Architecture, Bayes, Computation

Sometimes it seems as though each new step towards AI, rather than producing something which everyone agrees is real intelligence, merely reveals what real intelligence is not.

– Douglas Hofstadter, Gödel, Escher, Bach

In the first week of 9.13 The Human Brain, I was introduced to the now-ubiquitous concept of Marr’s three-levels approach to understanding the human mind. The top level is computational, expressing what outputs are to be inferred or computed from what inputs, e.g. recognizing the edges of an object; the intermediate level is algorithmic, describing conceptual routines that can be carried out to produce the desired results, e.g. the mathematics of a convolutional edge-detection filter; and the bottom level is implementation, showing physical hardware that can actually carry out the process, e.g. a group of neurons connected in the right way with the right activation parameters. The rest of the class focused mostly on the intermediate level and the tools we possess to study it, but recently I’ve been reading and thinking about the bottom level of hardware.

The substrate of human intelligence is made of neurons. And so far, the substrate on which we are trying to build computer intelligence is made mostly of transistors. As von Neumann pointed out over 60 years ago in The Computer and the Brain, the two are pretty different: neurons are inherently noisy and imprecise, being biological systems, and they work with a continuous range of potentials rather than the discrete ones and zeroes of binary logic, and communication between them is much slower than electrical impulses in a wire. Zooming out a little, the physical architecture of the brain involves billions of neurons interconnected in every which way with only a few centralized units for special functions, like the thalamus, which is contrasted with modern chip architecture with organized levels of memory, well-defined pipelined processing units, and a few communication buses between all the components. (Computer architecture also dates back to von Neumann (1953), interestingly.) Perhaps it is not surprising, then, that to create human-like intelligence, our research is in the direction of simulating human-like neural circuitry in digital hardware with neural networks, which end up consuming quite a lot more power than their native brain counterparts.

This begs the question, why not use electronic hardware that is more like neurons to begin with? Enter neuromorphic computer architecture. This paper (Neftci et al. 2013) is an example of how we would ‘program’ such a computer to perform a task like attending to a visual object on a screen and detecting its motion (quite a different paradigm of programming!), and more references to the structure of neuromorphic hardware can be found in its bibliography. I don’t find it hard to believe that a good set of abstractions on top of such hardware, whether engineered for our convenience or adopted from discoveries in the operation of the brain, can make it much easier to efficiently emulate processes of perception in electronics. I don’t know enough about cognitive processes outside of perception to say whether the same holds generally.

But there is mounting evidence that a deeper phenomenon is behind the intelligent behavior of a mess of neurons: Bayes’ Theorem (an excellent tutorial suitable for any level of math). It seems that the brain essentially holds a fairly detailed hypothesis of what the world around it is like, and all sensory input is counted as evidence in incrementally updating this hypothesis, and what a person perceives is heavily colored by the hypothesis/expectation of the surroundings unless sensory input is startling enough that the hypothesis breaks down. For example, your skin is (probably) feeling the weight of your clothes constantly, which your brain ignores under the hypothesis that it is generally what clothes feel like. If you shift position, the sensory input changes, but it still fits the hypothesis very well; you probably don’t notice that anything changed at all, you still have the same percept of wearing clothes. However, if a bug drops onto your back, you are suddenly aware of the weight and crawling movement, because that is definitely not how clothes typically feel—the hypothesis that there is nothing else on your body except clothes is abandoned. This exchange between mental models and sensory input, driven by Bayesian updates of the likelihood of various events and hypotheses, is conjectured to be a large component of the workings of the mind.

Explicitly modelling the real world in a Bayesian way is hard. Consider this example from a talk I recently attended (which, incidentally, led to me writing this post). The speaker (Abby Liu) built a map tool that starts out in a view of, say, the entire United States, and the user wants to end up in a view of, say, Austin. The traditional way to do this is that the user does a sequence of zooming and panning actions that the map executes. The speaker asked, what if the map actively tried to figure out where the user wanted to go? So the map starts with some prior probability distribution about which cities the user is most likely to want to go to, say, proportional to population, and instead of the map responding to the user’s inputs, we model it as the user responding to the map’s questions: “If I (the map) zoom in to the Northeast, what do you (the user) do?” – “I pan west.” – “Ah, so I have inferred that you do not want to look at a Northeast city.” And so on until the city is reached, the map keeps updating its probability distribution for where the user wants to go, and asks questions (by moving the map) that maximize the information gain at each step. One of the major challenges the speaker mentioned was that this is currently only practical for a small, discrete range of user actions (say, panning in one of eight compass directions) and a small, discrete hypothesis space (say, only major cities) because doing Bayesian updates for more general cases is mathematically tricky and computationally expensive. Imagine maintaining a probability distribution over every block in the country, and updating for every tiny motion of the user’s cursor…

Again, this begs the question, why not use hardware that works with probability distributions and Bayesian updates to begin with, instead of trying to emulate that with finicky binary bits and logic gates? This time my ‘original idea’ is not that far behind the research curve! Here are some papers describing various approaches to Bayesian hardware:

