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c6025c36-8841-4f6c-8ac7-697ba5d9f84f | trentmkelly/LessWrong-43k | LessWrong | Stupid Questions June 2015
This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Don't be shy, everyone has holes in their knowledge, though the fewer and the smaller we can make them, the better.
Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.
To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag. |
f51c7da5-2e3b-4115-b734-d770fa850416 | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Problems Involving Abstraction?
I'm working on a post of examples for how to formulate problems involving abstraction (using the [abstraction formulation here](https://www.lesswrong.com/s/ehnG4mseKF6xALmQy/p/vDGvHBDuMtcPd8Lks)). This isn't going to solve problems, just show how to set them up mathematically.
To that end, I'd like to hear particular problems people are interested in which intuitively seem to involve abstraction. Examples of the sort of thing I have in mind:
* Humans generally seem to care about abstract objects, not individual atoms, so it seems like abstraction should be relevant to [impact measures](https://www.lesswrong.com/tag/impact-measures). How would we formalize that?
* Humans can figure out what a new word means with ridiculously few examples, suggesting that we already have some "latent space" with a simple representation of the-class-of-things-corresponding-to-the-new-word. That sounds like it has something to do with abstraction. What's going on there?
* The sort of "maps" we use in the real world (like street maps, for instance) are lossy, abstract representations of the territory (i.e. streets). How can we usefully formulate map-territory correspondence for such abstract maps? Is possible for a system to use its abstract map to [recognize flaws in its own abstract-map-making process](https://www.lesswrong.com/posts/46qnWRSR7L2eyNbMA/the-lens-that-sees-its-flaws)?
There is a high chance that your request (or at least something very similar to it) will be incorporated in the post. So, what examples would people like to see? |
d03dc53d-290d-4486-804e-5eebcb53a4ac | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Defining AI wireheading
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What does it mean for an AI to wirehead its reward function? We're pretty clear on what it means for a human to [wirehead](https://wiki.lesswrong.com/wiki/Wireheading) - artificial stimulation of part of the brain rather than genuine experiences - but what does it mean for an AI?
We have a lot of examples of wireheading, especially in informal conversation (and some specific prescriptive examples which I'll show later). So, given those examples, can we define wireheading well - [cut reality at its joints](https://www.lesswrong.com/posts/d5NyJ2Lf6N22AD9PB/where-to-draw-the-boundary)? The definition won't be - and can't be - perfectly sharp, but it should allow us to have clear examples of what is and what isn't wireheading, along with some ambiguous intermediate cases.
Intuitive examples
------------------
Suppose we have a weather-controlling AI whose task is to increase air pressure; it gets a reward for so doing.
What if the AI directly rewrites its internal reward counter? Clearly wireheading.
What if the AI modifies the input wire for that reward counter? Clearly wireheading.
What if the AI threatens the humans that decide on what to put on that wire? Clearly wireheading.
What if the AI takes control of all the barometers of the world, and sets them to record high pressure? Clearly wireheading.
What if the AI builds small domes around each barometer, and pumps in extra air? Clearly wireheading.
What if the AI fills the atmosphere with CO₂ to increase pressure that way? Clearly wire... actually, that's not so clear at all. This doesn't seem a [central example](https://www.lesswrong.com/posts/yCWPkLi8wJvewPbEp/the-noncentral-fallacy-the-worst-argument-in-the-world) of wireheading. It's a failure of alignment, yes, but it doesn't seem to be wireheading.
Thus not every example of [edge](https://arbital.com/p/edge_instantiation/) or [perverse](https://en.wikipedia.org/wiki/AI_control_problem#The_problem_of_perverse_instantiation:_%22be_careful_what_you_wish_for%22) instantiation is an example of wireheading.
Prescriptivist wireheading, and other definitions
-------------------------------------------------
A lot of posts [and](https://arxiv.org/pdf/1605.03143.pdf) [papers](https://arxiv.org/pdf/1705.08417.pdf) (including [some of mine](https://www.lesswrong.com/posts/b8HauRWrjBdnKEwM5/rigging-is-a-form-of-wireheading)) take a prescriptivist approach to wireheading.
They set up a specific situation (often with a causal diagram), and define a particular violation of some causal assumptions as wireheading (eg "if the agent changes the measured value X without changing the value of α, which is being measured, that's wireheading").
And that is correct, as far as it goes. But it doesn't cover all the possible examples of wireheading.
Conversely, [this post](https://www.lesswrong.com/posts/aMXhaj6zZBgbTrfqA/a-definition-of-wireheading) defines wireheading as a divergence between a true utility and a substitute utility (calculated with respect to a model of reality).
This is too general, almost as general as saying that every [Goodhart curse](https://arbital.com/p/goodharts_curse/) is an example of wireheading.
Note, though, that the converse is true: every example of wireheading *is* a Goodhart curse. That's because every example of wireheading is maximising a proxy, rather than the intended objective.
The definition
--------------
The most intuitive example of wireheading is that there is some property of the world that we want to optimise, and that there is some measuring system that estimates that property. If the AI doesn't optimise the property, but instead takes control of the measuring system, that's wireheading (bonus points if the measurements the AI manipulates go down an actual wire).
This re-emphasises that "[wireheading is in the eye of the beholder](https://www.lesswrong.com/posts/BvctuKocyWR4YYea3/wireheading-is-in-the-eye-of-the-beholder)": if our true goal is actually the measuring system (maybe our AI is in competition with another one to maximise a score in a game, and we really don't care how it does this), then there will be no wireheading, just an AI following a correct objective.
Thus wireheading is always a failure of some (implicit or explicit) goal; thus every example of wireheading is a failure of value alignment, though the converse is not true.
Also key to the definition is the fact that the measuring system is, in some sense "much smaller" than whatever property of the system it is measuring. Pumping out CO₂ is not the correct instantiation of some goal along the lines of "increase air pressure so humans enjoy better weather"; but nor is it merely manipulating the measurement of that goal.
The definition
--------------
Thus we can define wireheading as:
* Given some implicit goal G, an agent wireheads if, instead of moving towards G, it manipulates some *narrow* measurement channel that is intended to measure G, but will fail to do so after the agent's manipulation.
The difference with the prescriptivist approach is that the measurement channel is not specified; instead, we ask whether we can usefully characterise some feature of the setup as a "narrow measurement channel", and then apply the definition.
This can be seen as a particular [failure of abstraction](https://www.lesswrong.com/posts/Lb3xCRW9usoXJy9M2/platonic-rewards-reward-features-and-rewards-as-information): the abstract goal G was collapsed to the output of the measurement channel.
Examples, counter-examples, and edge cases
------------------------------------------
Under this definition, all the intuitive examples of wireheading above fit: the measurement channel the AI takes over (its internal counter, the wire going into it, the statements made by humans, the barometers, the immediate environments of the barometers) is always much smaller than the whole atmosphere, which was its intended goal.
And that's why the CO₂ example isn't wireheading: the AI is doing a massive manipulation of the world, on the same scale as its intended goal; it isn't just manipulating the measurement channel[[1]](#fn-oJdC8GSuRYeGynsAY-1).
The case of the domes around the barometers is an interesting one to consider. Obviously, if the AI put a dome around the planet and pumped in extra air, this wouldn't count as wireheading. Thus, we can imagine the domes growing bigger and merging, thus giving a smooth transition from "clearly wireheading" to "clearly not wireheading", and showing that ambiguous cases must exist.
We can also produce examples of Goodhart curse that are not wireheading. Take the practice of "[teaching to the test](https://en.wikipedia.org/wiki/Teaching_to_the_test)". In this case, there is a simple objective (the test results) and the school acts to optimise for that objective. However, in typical schools this is not wireheading; teaching to the test involves drilling students in specific skills, training them, and having them memorise certain facts. Though these are done specifically to pass the test, these are the kinds of actions that a teacher would undertake anyway. One can talk about how this "narrows" the intellect, but, except in extreme cases, this cannot be characterised as gaining control of a narrow measurement channel.
For an interesting edge case, consider the RL agent playing the game CoastRunners. As described [here](https://openai.com/blog/faulty-reward-functions/), the score-maximising agent misbehaved in an interesting way: instead of rushing to complete the level with the highest score possible, the agent instead found a way to boat in circles, constantly hitting the same targets and ever increasing its score.
Is that wireheading? Well, it's certainly Goodhart: there is a discrepancy between the implicit goals (got round the course fast, hitting targets) and the explicit (maximise the score). But do we feel that the agent has control of a "narrow" measurement channel?
I'd argue that it's probably not the case for CoastRunners. The "world" for this agent is not a particularly rich one; going round and round and hitting targets is what the agent is intended to do; it has just found an unusual way of doing so.
If, instead, this behaviour happened in some subset of a much richer game (say, SimCity), then we might see it more naturally as wireheading. The score there is intended to measure a wider variety of actions (building and developing a virtual city while balancing tax revenues, population, amenities, and other aspects of the city), so "getting a high score while going round in circles" is much closer to "controlling a measurement channel that is narrow (as compared to the implicit goal)" than in the CoastRunners situation.
But, this last example can illustrate the degree of judgement and ambiguity that can exist when identifying wireheading in some situations.
---
1. Note that the CO₂ example can fit with the definition of [this post](https://www.lesswrong.com/posts/aMXhaj6zZBgbTrfqA/a-definition-of-wireheading). One just needs to imagine that the agent's model does not specify the gaseous content of the air in sufficient detail to exclude a CO₂-rich air as a solution to the goal.
This illustrates that the definition used in that post doesn't fully capture wireheading. [↩︎](#fnref-oJdC8GSuRYeGynsAY-1) |
970d4bee-ac1b-4231-b771-5b7da1665fcd | trentmkelly/LessWrong-43k | LessWrong | Albert Hirschman on liberalism and nationalism
Albert Hirschman wrote Exit, Voice and Loyalty at the very dawn of neoliberalism and you can think of it as a warning before its time. His story is about what happens to a liberalism based on choice and nothing more.
Suppose that your brand of cornflakes isn’t good enough. The flakes are too soggy and flavourless. You have two remedies, says Hirschman: exit to a different brand, or exercise voice by writing a letter of complaint. The obvious choice here is exit, and indeed there are many brands of cornflakes, and no obvious market failure. But now suppose we extend this idea, to the market for train transport, say — or for political services. Then there might be a tension. Maybe train companies, or politicians, will only know how to improve their services if someone tells them. If so, and if everyone relies too much on exit, then we might end up in a world where everyone is always shuffling between different mediocre services. In particular, says Hirschman, “the presence of the exit alternative… can atrophy the development of the art of voice”. Exercising voice, by writing letters of complaint, or by taking part in democracy, is difficult and takes time to get good at. It is also often best done together. It’s hard to get a protest movement going if each member worries that everyone else is planning to exit.
And so even before neoliberalism and the sovereign consumer and a whole bunch of libertarians dreaming of the market in jurisdictions, Hirschman was — if you squint quite a bit — the world’s least likely post-liberal. Too much choice and you will become weak and selfish! Decadent! You won’t be able to act together!
Hirschman interpreting for the German general Anton Dostler at a 1945 war crimes trial. Hirschman was Jewish by background and had fought for the republicans in Spain, later helping to organize escape routes from occupied Europe to the US.
----------------------------------------
I visited Czernowitz (Ukrainian: Chernivtsi, Romanian: Cernăuți) in |
badbbac0-45eb-448e-b100-42e872ea8beb | trentmkelly/LessWrong-43k | LessWrong | D&D.Sci December 2022: The Boojumologist
This is an entry in the 'Dungeons & Data Science' series, a set of puzzles where players are given a dataset to analyze and an objective to pursue using information from that dataset.
STORY (Skippable)
You were saddled with debt, and despair, and regret;
But you left it behind to embark,
With a visiting ship who were planning a trip,
Hunting some strange sea-beasts they call . . . “Snark”?
(After climbing aboard and departing the shore,
Your life is if anything worse.
The grog makes you groggy; the sea makes you soggy;
The songs leave you thinking in verse.)
Snark-hunting, you find, is a peaceful pastime.
By now, every crew knows the way,
To - with ease! - guarantee their success and safety,
As they seek, and they lure, and they slay.
A single exception proves the above rule:
While with most Snarks, Snark-hunting’s a breeze,
If your Snark is a Boojum, it hunts you right back,
And slays you with similar ease.
When you learn the detail that a Snark-hunt can fail,
You speak with the crew and insist,
On wielding Science skills to discern which Snarks kill,
And becoming the ship’s Analyst.
(The Butcher and Baker and Barnacle-Scraper,
All tell you that simply won’t do.
“You’ll be telling us what is a Boojum, what’s not;
Thus, you’re our Boojumologist; true?”
When you meekly protest you’ve not earned that address,
They rebut: “As you clearly percieve,
The Bellman can’t ring, the Belter can’t sing,
And the Beaver knows not how to Beave.”)
Notwithstanding contention viz naming conventions,
Your task is as clear as could be:
Using data from trips made by similar ships,
Choose which Snarks to fight, which to flee.
DATA & OBJECTIVES
* You have a dataset of Snark sightings. This tells you whether a sighted Snark was hunted and whether – if hunted – it turned out to be a Boojum. (To clarify, the order of operations is “someone sights a Snark” “they take notes on its behavior and relay this information to the nearest Snark- |
81cf946a-2660-4b74-858a-e02a8c21b032 | trentmkelly/LessWrong-43k | LessWrong | List of notable people who believe in AI X-risk?
Remembering Scott's AI Researchers On AI Risk. The debate about AI is heating up, and a lot of people rely on authority figures for knowing what to believe. Do we have a current list of notable AI X-risk believers? Would there be value in compiling such a list? |
1214c95e-c1cc-40dc-8621-d6bb0433d11c | trentmkelly/LessWrong-43k | LessWrong | Quick Summaries of Two Papers on Kant and Game Theory
It's not obvious from a Darwinian point of view why people (even people who aren't related) often cooperate with one another. Researchers like Axelrod and Hamilton (1981) have used game theory to show that some strategies are both adaptive and favour cooperation. Though these projects may not say anything about how we should act towards one another[1], they do say something about why we do act the way we act. That's important because we'd like moral strategies (under whatever view we think is more plausible) to outcompete immoral strategies; if we notice that various game-theoretic situations tend to steer people away from sound ethics, that's both (1) an indication that these situations are to be avoided and (2) an invitation to think up situations that steer people towards better, more ethical strategies.
The rest of this post comprises summaries of two papers about this problem (specifically concerning my preferred ethics, Kantianism), plus some brief comments on each of them.
Laslier (2020): In the stag hunt game, two hunters choose to either hunt a stag or hunt hares – if both hunt the stag, they both get a very good payoff (the stag); if both hunt hares, they both get a bad payoff (few hares); but if one hunts the stag and the other hunts hares, the stag hunter gets a very bad payoff (nothing) and the hare hunter gets a good payoff (many hares). This game has two pure strategy Nash equilibria: the Pareto optimal outcome where both hunters hunt the stag, and the outcome where both hunters hunt hare. It also has one mixed strategy Nash equilibrium: both players hunt the stag with 2/3rds probability and hares with 1/3rd probability.
A "Kantian" player asks: "Which strategy is best if both players use it?"; the answer is to always hunt the stag. In a simple model, like that in Roemer (2019), a "Nasher" is a player that goes for one of the (non-Kantian) Nash equilibria. If a Nasher goes with the mixed strategy, they're outcompeted by the Kantian. If a Nasher alw |
10f0dba5-86a5-49fb-9c89-8f56a6e96f27 | trentmkelly/LessWrong-43k | LessWrong | Show LW: Diaspora Project Map and Preregistration Database
Hi everyone,
I and a handful of other folks have been compiling a list of different LessWrong Diaspora projects. I started it after someone asked me how they could volunteer for LessWrong and I didn't have a good answer for them.
It's optimized for people who are researching what sort of things have already been done in the community, or are looking for a place where they can help. The basic criteria for eligibility are:
* Being something I'm sure is a 'real' Diaspora project. i.e, started by a credible person who is a member of the Diaspora. (Membership is of course a little fuzzy, err on the side of inclusion.)
