RL and the Game of Mathematics

I recently gave a lecture on Alphago in a deep learning class.  I had of course heard of the recent accomplishments — learning in a matter of days, or even hours, and entirely from self-play, to become superhuman in go, chess and shogi.  I had not, however, appreciated the magnitude of this advance until I read the papers (Alphgo Zero and AlphaZero). Alphago Zero is about 1400 ELO points above Alphago Lee — the program that defeated Lee Sedol. The ELO scale is the one used to rate chess players — 200 ELO points corresponds to 75% chance of victory by the stronger player.  Measuring a huge gap like 1400 points requires using a sequence of versions of the program of increasing strength.  Also, Alphago Zero runs on a single machine with four GPUs.  Alphago Lee runs on a supercomputer.  Also, the algorithm underlying Alphago Lee is new and simpler with no rollouts or hand designed features.  Also, the same algorithm does the same trick in chess and shogi — quickly learning to be far superior to any previous program for these games.  Also, the introduction of a fundamentally new algorithm seems likely to open up yet more progress in RL. The new algorithm can be viewed as tree-search bootstrapping  applied to policy gradient.

Ten years ago Jonathan Schaeffer at the University of Alberta proved that checkers is a draw.  His group computed and stored full drawing strategies for each player.  This was listed by Science magazine as one of the ten greatest breakthroughs of 2007. The game of chess is also believed to be a draw, although computing full drawing strategies is completely infeasible.   Empirically, as the level of play increases draws become more common.  Computers have been super-human in chess for the last 20 years.  The prevalence of draws at super-human levels causes the ELO scale to break down. (In practice go games are never drawn.) The strongest computer program, stockfish, was thought to be unbeatable — equivalent to perfect play.  AlphaZero defeated Stockfish 25/50 games playing from white (and lost none) and defeated Stockfish 3/50 games from black (and lost none).

DeepMind’s success in RL is based on perfect simulation.  Perfect simulation is possible in Atari games and in board games such as go and chess.  Perfect simulation allows large-scale on-policy learning.  Large-scale on-policy learning is not feasible in the “game” of human dialogue (the Turing test).  We don’t know how to accurately simulate human dialogue.  Mathematics, on the other hand, can be formulated as a game with rules defined by a formal foundational logic.  This immediately raises the question of whether the recent dramatic results in RL for games might soon lead to super-human open-domain mathematicians.

The foundations of mathematics has been largely ignored by AI researchers.  I have always felt that the foundations — the rules of “correct thought” — reflect an innate cognitive architecture.  But independent of such beliefs, it does seem possible to formulate inference rules defining the general notion of proof accepted by the mathematics community.  For this we have ZFC or type theory.  More on this later.  But in addition to rules defining moves, we need an objective— a reward function.  What is the objective of mathematics?

I believe that a meaningful quantitative objective for mathematics must involve an appropriate formal notion of a mathematical ontology — a set of concepts and statement made in terms of them. It is easy to list off  basic mathematical concepts — groups, rings, fields, vector spaces, Banach spaces, Hilbert spaces, manifolds, Lie algebras, and many more.  For some concepts, such as the finite simple groups or the compact two manifolds, complete classification is possible — it is possible to give a systematic enumeration of the instances of the concept up to isomorphism. Formulating an objective in terms of a concept ontology requires, of course, being clear (formal) about what we mean by a “concept”.  This brings us to the importance of type theory.

Since the 1920’s (or certainly the 1950s) the mathematics community has generally accepted Zermello-Fraenkel set theory with the axiom of choice (ZFC) as the formal foundation.  However, the ontology of mathematics is clearly organized around the apparent fact that the instances of a concept are only classified up to the notion of isomorphism associated with that concept. The classification of finite simple groups or compact two manifolds is done up to isomorphism. The fact that each concept comes with an associated notion of isomorphism has never been formalized in the framework of ZFC.  To formalize concepts and their associated notion of isomorphism we need type theory.

I have been working for years on a set-theoretic type theory — a type theory fully compatible with the ZFC foundations.  Alternatively there is constructive type theory and its extension to homotopy type theory (HoTT).  I will not discuss the pros and cons of different type theories here except to say that I expect mathematicians will be more accepting of a type theory fully compatible with the grammar of informal mathematical language as well as being compatible with the currently accepted content of mathematics.

Type theory gives a set of formal rules defining the “well-formed” concepts and statements.  I believe that a meaningful objective for mathematics can then be defined by an a-prior probability distribution over type-theoretically well-formed mathematical questions.  The well-formedness constraint ensures that each question is meaningful. We do not want to ask if 2 is a member of 3. The objective is to be able to answer the largest possible fraction of these well-formed questions.

Of course the questions under investigation in human mathematics evolve over time and this evolution is clearly governed by a social process.  But I claim that this is consistent with the existence of an objective notion of mathematical significance.  A socially determined distribution over questions can be importance-corrected (in the statistical sense) in judging objective significance. The use of a sampling distribution different from the prior, which is then importance corrected, seems likely to be useful in any case.

