Deep learning and neural nets are all the rage, today, and have displaced symbolic AI systems in most applications. It’s commonly believed that the two approaches have nothing to do with each other; that they’re just completely different, and that’s that. But this is false: there are some profound similarities; they are not only variants of one-another, but, in a deep way, they have commonalities that render them essentially identical. This blog post attempts to explain how. It introduces a much longer and rather mathematically dense writeup articulating these ideas more precisely.

The clearest starting point for seeing this seems to be natural language, where neural net methods have made great strides, but have not surpassed traditional (symbolic) linguistic theory. However, once this similarity is understood, it can be ported over to other domains, including deep-learning strongholds such as vision. To keep the discussion anchored, and to avoid confusing abstractions, what follows will focus entirely on linguistics; it is up to you, the reader, to imagine other, more general settings.

The starting point for probabilistic approaches (including deep learning) is the Bayesian network: a probability P(x_{1},x_{2},…,x_{n}) of observing n events. For language, the x_{k} are taken to be words, and n is the length of the sentence, so that P(x_{1},x_{2},…,x_{n}) is the “probability” of observing the sequence x_{1},x_{2},…,x_{n} of words. The technical problem with this viewpoint is the explosively large space: if one limits oneself to a vocabulary of 10 thousand words (and many people don’t) and sentences of 20 words or less, that’s (10^{4})^{20}=10^{80} =2^{270} probabilities, an absurdly large number even for Jupiter-sized computers. If n is the length of this blog post, then it really seems impossible. The key, of course, is to realize that almost all of these probabilities are effectively zero; the goal of machine learning is to find a format, a representation for grammar (and meaning) that effortlessly avoids that vast ocean of zero entries.

Traditional linguistics has done exactly that: when one has a theory of syntax, one has a formulation that clearly states which sentences should be considered to be grammatically valid, and which ones not. The trick is to provide a lexis (a lookup table), and some fairly small number of rules that define how words can be combined; i.e. arranged to the left and right of one-another. You look up a word in the lexis (the dictionary) to find a listing of what other words are allowed to surround it. Try every possible combination of rules until you find one that works, where all of the words can hook up to one-another. For the purposes here, the easiest and the best way to visualize this is with Link Grammar, a specific kind of dependency grammar. All theories of grammar will constrain allowed syntax; but Link Grammar is useful for this blog post because it explicitly identifies words to the left, and words to the right. Each lexical entry is like a template, a fill-in-the-blanks form, telling you exactly what other words are allowed to the left, and to the right, of the given word. This left-right sequence makes it directly comparable to what neural-net approaches, such as Word2Vec or SkipGram do.

What does Word2Vec do? Clearly, the 2^{270} probabilities in a twenty-word sentence is overwhelming; one obvious simplification is to look only at N-grams: that is, to only look at the closest neighboring words, working in a window that is N words wide. For N=5, this gives (10^{4})^{5}=10^{20} =2^{65} which is still huge, but is bearable. When scanning actual text, almost all of these combinations won’t be observed; this is just an upper bound. In practice, a table of 5-grams fit in latter-day computer RAM. The statistical model is to map each N-gram to a vector, use that vector to define a Boltzmann distribution (P=exp(v⋅w)/Z), and then use gradient ascent (hill-climbing) to adjust the vector coefficients, so as to maximize the probability P.

How are Word2Vec and Link Grammar similar? Well, I hope that the above description of Link Grammar already planted the idea that each lexical entry is a lot like an N-gram. Each lexical entry tells you which words can appear to the right, and which words to the left of a given word. Its a bit less constrained than an N-gram: there’s no particular distance limitation on the dependency links. It can also skip over words: a lexical entry is more like a skip-gram. Although there is no distance limitation, lexical entries still have a small-N-like behavior, not in window size, but in attachment complexity. Determiners have a valency of 1; nouns a valency of 2 or 3 (a link to a verb, to an adjective, to a determiner); verbs a valency of 2, 3 or 4 (subject, object, etc.). So lexical entries are like skip-grams: the window size is effectively unbounded, but the size of the context remains small (N is small).

Are there other similarities? Yes, but first, a detour.