* Either a web presence or contact information.
Suggestions for additions should go in the comment box on that page. (I'll also accept them in this thread, but understand I'll probably only check back on it for the first three days or so.)
We've also put together a preregistration database. A preregistration database solicits plans for projects and experiments before their results are known. It helps with the problem where no one wants to talk about something that didn't work. Ideally it becomes part of a framework for helping people put together solid projects.
Right now that framework includes a Project Litmus Test that's meant to help people who want to do something not fail from the outset through poor planning. The current question set is based on what we think are common failure modes for Diaspora projects, but over time I'd like to refactor it based on empirical interaction with peoples real ideas. In the best case scenario a workshop would be developed that helps people go from concept to execution.
My hope is that such a framework could put a real dent in the chronic lack of coordination that you've already been told about a thousand times.
The Map
The Preregistation Database |
6287eecf-a4cd-490d-a398-4b1edd2f7847 | trentmkelly/LessWrong-43k | LessWrong | Book review: Human Compatible
Book review: Human Compatible, by Stuart Russell.
Human Compatible provides an analysis of the long-term risks from artificial intelligence, by someone with a good deal more of the relevant prestige than any prior author on this subject.
What should I make of Russell? I skimmed his best-known book, Artificial Intelligence: A Modern Approach, and got the impression that it taught a bunch of ideas that were popular among academics, but which weren't the focus of the people who were getting interesting AI results. So I guessed that people would be better off reading Deep Learning by Goodfellow, Bengio, and Courville instead. Human Compatible neither confirms nor dispels the impression that Russell is a bit too academic.
However, I now see that he was one of the pioneers of inverse reinforcement learning, which looks like a fairly significant advance that will likely become important someday (if it hasn't already). So I'm inclined to treat him as a moderately good authority on AI.
The first half of the book is a somewhat historical view of AI, intended for readers who don't know much about AI. It's ok.
Key proposals
Russell focuses a moderate amount on criticizing what he calls the standard model of AI, in which someone creates an intelligent agent, and then feeds it a goal or utility function.
I'm not too clear how standard that model is. It's not like there's a consensus of experts who are promoting it as the primary way to think of AI. It's more like people find the model to be a simple way to think about goals when they're being fairly abstract. Few people seem to be defending the standard model against Russell's criticism (and it's unclear whether Russell is claiming they are doing so). Most of the disagreements in this area are more about what questions we should be asking, rather than on how to answer the questions that Russell asks.
Russell gives a fairly cautious overview of why AI might create risks that are as serious as the risks gorillas face from h |
0e95fb9b-71c7-4a4a-8839-9a1d092fc558 | trentmkelly/LessWrong-43k | LessWrong | Modal SAT: Self Cooperation
Post 2 in in Modal SAT series. In this post, we show that SC Modal SAT is equivalent to Modal SAT.
For this, we need just need to prove the following theorem:
Theorem: If there exists a modal agent M such that Ci cooperates with M for each Ci∈C and such that M defects against Di for each Di∈D, then there exists an M′ which satisfies the above properties and cooperates with itself.
Proof: Let n be greater than the total number of boxes in agents in C and D. Consider the agents Xn and Xn+1 defined by Xi(B)=C↔¬□i⊥.
We define M′ by
1. M′(B)=C if ¬□n+1⊥ and □(¬□n⊥→B(CooperateBot)=C) and ¬□(¬□n−1⊥→B(CooperateBot)=C)
2. M′(B)=D if ¬□n+2⊥ and □(¬□n+1⊥→B(CooperateBot)=C) and ¬□(¬□n⊥→B(CooperateBot)=C)
3. M′(B)=C if ¬□n+3⊥ and □(¬□n+1⊥→B(Xn)=C) and □(¬□n+2⊥→B(Xn+1)=D)
4. M′(B)=M(B) otherwise
Rule 1 says that M′ cooperates with Xn, rule 2 says that M′ defects against Xn+1, and rule 3 says that M′ cooperates with anyone who (provably assuming ¬□n+1⊥) cooperates with Xn and (provably assuming ¬□n+2⊥)defects against Xn+1. Thus, M′ cooperates with itself.
For any bot B with fewer than n boxes, the conditions of 1, 2, and 3 are all false. For 1 and 2, this is because the actions of such bots against CoopearteBot stabilize by the time you assume ¬□n−1⊥. For 3, this is because these bots cannot distinguish between Xn and Xn+1.
Therefore M′ behaves the same as M on all inputs with fewer than n boxes, so M′ cooperates with every bot in C and defects against every bot in D.
□
Note that I was lazy here, and took way more longer to cooperate with myself than I had to. In principle, if there are only k bots that I need to consider (including bots I need to consider because they are referenced by bots I care about), then regardless of how many boxes are in each bot, It should be possible to achieve self cooperation within 2log2k worlds of the Kripke frame. That is, log2k worlds to identify a single bot that is distinguishable from all other bots, and another log2k worlds to |
1d58138d-1ffa-4b71-aa3b-0075531e0624 | trentmkelly/LessWrong-43k | LessWrong | "You'll Never Persuade People Like That"
Sometimes, when someone is arguing for some proposition, their interlocutor will reply that the speaker's choice of arguments or tone wouldn't be effective at persuading some third party.
This would seem to be an odd change of topic. If I was arguing for this-and-such proposition, and my interlocutor isn't, themselves, convinced by my arguments, it makes sense for them to reply about why they, personally, aren't convinced. Why is it relevant whether I would convince some third party that isn't here?
What's going on in this kind of situation? Why would someone think "You'll never persuade people like that" was a relevant reply?
"Because people aren't truthseeking and treat arguments as soldiers" doesn't seem like an adequate explanation by itself. It's true, but it's not specific enough: what particularly makes appeal-to-persuading-third-parties an effective "soldier"?
----------------------------------------
The bargaining model of war attempts to explain why wars are fought—and not fought; even the bitterest enemies often prefer to grudgingly make peace with each other rather than continue to fight.
That's because war is costly. If I estimate that by continuing to wage war, there's a 60% chance my armies will hold a desirable piece of territory, I can achieve my war objectives equally well in expectation—while saving a lot of money and human lives—by instead signing a peace treaty that divides the territory with the enemy 60/40.
If the enemy will agree to that, of course. The enemy has their own forecast probabilities and their own war objectives. There's usually a range of possible treaties that both combatants will prefer to fighting, but the parties need to negotiate to select a particular treaty, because there's typically no uniquely obvious "fair" treaty—similar to how a buyer and seller need to negotiate a price for a rare and expensive item for which there is no uniquely obvious "fair" price.
----------------------------------------
If war is bargai |
129074f6-688f-4511-bd24-0dfcf9ed8338 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | AI-Based Code Generation Using GPT-J-6B
Above is a link to an interesting post about synthetic code generation with a transformer model trained on *The Pile*, which contains a large chuck of GitHub and StackOverflow. Due to *CommonCrawl*'s deficiency in this area, the much smaller GPT-J-6B outperforms OpenAI’s largest publicly available GPT-3 models. The performance is impressive enough that one wonders how capable a 100+ billion parameter model trained on *The Pile* will be, let alone what an AlphaGo-level engineering effort towards the end of synthetic code generation would achieve.
As the *The Pile* was created to provide a dataset for 100 billion paramater+ models, we may not have to wait long. The examples in the post are clearly trivial, but I personally take this to be something of a fire alarm. I was not previously aware of how poorly-optimized GPT-3 was for code generation, and I have updated toward surprising gains in this area in the next few years.
I no longer consider agents with superhuman performance in competitive programming to be a ridiculous thing to pursue.
It is useful to remind myself of how shocked I would be to see such things in 2012. In 2012 I would have taken this as a sign that AGI was near.
Scenario-based planning postulates that one should predict symptoms emblematic of a given scenario and then robotically assume you are in said scenario once a sufficient number of these symptoms come to pass. I am unsure whether there is wisdom in this approach, but I find it a discomfiting line of thought. |
6f5c3414-8049-447a-a39c-0b6e9fa18d0c | trentmkelly/LessWrong-43k | LessWrong | Utility indifference and infinite improbability drives
In MIRI's paper on Corrigibility, we describe a version of Stuart Armstrong's utility indifference technique (see also this LessWrong post), and show that an agent using our version acts as if it believes it has a "magical" way of influencing the world, similar to the problem described in my post on Exploiting EDT---Eliezer calls this an "infinite improbability drive".
Stuart has pointed out to me that the version of utility indifference we describe in the paper is different in an important way from his original proposal, which involves conditioning on a low-probability stochastic event. If my current understanding is correct, Stuart's version is indeed immune to the infinite improbability drive, as long as there is no way for the agent to influence the stochastic event. However, this version can still be interpreted as giving the agent an incorrect probability distribution, and it seems like any agent with this property will be willing to accept bets which are clearly bad.
Model
The model of utility indifference I will use in this post is an instance of the one in the Corrigibility paper, with specific support for the stochastic event in Stuart's original proposal.
As in the paper, for simplicity we'll look at a toy world where all that happens is that the agent takes an action from a set A1, makes a single observation from a set O1, and takes a second action from a set A2, and we assume that our utility functions are specified in terms of these actions and observations (rather than some underlying states of the world).
Our world model is given by a function p(o;a1), which gives the probability of different observations o∈O given that the agent's first action is a1∈A1. In other words, p(⋅;a1) is a probability distribution over O, for each a1∈A1. The implicit idea is that these probability distributions are computed by causal counterfactuals (or something better).
We are given two utility functions, u(a1,o,a2) and v(a1,o,a2). When choosing its first action, |
d269d157-6aa3-4ebd-a735-761967d836ea | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Making a Difference Tempore: Insights from 'Reinforcement Learning: An Introduction'
> The safety of artifi cial intelligence applications involving reinforcement learning is a topic that deserves careful attention.
Foreword
========
Let's get down to business.
Reinforcement Learning
======================
1: Introduction
---------------
2: Multi-armed Bandits
----------------------
*Bandit basics, including nonstationarity, the value of optimism for incentivizing exploration, and upper-confidence-bound action selection.*
Some explore/exploit results are relevant to daily life - I highly recommend reading *[Algorithms to Live By: The Computer Science of Human Decisions](https://www.amazon.com/Algorithms-Live-Computer-Science-Decisions/dp/1627790365)*.
3: Finite Markov Decision Processes
-----------------------------------
*The framework.*
4: Dynamic Programming
----------------------
*Policy evaluation, policy improvement, policy iteration, value iteration, generalized policy iteration. What a nomenclative nightmare.*
5: Monte Carlo Methods
----------------------
*Prediction, control, and importance sampling.*
### Importance Sampling
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b, we then want to approximate the value function for the target policy .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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vπ we're trying to learn; in other words, the distribution of states we sample is different. This gives us a skewed picture of .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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vπ, so we must overcome this bias.
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.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
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@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
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@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
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π can (*i.e.*, .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
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.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
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.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
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.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
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.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
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@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
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∀a,s:π(a|s)>0⟹b(a|s)>0), we can overcome by adjusting the return .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
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.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
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ρt:T−1:=∏T−1k=tπ(Ak|Sk)b(Ak|Sk). This cleanly recovers .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
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.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
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.mjx-mlabeledtr {display: table-row}
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.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
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@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
vπ by the definition of expectation:
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.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
ρt:T(t)−1's can be arbitrarily large (suppose .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
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.51×10−10=.5×1010), so the variance of this estimator can get pretty big. To get an estimator whose variance converges to 0, try
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V(s):=∑t∈T(s)ρt:T(t)−1Gt∑t∈T(s)ρt:T(t)−1.### Death and Discounting
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1−γ probability to die on the next turn, so we discount rewards accordingly.
This intuition hugs pre-theoretic understanding much more closely; if you have just 80 years to live, you might dive that big blue sky. If, however, you imagine there's a non-trivial chance that humanity can be more than just a flash in the pan, you probably care more about the far future.
6: Temporal-Difference Learning
-------------------------------
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Q*-learning.*
### Learning TD Learning
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TD(0)⋆SARSAQ
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vπ. It helps you learn the value of a given policy, and is not used for control.
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@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
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@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
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@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
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@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
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@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
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@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
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qπ using the quintuples .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
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.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
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.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
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.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
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(S,A,R,S′,A′).