Of course I am not the only one to be considering deep learning for mathematical inference.  There have now been three meetings of the Conference on AI and Theorem Proving.  Many of the papers in this conference propose neural architectures for “premise selection” — the problem of selecting facts from a mathematical library that are likely to be useful for a given problem.  Christian Szegedy, a co-author of batch-normalization and a principal developer of Google’s inception network for image classification, has been attending this conference and working on deep learning for theorem proving [1,2,3].  I myself have attended the last two meetings and have found them useful in framing my own thinking on deep architectures for inference.

In summary, DeepMind’s dramatic recent success in RL raises the question of whether RL can produce a super-human player of the game of mathematics.  I hereby throw this out as a challenge to DeepMind and to the AI community generally.  A super-human open-domain mathematician would seem a major step toward AGI.





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The Role of Theory in Deep Learning

This blog post is inspired by the recent NIPS talk by Ali Rahimi and the response by Yann LeCun.  The issue is fundamentally the role of theory in deep learning.   I will start with some quotes from Rahimi’s talk.


Machine learning has become alchemy.

Alchemy worked [for many things].

But scientists had to dismantle two thousand years worth of alchemical theories.

I would like to live in a society whose systems are built on verifiable rigorous knowledge and not on alchemy.


Understanding (theoretical or otherwise) is a good thing.

[However, Rahimi’s statements are dangerous because] it’s exactly this kind of attitude that lead the ML community to abandon neural nets for over 10 years, despite ample empirical evidence that they worked very well in many situations.

I fundamentally agree with Yann that a drive for rigor can mislead a field.  Perhaps most dangerous is the drive to impress colleagues with one’s mathematical sophistication rather than to genuinely seek real progress.

But I would like to add my own spin to this debate.  I will start by again quoting Rahini:


[When a deep network doesn’t work] I think it is gradient descent’s fault.

Gradient descent is the cornerstone of deep learning. Gradient descent is a form of local search.  Here are some other examples of local search:

The evolution of the internal combustion engine from the 1790s through the twentieth century.

The evolution of semiconductor processes over the last fifty years of Moore’s law.

Biological evolution including the evolution of the human brain.

The evolution of mathematics from the ancient Greeks to the present.

The hours of training alphago(zero) takes to become the world’s strongest chess program through self play.

Local search is indeed mysterious.  But can we really expect a rigorous theory of local search that predicts or explains the evolution of the human brain or the historical evolution of mathematic knowledge?  Can we really expect to predict by some sort of second order analysis of gradient descent what mathematical theories will emerge in the next twenty years?  My position is that local search (gradient descent) is extremely powerful and fundamentally forever beyond any fully rigorous understanding.

Computing power has reached the level where gradient descent on a strong architecture on a strong GPU can only be understood as some form of very powerful general non-convex local search similar in nature to the above examples.  Yes, the failure of a particular neural network training run is a failure of gradient descent (local search).  But that observation provides very little insight or understanding.

A related issue is one’s position on the time frame for artificial general intelligence (AGI).  Will rigor help achieve AGI?  Perhaps even Rahini would find it implausible that a rigorous treatment of AGI is possible.  A common response by rigor-seekers is that AGI is too far away to talk about.  But I find it much more exciting to think we are close. I have written a blog post on the plausibility of near-term machine sentience.

I do believe that insight into architecture is possible and that such insight can fruitfully guide design.  LSTMs appeared in 1997 because of a “theoretical insight” about a way of overcoming vanishing gradients.  The understanding of batch normalization as a method of overcoming internal covariate shift is something I do feel that I understand at an intuitive level (I would be happy to explain it).  Intuitive non-rigorous understanding is the bread and butter of theoretical physics.

Fernando Pereira (who may have been quoting someone else) told me 20 years ago about the “explorers” and the “settlers”.  The explorers see the terrain first (without rigor) and the settlers clean it up (with rigor).  Consider calculus or Fourier analysis. But in the case of local search I don’t think the theorems (the settlers) will ever arrive.

Progress in general local search (AGI) will come, in my opinion, from finding the right models of computation — the right general purpose architectures — for defining the structure of “strong” local search spaces.  I have written a previous blog post on the search for general cognitive architectures.  Win or lose, I personally am going to continue to pursue AGI.

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Choice as a Natural Kind Term

This is a sequel to my previous post on determinism, free will and the existence of choice.  Here I want to consider the semantics of the word “choice” from the perspective of lexical semantics generally.  I will focus on the concept of a natural kind as formulated by Kripke, Putnam and Donnellan and which I encountered long ago in the introduction to the volume naming necessity and natural kinds.

The basic idea is that a word like “gold” or “lemon” is used meaningfully without any precise definition.  There is a substance that we call gold and the existence of gold was known, and gold was correctly referred to, long before the acceptance of the atomic theory.  Similarly one can meaningfully refer to “lemons” or “dogs” without knowing all the scientific facts and criteria that might be used in attempting to give a precise definition.  A term which is used meaningfully, and for practical purposes does have a well defined referent, but where the detailed nature of the referent is unknown, is called a natural kind term.