What happened to the 2^{270} probabilities? A symbolic theory of grammar, such as Link Grammar, is saying that nearly all of these are zero; the only ones that are not zero are the ones that obey the rules of the grammar. Consider the verb “throw” (“Kevin threw the ball”). A symbolic theory of grammar effectively sates that the only non-vanishing probabilities are those of the form P(x_{1},x_{2},…,x_{k}=noun,…,x_{m}=throw,…,x_{p}=object,…,x_{n}) and that all the others must be zero. Now, since nouns make up maybe 1/2 of all words (the subject and object are both nouns), this constraint eliminates 10^{4}x10^{4}/(2×2)= 2^{24} possibilities (shrinking 2^{270} to 2^{270-24} = 2^{246}). Nothing to sneeze at, given that its just one fairly simple rule. But this is only one rule: there are others, which say things like “singular count nouns must be preceded by a determiner” (so, “the ball”). These constraints are multiplicative: if the determiner is missing, then the probability is exactly zero. There are only a handful of determiners, so another factor of 10^{4}=2^{13 }is vaporized. And so on. A relatively small lexis quickly collapses the set of possibilities. Can we make a back of the envelope estimate? A noun-constraint eliminates half the words (leaving 5K of 10K possibilities). A determiner constraint removes all but 10 possibilities. Many grammatical classes have only a few hundred members in them (“throw” is like “hit”, but is not like “smile”). So, realistically, each x_{k} will have from 10 to 1000 possibilities, maybe about 100 possibilities “on average” (averages are dicey to talk about, as these are Zipfian distributions). Thus there are only about 100^{20} =10^{40} =2^{130} grammatically valid sentences that are 20 words long, and these can be encoded fairly accurately with a few thousand lexical entries.

In essence, a symbolic theory of grammar, and more specifically, dependency grammars, accomplish the holy grail of Bayesian networks: factorizing the Bayesian network. The lexical rules state that there is a node in the network, for example, P(x_{1},x_{2},…,x_{k}=noun,…,x_{m}=throw,…,x_{p}=object,…,x_{n}) and that the entire sentences is a product of such nodes: The probability of “Kevin threw the ball” is the product P(x_{1}=Kevin, x_{2}=verb, …, x_{n}) P(x_{1},x_{2}, …, x_{k}=noun, …, x_{m}=throw, …, x_{p}=object, …, x_{n}) P(x_{1},x_{2}, …, x_{k}=the, …, x_{m}=noun, …, x_{n}) P(x_{1},x_{2}, …, x_{n}=ball) Stitch them all together, you’ve got a sentence, and its probability. (In Link Grammar, -log P is called the “cost”, and costs are additive, for parse ordering.) To be explicit: lexical entries are exactly the same thing as the factors of a factorized Bayesian network. What’s more, figuring out which of these factors come to play in analyzing a specific sentence is called “parsing”. One picks through the lookup table of possible network factors, and wires them up, so that there are no dangling endpoints. Lexical entries are look like subsets of a graph: a vertex, and some dangling edges hanging from the vertex. Pick out the right vertexes (one per word), wire them together so that there are no dangling unconnected edges, and viola! One has a graph: the graph is the Bayesian network. Linguists use a different name: they call it a dependency parse.

The Word2Vec/SkipGram model also factorizes, in much the same way! First, note that the above parse can be written as a product of factors of the form P(word|context), the product running over all of the words in the sentence. For a dependency grammar, the context expresses the dependency relations. The Word2Vec factorization is identical; the context is simpler. In it’s most naive form, its just a bag of the N nearest words, ignoring the word-order. But the word-order is ignored for practical reasons, not theoretical ones: it reduces the size of the computational problem; it speeds convergence. Smaller values of N mean that long-distance dependencies are hard to discover; the skipgram model partly overcomes this by keeping the bag small, while expanding the size of the window. If one uses the skip-gram model, with a large window size, and also keep track of the word-order, and restrict to low valencies, then one very nearly has a dependency grammar, in the style of Link Grammar. The only difference is that such a model does not force any explicit dependency constraints; rather, they are implicit, as the words must appear in the context. Compared to a normal dependency grammar, this might allow some words to be accidentally double-linked, when they should not have been. Dependency grammar constraints are sharper than merely asking for the correct context. To summarize: the notions of context are different, but there’s a clear path from one to the other, with several interesting midway points.

The next obvious difference between Link Grammar and Word2Vec/SkipGram are the mechanisms for obtaining the probabilities. But this is naive: in fact, they are much more similar than it first appears. In both systems, the starting point is (conceptually) a matrix, of dimension WxK, with W the size of the vocabulary, and K the size of the context. In general, K is much much larger than W; for Word2Vec, K could get as large as W^{N} for a window size of N, although, in practice, only a tiny fraction of that is observed. Both Link Grammar and Word2Vec perform approximate matrix factorization on this matrix. The way the approximations are done is different. In the case of Word2Vec, one picks some much smaller dimension M, typically around M=200, or maybe twice as large; this is the number of “neurons” in the middle layer. Then, all of W is projected down to this M-dimensional space (with a linear projection matrix). Separately, the K-dimensional context is also projected down. Given a word, let its projection be the (M-dimensional) vector u. Given a context, let it’s projection be the vector v. The probability of a word-in-context is given by a Boltzmann distribution, as exp(u⋅v)/Z where u⋅v is the dot product and Z is a scale factor (called the “partition function“). The basis elements in this M-dimensional space have no specific meaning; the grand-total vector space is rotationally invariant (only the dot product matters, and dot products are scalars).