> As in all on-policy methods, we continually estimate .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
> .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
> .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
> .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
> .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
> .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
> .mjx-numerator {display: block; text-align: center}
> .mjx-denominator {display: block; text-align: center}
> .MJXc-stacked {height: 0; position: relative}
> .MJXc-stacked > \* {position: absolute}
> .MJXc-bevelled > \* {display: inline-block}
> .mjx-stack {display: inline-block}
> .mjx-op {display: block}
> .mjx-under {display: table-cell}
> .mjx-over {display: block}
> .mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-stack > .mjx-sup {display: block}
> .mjx-stack > .mjx-sub {display: block}
> .mjx-prestack > .mjx-presup {display: block}
> .mjx-prestack > .mjx-presub {display: block}
> .mjx-delim-h > .mjx-char {display: inline-block}
> .mjx-surd {vertical-align: top}
> .mjx-mphantom \* {visibility: hidden}
> .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
> .mjx-annotation-xml {line-height: normal}
> .mjx-menclose > svg {fill: none; stroke: currentColor}
> .mjx-mtr {display: table-row}
> .mjx-mlabeledtr {display: table-row}
> .mjx-mtd {display: table-cell; text-align: center}
> .mjx-label {display: table-row}
> .mjx-box {display: inline-block}
> .mjx-block {display: block}
> .mjx-span {display: inline}
> .mjx-char {display: block; white-space: pre}
> .mjx-itable {display: inline-table; width: auto}
> .mjx-row {display: table-row}
> .mjx-cell {display: table-cell}
> .mjx-table {display: table; width: 100%}
> .mjx-line {display: block; height: 0}
> .mjx-strut {width: 0; padding-top: 1em}
> .mjx-vsize {width: 0}
> .MJXc-space1 {margin-left: .167em}
> .MJXc-space2 {margin-left: .222em}
> .MJXc-space3 {margin-left: .278em}
> .mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
> .mjx-line-box-test {display: table!important}
> .mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
> .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
> .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
> .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
> .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
> .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
> .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
> .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
> .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
> .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
> .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
> .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
> .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
> .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
> .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
> .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
> .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
> .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
> .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
> .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
> .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
> .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
> .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
> .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
> .MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
> .MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
> @font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
> @font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
> @font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
> @font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
> @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
> @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
> @font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
> @font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
> @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
> @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
> @font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
> @font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
> @font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
> @font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
> @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
> @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
> @font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
> @font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
> @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
> @font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
> @font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
> @font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
> @font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
> @font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
> @font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
> @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
> qπ for the policy .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
> .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
> .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
> .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
> .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
> .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
> .mjx-numerator {display: block; text-align: center}
> .mjx-denominator {display: block; text-align: center}
> .MJXc-stacked {height: 0; position: relative}
> .MJXc-stacked > \* {position: absolute}
> .MJXc-bevelled > \* {display: inline-block}
> .mjx-stack {display: inline-block}
> .mjx-op {display: block}
> .mjx-under {display: table-cell}
> .mjx-over {display: block}
> .mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-stack > .mjx-sup {display: block}
> .mjx-stack > .mjx-sub {display: block}
> .mjx-prestack > .mjx-presup {display: block}
> .mjx-prestack > .mjx-presub {display: block}
> .mjx-delim-h > .mjx-char {display: inline-block}
> .mjx-surd {vertical-align: top}
> .mjx-mphantom \* {visibility: hidden}
> .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
> .mjx-annotation-xml {line-height: normal}
> .mjx-menclose > svg {fill: none; stroke: currentColor}
> .mjx-mtr {display: table-row}
> .mjx-mlabeledtr {display: table-row}
> .mjx-mtd {display: table-cell; text-align: center}
> .mjx-label {display: table-row}
> .mjx-box {display: inline-block}
> .mjx-block {display: block}
> .mjx-span {display: inline}
> .mjx-char {display: block; white-space: pre}
> .mjx-itable {display: inline-table; width: auto}
> .mjx-row {display: table-row}
> .mjx-cell {display: table-cell}
> .mjx-table {display: table; width: 100%}
> .mjx-line {display: block; height: 0}
> .mjx-strut {width: 0; padding-top: 1em}
> .mjx-vsize {width: 0}
> .MJXc-space1 {margin-left: .167em}
> .MJXc-space2 {margin-left: .222em}
> .MJXc-space3 {margin-left: .278em}
> .mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
> .mjx-line-box-test {display: table!important}
> .mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
> .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
> .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
> .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
> .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
> .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
> .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
> .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
> .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
> .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
> .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
> .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
> .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
> .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
> .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
> .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
> .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
> .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
> .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
> .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
> .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
> .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
> .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
> .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
> .MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
> .MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
> @font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
> @font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
> @font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
> @font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
> @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
> @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
> @font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
> @font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
> @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
> @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
> @font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
> @font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
> @font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
> @font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
> @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
> @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
> @font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
> @font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
> @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
> @font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
> @font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
> @font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
> @font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
> @font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
> @font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
> @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
> π, and at the same time change .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
> .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
> .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
> .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
> .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
> .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
> .mjx-numerator {display: block; text-align: center}
> .mjx-denominator {display: block; text-align: center}
> .MJXc-stacked {height: 0; position: relative}
> .MJXc-stacked > \* {position: absolute}
> .MJXc-bevelled > \* {display: inline-block}
> .mjx-stack {display: inline-block}
> .mjx-op {display: block}
> .mjx-under {display: table-cell}
> .mjx-over {display: block}
> .mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-stack > .mjx-sup {display: block}
> .mjx-stack > .mjx-sub {display: block}
> .mjx-prestack > .mjx-presup {display: block}
> .mjx-prestack > .mjx-presub {display: block}
> .mjx-delim-h > .mjx-char {display: inline-block}
> .mjx-surd {vertical-align: top}
> .mjx-mphantom \* {visibility: hidden}
> .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
> .mjx-annotation-xml {line-height: normal}
> .mjx-menclose > svg {fill: none; stroke: currentColor}
> .mjx-mtr {display: table-row}
> .mjx-mlabeledtr {display: table-row}
> .mjx-mtd {display: table-cell; text-align: center}
> .mjx-label {display: table-row}
> .mjx-box {display: inline-block}
> .mjx-block {display: block}
> .mjx-span {display: inline}
> .mjx-char {display: block; white-space: pre}
> .mjx-itable {display: inline-table; width: auto}
> .mjx-row {display: table-row}
> .mjx-cell {display: table-cell}
> .mjx-table {display: table; width: 100%}
> .mjx-line {display: block; height: 0}
> .mjx-strut {width: 0; padding-top: 1em}
> .mjx-vsize {width: 0}
> .MJXc-space1 {margin-left: .167em}
> .MJXc-space2 {margin-left: .222em}
> .MJXc-space3 {margin-left: .278em}
> .mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
> .mjx-line-box-test {display: table!important}
> .mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
> .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
> .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
> .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
> .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
> .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
> .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
> .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
> .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
> .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
> .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
> .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
> .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
> .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
> .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
> .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
> .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
> .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
> .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
> .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
> .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
> .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
> .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
> .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
> .MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
> .MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
> @font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
> @font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
> @font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
> @font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
> @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
> @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
> @font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
> @font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
> @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
> @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
> @font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
> @font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
> @font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
> @font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
> @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
> @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
> @font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
> @font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
> @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
> @font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
> @font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
> @font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
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> @font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
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> @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
> @font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
> @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
> π toward greediness with respect to .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
> .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
> .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
> .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
> .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
> .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
> .mjx-numerator {display: block; text-align: center}
> .mjx-denominator {display: block; text-align: center}
> .MJXc-stacked {height: 0; position: relative}
> .MJXc-stacked > \* {position: absolute}
> .MJXc-bevelled > \* {display: inline-block}
> .mjx-stack {display: inline-block}
> .mjx-op {display: block}
> .mjx-under {display: table-cell}
> .mjx-over {display: block}
> .mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-stack > .mjx-sup {display: block}
> .mjx-stack > .mjx-sub {display: block}
> .mjx-prestack > .mjx-presup {display: block}
> .mjx-prestack > .mjx-presub {display: block}
> .mjx-delim-h > .mjx-char {display: inline-block}
> .mjx-surd {vertical-align: top}
> .mjx-mphantom \* {visibility: hidden}
> .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
> .mjx-annotation-xml {line-height: normal}
> .mjx-menclose > svg {fill: none; stroke: currentColor}
> .mjx-mtr {display: table-row}
> .mjx-mlabeledtr {display: table-row}
> .mjx-mtd {display: table-cell; text-align: center}
> .mjx-label {display: table-row}
> .mjx-box {display: inline-block}
> .mjx-block {display: block}
> .mjx-span {display: inline}
> .mjx-char {display: block; white-space: pre}
> .mjx-itable {display: inline-table; width: auto}
> .mjx-row {display: table-row}
> .mjx-cell {display: table-cell}
> .mjx-table {display: table; width: 100%}
> .mjx-line {display: block; height: 0}
> .mjx-strut {width: 0; padding-top: 1em}
> .mjx-vsize {width: 0}
> .MJXc-space1 {margin-left: .167em}
> .MJXc-space2 {margin-left: .222em}
> .MJXc-space3 {margin-left: .278em}
> .mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
> .mjx-line-box-test {display: table!important}
> .mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
> .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
> .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
> .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
> .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
> .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
> .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
> .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
> .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
> .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
> .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
> .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
> .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
> .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
> .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
> .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
> .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
> .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
> .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
> .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
> .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
> .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
> .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
> .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
> .MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
> .MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
> @font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
> @font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
> @font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
> @font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
> @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
> @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
> @font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
> @font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
> @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
> @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
> @font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
> @font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
> @font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
> @font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
> @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
> @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
> @font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
> @font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
> @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
> @font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
> @font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
> @font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
> @font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
> @font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
> @font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
> @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
> qπ.
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.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
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.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
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.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
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@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
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@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
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qπ.
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@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
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π and then use the maximal action value at the next state in your TD target term.
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.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
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.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
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.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
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@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
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@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
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@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
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maximization bias
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Q-learning.
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Φ(.8)100≈4.6×10−11. The more actions you have, the higher the probability is that at the maximum is just an outlier. See: [regression toward the mean and mutual funds](https://en.wikipedia.org/wiki/Regression_toward_the_mean#Importance).
To deal with this, we toss another .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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Q-learners are unlikely to be biased in the same way. For some reason, this [wasn't discovered until 2010](https://papers.nips.cc/paper/3964-double-q-learning).
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n-step Bootstrapping
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n*-step everything.*
8: Planning and Learning with Tabular Methods
---------------------------------------------
*Models, prioritized sweeping, expected vs. sample updates, MCTS, and rollout algorithms.*
### Roles Models Play
*Distribution* models include the full range of possible futures and their probabilities. For example, a distribution model for two fair coins: .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
{HH: .25, HT: .5, TT: .25}.
*Sample* models just let you, well, sample. .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
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HH, HT, HT, HT, HH, TT, HH, HT... You could sample thousands of times and gain a high degree of confidence that the above distribution model is generating the data, but this wouldn't be your *only* hypothesis (granted, it might have the lowest Kolmogorov complexity).
Note that a distribution model also can function as a sample model.
9: On-policy Prediction with Approximation
------------------------------------------
*We finally hit the* good stuff*: value-function approximators and stochastic-/semi-gradient methods.*
10: On-policy Control with Approximation
----------------------------------------
11: Off-policy Methods with Approximation
-----------------------------------------
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deadly triad *of function approximation, bootstrapping, and off-policy training converge to sow d̴iv̢erge͏nc̸e͟ and i͟nstab̕il̡i̡t̷y in your methods. Metal.*
12: Eligibility Traces
----------------------
*I was told to skip this chapter; the first half was my favorite part of the book.*
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.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
TD(λ) uses a backwards-view eligibility trace to update features, which I find elegant. Suppose you have some feature extraction function .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
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λ=1 recovers Monte-Carlo return methods.
When I was a kid, there was a museum I loved going to - it had all kinds of wacky interactive devices for kids. One of them took sounds and "held them in the air" at a volume which slowly decreased as a function of how long ago the sound was made. State features are sounds, and volume is eligibility.
13: Policy Gradient Methods
---------------------------
*The policy gradient theorem, REINFORCE, and actor-critic.*
14: Psychology
--------------
*Creating a partial mapping between reinforcement learning and psychology.*
### Mental, Territorial
There was a word I was looking for that "mental model" didn't quite seem to fit: "the model with respect to which we mentally simulate courses of action". CFAR's "inner sim" terminology didn't quite map either, as to me, that points to the *system-in-motion* more than *that-on-which-the-system-runs*. The literature dubs this a cognitive map.
> Individual place cells within the hippocampus correspond to separate locations in the environment with the sum of all cells contributing to a single map of an entire environment. The strength of the connections between the cells represents the distances between them in the actual environment. The same cells can be used for constructing several environments, though individual cells' relationships to each other may differ on a map by map basis. The possible involvement of place cells in cognitive mapping has been seen in a number of mammalian species, including rats and macaque monkeys. Additionally, in a study of rats by Manns and Eichenbaum, pyramidal cells from within the hippocampus were also involved in representing object location and object identity, indicating their involvement in the creation of cognitive maps.
> Wikipedia, *[Cognitive map](https://en.wikipedia.org/wiki/Cognitive_map)*
In light of [my work on whitelisting](https://www.lesswrong.com/posts/H7KB44oKoSjSCkpzL/worrying-about-the-vase-whitelisting), I've been thinking about why we're so "object-oriented" in our mental lives. An off-the-cuff hypothesis: to better integrate with the rest of our mental models, the visual system directly links up to our object schema. One such object is then recognized and engraved as a discrete "thing" in our map. Hence, we emotionally "know" that the world "really is" made up of objects, and isn't just a collection of particles.
> Most of the information used by people for the cognitive mapping of spaces is gathered through the visual channel (Lynch, 1960).
[Lahav and Mioduser's research](http://playpen.icomtek.csir.co.za/~acdc/assistive%20devices/Artabilitation2008/archive/2004/papers/2004_S04_N4.pdf) somewhat supports this idea, suggesting that blind people not only have lower-fidelity and more declarative (as opposed to procedural / interactive) cognitive maps, they're also less likely to provide object-to-object descriptions.
*Epistemic status: I made a not-obviously-wrong hypothesis and found two pieces of corroborating evidence. If anyone who knows more about this wants to chime in, please do!*
15: Neuroscience
----------------
*The reward prediction error hypothesis, dopamine, neural actor-critic, hedonistic neurons, and addiction.*
16: Applications and Case Studies
---------------------------------
*From checkers to checkmate.*
17: Frontiers
-------------
*Temporal abstraction, designing reward signals, and the future of reinforcement learning. In particular, the idea I had for having a whitelist-enabled agent predict expected object-level transitions is actually one of the frontiers: general value functions. Rad.*
### Pandora's AI Boxing
> The rapid pace of advances in AI has led to warnings that AI poses serious threats to our societies, even to humanity itself.
This chapter talks a fair amount about the serious challenges in AI alignment (not sure if you all have heard of that problem), which is heartening.
> As to safety, hazards possible with reinforcement learning are not completely di fferent from those that have been managed successfully for related applications of optimization and control methods.
I'm not sure about that. Admittedly, I'm not as familiar with those solutions, but the challenge here seems to be of a qualitatively different caliber. Conditional on true AI's achievement, we'd want to have extremely high confidence that it pans out *before* we flip the switch. The authors acknowledge that
> it may be impossible for the agent to achieve the designer's goal no matter what its reward signal is.
I don't think it's impossible, but it's going to be extremely hard to get formal probabilistic guarantees. I mean, if you don't know an agent's rationality, [you can't learn their utility function](https://www.lesswrong.com/posts/rtphbZbMHTLCepd6d/humans-can-be-assigned-any-values-whatsoever). If you do know their rationality but not their probability distribution over outcomes, [you still can't learn their utility function](<em>h</em>ttps://www.lesswrong.com/posts/kYgWmKJnqq8QkbjFj/bayesian-utility-representing-preference-by-probability).
This is just *one* of the [many](https://arbital.com/p/advanced_safety/) [open problems in alignment theory](https://arbital.com/p/taskagi_open_problems). If that's not enough, there are always the unknown unknowns...
Final Thoughts
--------------
I read the "nearly-complete" draft of the second edition, available [here](http://incompleteideas.net/book/the-book-2nd.html). It was pretty good, but I did find most of the exercises either too easy or requiring considerable programming investment to set up the environment described. The former makes sense, as I've already taken a course on this, and I'm probably a bit above the introductory level.