Of course we can scientifically investigate the nature of gold or lemons.  This does not mean that the referent can be changed.  The term gold had a well defined referent in roman times and it would be scientific folly to investigate the nature of gold by changing the meaning (or referent) of the term.

Now the issue here is whether the term “choice”, like the term “gold”, has a meaning by virtue of its colloquial usage.  To quote Wittgenstein, the meaning is the use.  It would be scientific folly to study the nature of “choice” by redefining the word to mean something else.  In particular, consider the claim that a deterministic computer does not have choice.  One should recognize that when we talk about choices we are referring to something, just as the word “gold” refers to something.  I claim that statements about the nature of choice can be true or false in the same way that statements about the nature of gold can be true or false.  Furthermore, I claim that the statement that a deterministic computer cannot have choice is scientifically false in the same way that the statement that gold is a compound (like salt) is scientifically false.  Does alphago make choices?

But if “choice”, like “gold”, is a natural kind term, what are choices really?  The notion of choice seems intimately related to the computational process of decision making.  Choices are the options considered in decision making.  The notion of choice, or option, seems central to decision theory and central to the understanding of any decision making process whether or not that process is truly stochastic or is classically deterministic.  The sentence “I decided to cancel the interview” seems to have a clear (if colloquial) meaning concerning some internal cognitive decision making process.  The statement that an option existed and a decision was made seems perfectly compatible with a deterministic decision making process (as in a chess program).

Furthermore, whatever choices are, it seems that a sentence of the form “x was an option” has truth conditions — it can be true in some situations and not in others.  The truth conditions may be complex and subtle, but the truth conditions, whatever they are, seem orthogonal to the issue of whether the decision making computation is deterministic.

It seems to me that “being free to choose” cannot be distinguished from “having options”.  The deterministic chess program does have options — the set of legal moves exists and defines the options.  We humans largely understand when we “have options” which we appropriately take to be synonymous with “being free to choose”.  These words refer to something real just as “gold” refers to something real.

The truth conditions of “x was an option” seem related to the truth conditions of counterfactuals such as “if I had chosen to do x then y would have happen”.  Such statements have clear meanings in the case of formal games. Such statements presumably also have truth conditions in normal human circumstances.

Choices are, by definition, the things people refer to when they talk about choices and options.  We can investigate what these things (choices) are, but we should avoid trying to redefine the referent of this natural kind term.  The meaning is the use.
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CTC and the EG Algotithm: Discrete Latent Choices without Reinforcement Learning

Section 20.9.1 of Goodfellow, Bengio and Courville is titled “Back-Propagating through Discrete Stochastic Operations”.  It discusses the REINFORCE algorithm which requires sampling sequences of discrete latent choices.  It is well known that sampling choices yields a high variance gradient estimate.  While such sampling seems necessary for reinforcement learning applications (such as games) there are other settings where one can train discrete latent choices using a closed form gradient and avoid high variance choice sampling.  This is done in the CTC algorithm (Connectionist Temporal Classification, Graves et al. 2006).  CTC is arguably the currently dominant approach to speech recognition.  It is mentioned in Goodfellow et al. in the section on speech recognition but not in the general context of training discrete latent choices.

This post is also a sequel to the previous post on variational auto-encoders and EM.  In CTC, as in various other models, we are interested in a marginal distribution over latent variables.  In this case we have training pairs (x, y) and are interested in maximizing the marginal distribution.

\Theta^* = \mathrm{argmax}_\Theta\;  E_{(x,y)}\;\ln P_\Theta(y|x)

= \mathrm{argmax}_\Theta \; E_{(x,y)}\;\ln\; \sum_z \;P_\Theta(y,z|x)

In CTC x is a speech signal, y is a (time-compressed) phonetic transcription, and z is a sequence of latent “emission decisions”.  This is described in more detail below.

The previous post noted that both VAEs and EM can be phrased in terms of a unified objective function involving an “encoder distribution” P_\Psi(z|x,y).  Here we have made everything conditional on an input x.

\Psi^*,\Theta^* = \mathrm{argmax}_{\Psi,\Theta}\;E_{(x,y)}\;{\cal L}(\Psi,\Theta,x,y)

{\cal L}(\Psi,\Theta,x,y) = E_{z \sim P_\Psi(z|x,y)}\; \ln P_\Theta(z,y|x) + H(P_\Psi(z|x,y))\;\;\;(1)

= \ln\;P_{\Theta}(y|x) - KL(P_\Psi(z|x,y),P_{\Theta}(z|x,y))\;\;\;(2)

The equivalence between (1) and (2) is derived in the previous post.  In EM one performs alternating optimization where (2) is used to optimize the encoder distribution P_\Psi(z|x,y) while holding the model distribution P_\Theta(y,z|x) fixed, and (1) is used to optimize model distribution P_\Theta(z,y) while holding the encoder distribution P_\Psi(z|x,y) fixed.  In EM each of these optimizations is done exactly in closed form yielding an exact alternating maximization algorithm.  In CTC the E step optimization is computed exactly in closed form but the M step is replaced by an exact closed form calculation of the gradient of (1) with respect to \Theta.  I will call this the EG algorithm (expected gradient).

