve the linear illustration. See our paper for extra particulars.

All in all, this outcome offers one of many first full closed-form theories of function studying in a minimal but related pure language process. On this sense, we consider our work is a vital step ahead within the broader challenge of acquiring sensible analytical options describing the efficiency of sensible machine studying algorithms.

Study extra about our work: Hyperlink to full paper

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This put up initially appeared on Dhruva Karkada’s weblog.