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.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
TD(0)'s convergence (p. 206-207) is underspecified:
> In general, .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
> .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
> .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
> .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
> .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
> .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
> .mjx-numerator {display: block; text-align: center}
> .mjx-denominator {display: block; text-align: center}
> .MJXc-stacked {height: 0; position: relative}
> .MJXc-stacked > \* {position: absolute}
> .MJXc-bevelled > \* {display: inline-block}
> .mjx-stack {display: inline-block}
> .mjx-op {display: block}
> .mjx-under {display: table-cell}
> .mjx-over {display: block}
> .mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-stack > .mjx-sup {display: block}
> .mjx-stack > .mjx-sub {display: block}
> .mjx-prestack > .mjx-presup {display: block}
> .mjx-prestack > .mjx-presub {display: block}
> .mjx-delim-h > .mjx-char {display: inline-block}
> .mjx-surd {vertical-align: top}
> .mjx-mphantom \* {visibility: hidden}
> .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
> .mjx-annotation-xml {line-height: normal}
> .mjx-menclose > svg {fill: none; stroke: currentColor}
> .mjx-mtr {display: table-row}
> .mjx-mlabeledtr {display: table-row}
> .mjx-mtd {display: table-cell; text-align: center}
> .mjx-label {display: table-row}
> .mjx-box {display: inline-block}
> .mjx-block {display: block}
> .mjx-span {display: inline}
> .mjx-char {display: block; white-space: pre}
> .mjx-itable {display: inline-table; width: auto}
> .mjx-row {display: table-row}
> .mjx-cell {display: table-cell}
> .mjx-table {display: table; width: 100%}
> .mjx-line {display: block; height: 0}
> .mjx-strut {width: 0; padding-top: 1em}
> .mjx-vsize {width: 0}
> .MJXc-space1 {margin-left: .167em}
> .MJXc-space2 {margin-left: .222em}
> .MJXc-space3 {margin-left: .278em}
> .mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
> .mjx-line-box-test {display: table!important}
> .mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
> .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
> .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
> .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
> .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
> .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
> .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
> .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
> .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
> .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
> .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
> .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
> .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
> .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
> .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
> .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
> .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
> .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
> .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
> .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
> .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
> .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
> .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
> .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
> .MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
> .MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
> @font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
> @font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
> @font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
> @font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
> @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
> @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
> @font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
> @font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
> @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
> @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
> @font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
> @font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
> @font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
> @font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
> @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
> @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
> @font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
> @font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
> @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
> @font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
> @font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
> @font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
> @font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
> @font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
> @font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
> @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
> wt will be reduced toward zero whenever .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
> .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
> .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
> .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
> .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
> .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
> .mjx-numerator {display: block; text-align: center}
> .mjx-denominator {display: block; text-align: center}
> .MJXc-stacked {height: 0; position: relative}
> .MJXc-stacked > \* {position: absolute}
> .MJXc-bevelled > \* {display: inline-block}
> .mjx-stack {display: inline-block}
> .mjx-op {display: block}
> .mjx-under {display: table-cell}
> .mjx-over {display: block}
> .mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-stack > .mjx-sup {display: block}
> .mjx-stack > .mjx-sub {display: block}
> .mjx-prestack > .mjx-presup {display: block}
> .mjx-prestack > .mjx-presub {display: block}
> .mjx-delim-h > .mjx-char {display: inline-block}
> .mjx-surd {vertical-align: top}
> .mjx-mphantom \* {visibility: hidden}
> .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
> .mjx-annotation-xml {line-height: normal}
> .mjx-menclose > svg {fill: none; stroke: currentColor}
> .mjx-mtr {display: table-row}
> .mjx-mlabeledtr {display: table-row}
> .mjx-mtd {display: table-cell; text-align: center}
> .mjx-label {display: table-row}
> .mjx-box {display: inline-block}
> .mjx-block {display: block}
> .mjx-span {display: inline}
> .mjx-char {display: block; white-space: pre}
> .mjx-itable {display: inline-table; width: auto}
> .mjx-row {display: table-row}
> .mjx-cell {display: table-cell}
> .mjx-table {display: table; width: 100%}
> .mjx-line {display: block; height: 0}
> .mjx-strut {width: 0; padding-top: 1em}
> .mjx-vsize {width: 0}
> .MJXc-space1 {margin-left: .167em}
> .MJXc-space2 {margin-left: .222em}
> .MJXc-space3 {margin-left: .278em}
> .mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
> .mjx-line-box-test {display: table!important}
> .mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
> .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
> .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
> .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
> .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
> .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
> .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
> .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
> .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
> .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
> .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
> .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
> .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
> .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
> .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
> .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
> .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
> .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
> .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
> .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
> .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
> .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
> .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
> .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
> .MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
> .MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
> @font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
> @font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
> @font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
> @font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
> @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
> @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
> @font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
> @font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
> @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
> @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
> @font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
> @font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
> @font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
> @font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
> @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
> @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
> @font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
> @font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
> @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
> @font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
> @font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
> @font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
> @font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
> @font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
> @font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
> @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
> A is positive de finite, meaning .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
> .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
> .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
> .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
> .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
> .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
> .mjx-numerator {display: block; text-align: center}
> .mjx-denominator {display: block; text-align: center}
> .MJXc-stacked {height: 0; position: relative}
> .MJXc-stacked > \* {position: absolute}
> .MJXc-bevelled > \* {display: inline-block}
> .mjx-stack {display: inline-block}
> .mjx-op {display: block}
> .mjx-under {display: table-cell}
> .mjx-over {display: block}
> .mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-stack > .mjx-sup {display: block}
> .mjx-stack > .mjx-sub {display: block}
> .mjx-prestack > .mjx-presup {display: block}
> .mjx-prestack > .mjx-presub {display: block}
> .mjx-delim-h > .mjx-char {display: inline-block}
> .mjx-surd {vertical-align: top}
> .mjx-mphantom \* {visibility: hidden}
> .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
> .mjx-annotation-xml {line-height: normal}
> .mjx-menclose > svg {fill: none; stroke: currentColor}
> .mjx-mtr {display: table-row}
> .mjx-mlabeledtr {display: table-row}
> .mjx-mtd {display: table-cell; text-align: center}
> .mjx-label {display: table-row}
> .mjx-box {display: inline-block}
> .mjx-block {display: block}
> .mjx-span {display: inline}
> .mjx-char {display: block; white-space: pre}
> .mjx-itable {display: inline-table; width: auto}
> .mjx-row {display: table-row}
> .mjx-cell {display: table-cell}
> .mjx-table {display: table; width: 100%}
> .mjx-line {display: block; height: 0}
> .mjx-strut {width: 0; padding-top: 1em}
> .mjx-vsize {width: 0}
> .MJXc-space1 {margin-left: .167em}
> .MJXc-space2 {margin-left: .222em}
> .MJXc-space3 {margin-left: .278em}
> .mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
> .mjx-line-box-test {display: table!important}
> .mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
> .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
> .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
> .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
> .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
> .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
> .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
> .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
> .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
> .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
> .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
> .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
> .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
> .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
> .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
> .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
> .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
> .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
> .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
> .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
> .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
> .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
> .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
> .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
> .MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
> .MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
> @font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
> @font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
> @font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
> @font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
> @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
> @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
> @font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
> @font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
> @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
> @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
> @font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
> @font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
> @font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
> @font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
> @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
> @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
> @font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
> @font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
> @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
> @font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
> @font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
> @font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
> @font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
> @font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
> @font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
> @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
> y⊤Ay>0 for real vector .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
> .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
> .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
> .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
> .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
> .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
> .mjx-numerator {display: block; text-align: center}
> .mjx-denominator {display: block; text-align: center}
> .MJXc-stacked {height: 0; position: relative}
> .MJXc-stacked > \* {position: absolute}
> .MJXc-bevelled > \* {display: inline-block}
> .mjx-stack {display: inline-block}
> .mjx-op {display: block}
> .mjx-under {display: table-cell}
> .mjx-over {display: block}
> .mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-stack > .mjx-sup {display: block}
> .mjx-stack > .mjx-sub {display: block}
> .mjx-prestack > .mjx-presup {display: block}
> .mjx-prestack > .mjx-presub {display: block}
> .mjx-delim-h > .mjx-char {display: inline-block}
> .mjx-surd {vertical-align: top}
> .mjx-mphantom \* {visibility: hidden}
> .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
> .mjx-annotation-xml {line-height: normal}
> .mjx-menclose > svg {fill: none; stroke: currentColor}
> .mjx-mtr {display: table-row}
> .mjx-mlabeledtr {display: table-row}
> .mjx-mtd {display: table-cell; text-align: center}
> .mjx-label {display: table-row}
> .mjx-box {display: inline-block}
> .mjx-block {display: block}
> .mjx-span {display: inline}
> .mjx-char {display: block; white-space: pre}
> .mjx-itable {display: inline-table; width: auto}
> .mjx-row {display: table-row}
> .mjx-cell {display: table-cell}
> .mjx-table {display: table; width: 100%}
> .mjx-line {display: block; height: 0}
> .mjx-strut {width: 0; padding-top: 1em}
> .mjx-vsize {width: 0}
> .MJXc-space1 {margin-left: .167em}
> .MJXc-space2 {margin-left: .222em}
> .MJXc-space3 {margin-left: .278em}
> .mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
> .mjx-line-box-test {display: table!important}
> .mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
> .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
> .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
> .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
> .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
> .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
> .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
> .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
> .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
> .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
> .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
> .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
> .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
> .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
> .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
> .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
> .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
> .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
> .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
> .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
> .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
> .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
> .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
> .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
> .MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
> .MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
> @font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
> @font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
> @font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
> @font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
> @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
> @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
> @font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
> @font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
> @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
> @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
> @font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
> @font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
> @font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
> @font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
> @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
> @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
> @font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
> @font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
> @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
> @font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
> @font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
> @font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
> @font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
> @font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
> @font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
> @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
> @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
> @font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
> y.
An additional precondition: .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
y can't be the zero vector.
> For a key matrix of this type, positive definiteness is assured if all of its columns sum to a nonnegative number.
Unless I totally missed what "this type" entails, this is false if taken at face value:
.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
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(−255−2)has nonnegative column sums and is also indefinite, having eigenvalues of 3 and -7.
However, the claim *is* true in the subtle way they use it - they're *actually* showing that since the matrix is symmetric, real, and strictly diagonally dominant with positive diagonal entries, it's also positive definite. This could be made clearer.
In all, reading this book was definitely a positive experience.
Forwards
========
I'll be finishing *Analysis II* before moving on to Jaynes's *Probability Theory* in preparation for a class in the fall.
Dota 2
------
Recently, OpenAI [made waves with their OpenAI Five Dota 2 bot](https://blog.<em>openai.com/o</em>penai-five/). To REINFORCE what I just learned and solidified, I might make a post in the near future breaking down how *Five* differs from the *Alpha(Go) Zero* approach, quantifying my expectations for *The International* for [calibration](https://en.wikipedia.org/wiki/Brier_score).
No Longer a Spectator
---------------------
Four months and one week ago, [I started my journey](https://www.lesswrong.com/s/KGYLvTqFiFE2CpHfJ/p/WPtdQ3JnoRSci87Dz) through the MIRI reading list. In those dark days, attempting a proof induced a stupor similar to that I encountered approaching a crush in grade school, my words and thoughts leaving me.
Six textbooks later and with a *little* effort, I'm able to prove things like the convergence of Monte Carlo integration to the Riemann integral, threading together lessons from *All of Statistics* and *Analysis I*;target in mind, words and thoughts remaining firmly by my side.
> The rapid pace of advances in artifi cial intelligence has led to warnings that arti ficial intelligence poses serious threats to our societies, even to humanity itself. The renowned scientist and artifi cial intelligence pioneer Herbert Simon anticipated the warnings we are hearing today...
> He spoke of the eternal conflict between the promise and perils of any new knowledge, reminding us of the Greek myths of Prometheus, the hero of modern science, who stole fire from the gods for the benefit of mankind, and Pandora, whose box could be opened by a small and innocent action to release untold perils on the world.
> Simon urged us to recognize that as designers of our future and *not mere spectators*, the decisions *we* make can tilt the scale in Prometheus' favor.
---
*If you are interested in working together to learn MIRI-relevant math, if you have a burning desire to knock the alignment problem down a peg, if you're in a* scale-tilting *mood - I would be more than happy to work with you. [Messaging me](https://www.lesswrong.com/users/turntrout) may also net you an invitation to the MIRIx Discord server.* |
ea482daf-84b3-4ec5-bbd5-f25ee0629218 | trentmkelly/LessWrong-43k | LessWrong | Elasticity of Wheat Supply?
I've been hearing concerns about famine stemming from the war in Ukraine. When thinking about how bad that situation is likely to be, the most important thing I don't know is the price elasticity of wheat: how much more wheat do farmers plant, given some increase in price? This is the sort of thing which I expect economists have already measured pretty well. Does anyone know of a good source? |
8af1aeb3-ff24-4595-a0cf-ae7195795cbb | trentmkelly/LessWrong-43k | LessWrong | Coronavirus: California case growth
In this post, I try to understand the case growth rate for coronavirus cases in California, and try to address questions such as:
* How long will the case count continue to grow?
* At what level will the case count stabilize?
* To what extent will we be able to infer from the data whether level 2 restrictions were sufficient, or level 3 restrictions were necessary, to stop or significantly slow down case growth? The "level 2" and "level 3" jargon are from my previous post.
Skip to answers and lessons for my (incomplete and tentative) answers.
NOTE: My original post was based on data till 2020-03-27 (row 17 in the spreadsheet). On 2020-04-01 (April 1, 2020) I made edits to this post of two kinds:
* Language improvements, based on feedback in the comments, without any substantive changes to the model.
* Addenda at the end of some sections giving updates based on data seen since I originally published the post. I used data till 2020-03-31 (row 21).
A simple model from true currently-or-eventually-symptomatic cases to confirmed cases to deaths or recoveries
The model
For simplicity, I will use the shorthand "true currently-or-eventually-symptomatic cases" only for cases where a person is already infected and will eventually become symptomatic (so this will include both currently symptomatic cases and cases that are presymptomatic, i.e., will become symptomatic later). I expect that most asymptomatic cases (i.e., cases that never become symptomatic) won't get diagnosed, and therefore won't count in the number of confirmed cases either, so this seems a reasonable approximation for the model I will present below. However, if incorrect, this could cause estimates to be off by a factor of two or more, depending on the fraction of cases that are asymptomatic.
The simplistic model identifies the following flow:
1. Get infected
2. Start showing symptoms
3. Get a test
4. Get test results
5. Recover or die
Technically, 5 can happen before 3 or 4; the logical |
90145757-99c0-4ab9-b15d-d2c1f51675bd | trentmkelly/LessWrong-43k | LessWrong | AI #39: The Week of OpenAI
The board firing Sam Altman, then reinstating him, dominated everything else this week. Other stuff also happened, but definitely focus on that first.
TABLE OF CONTENTS
Developments at OpenAI were far more important than everything else this read. So you can read this timeline of events over the weekend, and this attempt to put all the information together.
1. Introduction.
2. Table of Contents.
3. Language Models Offer Mundane Utility. Narrate your life, as you do all life.
4. Language Models Don’t Offer Mundane Utility. Prompt injection unsolved.
5. The Q Continuum. Disputed claims about new training techniques.
6. OpenAI: The Saga Continues. The story is far from over.
7. Altman Could Step Up. He understands existential risk. Now he can act.
8. You Thought This Week Was Tough. It is not getting any easier.
9. Fun With Image Generation. A few seconds of an Emu.
10. Deepfaketown and Botpocalypse Soon. Beware phone requests for money.
11. They Took Our Jobs. Freelancers in some areas are in trouble.
12. Get Involved. Dave Orr hiring for DeepMind alignment team.
13. Introducing. Claude 2.1 looks like a substantial incremental improvement.
14. In Other AI News. Meta breaks up ‘responsible AI’ team. Microsoft invests $50b.
15. Quiet Speculations. Will deep learning hit a wall?
16. The Quest for Sane Regulation. EU AI Act struggles, FTC AI definition is nuts.
17. That Is Not What Totalitarianism Means. People need to cut that claim out.
18. The Week in Audio. Sam Altman, Yoshua Bengio, Davidad, Ilya Sutskever.
19. Rhetorical Innovation. David Sacks says it best this week.
20. Aligning a Smarter Than Human Intelligence is Difficult. Technical debates.
21. People Are Worried About AI Killing Everyone. Roon fully now in this section.
22. Other People Are Not As Worried About AI Killing Everyone. Listen to them.
23. The Lighter Side. Yes, of course I am, but do you even hear yourself?
LANGUAGE MODELS OFFER MUNDANE UTILITY
GPT-4-Turbo sub |
916a1884-763f-4054-a9f8-c05b08b24546 | trentmkelly/LessWrong-43k | LessWrong | An AI Manhattan Project is Not Inevitable
Early last month, Leopold Aschenbrenner released a long essay and podcast outlining his projections for the future of AI. Both of these sources are full of interesting arguments and evidence, for a comprehensive summary see Zvi’s post here. Rather than going point by point I will instead accept the major premises of Leopold’s essay but contest some of his conclusions.
So what are the major premises of his piece?
1. There will be several orders of magnitude increase in investment into AI. 100x more spending, 100x more compute, 100x more efficient algorithms, and an order of magnitude or two gains from some form of “learning by doing” or “unhobbling” on top.
2. This investment scale up will be sufficient to achieve AGI. This means the models on the other side of the predicted compute scale up will be able to automate all cognitive jobs with vast scale and speed.
3. These capabilities will be essential to international military competition.
All of these premises are believable to me and well-argued for in Leopold’s piece.
Leopold contends that these premises imply that the national security state will take over AI research and the major data centers, locking down national secrets in a race against China, akin to the Manhattan project.
> Ultimately, my main claim here is descriptive: whether we like it or not, superintelligence won’t look like an SF startup, and in some way will be primarily in the domain of national security.
>
> By late 26/27/28 … the core AGI research team (a few hundred researchers) will move to a secure location; the trillion-dollar cluster will be built in record-speed; The Project will be on.
The main problem is that Leopold’s premises can be applied to conclude that other technologies will also inevitably lead to a Manhattan project, but these projects never arrived. Consider electricity. It's an incredibly powerful technology with rapid scale up, sufficient to empower those who have it far beyond those who don’t and it is essential |
c60c1e20-eaa7-4a9d-a63d-84c735cc437a | trentmkelly/LessWrong-43k | LessWrong | Open Thread, February 15-29, 2012
If it's worth saying, but not worth its own post (even in Discussion), then it goes here. |
5de749a0-f642-4dee-aa59-14f14b796678 | trentmkelly/LessWrong-43k | LessWrong | A few questions on International Rationality
Disclaimer: I'm still fairly new here, and though I did use the search bar it's entirely possible this has been discussed before. Just point me in the right direction if this is so.