Example: A Deep HMM Language Model: A deep language model is typically an auto-regressive RNN1.

Screen Shot 2017-11-01 at 10.12.54 AM.png

[Andrej Karpathy]

Here the model stochastically generates a word and then uses the generated word as input in computing the next hidden state.  In contrast to a classical finite-state HMM, the hidden state is determined by the observable word sequence.  In a classical HMM many different hidden state sequences can generate the same word sequence — the hidden states are latent.  We can introduce discrete latent hidden states H_0, \ldots, H_T such that the full hidden state is a pair (h_t, H_t) where h_t is an RNN state vector determined by observable words and H_t is a latent discrete state not determined by the observable sequence.  We allow the discrete hidden state transition probabilities to depend on the hidden vector of an RNN.  In equations we have

h_{t+1} = \mathrm{RNNcell}_\Theta(h_t, e(w_t))

P_\Theta(H_{t+1} \;|\;w_1,\ldots,\;w_t,\;H_1,\ldots,\;H_t) = P_\Theta(H_{t+1}\;|\;h_{t+1},e(H_t))

P_\Theta(w_{t+1} \;|w_1,\ldots,w_t,\;H_1,\ldots,\;H_t) = P_\Theta(w_{t+1}\;|\;h_{t+1},e(H_{t+1})).

Here \Theta is the system of network parameters and e(w) and e(H) are vector embeddings of the word w and discrete state H. This model allows P_\Theta(w_1,\ldots,w_T) to be computed by dynamic programming over the possible hidden state sequences using the forward-backward procedure.  The gradient of the probability can also be computed in closed form and we can do SGD without sampling the discrete hidden states.  For dynamic programming to work it is important that the state vector h_t is determined by h_0 and the observed words w_1,\ldots,w_{t-1} and is not influenced by the hidden state sequence H_1,\ldots,H_{t-1}.

The general case: The general EG algorithm alternates an E step, which is a exact optimization as in EM, and a G step in which we take a gradient step in \Theta defined by computing the gradient of (1):

\nabla_\Theta {\cal L}(\Psi,\Theta,y) = E_{(x,y)}\;E_{z \sim P_\Psi(z|x,y)}\; \nabla_\Theta\;\ln\;P_\Theta(y,z|x)\;\;\;\;\;(3)

By (2), and the fact that that the gradient of the KL divergence is zero when P_\Psi(z|y) = P_\Theta(z|y), for the G step we also have

\nabla_\Theta {\cal L}(\Psi, \Theta,y) = \nabla_\Theta \ln P_\Theta(y|x)\;\;\;(4)

Combining (3) and (4) we see that that the gradient defined by (3) in the G step equals the gradient of the top level objective.

We now consider the case where P_\Theta(z,y|x) is defined by a generative process where the pair (z,y) is generated stochastically by a series of discrete decisions.  Each decision is assumed to be a draw from a certain probability distribution. I will write Q \leadsto j for the decision (event) of drawing j from distribution Q .  Every decision made in the above HMM has the form

P_\Theta(H|h_{t+1},e(H^i)) \leadsto H^j

where H^i and H^j are particular discrete hidden states.  For a sequence of length T and for n discrete hidden states we have a total of Tn^2 possible choice events and these choice events form the edges of an HMM trellis.  Note that it is important that h_t does not depend on the patch through the the trellis.

But getting back to the general case let {\cal C}(z,y|x) be the choices made in generating (z,y) under the model P_\Theta(z,y|x).  We then have

\ln P_\Theta(z,y|x) = \sum_{(Q \leadsto  j) \in {\cal C}(z,y|x)}\;\ln P(j\;|\;Q).

We now have

\nabla_\Theta \ln P_\Theta (y|x) =  E_{z \sim P_\Theta(z|x,y)}\;\nabla_\Theta \;\sum_{(Q \leadsto j) \in {\cal C}(y,z|x)} \;\; \ln P(j\;|\;Q)\;\;\;\;(5)

Now let {\cal C}(y|x) = \bigcup_z \;{\cal C}(y,z|x).  In the HMM example {\cal C}(y|x) is the set of all edges in the HMM trellis while {\cal C}(z,y|x) is one particular path through the trellis.  Also let 1((Q\leadsto j) \in {\cal C}(z,y|x)) be the indicator function that the choice Q \leadsto j occurs when generating (z,y) from x. We can now rewrite (5) as

\nabla_\Theta \ln P_\Theta(y|x) = \sum_{(Q \leadsto j) \in{\cal C}(y|x)} \left(E_{z \sim P_\Theta(z|x,y)} \; 1((Q \rightarrow j)\in{\cal C}(y,z|x))\right)\; \nabla_\Theta\;\ln P_\Theta(j|Q)

The expectation E_z 1((Q \leadsto j) \in {\cal C}(y,z|x)) are assumed to be computable by dynamic programming.  In the HMM example these are the edge probabilities in the HMM lattice.