The primary task for Word2Vec/SkipGram is to discover the two projection matrices. This can be done by gradient ascent (hillclimbing), looking to maximize the probability. The primary output of Word2Vec are the two projection matrices: one that is WxM-dimensional, the other that is MxK-dimensional. In general, neither of these matrices are sparse (that is, most entries are non-zero).

Link Grammar also performs a dimensional reduction, but not quite exactly by using projection matrices. Rather, a word can be assigned to several different word-categories (there are *de facto* about 2300 of these in the hand-built English dictionaries). Associated with each category is a list of dozens to thousands of “disjuncts” (dependency-grammar dependencies), which play the role analogous to “context”. However, there are far, far fewer disjuncts than there are contexts. This is because every (multi-word) context is associated with a handful of disjuncts, in such a way that each disjunct stands for hundreds to as many as millions of different contexts. Effectively, the lexis of Link Grammar is a sparse CxD-dimensional matrix, with C grammatical categories, and D disjuncts, and most entries in this CxD dimensional matrix being zero. (The upper bound on D is L^{V}, where L is the number of link types, and V is the maximum valency – about 5 or 6. In practice, D is in the tens of thousands.) The act of parsing selects a single entry from this matrix for each word in the sentence. The probability associated to that word is exp(-c) where c is the “cost”, the numerical value stored at this matrix entry.

Thus, both systems perform a rather sharp dimensional reduction, to obtain a much-lower dimensional intermediate form. Word2Vec is explicitly linear, Link Grammar is not exactly. However (and this is important, but very abstract) Link Grammar can be described by a (non-symmetric) monoidal category. This category is similar to that of the so-called “pregroup grammar”, and is described in a number of places (some predating both Link Grammar an pregroup grammar). The curious thing is that linear algebra is also described by a monoidal category. One might say that this “explains” why Word2Vec works well: it is using the same underlying structural framework (monoidal categories) as traditional symbolic linguistics. The precise details are too complex to sketch here, and must remain cryptic, for now, although they are open to those versed in category theory. The curious reader is encouraged to explore category-theoretic approaches to grammar, safe in the understanding that they provide a foundational understanding, no matter which detailed theory one works in. At the same time, the category-theoretic approach suggests how Word2Vec (or any other neural-net or vector-based approach to grammar) can be improved upon: it shows how syntax can be restored, with the result still looking like a funny/unusual kind of sparse neural-net. These are not conflicting approaches; they have far far more in common than meets the eye.

A few words about word-senses and semantics are in order. It has been generally observed that Word2Vec seems to encode “semantics” in some opaque way, in that it can distinguish different word-senses, based on context. The same is true for Link Grammar: when a word is used in a specific context, the result of parsing selects a single disjunct. That disjunct can be thought of as a hyper-fine grammatical category; but these are strongly correlated with meaning. Synonyms can be discovered in similar ways in both systems: if two different words all share a lot of the same disjuncts, they are effectively synonymous, and can be used interchangeably in sentences.

Similarly, given two different words in Word2Vec/SkipGram, if they both project down to approximately the same vector in the intermediate layer, they can be considered to be synonymous. This illustrates yet another way that Link Grammar and Word2Vec/SkipGram are similar: the list of all possible disjuncts associated with a word is also a vector, and, if two words have almost co-linear disjunct vectors, they are effectively synonymous. That is, disjunct-vectors behave almost exactly like neuron intermediate-layer vectors. They encode similar kinds of information into a vector format.

This is also where we have the largest, and the most important difference between neural net approaches, and Link Grammar. In the neural net approach, the intermediate neuron layer is a black box, completely opaque and un-analyzable, just some meaningless collection of floating-point numbers. In Link Grammar, the disjunct vector is clear, overt, and understandable: you can see exactly what it is encoding, because each disjunct tells you exactly the syntactic relationship between a word, and its neighbors. This is the great power of symbolic approaches to natural-language: they are human-auditable, human understandable in a way that neural nets are not. (Currently; I think that what this essay describes is an effective sketch for a technique for prying open the lid of the black box of neural nets. But that’s for a different day.)

The arguments presented above are worked out in greater detail, with all the mathematical trimmings, in this linked PDF. The commonality, by means of category theory, is explored in a different paper, on sheaves.

Many thanks and appreciation to Hanson Robotics for providing the time to think and write about such things. (Much much earlier and kludgier versions of these ideas have been seen in public, in older robot demos.)

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Symbolic and Neural Nets: Two Sides of the Same Coin