While reading about 4chan's Japanese progenitor website, it occurred to me that I know nothing about the state of rationality in the non-English-speaking world, and more specifically the non-English-speaking internet. Is there a Russian version of SIAI? A Japanese Less Wrong? What about Korean Robin Hansons and Eliezer Yudkowskys?
If we take Religion as any indication of irrationality then America should be one of the least rational countries in the world. So if there are like-minded individuals out there speaking in languages we don't know, are we doing anything to collaborate with them? Do they have their own sequences and their own HPMORs which we could be reading?
And if there are no Singularitarian, Cryonics-Supporting, Utilitarianism-Advocating websites for the majority of the human race, isn't that a huge deal? Aren't Europeans and Asians more likely to be open to rationality, if only because of their atheism? If we want Friendly-AI to be developed, should we be translating the sequences into Chinese and Hindu as quickly as possible? |
769ede74-f3f5-4572-bf50-aaef97e55d23 | StampyAI/alignment-research-dataset/blogs | Blogs | Investigation into the relationship between neuron count and intelligence across differing cortical architectures
Survey participants (*n = 83*) were given anonymized descriptions of behavior in the wild for four animals: one bird species and one primate species with a similar neuron count, and one bird species and one primate species with twice as many neurons. Participants judged the two large-brained animals to display more intelligent behavior than the two smaller-brained animals on net, due to the large-brained animals’ substantial tool use being seen as a strong sign of intelligence, next to the small-brained animals absence of tool use. Other results were mixed. Participants did not judge either primates or birds to display more intelligent behavior.
1. Background
-------------
The existence of a correlation between brain size and intelligence across animal species is well-known (Roth & Dicke, 2005). Less clear is the extent to which brain size–in particular, neuron count–is responsible for differences in cognitive abilities between species. Here, we investigate one possible factor, the tissue organization of the cerebral cortex, by comparing cognitive abilities of animals with differing cortical architectures.
Primates make a natural target for comparison, since their intelligence has already been extensively studied. Additionally, comparing primate cognitive abilities to taxa that are farther from the human line may allow us to either confirm or deny the existence of a hard step for the evolvability of intelligence between primates and their last common ancestor with other large-brained animals (Shulman & Bostrom, 2012)[1](https://aiimpacts.org/investigation-into-the-relationship-between-neuron-count-and-intelligence-across-differing-cortical-architectures/#easy-footnote-bottom-1-1294 "</span><span style=\"font-weight: 400;\">Due to the observer selection effect, the fact that the particular evolutionary line containing humans and directly related species (ie, primates) lead to high levels of intelligence is not sufficient evidence that intelligence is not hard for evolution to produce; another line evolving intelligence, independent of ourselves, represents much stronger evidence. (See Shulman & Bostrom 2012.)</span><span style=\"font-weight: 400;\">"). Although some informal comparisons with other animals have been made, so far there have been few attempts to make detailed or quantitative comparisons between primate and non-primate intelligence.
There is only one extant alternative to primate cerebral architecture which has scaled to a similar size in terms of neuron count, that of birds, a lineage which diverged from our last common ancestor over 300 million years ago (for neuron counts of species across several lineages, see [here](https://en.wikipedia.org/wiki/List_of_animals_by_number_of_neurons)). Avian cortical architecture appears strikingly different from primates and indeed all mammals (see 1.1). However, compared to primates, radically less research effort has gone into investigating bird intelligence in a way that would enable comparison with other species. Therefore, in addition to theoretical difficulties (see 1.3), we also face the practical difficulty of comparing bird and primate intelligence without the aid of a rich psychometric literature, as exists for humans. Despite this difficulty, we believe that the comparison is nonetheless worthwhile, as it could give us insight into the flexibility of possible solutions to the problem of intelligence, given “hardware” of sufficient size.
For instance, if primates performed especially well relative to their absolute number of brain neurons or brain energy budget, this might indicate that primate cortical architecture (or some other systematic difference between primate and avian brains) was especially well-suited to producing intelligence. Furthermore, it would suggest that the evolution of biological intelligence faced design-related bottlenecks moreso than energy- or “hardware” bottlenecks. Likewise, if bird and primate architectures perform similarly despite different organization, this at the very least would indicate that the space of “wetware” architectures that lent themselves to the successful implementation of intelligence was larger than one. More speculatively, it could be taken as a sign that working brain architectures are fairly easy to come by, given a sufficient number of neurons and/or a sufficiently high brain energy budget.
### 1.1 Mammalian vs avian brains: Similarities and differences
[](http://aiimpacts.org/wp-content/uploads/2019/02/fig1-animal-survey-page.jpg)**Figure 1**. Image by [J. Arthur Thomson](https://commons.wikimedia.org/w/index.php?curid=9943793)
The usefulness of the comparison between birds and primates relies on the degree to which the same resources (a particular quantity of brain neurons) are arranged differently. At a glance, the majority of tissue in the avian and primate brains appears to be quite different, as the structure which evolved after the divergence point 300 million years ago–the cerebral cortex–occupies ~80% of the volume of both avian and primate brains. However, there is nonetheless a great deal of overlap in non-cerebral structures, and there is even reason to believe that the cerebral cortex has more commonality between bird and primate than might naively be expected (Kaas, 2017).
In the central nervous system, the common structures shared by mammals and birds include the spinal cord, the hindbrain, and the midbrain. These regions are primarily responsible for non-cognitive processes such as autonomic, sensorimotor, and circadian functions. Although each of these structures underwent changes to accommodate differences in body plan, environment, and niche, they are overall quite similar. Additionally, they have an unambiguously homologous (that is, similar by virtue of common descent) relationship in birds and mammals (Güntürkün, Stacho, & Strockens, 2017).
Atop the midbrain sits the forebrain, in particular the telencephalon, which is evolution’s most recent addition and the region which displays the most novel properties. The lower portion of the forebrain (the basal ganglia) is likely homologous between birds and mammals, but beyond this point the architectures diverge markedly. This uppermost layer is known as the pallium, or more commonly as the cerebral cortex in mammals.
Most of the mammalian cerebral cortex can be classed as neocortex. Neocortex spans six horizontally-oriented [layers](https://en.wikipedia.org/wiki/Cerebral_cortex#Layers), with neurons organized into vertical columns, which may both interact with adjacent columns, and also send efferents (outgoing fibers) to distant columns or even locations farther afield in the nervous system. (However, some areas of mammalian cerebral cortex, such as parts of the hippocampus, have only three or four cell layers.) In contrast, the analogue to our neocortex in birds–the pallium–contains no layers or columns, and neurons are instead organized into nuclei. The extent to which the neocortex and the avian pallium are elaborations on pre-existing structures (and therefore homologous), versus de novo inventions of early mammals/birds, is still debated (Puelles et al., 2017). However, it is interesting to note that the most abundant type of neuron in mammalian cerebral cortex, the excitatory pyramidal cell, is also common in the avian pallium, having originated in an early vertebrate ancestor (Naumann & Laurent, 2017).
The most immediately obvious difference between mammalian brains and avian brains is their size. For an animal adapted for flight, bulk would have been particularly costly, and this pressure probably forced neurons to become smaller and more tightly packed, resulting in a small brain dense with neurons (Olkowicz et al., 2016). However, neurons in mammal brains are both large relative to comparably sized bird brains, and also scale with the size of the brain. The only mammalian order exempt from this neuron scaling rule is primates (Herculano-Houzel, Collins, Wong, & Kaas, 2007). Therefore, although they still possess larger neurons than those of birds, primates were able to increase neuron count relatively efficiently through brain size increases, and are less constrained than birds with regard to size and weight limits.
Although it was reasoned that larger neurons would be more energetically expensive due to the maintenance cost of neurons even at rest, this has not been borne out empirically. At least in mammals, the per-neuron energy budget appears to be relatively constant within brain structures, and does not vary as a function of cell size (Herculano-Houzel, 2011). This finding has not been verified in birds, however the commonality of cell types across mammalian and avian brains suggests that it is likely true for birds as well. Interestingly, neuronal energy budget appears to differ substantially between brain structures: energy consumption by cerebral neurons, which are predominantly pyramidal cells, is an order of magnitude higher than that of cerebellar neurons, which are predominantly small granule cells.
This may have functional relevance for the final notable difference between primate and avian brains, the relative size of certain brain regions. While both bird and mammal brains are dominated volumetrically by the telencephalon (including the cerebral cortex/pallium), only in birds are the majority of neurons contained within this structure. In mammals, the densely-packed cerebellum expanded in tandem with the cerebrum,[2](https://aiimpacts.org/investigation-into-the-relationship-between-neuron-count-and-intelligence-across-differing-cortical-architectures/#easy-footnote-bottom-2-1294 "</span>Note that there are several overlapping anatomical terms here: the cerebrum, a mammalian structure, encompasses the cerebral cortex, the folded gray tissue visible on the outside of the lobes, and the connective white matter below it. The analogue in birds is the pallium. Below the cerebrum/pallium are the basal ganglia, and these structures collectively make up the telencephalon (the latest developing embryonic structure, and a part of the forebrain).") while this structure remained relatively small in birds.
This is a topic of some curiosity, since the cerebellum was previously thought to simply control motor processes. The observation that it scaled proportionally to brain size may have contributed to the popularity of the “encephalization quotient”, based on the notion that the amount of brain tissue required to control a body scales with the size of the body. However, more recent findings suggest a [broader role](https://en.wikipedia.org/wiki/Cerebellum#Function) for the cerebellum in humans, including in cognitive functions. If the cerebellum made a substantial contribution to cognition, it would call to mind several possible scenarios.
It’s possible that after it was no longer useful to improve motor control, developmental or other constraints made changing the brain’s scaling rules to de-emphasize the cerebellum costly. Instead of reassigning the brain’s volume budget, perhaps cerebellar tissue was repurposed to serve cognitive functions which had been pushed out of the cerebrum, a structure which had already become crowded enough to resort to lateralizing functions (relegating certain domains, like language, to one side of the brain exclusively, in contrast to the default in animals of bilateral function). Since the cerebrum and cerebellum are extremely cytoarchitecturally dissimilar, sharing neither cell types nor organization, this would be evidence of generality of function across different neural tissue types. Indeed, it would be more impressive than if bird and mammal cortex were functionally equivalent, since a mammal’s cerebellum bears far less resemblance to its neocortex than its neocortex does to a bird’s pallium.
Alternatively, birds may lack some novel functions which emerged in mammals as the result of the expanding cerebellum. Finally, the most disheartening possibility is that the extra cerebellar tissue in large-brained mammals represents an inferior allocation of brain tissue.
### 1.2 Common models of brain-based intelligence differences between species
Historically, there was much popular support for the idea that differences in brain size tracked differences in intelligence between species. Several variations on this theme have also built a following in the past century, including encephalization quotient, brain-to-body ratio, and neuron count. These could be called the “More is Better” class of models, where increases in intelligence across species are attributed to greater absolute amounts of brain tissue, neurons, synapses, etc, or to greater amounts relative to some expected amount.
Although among these models the most parsimonious currently appears to be neuron count (see [here](https://docs.google.com/document/d/1xBZbgz4hY4F31o52SHquERvhnr99dMDhIHDeS9xmyok/edit?usp=sharing) and Herculano-Houzel 2009), the intuitively appealing “relative size” models–encephalization quotient and brain-to-body ratio–may still have heuristic value in distinguishing between similarly-sized brains, despite lacking mechanistic explanatory power. This is because a relatively large investment in brain tissue compared to body size would imply stronger selection pressure for intelligence. However, in this case, the likely mechanism of the cognitive advantage falls under the next category.
The other class of models could be called “Structural Improvements”, where intelligence increases are attributed to improvements in brain architecture. At a gross brain level, the most popular of these models implicates the size of the forebrain, relative to the rest of the brain. Other possibilities in this space include tissue-level properties (such as whether cells are arranged into layers or nuclei), as well as much finer cytoarchitectural adjustments, altered developmental processes, functional properties of neurons, and features like gyrification (cortical folding).
While it’s certainly the case that both quantitative and qualitative changes factored into the development of higher intelligence, the degree to which one or the other explains the variance between species is not well understood. This uncertainty is due in part to the difficulty of measuring animal intelligence across a collection of species diverse enough to differ in both quantitative and qualitative brain characteristics. (Additionally, our understanding of qualitative interspecific differences that are less apparent than the architectural differences we focus on here is currently rather poor.) Such a set of animal species would tend to vary not simply in characteristics related to intelligence, but also in body plan, physical abilities, temperament, accessibility for human study, and the evolutionary pressures favoring intelligence in the species.
The nature of the intelligence construct adds a further layer of obscurity. While the general factor (*g*) is well-accepted among intelligence researchers with regard to humans (Carroll, 1997), the body of evidence in non-humans–and especially in non-primates–is small and somewhat conflicting (Burkart, Schubiger, & van Schaik, 2017). Furthermore, it’s likely that assumptions of generality hold less well in animals with low cognitive capacity (for instance, in insects).
### 1.3 Previous attempts to measure primate and avian intelligence
Our knowledge of primate intelligence is primarily informed by a diverse body of laboratory tasks that attempt to measure various aspects of cognition. While any particular task is likely to be a relatively weak signal of overall intelligence on its own, combining this result with the results of dissimilar tasks will tend to improve the measure, as has been found in human intelligence testing. Very few studies have attempted to administer such a battery of intelligence tasks at the level of an individual non-human subject; however, a ‘species-level battery’ may be assembled from the single-task results that do exist. Especially when this ‘species-level battery’ is based on a small number of tests, care must be taken to ensure that the procedures for administering tasks were the same across species. Luckily, the large amount of primate cognition research conducted in the last century allows the construction of a battery according to these criteria. The measurement of primate intelligence is discussed further [here](https://docs.google.com/document/d/1xBZbgz4hY4F31o52SHquERvhnr99dMDhIHDeS9xmyok/edit?usp=sharing).
In comparison with primates, the collection of cognitive tests that have been administered to bird species is disappointingly sparse. There are few examples of directly comparable tasks that have been administered to multiple species, preventing the construction of a battery from laboratory tasks. Even rarer are tasks that would enable comparison between primate species and bird species.
An alternative methodology that has been validated in primates is based on observations of behavior in the wild. Because the cognitive abilities displayed in the laboratory are likely the result of behavioral adaptations to challenging physical or social environments, it stands to reason that certain species-typical behaviors should correlate with the average intelligence of the species; that is, species that act intelligent in the lab should act intelligent in the field. This approach was used by Reader and colleagues (2011), who found that the number of reports citing instances of several types of behavior (eg tool use, social learning) correlated with each other, supporting the existence of a general factor of intelligence in primates. Furthermore, these results correlated with the results of the laboratory test battery discussed above at 0.7.
2. Estimating animal intelligence by survey: Methods
----------------------------------------------------
Rather than conducting a comprehensive behavioral review across many genera, as Reader and colleagues did (see 1.3), we restricted our analysis to a small set of primates and birds which were matched for total neuron count. We then gathered behavioral observations from the academic literature on each species, attempting to draw evidence from all plausibly relevant domains of animal life, and used these to construct a questionnaire for ranking animal intelligence. This was then given to a small, non-random pilot sample, as well as a larger sample of Mechanical Turk workers. In addition to apparent difficulty of behaviors in several behavioral domains, participants were asked to rank the relevance of behavioral domains to intelligence, and this ranking was used to weight the within-domain scores. Where possible, we removed features of descriptions which would have identified an animal as a bird or a primate.
Although far below the standard demanded of well-validated measures of intelligence,[3](https://aiimpacts.org/investigation-into-the-relationship-between-neuron-count-and-intelligence-across-differing-cortical-architectures/#easy-footnote-bottom-3-1294 "(<i>Reviewer’s note) </i>In AI, the current state of the art for estimating distance-to-AGI is to look at the capabilities of various AI systems and use intuition to make a guess at how intelligent they are compared to the imagined AGI. In comparison to this, the methodology shown here is an improvement.") we believe that the aggregated judgments of survey participants can offer some information about an agent’s intelligence due to the moderate correlation of peer-rated intelligence with measured IQ within humans. For instance, Bailey and Mettetal (1977) found that spouses’ ratings had a correlation of 0.6 with scores on the Otis Quick Scoring Test of Mental Ability, while Borkenau and Liebler (1993) found that acquaintances’ ratings had a correlation of 0.3 with test scores. Most impressively, they also found that strangers shown a short video of a subject reading from a script gave ratings of the subject’s intelligence that correlated at 0.38 with the subject’s actual test scores.