CTC: In CTC we are given a sequence x_1, \ldots, x_T (typically acoustic feature vectors sampled at 100 frames per second) and we must output a sequence of symbols (typically letters or phonetic symbols) y_1, \ldots, y_N with N \leq T and typically N << T.  More precisely, for network parameters \Theta the model defines P_\Theta(y_1,\ldots,y_N|x_1,\ldots,x_T).  However, in the model there is a latent sequence \hat{y}_1,\ldots,\hat{y}_T where each \hat{y}_T is either an output symbol or the special symbol \bot.  The sequence y_1,\ldots,y_N is derived from \hat{y}_1,\ldots,\hat{y}_T by removing the occurrences of \bot. The model can be written as

h_{t+1} = \mathrm{RNNcell}_\Theta(h_t, x_t)

P\Theta(\hat{y}_{t+1}|x_1,\ldots,x_t) = P_\Theta(\hat{y}_{t+1}|h_{t+1})

We can use the EG algorithm to compute the gradient of \ln P_\Theta(y_1,\ldots,y_N|x_1,\ldots,x_T) where EG sums over all possible sequences \hat{y}_1,\ldots,\hat{y}_T that reduce to y_1,\ldots,y_N.  I leave it to the reader to work out the details of the dynamic programming table. For the dynamic programming to work it is important that h_t does not depend on the latent sequence \hat{y}_1,\ldots,\hat{y}_T.


  1. A week after this post was originally published Gu et al. published a non-autoregressive language model.
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I recently realized the connection between the expectation maximization algorithm (EM) and variational autoencoders (VAE).  Both optimize the same objective function where VAE performs gradient descent based on a sampling estimate of the gradient while EM performs exact alternating maximization in models where this is possible.  It turns out that the alternating optimization view of EM is described in section 9.4 of Bishop’s 2006 text on machine learning and also in section 19.2 of the text by Goodfellow, Bengio and Courville.   I had overlooked this connection between EM and VAEs, and it seems important, so here is a blog post that helps me, at least, organize my thinking.

First some background.

The expectation maximization (EM) algorithm is a time-honored cornerstone of machine learning.  Its general formulation was given in a classic 1977 paper by Dempster, Laird and Rubin, although many special cases had been developed prior to that.  The algorithm is used for estimating unknown parameters of a probability distribution involving latent variables.  The algorithm is usually introduced by looking at the special case of modeling a collection of points by a mixture of Gaussians.  Here we are given points where the model assumes that each point came from an unlabeled component of the mixture (the component associated with the point is a latent variable). Given the points one must estimate a component weight, a mean and a covariance matrix for each component of the mixture.

Screen Shot 2017-10-02 at 10.06.41 AM.png

This is a non-convex optimization problem. EM iteratively re-estimates the parameters with a guarantee that for each iteration, unless one is already at a parameter setting with zero probability gradients (a stationary point), the probability of the points under the model improves.  There are numerous important special cases such as learning the parameters of a hidden Markov model or a probabilistic context free grammar.

Variational autoencoders (VAEs) (tutorial) were introduced in 2014 by Kingma and Welling in the context of deep learning.  Autoencoding is related to compression.  JPEG compresses an image into an encoded form that uses less memory but can then be decompressed to get the image back with some loss of resolution.

Screen Shot 2017-10-02 at 10.09.13 AM.png
x \hspace{1.0in}P_\Psi(z|x) \hspace{1.0in} z \hspace{1.0in} P_\Theta(z,x)\hspace{1.0in}x'

[Kevin Franz]

Information theoretically, compression is closely relate to distribution modeling. Shannon’s source coding theorem states that for any probability distribution, such as the distribution on “natural” images, one can (block) code the elements using a number of bits per element equal to the information-theoretic entropy of the distribution.  In the above figure the average number of bits in the compressed form in the middle should be equal to the entropy of the probability distribution over images.

VAEs, however, do not directly address the compression problem.  Instead they focus on fitting the parameters \Theta of the distribution P_\Theta(z,x) so as to maximize the marginal probability P_\Theta(x) of a given set x of samples (such as a set of points or a set of images). The encoder is just a tool to make this parameter fitting more efficient. The problem being solved by a VAE is the same as the problem being solved by EM — fitting the parameters of a probability distribution to given data where the model includes latent variables not specified in the data.

VAE = EM. Both EM and VAEs are formulated in terms of the following objective function on the two distributions P_\psi(z|y) (the encoder) and P_\Theta(y,z) (the model defining P(z) and P(y|z)).

\Psi^*,\Theta^* = \mathrm{argmax}_{\Psi,\Theta}\;\sum_y\;{\cal L}(\Psi,\Theta,y)

{\cal L}(\Psi,\Theta,y) = E_{z \sim P_\Psi(z|y)}\; \ln P_\Theta(z,y) + H(P_\Psi(z|y))\;\;\;(1)

= \ln\;P_{\Theta}(y) - KL(P_\Psi(z|y),P_{\Theta}(z|y))\;\;\;(2)

Here y is summed over training data. The equivalence of  (1) and (2) is important as they provide different outlooks on the objective.   The equivalence is implied by the following observation.