  • Bayes with memristors (Serb et al. 2017). A memristor is the elusive (until this millenium) ‘missing element’ of electronics, the others being resistors, capacitors, and inductors, postulated in 1971 by Leon Chua (not von Neumann this time, honestly that would have been too much). As far as I understand (which is not very far at all), a memristor’s resistance depends on how much current has passed through it, so it can be ‘set’ to ‘store’ a desired resistance than can then be ‘recalled’ by applying small voltages across it and measuring current flow. Serb et al.’s theoretical hardware does a sort of matrix multiplication between a vector of priors and a matrix of conditional probabilities, using memristors as the multiplicative elements, and you should read the (short and mostly simple) paper to understand how this works.
  • Bayes with stochastic computing (Friedman et al. 2016), and a more implementation-heavy paper (Thakur et al. 2016) with a different approach that I didn’t really understand. Stochastic computing is a computing paradigm in sharp contrast to standard digital electronics, because it relies fundamentally on randomness to function: a number p between 0 and 1 is expressed as an infinite random bitstream in which each bit independently has a p chance of being a 1; two numbers can be multiplied by bitwise ANDing their bitstreams; two numbers can be averaged by alternating between their bitstreams; etc. Naturally, the result of a computation continually becomes more precise as more bits are processed, but the initial bits provide a good rough approximation as long as they are truly random. C.f. traditional adders that must start with the least significant bits and produce the entire answer in one go at the end. To implement Bayesian inference, the first paper uses an electronic component called a Muller C-element that again has a ‘memory’ of sorts, and corresponds very well to the fundamental statement of Bayes’ Theorem. (In stochastic computing this memory creates ‘autocorrelated’ patterns in the random bit streams (now they remember their own history and aren’t independent anymore?) that requires some clever math to fix, which is beyond me but not beyond the references in the paper.)
  • Bayes with software optimization (Murray 2013). The author creates a software library that compiles a specification of a Bayesian inference model into machine code, optimized for the specific high-performance hardware being used, such as a GPU. This is not as much a change in hardware substrate as writing software really close to that substrate to optimize how it is used.

An aside about stochastic computing: isn’t it curious that neurons are also stochastic, in a way? A passive neuron doesn’t just sit there, it merely fires (randomly) at a much lower rate than an activated neuron. On receiving new data, neural computation must ‘settle’ into a decision after a few moments of activation spike chain reactions. Hmmm…

Another aside about stochastic computing: it reminded me of Marshall, a robust real number computation framework (and a theoretical exposition). Today’s floating-point numbers give rise to today’s infamous floating-point errors, and with digital computers this is in some sense an unavoidable problem, so Marshall bypasses it by representing a real number as the set of all numbers smaller than it paired with the set larger than it. As the computation proceeds, the range can be made smaller/more precise, until it is small enough for the required application—and Marshall does this in a proven-to-be-correct way. Of course, this is very different from stochastic computation, but in my head they have the same flavor of increasing precision and ensuring robustness (tiny changes in input don’t cause unreasonable changes in output).

In what other ways can the substrate of computation be changed? Although this is not what quantum computers do, what about exploiting the quantum mechanical fact that atoms are really and truly made up of probability distributions (albeit complex ones), which could be manipulated to perform computations much more directly than trying to mimic the distributions in bits? What about using light waves as the fundamental unit of computation, expressing quantities as frequencies, multiplication as superposition, addition as amplitude modulation, …?

But in all the discussion above we have been working towards some nebulous ‘intelligence’ that we presumably will recognize once we build it. I’d like to ask the dual question: using this range of computation substrates, what other types of intelligence can we create? Recognizing an intelligence that is not sufficiently ‘human-like’ could be extremely challenging, because we as humans have never done it before. Eliezer Yudkowsky makes some very good points about this in his article about what it really means to imagine an alien intelligence that I very much recommend reading. To quote, “The only reason you can try at all to grasp anything as physically complex and poorly understood as the brain of another human being, is that you configure your own brain to imitate it. […] [But] minds that feel emotions you’ve never felt yourself, or that fail to feel emotions you would feel? […] I can tell you to imagine an alien that grew up in universe with four spatial dimensions, instead of three spatial dimensions, but you won’t be able to reconfigure your visual cortex to see like that alien would see.”

Yudkowsky’s favorite example of an alien intelligence is natural selection. It, as much as ‘it’ can be considered an entity, does not operate with a ‘purpose’, no matter how much most humans thinking about evolution attribute a purpose to it; genes that happen to mutate to produce incrementally beneficial changes, just statistically happen to proliferate, regardless of whether you think the purpose of our eyes is to help us see. And it takes a great deal of effort and training to be able to reason about natural selection processes correctly, without anthropomorphizing.

In the same vein, consider the ‘intelligence’ of a mobile phone, that can calculate bigger numbers faster than any human, etc. etc. “Ah, but it is not intelligent”, you say, “because it cannot do X Y Z things that a human can.” Yet it takes years of training to be able to understand how it works and write software that can use its full potential. To most people, its intelligence is as incomprehensible as is the intelligence of evolution; the student biologist arguing for why evolution ‘should have’ selected for a certain mutation is equivalent to the student programmer writing comments explaining the intended function of their code and complaining that the compiler seems to ignore them.

What other intelligences exist already, on whatever substrate, that we can learn from? We’ve tried simulating neurons in digital electronics; can we gain anything from simulating boolean logic in neurons? Some things could be fundamentally difficult for one substrate yet trivially easy for another, just like some computational constructs are easy in one programming language and tedious in another. What can we learn from and about these differences?

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