The problem of rating human intelligence from impressions is in some ways quite a different one from the rating of an unfamiliar species. One factor that could potentially make judgment of humans easier is that human society rewards intelligence by conferring certain forms of status differentially on those who display greater cognitive ability, in ways that are legible to both close associates (ie spouses) and total strangers. This means that individual raters are already benefiting from the aggregated judgments of many past raters (indeed, these positional signals may constitute the majority of evidence in low information situations like acquaintanceship). Additionally, humans have a natural point of reference for the behavior of other humans, and this familiarity probably allows much more accurate comparisons.
However, judgment of other humans may also suffer from several disadvantages that judgment of nonhuman animals does not. Because humans in the same social group often occupy a relatively narrow range of the intelligence distribution, raters are asked to distinguish between differences in behavior that are small in absolute terms. For example, in the studies cited above, samples were drawn from college populations, which are famously range-restricted. Furthermore, raters of humans likely do not have the full range of behavior available to draw evidence from when considering strangers, acquaintances, or even spouses. In contrast, we attempted to capture all potentially relevant behavioral domains in data collection for our survey. Finally, as each others’ main social competitors, humans probably have stronger conflicts of interest in evaluating the intelligence of other humans, and thus may be disincentivized to make completely honest judgments.
Overall, we expect our methodology to produce weaker results than what is possible for raters of human subjects, but not radically so. It should be noted that, because of the scarcity of psychometric data for the species studied, we were not able to verify a correlation with other measures of intelligence. However, it would be possible to validate some version of this methodology with species for which psychometric data does exist (see 4.2).
### 2.1 Study object selection
We chose to study four animals: one larger-brained specimen of each of bird and primate, and one smaller-brained specimen of each. Having already established a strong relationship between brain size and intelligence within architecture types (see [here](https://docs.google.com/document/d/1xBZbgz4hY4F31o52SHquERvhnr99dMDhIHDeS9xmyok/edit?usp=sharing)), varying both architecture type and size allowed us to consider the degree to which one architecture type consistently outperformed the other–for instance, if the smaller version of one architecture outperformed both smaller and larger versions of the other architecture, this would more strongly suggest superiority due to structure than would a performance difference in two architectures of similar size.
Since we were limited to only those species in which [neuron count is known](https://en.wikipedia.org/wiki/List_of_animals_by_number_of_neurons), and where there is overlap between birds and primates, we had only five primates to choose from, three of which had few instances of behavioral reports (the Northern greater galago, *Otolemur garnettii*; the common marmoset, *Callithrix jacchus*; and the gray mouse lemur, *Microcebus murinus*).
Of the remaining primates, the squirrel monkey (*Saimiri sciureus*) was the larger-brained, with 3.2 billion neurons. Only one bird, the blue and yellow macaw (*Ara ararauna*) was reported as having a similarly large number of neurons, at 3.1 billion. The smaller-brained primate, the owl monkey (*Aotus trivirgatus*) has less than half this number of neurons at 1.5 billion, and was matched by both the grey parrot (*Psittacus erithacus*) at 1.6 billion and a corvid, the rook (*Corvus frugilegus*), at 1.5 billion. Because of the close evolutionary relationship between the two selected primates (~30 million years divergence time for *Saimiri* and *Aotus*, according to [TimeTree](http://www.timetree.org/)), we chose to focus on the parrots, who share a similar evolutionary relationship (~30 million years divergence time for *Ara* and *Psittacus*, versus ~80 million years for *Ara* and *Corvus*).
It was expected that the factor of two difference in neuron count between the larger- and smaller-brained samples would be substantial enough to provide some signal despite the noisy nature of behavioral data and analysis, without being so enormous as to render the results trivial. Supposing the relationship between intelligence and neuron count scaled logarithmically, the difference between our sample would be somewhat smaller than the difference between humans and chimpanzees, who differ by a factor of three. (In absolute terms, the neuron count difference is more comparable to neuron count differences between individual humans.) However, it is worth noting that, in our analysis of primate intelligence from lab tests, a factor of two difference was approximately the lower bound for reliably producing a difference in measured intelligence.
Because the features of a single species are often studied unevenly, we improved our coverage of the behavioral spectrum by broadening data collection to include all species in a genus. This is a common practice in the study of animal behavior, generally poses fewer problems than groupings at higher taxa, and prevented us from having to search multiple species names in cases where these had changed in the last century. Furthermore, although brain sizes varied somewhat within genera, the size distribution of the smaller-brained genera (*Aotus* and *Psittacus*) had little to no overlap with that of the larger-brained genera (*Saimiri* and *Ara*). Species in each genus with available brain size data are shown in the table below. It is probably the case that not all species listed in the table were represented in our data, and that some species were overrepresented within their genus, however in many cases the exact species was not specified in the source.
| | | |
| --- | --- | --- |
| **Genus** | **Species/sample** | **Brain mass (g)** |
| *Aotus* | *trivirgatus* (n = 2) | 15.7 |
| | *trivirgatus* (other sources, n = 288) | 17.2 (SD = 1.6) |
| | *azarai* (n = 6) | 21.1 |
| | *lemurinus* (n = 34) | 16.8 |
| *Saimiri* | *sciureus* (n = 2) | 30.2 |
| | *sciureus* (other sources, n = 216) | 24.0 (SD = 2.0) |
| | *boliviensis* (n = 3) | 25.7 |
| | *oerstedii* (n = 81) | 21.4 |
| *Psittacus* | *erithacus* (Olkowicz sample, n = 2) | 8.8 |
| | *erithacus* (other sources, n = 1) | 6.4 |
| *Ara* | *ararauna* (Olkowicz sample, n = 1) | 20.7 |
| | *ararauna* (other sources, n = 20) | 17.0 |
| | *chloropterus* (n = 7) | 22.2 |
| | *hyacinthus* (n = 12) | 25.0 |
| | *rubrogenys* (n = 4) | 12.1 |
### 2.2 Behavioral data collection
For each genus, we searched English language journals for behavioral observations demonstrating learning, behavioral flexibility, problem-solving, social communication, and other traits that imply intelligence. We excluded observations that involved training or interaction with humans (such as [the Alex studies](https://en.wikipedia.org/wiki/Alex_(parrot))).
A problematic element of this type of behavioral study is the disproportionate research effort focused on certain species over others, and in certain domains of behavior. While none of the animals studied had an especially large representation in the literature, *Aotus*, *Ara* and *Psittacus* were generally less well represented than *Saimiri*. In the case of *Psittacus*, a very large proportion of our data was drawn from two sources by a single author. Additionally, conventions regarding the way in which behavior was studied and which details of behavior were considered salient seemed to differ somewhat between ornithologists and primatologists. For instance, while the vocal repertoire and functional significance of vocalizations were frequently a topic of great interest to primatologists, at least in our sample, vocal communication was given a much more casual treatment by ornithologists. Therefore, our data may cause primates and birds to appear to have more qualitative differences in cognitive ability than actually exist.
In our analysis, we make no explicit attempt to correct for these differences in research effort, but do indicate areas of disproportionately high or low coverage of a species, and recommend that the reader bear these in mind when interpreting our results.
After collection, the behavioral observations were sorted into eight functional categories, including three which primarily involved interaction with the environment (tool use, navigation/range, and shelter selection), and five involving social interaction (group dynamics, mate dynamics, care of young, play, and predation prevention). For the accompanying data for each genus, see [S1](https://docs.google.com/document/d/1rVBHRFtIZCb0rh84O5D0iUKBCa2T4BB1OYBpl3OY9Os/edit?usp=sharing). Below are full descriptions of the eight behavioral categories.
#### 2.2.1 Tool use
Tool use involves the manipulation of an intermediate object to affect a final object. In more sophisticated instances of this behavior, the intermediate object is modified from its original form to better serve its intended purpose. Some degree of tool use is widely reported among great apes and certain corvids, and is seldom seen in “lower” animals (Smith & Bentley-Condit, 2010). Tool use may draw on cognitive abilities such as planning, means-end reasoning, spatial or mechanical reasoning, and creativity. (However, it cannot be assumed that apparent tool use demonstrates any of these abilities–some simple animals can use objects as “tools” in a highly inflexible, presumably hard-coded way which requires no learning.)
Despite an extensive search, examples of tool use in the wild (or a wild-mimicking environment) were not found for either *Aotus* or *Psittacus*. However, since at least one of these animals (*Psittacus*) can display tool-using behaviors in environments with frequent human contact (for instance, in a laboratory or pet environment) (Janzen, Janzen, & Pond, 1976), it’s unlikely that that these animals have no capacity at all for developing tool use. Therefore, other explanations for the lack of tool use in the wild should be considered. For one, both species are somewhat more neophobic than *Saimiri* and *Ara*, and thus are less likely to interact with unfamiliar objects frequently enough to develop a use for them. Furthermore, both species are substantially less well-studied in the wild than *Saimiri* (but not *Ara*), and may simply use tools too infrequently or inconspicuously to be noticed.
However, because of its relative rarity, spontaneous tool use is often taken to be “absent until proven present” in an animal species, and we have adhered to this convention in the present study. Readers who disagree with this approach may regard the scores of *Aotus* and *Psittacus* on this metric as a lower bound.
#### 2.2.2 Navigation/range
The range and territory size of an animal are how far it typically travels on a day-to-day basis, and the total area in which its ranging happens, respectively. Since an animal that travels more distantly will encounter more different environments than one that travels less distantly, larger ranges or territory sizes could signal more behavioral flexibility. Additionally, large ranges or variable routes may be more taxing on memory.
Relatively little information was available in this category for *Ara* and *Psittacus*. One might also expect that the skills required for navigation on land would differ substantially from those required for air navigation. In the final version of the survey, we consolidated this category with the following category.
#### 2.2.3 Shelter selection
Where an animal chooses to rest or nest is one of the most frequent decisions it makes, and for prey animals may be one of the more important for survival. When searching for shelter, some optimization criteria may place large demands on perceptual or planning abilities, or on memory.
In the final version of the survey, we consolidated this category with the category above. While neither category alone was judged by participants to contain a large amount of evidence for intelligence, we hoped that combining the two would improve the signal and balance a survey heavy on social behaviors.
#### 2.2.4 Group dynamics
The dynamics of group interaction vary dramatically between species, and frequently even within species in different geographic locations. Social group size of non-herding animals (that is, animals that do not affiliate with conspecifics merely to reduce predation risk) is thought to be correlated with intelligence, and some theories of the evolution of higher intelligence implicate social competition or cooperation as a primary driver (Dunbar, 1998). Furthermore, the range and flexibility of an animal’s vocal or visual communication may indicate the level of complexity of the species’ social life. Often, animals that have close or important relationships with their conspecifics engage in social grooming behaviors.
Due to the amount and complexity of evidence that fell into this category, it was particularly difficult to consolidate these behaviors into a truly representative description of each species. In the final version of the survey, this category was consolidated into a new category, “Social dynamics”.
#### 2.2.5 Mate dynamics
Mate dynamics includes sexual and pair bonding behavior, as well as behaviors relevant to sexual competition. Some examples of behavior that falls into this category are courtship behaviors, social grooming between mates, and joint territorial displays. Some pairbonded animals, particularly birds, engage in the majority of their social interactions with a mate, rather than with group members (Luescher, 2006).
In the final version of the survey, this category was consolidated into the category “Social dynamics”.
#### 2.2.6 Care of young
As well as being an important social relationship in some species of animals, parent/offspring interaction during development generally holds clues about the degree to which learning influences an animal’s behavior, as well as whether an animal participates in social learning (that is, learning by mimicry or emulation of conspecifics) or trial-and-error learning. Longer development times and higher parental investment typically correlate with learning ability in a species.
*Aotus* was not included in this comparison due to a lack of information. *Psittacus* and *Ara* had very poor representation in the literature compared to *Saimiri*. However, the category was retained due to its consistently high rating on the importance score.
#### 2.2.7 Play
Play behavior is essentially a nonfunctional, simulated version of a functional behavior found in adult animals’ usual repertoire, and is more often seen in juvenile animals. Play probably exists to facilitate learning and practice of necessary skills, especially social ones. Play fighting is a very common form of play in social species.
Participants in our early Mechanical Turk sample did not find this category very informative, and indeed it is more a correlate of (or precursor to) intelligent behavior than intelligent in itself. It was therefore removed from the final version of the survey, although some details were preserved in the “Social dynamics” category.
#### 2.2.8 Predation prevention
Animals evade predation through individual precautionary actions, threat signalling, and sometimes group coordination. Since offspring are both highly valuable and also more vulnerable to predation, much of the behavior in this category centers around defense of the nest. Associations between threat types and the amount of alarm appropriate may be learned to a greater or lesser degree in different species, as well as the proper form of the threat signal in the animal’s social group. Furthermore, threats may be classed into few or many types, facilitating greater or lesser nuance in response actions.
Participants in our early Mechanical Turk sample did not find this category very informative, and it was not easily subsumable into “Social dynamics”, so this category was struck from the final version of the survey.
### 2.3 Survey construction and procedures
We synthesized the reports from each category into a representative summary of a species’ behavior in that domain. Where possible, this included any details that might indicate the degree to which behaviors were learned, demonstrated flexibility across different environmental conditions, or were apparently supported by particular cognitive strategies. The summaries were then used to construct a questionnaire which asked participants to rate the apparent intelligence of behaviors against other behaviors in that same category. Afterward, participants were asked which categories they thought contained the most evidence about intelligence, on a scale of one to five. The questionnaire was given to a small random sample of Mechanical Turk workers (*n = 12*), as well as a small nonrandom panel composed of myself, Paul Christiano, Finan Adamson, Carl Shulman, Chris Olah and Katja Grace. Later, the questionnaire was condensed into four sections (tool use, navigation/shelter selection, social dynamics, and care of young) and given to a larger sample of Mechanical Turk workers (*n = 104*).
Because the term “intelligence” is somewhat value-laden and tends to have many idiosyncratic meanings attached to it, we chose to use the word “cognitive complexity” in its place. The hope was that this would reduce conflation with “rationality” or “adaptiveness”, which are both common lay misunderstandings of the term. We also attempted to reduce bias in survey responses by blinding participants to properties not directly relevant to the behaviors being described (including brain size and, wherever possible, membership in the bird or primate class).
#### 2.3.1 Pilot survey
The pilot survey included all eight categories of behavior, as well as longer and more detailed summaries. Mechanical Turk participants were selected through the platform [Positly](https://www.positly.com/), and the survey was administered using [Google Forms](https://www.google.com/forms/about/). Participants were asked to rate the behaviors presented on a 10 point scale against others in the same category, *not* against behaviors that had been presented in previous categories, and were given the option of providing commentary. Participants were also asked to rate categories against each other for evidence of intelligence on a five point scale. All questions from this version can be found in [S1](https://docs.google.com/document/d/1rVBHRFtIZCb0rh84O5D0iUKBCa2T4BB1OYBpl3OY9Os/edit?usp=sharing), and participant responses can be found in [S2](https://docs.google.com/spreadsheets/d/11aZRnx9z-4LHPhi4DQncegx8kLV1GkV8AFMidhshtMk/edit?usp=sharing).
Mechanical Turk data from this round of the survey was used to inform the abridgment of the final version. In particular, we removed or consolidated sections that had been rated by participants as less important, and adjusted the wording or level of detail on questions that seemed unclear to participants.