E_{z \sim P_\Psi(z|y)} \;\ln P_\Theta(z,y)
= E_{z \sim P_\Psi(z|y)} \;\ln P_\Theta(y)P_\Theta(z|y)
= E_{z \sim P_\Psi(z|y)}\;\ln P_\Theta(y) + \;E_{z \sim P_\Psi(z|y)}\;\ln P_\Theta(z|y)
= \ln P_\Theta(y) + \;E_{z \sim P_\Psi(z|y)}\;\ln P_\Theta(z|y)

EM can be defined as alternating optimization of \sum_y {\cal L}(\Psi,\Phi,y) where (2) supports the optimization of \Psi (the E step) and (1) supports the optimization of \Theta (the M step).  The EM algorithm applies to models where both the E step and the M step can be solved efficiently exactly and for such models each M step is guaranteed to improve \sum _y \ln P_\Theta(y).  For these models this alternating optimization is typically far superior to any form of gradient descent on \sum_y \ln P_\Theta(y).

VAEs are used in models where the alternating optimization cannot be done efficiently in closed form.  A VAE performs gradient descent on (1) where the gradient is estimated by sampling z from the encoder P_\Psi(z|y).  This typically involves a “reparameterization trick”.  One represents P_\Psi(z|y) by taking z = g_\Psi(y,\eta) where \eta is a random variable of a fixed distribution. We can then rewrite (1) as

{\cal L}(\Psi,\Theta,y) = E_{\eta}\; \ln P_\Theta(g_\Psi(y,\eta),y) + H(P_\Psi(g_\Psi(y,\eta)|y))\;\;\;(3)

It is common to take P(\eta)P_\Psi(z|y), P_\Theta(z) and P_\Theta (y|z) to all be high dimensional Gaussians each represented by a mean and a (diagonal) covariance . The noise variable \eta is taken to be a fixed zero mean isotropic Gaussian while P_\Psi(z|y), P_\Theta(z) and P_\Theta (y|z) have means and covariances computed by deep networks.  We can now calculate the gradient of (3) for any given sample of \eta.

It is not clear whether the term “variational auto encoder” should imply the use of Gaussians.  Goodfellow et al. seem to take the position that VAEs are defined by gradient descent on (1) where the gradient is estimated by sampling z from P_\Psi(z|y).  This does not necessarily involve Gaussians.  A version of HMM-VAEs has recently been formulated which includes discrete latent HMM states.  However the observed variable y is still continuous and Gaussians are still involved.  Presumably one can formulate VAEs where both y and z are structured discrete variables, such a language model with discrete latent states.  These VAEs would presumably not involve Gaussians.

Other approaches to optimizing {\cal L}(\Psi,\Theta,y) are possible. The speech recognition algorithm CTC uses an exact E step but a gradient M step and achieves gradient descent on deep networks with discrete latent variables but without latent variable sampling.  See the following blog post for a discussion of the CTC approach which we might call the EG (expected gradient) algorithm.

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Deep Meaning Beyond Thought Vectors

I ended my last post by saying that I might write a follow-up post on current work that seems to exhibit progress toward natural language understanding.   I am going to discuss a couple sampled papers but of course these are not the only promising papers.  I just want to give some examples to argue that significant progress is being made and that there is no obvious limit. I will start with the following quotation which has gained considerably currency lately.

“You can’t cram the meaning of a whole %&!$ing sentence into a single $&!*ing vector!”

Ray Mooney

This quotation seems to be becoming a legacy similar to that of Jelinek’s quotation that “every time I fire a linguistic our speech recognition error rate improves”. However, while the underlying phenomenon pointed to by Jelinek — the dominance of learning over design — has been controversial over the years, I suspect that there is already a large consensus that Mooney is correct — we need more than “thought vectors”.

Meaning as a sequence of vectors: Attention

The first step beyond representing a sentence as a single vector is to represent a sentence as a sequence of vectors. The attention mechanism of machine translation systems essentially does this. It takes the meaning of the input sentence to be the sequence of hidden state vectors of an RNN (LSTM or GRU).  As the translation is generated the attention mechanism extracts information from interior of the input sentence.  There is no clear understanding of what is being represented, but the attention mechanism is now central to high performance translation systems such as that recently introduced by Google.  Attention is old news so I will say no more.

Meaning as a sequence of vectors: Graph-LSTMs

Here I will mention two very recent papers.  The first is a paper that appeared at ACL this year by Nanyun Peng, Hoifung Poon, Chris Quirk, Kristina Toutanova, Wen-tau Yih, entitled “Cross-Sentence N-ary Relation Extraction with Graph LSTMs“.  This paper is on relation extraction  — extracting gene-mutation-drug triples from the literature on cancer mutations.  However, here I am focused on the representation used for the meaning of text.  As in translation, text is converted to a sequence of vectors with a vector for each position in the text.  However, this sequence is not computed by a normal LSTM or GRU.  Rather the links of a dependency parse are added as edges between the positions in the sentence resulting in a graph that includes both the dependency edges and the “next-word” edges as in the following figure from the paper.Screen Shot 2017-08-31 at 3.30.47 PM.png

The edges are divided into those that go left-to-right and those that go right-to-left as shown in the second part of the figure.  Each of these two edge sets forms a DAG (a directed acyclic graph).  Tree LSTMs immediately generalize to DAGS and I might prefer the term DAG-LSTM to the term Graph-LSTM used in the title.  They then run a DAG-LSTM from left to right over the left-to-right edges and a DAG-LSTM from right-to-left over the right-to-left edges and concatenate the two vectors at each position.  The dependency parse is provided by an external resource.  Importantly, different transition parameters are used for different edge types — the “next-word” edge parameters are different from the “subject-of” edge parameters which are different from the “object-of” edge prameters.

A startlingly similar but independent paper was posted on arXiv in March by Bhuwan Dhingra, Zhilin Yang, William W. Cohen and Ruslan Salakhutdinov, entitled “Linguistic Knowledge as Memory for Recurrent Neural Networks“.  This paper is on reading comprehension but here I will again focus on the representation of the meaning of text.  They also add edges between the positions in the text.  Rather than add the edges of a dependency parse they add coreference edges and edges for hyponym-hypernym relations as in the following figure from the paper.

Screen Shot 2017-08-31 at 3.45.46 PM.png

As in the previous paper, they then run a DAG-RNN (a DAG-GRU) from left-to-right and also right-to-left.  But they use every edge in both directions with different parameters for different types of edges and different parameters for the left-to-right and right-to-left directions of the same edge type.  Again the coreference edges and hypernym/hyponym edges are provided by an external resource.

Meaning as a sequence of vectors: Self-Attention

As a third pass I will consider self attention as developed in “A Structured Self-attentive Sentence Embedding” by Lin et al. (IBM Watson and the University of Montreal) and “Attention is all you need” by Vaswani et al. (Google Brain, Google Research and U. of Toronto).  Although only a few months old these papers are already well known.  Unlike the Graph-LSTM papers above, these papers learn graph structure on the text without the use of external resources.  They also do not use any form of RNN (LSTM or GRU) but rely entirely on learned graph structure. The graph structure is created through self-attention.  I will take the liberty of some simplification —I will ignore the residual skip connections and various other details.  We start with the sequence of input vectors.  For each position we apply three matrices to get the three different vectors — a key vector, a query vector, and a value vector.  For each position we take the query vector at that position inner product the key vector at every other position to get an attention weighting (or set of weighted edges) from that position to the other positions in the sequence (including itself).  We then then weight the value vectors by that weighting and pass the result through a  feed forward network to get a vector at that position at the next layer.  This is repeated for some number of layers.  The sentence representation is the sequence of vectors in the last layer.

The above description ignores the multi-headed feature described in both papers.  Rather than just compute one attention graph they compute compute several different attention graphs each of which is defined by different parameters.  Each of these attention graphs might be interpreted as a different type of edge, such as the agent semantic role relation between event verbs and their agents. But the interpretations of deep models is always difficult.  The following figure from Vaswani et al. shows the edges from the word “making” where different colors represent different edge types.  Although the sentence is shown twice, the edges are actually between the words of a single sequence at a single layer of the model.

Screen Shot 2017-08-31 at 6.13.14 PM.pngThe following figure shows one particular edge type which is possibly interpretable as a coreference relation.

Screen Shot 2017-08-31 at 6.21.40 PM.png

We might call these networks self-attention networks (SANs). Note that they are sans LSTMs and sans CNNs :).

Meaning as an embedded knowledge graph

I want to complain at this point that you can’t cram the meaning of a bleeping sentence into a bleeping sequence of vectors.  The graph structures on the positions in the sentence used in the above models should be exposed to the user of the semantic representation.  I would take the position that the meaning should be an embedded knowledge graph — a graph on embedded entity nodes and typed relations (edges) between them.  A node representing an event can be connected through edges to entities that fill the semantic roles of the event type (a neo-Davidsonian representation of events in a graph structure).

One paper in this direction is “Dynamic Entity Representations in Neural Language Models” by Yangfeng Ji, Chenhao Tan, Sebastian Martschat, Yejin Choi and Noah A. Smith (EMNLP 2017). This model jointly learns to identify mentions, coreference, and entity embeddings (a vector representing the object referred to).

Another related paper is “Neural Symbolic Machines: Learning Semantic Parsers on Freebase with Weak Supervision” by Chen Lian, Jonathan Berant, Quoc Le, Kenneth D. Forbus and Ni Lao which appeared at ACL this year.  This paper addresses the problem of natural language question answering using freebase.  But again I will focus on the representation of the input sentence (question in this case). They convert the input question to a kind of graph structure where each node in the graph denotes a set of freebase entities.  An initial set of nodes is constructed by an external linker that links mentions in the question, such as “city in the United states”, to a set of freebase entities. A sequence-to-sequence model is then used to introduce additional nodes each one of which is defined by a relation to previously introduced nodes. This “graph” can be viewed as a program with instructions of the form n_i = op(R, n_j) where n_i is a newly defined node, n_j is a previously defined node, R is a database relation such as “born-in” and op is one of “hop”, “argmax”, “argmin”, or “filter”.  For example

n_1 = US-city;  n_2 = hop(born-in, n_1);  n_3 = argmax(population-of, n_1)

defines n_1 as the set of US cities, n_2 as the set of people born in a US city and n_3 as the largest US city.  Answering the input question involves running the generated program on freebase.  The sequence to sequence model is trained from question-answer pairs without any gold semantic graph output sequences. They use some fancy tricks to get REINFORCE to work. But the point here is that semantic parsing of this type converts input text to a kind of graph structure with embedded nodes (embedded by the sequence-to-sequence model). Another important point is that constructing the semantics involves background knowledge — freebase in this case.  Language is highly referential and “meaning” must ultimately involve reference resolution — linking to a knowledge base.