#### 2.3.2 Final survey
The final version of the survey included four categories: tool use, navigation/shelter selection, social dynamics, and care of young. Social dynamics collapsed group dynamics, mate dynamics, and play. This version of the survey was administered via [GuidedTrack](https://www.guidedtrack.com/), and added mandatory wait times to pages as well as a free response question assessing comprehension of the task instructions. Analysis was restricted to participants who were not rated as having poor comprehension (*n = 77*). All questions in this version can be found in [S1](https://docs.google.com/document/d/1rVBHRFtIZCb0rh84O5D0iUKBCa2T4BB1OYBpl3OY9Os/edit?usp=sharing), and participant responses can be found in [S2](https://docs.google.com/spreadsheets/d/11aZRnx9z-4LHPhi4DQncegx8kLV1GkV8AFMidhshtMk/edit?usp=sharing).
3 Estimating animal intelligence by survey: Results
---------------------------------------------------
#### 3.0.1 Pilot survey
We will present only the results from the small panel here, however the full data from this section can be found in the supplementary file.
Tool use, Group dynamics, and Play emerged as the most important categories, according to participant rating, with Navigation & range and Shelter selection rated as least important. Across most categories, especially those rated as more important, there was strong agreement that *Samiri*, *Ara* and *Psittacus* outranked *Aotus*. There was also reasonably good agreement that *Saimiri* and *Ara* outranked *Psittacus*. Finally, *Saimiri* generally outranked *Ara*, though the effect was less strong than in the other comparisons.
[](http://aiimpacts.org/wp-content/uploads/2019/02/fig6-animal-survey-page.png)
[](http://aiimpacts.org/wp-content/uploads/2019/02/fig5-animal-survey-page.png) [](http://aiimpacts.org/wp-content/uploads/2019/02/fig4-animal-survey-page.png) [](http://aiimpacts.org/wp-content/uploads/2019/02/fig3-animal-survey-page.png)
**Figure 2:** *Fields without scores (“Care of young” for Aotus) indicate that insufficient data was found to compose a behavioral description for that animal.*
Given this data, participants appeared to find our small-brained primate, *Aotus*, to display the least intelligent behavior, and found our large-brained primate, *Saimiri*, to display the most intelligent behavior, although within a similar range to our large-brained bird, *Ara*.
#### 3.0.2 Final survey
Among all four categories, participants reported that our descriptions of tool use provided the most evidence for intelligence, especially compared to the least informative category (Navigation and shelter selection). This aligned well with the pattern of answers within the category of Tool use, where there was strong agreement among participants on the rank order of Tool use behaviors, and the differences between Tool use behavior means were the largest of any category. The two larger-brained genera, *Saimiri* and *Ara*, were clear winners in this case, with participants reporting no significant difference between these two.
Social dynamics and Care of young were not clearly distinguishable from each other by importance rating, however participants responded quite differently to the evidence presented in these categories. All included genera (*Saimiri*, *Ara* and *Psittacus*) obtained about the same average score for Care of young, with no significant differences between them. However, for Social dynamics there were clear differences between the smaller-brained genera, *Aotus* and *Psittacus*, as well as the larger-brained bird and smaller-brained primate. Considering the borderline-significant comparison between *Saimiri* and *Ara* in this category (*p=0.06*), it would appear that participants rated birds slightly higher overall than primates on Social dynamics. Finally, Navigation and shelter selection was judged least important, but there were nonetheless clear differences in behavior scores between birds and primates, with primates outscoring birds, and no significant differences between sizes.
| | | | | | | |
| --- | --- | --- | --- | --- | --- | --- |
| | | | Differences in means | | | |
| Tool use vs Navigation / Shelter selection | Tool use vs Social dynamics | Tool use vs Care of young | Navigation / Shelter selection vs Social dynamics | Navigation / Shelter selection vs. Care of young | Social dynamics vs Care of young | |
| **1.0 +-0.2 (p<0.001)** | **0.6 +-0.2 (p<0.001)** | **0.8 +-0.2 (p<0.001)** | **-0.4 +-0.2 (p<0.01)** | -0.2 +-0.2 (p=0.13) | 0.2 +-0.2 (p=0.32) | |
| | | | | | | |
| --- | --- | --- | --- | --- | --- | --- |
| | Saimiri vs Ara | Saimiri vs Aotus | Saimiri vs Psittacus | Ara vs Aotus | Ara vs Psittacus | Aotus vs Psittacus |
| Tool use | 0.1 +-0.4 (p=0.79) | **3.7 +-0.4 (p<0.001)** | (see Saimiri vs Aotus) | **3.6 +-0.4 (p<0.001)** | (see Ara vs Aotus) | NA |
| Navigation / Shelter selection | **1.0 +-0.4 (p<0.01)** | -0.5 +-0.4 (p=0.29) | 0.5 +-0.4 (p=0.13) | **-1.5 +-0.4 (p<0.001)** | -0.5 +-0.3 (p=0.15) | **1.0 +-0.4 (p<0.01)** |
| Social dynamics | -0.8 +-0.4 (p=0.06) | 0.6 +-0.4 (p=0.16) | -0.3 +-0.4 (p=0.51) | **1.4 +-0.4 (p<0.001)** | 0.5 +-0.4 (p=0.19) | **-0.9 +-0.4 (p<0.01)** |
| Care of young | 0.1 +-0.3 (p=0.83) | Not measured | -0.3 +-0.3 (p=0.38) | Not measured | -0.4 +-0.3 (p=0.27) | Not measured |
[](http://aiimpacts.org/wp-content/uploads/2019/02/fig7-animal-survey-page.png)
[](http://aiimpacts.org/wp-content/uploads/2019/02/fig8-animal-survey-page.png)[](http://aiimpacts.org/wp-content/uploads/2019/02/fig9-animal-survey-page.png) [](http://aiimpacts.org/wp-content/uploads/2019/02/fig11-animal-survey-page.png)
**Figure 3:** *Fields without scores (“Care of young” for Aotus) indicate that insufficient data was found to compose a behavioral description for that animal.*
Overall, participants in this sample seemed to find the largest and most important differences between the two large- and two small-brained animals, not between the two primates and two birds. However, they did rate the birds slightly higher on social behaviors, while the primates were rated slightly higher on Navigation and shelter selection.
It’s possible that since the Tool use section compared instances of a behavior with the absence of a similar behavior, differences in scoring may have been inflated, relative to comparison between a tool-using behavior and an unrelated behavior in a non-tool using animal. Indeed, it is probable that the non-tool using animals in our sample have some problem-solving behavior akin to tool use in their repertoire, which was simply subtle enough to go unremarked upon by investigators. This sort of behavior could be seen as a precursor to the development of spontaneous complex tool use, and is probably what enables captive *Psittacus* to learn to solve tool-type problems in a laboratory setting. It is nonetheless striking that both larger-brained genera had strong evidence of spontaneous tool use, being either a regular component of its day-to-day life or an impressively novel use for an unfamiliar object, while no reports of the smaller-brained genera in the wild mentioned comparable problem-solving behaviors.
4 Discussion
------------
### 4.1 Conclusion
In all iterations, we found the survey method of estimating animal intelligence to be quite noisy, without strong agreement on the importance of some categories, or on the rankings of species within some categories. This is unsurprising, since participants were given descriptions of behaviors stripped of much potentially relevant context, in the interest of time, and were not experts in either intelligence or animal behavior. However, there was broad agreement between our participants in both versions of the survey on some high-level conclusions, namely: a) that tool use as presented was a particularly important source of evidence; and b) that, when rankings were weighted by importance as judged by participants, the two larger-brained animals outscored to the two smaller-brained animals.
Because of the small number of genera represented in our survey, it is difficult to draw strong conclusions about the relative contributions of neuron count, architecture, and other factors to intelligence. However, our data do not support the hypothesis that one tissue architecture is greatly superior to the other as a rule, and weakly supports the hypothesis that birds and primates with similar neuron numbers have similar cognitive abilities. In particular, given the behaviors described in our survey, participants were not able to systematically distinguish the two birds from the two primates across all categories, but were substantially more able to distinguish the small-brained animals from those with twice as many brain neurons.
We also did not see strong evidence of specialized intelligence that differed between the groups. That is, the two birds in our study seemed not clearly better or worse at any particular kinds of cognitively-demanding behaviors than the two primates. However, this is not a claim that none of the *species* involved have specialized abilities. We could easily imagine it being the case, for example, that if one were to place an owl monkey brain or a grey parrot brain in the body of an ostrich, both would perform similarly well at the *cognitive* challenges presented by ostrich life, while an owl monkey brain would not do nearly as well as a grey parrot brain at living the life of a grey parrot.
### 4.2 Implications and future directions
We hope our suggestive–if inconclusive–results spark greater interest in the highly neglected field of comparative animal intelligence. In particular, the further development and use of validated protocols for animal intelligence measurement seems to be a significant bottleneck to further progress. Furthermore, the gold standard of human psychometrics may not be a feasible model for animal intelligence measurement, given the prohibitive expense an analogous program in animals would incur (if traditional psychometric methods could even be applied usefully to most animals).
Our surveying method may represent an inexpensive alternative that can produce useable if imperfect results. Although we believe it has reasonably good theoretical support, the method is nonetheless unvalidated and would surely require refinement. To that end, future studies may consider applying our method to species where the rank order is more certain, such as humans and chimpanzees, or the collection of primate species that have been compared by a psychometric battery (see [here](https://docs.google.com/document/d/1xBZbgz4hY4F31o52SHquERvhnr99dMDhIHDeS9xmyok/edit?usp=sharing)).
With regard to the question of avian and primate per-neuron intelligence, our result has limited generalizability due to the small number of genera represented. Even within a broad architecture type, species may still vary in brain characteristics that are relevant to intelligence, and we might expect larger evolutionary distances within Primates or Aves to be reflected in brain differences. Idiosyncratic selective pressures of certain niches likely also have an impact here. In future, it may be fruitful to compare other orders of bird, such as Passeriformes (and especially Corvidae), with primates. As a particularly evolutionarily recent clade made up of strong ecological generalists, Corvidae might have developed structural improvements allowing them to excel in tool use and other cognitive abilities relative to other animals in their brain size class, and indeed there are at least many anecdotal reports of spontaneous tool use in wild corvids. There may also be interesting brain structure differences between New World primates, like the two represented in this study, and Old World primates.
Several limitations to the applicability of any bird-primate comparisons to the broader question surrounding architecture flexibility should be noted. Firstly, all brain structures other than the cerebral cortex are shared between birds and primates. Although these structures only account for a minority of brain volume, they could nonetheless perform some important precursor function to higher processing, such that an animal with a differently organized version could not perform as well cognitively, no matter their cortical architecture. This possibility seems less likely in light of the existence of cognitively advanced cephalopods like octopi, who are not vertebrates and therefore do not have a spinal cord or any other brain structures in common with birds and mammals.
Another issue pertains to scaling. While bird architectures clearly have the capacity to scale to the size of the smaller primate brains, no larger bird architectures have yet developed. This could be due to a number of limiting factors, including size limits imposed by the need to fly, a lack of adjacent niches that would support larger brains, or inherent randomness in the trajectory of brain evolution across lineages. However, it could also represent an upper bound on the scalability of bird-type cortical architecture.
5 Contributions
---------------
*Research, analysis and writing were done by Tegan McCaslin. Editing and feedback were provided by Katja Grace and Justis Mills. Feedback was provided by Daniel Kokotajlo and Carl Shulman.*
6 Bibliography
--------------
Bailey, R. C., & Mettetal, G. W. (1977). PERCEIVED INTELLIGENCE IN MARRIED PARTNERS. *Social Behavior and Personality: An International Journal*, 5(1), 137–141.<https://doi.org/10.2224/sbp.1977.5.1.137>
Borkenau, P., & Liebler, A. (1993). Convergence of stranger ratings of personality and intelligence with self-ratings, partner ratings, and measured intelligence. *Journal of Personality and Social Psychology*, *65*(3), 546–553.<https://doi.org/10.1037/0022-3514.65.3.546>
Burkart, J. M., Schubiger, M. N., & van Schaik, C. P. (2017). The evolution of general intelligence. *Behavioral and Brain Sciences*, *40*.<https://doi.org/10.1017/S0140525X16000959>
Carroll, J. B. (1997). Psychometrics, intelligence, and public perception. *Intelligence*, *24*(1), 25–52.<https://doi.org/10.1016/S0160-2896(97)90012-X>
Dunbar, R. I. M. (1998). The social brain hypothesis. *Evolutionary Anthropology: Issues, News, and Reviews*, *6*(5), 178–190. [https://doi.org/10.1002/(SICI)1520-6505(1998)6:5<178::AID-EVAN5>3.0.CO;2-8](https://doi.org/10.1002/(SICI)1520-6505(1998)6:5%3C178::AID-EVAN5%3E3.0.CO;2-8)
Güntürkün, O., Stacho, M., & Strockens, F. (2017). The brains of reptiles and birds. In J. H. Kaas (Ed.), *Evolution of Nervous Systems* (2nd ed., Vol. 1, pp. 171–221). Oxford, United Kingdom: Academic Press.
Herculano-Houzel, S., Collins, C. E., Wong, P., & Kaas, J. H. (2007). Cellular scaling rules for primate brains. *Proceedings of the National Academy of Sciences*, *104*(9), 3562–3567.<https://doi.org/10.1073/pnas.0611396104>
Herculano-Houzel, Suzana. (2009). The human brain in numbers: a linearly scaled-up primate brain. *Frontiers in Human Neuroscience*, *3*.<https://doi.org/10.3389/neuro.09.031.2009>
Herculano-Houzel, Suzana. (2011). Scaling of Brain Metabolism with a Fixed Energy Budget per Neuron: Implications for Neuronal Activity, Plasticity and Evolution. *PLoS ONE*, *6*(3), e17514.<https://doi.org/10.1371/journal.pone.0017514>
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Kaas, J. H. (Ed.). (2017). *Evolution of Nervous Systems* (2nd ed.). Oxford, United Kingdom: Academic Press.
Luescher, A. U. (Ed.). (2006). *Manual of parrot behavior* (1st ed). Ames, Iowa: Blackwell.
Naumann, R., & Laurent, G. (2017). Function and evolution of the reptilian cerebral cortex. In J. H. Kaas (Ed.), *Evolution of Nervous Systems* (2nd ed., Vol. 1, pp. 491–518). Oxford, United Kingdom: Academic Press.
Olkowicz, S., Kocourek, M., Lučan, R. K., Porteš, M., Fitch, W. T., Herculano-Houzel, S., & Němec, P. (2016). Birds have primate-like numbers of neurons in the forebrain. *Proceedings of the National Academy of Sciences*, *113*(26), 7255–7260.<https://doi.org/10.1073/pnas.1517131113>
Puelles, L., Sandoval, J., Ayad, A., del Corral, R., Alonso, A., Ferran, J., & Martinez-de-la-Torre, M. (2017). The pallium in reptiles and birds in light of the updated tetrapartite pallium model. In J. H. Kaas (Ed.), *Evolution of Nervous Systems* (2nd ed., Vol. 1, pp. 519–555). Oxford, United Kingdom: Academic Press.
Reader, S. M., Hager, Y., & Laland, K. N. (2011). The evolution of primate general and cultural intelligence. *Philosophical Transactions of the Royal Society B: Biological Sciences*, *366*(1567), 1017–1027.<https://doi.org/10.1098/rstb.2010.0342>
Reiner, A., Yamamoto, K., & Karten, H. J. (2005). Organization and evolution of the avian forebrain. *The Anatomical Record Part A: Discoveries in Molecular, Cellular, and Evolutionary Biology*, *287A*(1), 1080–1102.<https://doi.org/10.1002/ar.a.20253>
Smith, & Bentley-Condit, V. (2010). Animal tool use: current definitions and an updated comprehensive catalog. *Behaviour*, *147*(2), 185-32A.<https://doi.org/10.1163/000579509X12512865686555>
Roth, G., & Dicke, U. (2005). Evolution of the brain and intelligence. *Trends in Cognitive Sciences*, *9*(5), 250–257.<https://doi.org/10.1016/j.tics.2005.03.005>
Shulman, C., & Bostrom, N. (2012). How Hard Is Artificial Intelligence? Evolutionary Arguments and Selection Effects. *Journal of Consciousness Studies*, *19*(7–8), 103–130. |
7c6e2e57-d21c-4942-ac42-22d2008a0cd1 | trentmkelly/LessWrong-43k | LessWrong | King Under The Mountain: Adventure Log + Soundtrack
With the help of many dedicated Less Wrongers (players muflax, Karl, Charlie, and Emile; musicians Mike Blume and Alicorn, technical support Ari Rahikkala) we have successfully completed what is, as far as I know, the first actual Dungeons and Discourse adventure anywhere. Except we're not calling it that, because I don't have the rights to use that name. Though it's not precisely rationality related, I hope it is all right if I post a summary of the adventure by popular demand.