What’s next?

A nice survey of semantic formalisms is given in “The State of the Art in Semantic Representation” by Omri Abend and Ari Rappoport which appeared at this ACL.  A fundamental issue here is the increasing dominance of learning over design.  I believe that in the near term it will not be possible to separate semantic formalisms from learning architectures.  (In the long term there is the foundations of mathematics …) If the target semantic formalism is a neo-Davidsonian embedded knowledge graph, then this formalism must be unified with some learning architecture. The learning should be done in the presence of background knowledge — both semantic and episodic long term memory.  Background knowledge could itself be an embedded knowledge graph.  The paper “node2vec: Scalable Feature Learning for Networks” by Grover and Leskovec gives a method of embedding a given (unembedded) knowledge graph. But it ultimately seems better to generate the embedding jointly with acquiring the graph.

I have long believed that the most promising cognitive (learning) architecture is bottom-up logic programming.  Bottom-up (forward chaining) logic programming has a long history in the form of production systems, datalog, and Jason Eisner’s Dyna programming language.  There are strong theoretical arguments for the centrality of bottom-up logic programming. I have heard rumors that there is currently a significant effort at DeepMind based on deep inductive logic programming (Deep ILP) for bottom-up programs and that a paper will appear in the next few months.  I have high hopes …

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The Plausibility of Near-Term Machine Sentience.

When should we expect “operational sentience” — the point where the most effective way to interact with a machine is to assume it is sentient — to assume that it understands what we tell it.  I want to make an argument that near-term machine sentience in this sense is plausible where near-term means, say, ten years.

Deep learning has already dramatically improved natural language processing. Translation, question answering, parsing, and linking have all improved. A fundamental question is whether the recent advances in translation and linking provide evidence that we are getting close to “understanding”. I do not want to argue that we are getting close,
but rather just that we don’t know and that near-term sentience is “plausible”.

My case is based on the idea that paraphrase, entailment, linking, and structuring may be all that is needed. Paraphrase is closely related to translation — saying the same thing a different way or in a different language.  There has been great progress here. Entailment is determining if one statement implies another.  Classical logic was developed as model of entailment. But natural language entailment is far to slippery to be modeled by any direct application of formal logic. I interpret the progress in deep learning applied to NLP question answering as evidence for progress in entailment. Entailment is also closely related to paraphrase (no paraphrase is precisely meaning preserving) and the progress in translation seems related to potential progress in entailment. “Linking” means tagging natural language phrases with the database entries that the phrase is referring to.  The database can be taken to be freebase or wikidata or any knowledge graph. This is related to the philosophical problem of “reference” – what is our language referring to. But simply tagging phrases with database entries, such as “Facebook” or “the US constitution“, does not seem philosophically mysterious. I am using the term “structuring” to mean knowledge base population (KBP) – populating a knowledge base from unstructured text. Extracting entities and relationships (building a database) from unstructured text does not seem philosophically mysterious. It seems plausible to me that paraphrase, entailment, linking, and KBP will all see very significant near-term advances based on deep learning. The notions of “database” and “inference rule” (as in datalog) presumably have to be blended with the distributed representations of deep learning and integrated into “linking” and “structuring”. This seems related to memory networks with large memories. But I will save that for another post.

The plausibility of near-term machine sentience is supported by the plausibility that
language understanding is essentially unrelated to, and independent of, perception and action other than inputing and outputting language itself. There is a lot of language data out there. I have written previous blog posts against the idea of “grounding” the semantics of natural language in non-linguistic perception and action or in the physical position of material in space.

Average level human natural language understanding may prove to be easier than, say, average level human vision or physical coordination. There has been evolutionary
pressure on vision and coordination much longer than there has been evolutionary pressure on NLP understanding. For me the main question is how close are we to effective paraphrase, entialment, linking and structuring.  NLP researchers are perhaps the best AI practitioners to comment this blog post. I believe that Gene Charniak, a pioneer of machine parsing, believes that machine NLP understanding is at least a hundred years off. But I am more interested in laying out concrete arguments, one way or the other, than in taking opinion polls. Deep learning may have the ability to automatically handle the hundreds of thousands of linguistic phenomenon that seem to exist in, say, English. People learn language. Is there some reasoned argument that this cannot work within a decade?

Maybe at some point I will write a longer post outlining what current work seems to me to be on the path to machine sentience.

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