Also, at some point it turned into a musical. The first half of the songs are only available as lyrics at the moment, but Alicorn and MBlume very kindly produced the second half as real music, which I've uploaded to YouTube and linked at the bottom of this post (skip to it now).
THE ADVENTURE
BACKGROUND
The known world has many sects and religions, but all contain shadowy legends of two primeval deities: Sophia, Goddess of Wisdom; and Aleithos, God of Truth. When Sophia announced her plan to create thinking, rational beings, Aleithos objected, declaring that they would fall into error and produce endless falsehoods. Sophia ignored her brother's objections and created humankind, who named the world after their goddess-mother. But Aleithos' fears proved well-founded: humankind fell into error and produced endless falsehoods, and their clamor drove the God of Truth insane.
The once mighty Aleithos fell from heaven, and all of his angelic servants turned into Paradox Beasts, arachnoid monstrosities that sought and devoured those who challenged the laws of logic. Over centuries, most of the Paradox Beasts were banished, but Aleithos himself remained missing. And though thousands of seekers set off to all the corners of the world in search of Truth, the Mad God keeps his own counsel, if He still even exists at all.
The Truth God's madness had one other effect; the laws of physics, once inviolable, turned fluid, and those sufficiently advanced in the study of Truth gained apparently magical a |
3940e4ec-d2ef-4694-b42e-9274f407cd75 | trentmkelly/LessWrong-43k | LessWrong | Big-endian is better than little-endian
This is a response to the post We Write Numbers Backward, in which lsusr argues that little-endian numerical notation is better than big-endian.[1] I believe this is wrong, and big-endian has a significant advantage not considered by lsusr.
Lsusr describes reading the number "123" in little-endian, using the following algorithm:
* Read the first digit, multiply it by its order of magnitude (one), and add it to the total. (Running total: ??? one.)
* Read the second digit, multiply it by its order of magnitude (ten), and add it to the total. (Running total: ??? twenty one.)
* Read the third digit, multiply it by its order of magnitude (one hundred), and add it to the total. (Arriving at three hundred and twenty one.)
He compares it with two algorithms for reading a big-endian number. One is using the same process as for a little-endian number, but from right to left. I agree with him that this is worse than the little-endian algorithm, because it is easier to read a number in the same direction as the text that surrounds it, which in English is from left to right.
The other big-endian algorithm is the one I observe myself as usually using. For "321", it is:
* Count the digits (three), and convert that into an order of magnitude (one hundred). (Running total: ??? hundred ???.)
* Read the first digit, multiply it by its order of magnitude (one hundred), and add it to the total. (Running total: three hundred ???.)
* Read the second digit, multiply it by its order of magnitude (ten), and add it to the total. (Running total: three hundred and twenty ???.
* Read the third digit, multiply it by its order of magnitude (one), and add it to the total. (Arriving at three hundred and twenty one.)
The point raised by lsusr against the big-endian algorithm is that we must count a number's digits before we can start reading them. He doesn't say explicitly why he dislikes this, but I can see three reasons:
1. It makes the algorithm more complex.
2. It makes the algori |
db5ea800-93b7-4f18-97bd-1f2d1cdf53a1 | trentmkelly/LessWrong-43k | LessWrong | Effective diversity
1. Two diversity problems
Here are two concerns sometimes raised in Effective Altruist circles:
1. Effective Altruists are not very diverse—they are disproportionately male, white, technically minded, technically employed, located in a small number of rich places, young, smart, educated, idealistic, inexperienced. Furthermore, because of this lack of diversity, the community as a whole will fail to know about many problems in the world. For instance, problems that are mostly salient if you have spent many years in the world outside of college, if you live in India, if you work in manufacturing, if you have normal human attitudes and norms.
2. When new people join the Effective Altruism community and want to dedicate a lot of their efforts to effectively doing good, there is not a streamlined process to help them move from any of a wide range of previous activities to an especially effectively altruistic project, even if they really want to. And it gets harder the further away the person begins from the common EA backgrounds: a young San Franciscan math PhD working in tech and playing on the internet can move into research or advocacy or a new startup more easily than an experienced high school principal with a family in Germany can, plus it’s not even clear that the usual activities are a good use of that person’s skills. So less good is done, and furthermore less respect is accorded to such people than they arguably deserve, and so more forbearance is required on their part to stick around (possibly leading to problem 1).
2. Two divergent evaluations of diversity
These concerns are kind of opposite. Which suggests that if they ran into each other they might explode and disappear, at least a bit.
The first concern is based on a picture where people doing different things from the rest of the EA community are an extremely valuable asset, worthy of being a priority to pursue (with effort that could go to figuring out how to stop AI, or paying for malaria nets, |
e52cccbd-fe64-4a9d-a2e6-354120e97fca | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | Join the interpretability research hackathon
**TLDR**; Participate online or in-person in London, Aarhus, and Tallinn on the weekend 11th to 13th November in a fun and intense [AI safety research hackathon](https://alignmentjam.com) focused on interpretability research. We invite mid-career professionals to join but it is open for everyone (also no-coders) and we will create starter code templates to help you kickstart your team’s projects. [**Join here**](https://itch.io/jam/interpretability).
Below is an FAQ-style summary of what you can expect *(navigate it with the table of contents on the left)*.
What is it?
-----------
The Interpretability Hackathon is a **weekend-long event** where you participate in teams (1-6) to create interesting and fun research. You submit a PDF report that summarizes and discusses your findings in the context of AI safety. These reports will be judged by our panel and you can win up to $1,000!
It runs from **11th Nov to 13th Nov** (in two weeks) and you’re welcome to join for a part of it (see further down). We get an interesting talk by an expert in the field and hear more about the topic.
Everyone can participate and we encourage you to join especially if you’re **considering AI safety from another career**. We prepare templates for you to start out your projects and you’ll be surprised what you can accomplish in just a weekend – especially with your new-found friends!
*Read more about how to join, what you can expect, the schedule, and what previous participants have said about being part of the hackathon below.*
Where can I join?
-----------------
You can join the event both in-person and online but everyone needs to make an account and join the jam on the [itch.io page](https://itch.io/jam/interpretability).
The in-person locations include the LEAH offices in London right by UCL, Imperial, King’s College, and London School of Economics ([link](https://www.facebook.com/events/3461487534127612?ref=newsfeed)); Aarhus University in Aarhus, Denmark ([link](https://www.facebook.com/events/784971572582951)), and Tallinn, Estonia ([link](https://www.facebook.com/events/669822281240336/)). The virtual event space is on GatherTown ([link](https://app.gather.town/app/Yhi4XYj0zFNWuUNv/EA%20coworking%20and%20lounge?spawnToken=Zf4qz-a921GBMGU_)).
Everyone should join the Discord to ask questions, see updates and announcements, find team members, and more. [**Join here**](https://discord.gg/3PUSbdS8gY).
What are some examples of interpretability projects I could make?
-----------------------------------------------------------------
You can check out a bunch of interesting, smaller interpretability project ideas on [AI Safety Ideas](https://aisafetyideas.com/?sort=created_at&categories=Interpretability+%26+Explainability) such as [reconstructing the input from neural activations](https://aisafetyideas.com/?idea=129), [evaluating the alignment tax of interpretable models](https://aisafetyideas.com/?sort=created_at&categories=Interpretability+%26+Explainability&idea=15), or [making models’ uncertainty interpretable](https://aisafetyideas.com/project/mechanistic-interpretability?idea=95).
Other examples of practical projects can be to find new ways to visualize features in language models, such as Anthropic has been working on, distilling mechanistic interpretability research, create a demo for a much more interpretable language model, or [map out how a possible interpretable AGI might look with our current lens](https://aisafetyideas.com/?idea=130).
You can also do projects in explainability about how much humans understand why the outputs of language models look the way they do, how humans see attention visualizations, or maybe even the interpretability of humans themselves and take inspiration from the brain and neuroscience.
Also [check out the results](https://www.lesswrong.com/posts/5DsHZidaShW5EM9rz/results-from-the-language-model-hackathon) from the last hackathon to see what you might accomplish during just one weekend. The judges were really quite impressed with the full reports given the time constraint! You can also read the complete projects [here](https://alignmentjam.com/jams#:~:text=Language%20Model%20Hackathon).
### Inspiration
* Redwood Research's interpretability tools: <http://interp-tools.redwoodresearch.org/>
* The activation atlas: <https://distill.pub/2019/activation-atlas/>
* The Tensorflow playground: <https://playground.tensorflow.org/>
* The Neural Network Playground (train simple neural networks in the browser): <https://nnplayground.com/>
* Visualize different neural network architectures: <http://alexlenail.me/NN-SVG/index.html>
Digestible research
* [Distill publication on visualizing neural network weights](https://distill.pub/2020/circuits/visualizing-weights/)
* Andrej Karpathy's ["Understanding what convnets learn"](https://cs231n.github.io/understanding-cnn/)
* [Looking inside a neural net](https://ml4a.github.io/ml4a/looking_inside_neural_nets/)
You can also see more on [the resources page](https://alignmentjam.com/hackathon/interpretability).
Why should I join?
------------------
There’s loads of reasons to join! Here are just a few:
* See how fun and interesting AI safety can be
* Free food the whole weekend!
* Get to know new people who are into ML safety and interpretability
* Win up to $1,000, helping you towards your next Armani handbag
* Get practical experience with ML safety research
* Show the AI safety labs what you are able to do to increase your chances at some [amazing jobs](https://agisf.com/opportunities)
* Get a cool certificate that you can show your friends and family
* Acquire yourself a neat sticker
* Figure out what all these San Franciscan EAs are up to
* Get some proof that you’re actually really cool so you can get that grant to pursue AI safety research that you always wanted
* And many many more… [Come along](https://itch.io/jam/interpretability)!
What if I don’t have any experience in AI safety?
-------------------------------------------------
**Please**[**join**](https://itch.io/jam/interpretability)**!** This can be your first foray into AI and ML safety and maybe you’ll realize that it’s not that hard. Even if you don’t find it particularly interesting, this time you might see it in a new light!
There’s a lot of pressure from AI safety to perform at a top level and this seems to drive some people out of the field. We’d love it if you consider joining with a mindset of fun exploration and get a positive experience out of the weekend.
What is the agenda for the weekend?
-----------------------------------
The schedule runs from 6PM CET / 9AM PST Friday to 7PM CET / 10AM PST Sunday. We start with an introductory talk and end with an awards ceremony. Join the public ICal [here](https://calendar.google.com/calendar/u/0?cid=ZjViYmMzNjlhNDFmZjg5MmY5ZTkxOWJjMGVkMmFlNjRmOTBjMmVjNTMzZDVhMTZhMWNkMjY4ZjU1M2JhMTBlY0Bncm91cC5jYWxlbmRhci5nb29nbGUuY29t).
| | |
| --- | --- |
| CET / PST | |
| Fri 6 PM / 9 AM | Introduction to the hackathon, what to expect, and a talk from an expert. Splitting into teams. |
| Fri 7:30 PM / 10:30 AM | Hacking begins! Free dinner. |
| Sun 2 PM / 5 AM | Final submissions have to be finished. Judging begins. |
| Sun 6 PM / 9 AM | Award ceremony starts and the winning projects are presented. |
| Sun 7 PM / 10 AM | Socializing and dinner. |
I’m busy, can I join for a short time?
--------------------------------------
As a matter of fact, **we encourage you to join** even if you only have a short while available during the weekend!
For the last hackathon, the average work amount was 17 hours and a couple of participants only spent a few hours on their projects as they joined Saturday. Another participant was at an EA retreat at the same time and even won a prize!
**So** **yes,**you can both join without coming to the beginning or end of the event, and you can submit research even if you’ve only spent a few hours on it. We of course still encourage you to come for the intro ceremony and join for the whole weekend.
Wow this sounds fun, can I also host an in-person event with my local AI safety group?
--------------------------------------------------------------------------------------
**Definitely!** It might be hard to make it for this hackathon but we encourage you to join our team of in-person organizers around the world for the next hackathon that we expect to run in **mid December** as a Christmas special!
You can read more about what we require [here](https://alignmentjam.com/running) and the possible benefits it can have to your local AI safety group [here](https://alignmentjam.com/why). **Sign up as a host on the button on**[**this page**](https://alignmentjam.com/running).
What have previous participants said about this hackathon?
----------------------------------------------------------
| | |
| --- | --- |
| “You always learn a lot just going all into a project and it feels like an achievement when you come out the other side and have something cool to show. Definitely worth participating in.” | “The hackathon was a lot of fun to participate in, everyone was great and it was really interesting to spend the time figuring out what kind of risks are associated with language models and when they might occur.” |
| “Intro texts from Buck abt being an AI experimental psychologist really sparked my curiosity and made me more confident in my ability to actually generate good work (I wasn't really into AI safety before the hackathon).” | “I enjoyed the hackathon very much. The task was open-ended, and interesting to manage. I felt like I learned a lot about the field of AI safety by exploring the language models during the hackathon.” |
| “It was a fun way to spend the weekend! I think one of the best parts about it is working in teams - you learn some cooperation skills, as well as get to consider things that you maybe never even thought of before” | |
Where can I read more about this?
---------------------------------
* [The information page](https://alignmentjam.com/resources)
* [Hackathon page](https://itch.io/jam/interpretability)
* [The Alignment Jam website](https://alignmentjam.com)
* [Inspiration list for the interpretability hackathon](https://alignmentjam.com/hackathon/interpretability)
* [The Discord server](https://discord.gg/3PUSbdS8gY)
* [In-person and online events](https://alignmentjam.com/jams)
+ [The GatherTown hacking space](https://app.gather.town/app/Yhi4XYj0zFNWuUNv/EA%20coworking%20and%20lounge?spawnToken=Zf4qz-a921GBMGU_)
+ [London event page](https://www.facebook.com/events/3461487534127612?ref=newsfeed)
+ [Tallinn event page](https://www.facebook.com/events/669822281240336/)
+ [Aarhus event page](https://www.facebook.com/events/784971572582951)
* [The previous hackathon](https://itch.io/jam/llm-hackathon)
+ [Results from the previous hackathon](https://www.lesswrong.com/posts/5DsHZidaShW5EM9rz/results-from-the-language-model-hackathon)
+ [The previous hackathon’s announcement post](https://forum.effectivealtruism.org/posts/npm6mrJQzungTLsKj/black-box-investigations-research-hackathon)
+ [Starter code templates and inspiration list for the previous hackathon](https://alignmentjam.com/hackathon/llm)
Again, sign up here by [clicking “Join jam”](https://itch.io/jam/interpretability) and read more about the hackathons more generally [here](https://alignmentjam.com).
Godspeed, research jammers! |
5385ab36-0e25-4da4-994f-6ea58769d5ff | trentmkelly/LessWrong-43k | LessWrong | Meditation, insight, and rationality. (Part 2 of 3)
This post is the second in a series; you can read the first here.
I have already given a brief overview of what the goal of a particular style of meditation is, and why some individuals in this community might find it beneficial to pursue. The basic structure of this article will be as follows: a brief restatement of my major claims, a highly abridged history of meditation in one Buddhist-associated tradition and of models of the path towards enlightenment (from its ancient Buddhist roots to the modern day), and then a short-but-explicit set of instructions which an interested individual can use to see for themselves whether this style of meditation leads to what I have claimed it does.
My basic claims from Part 1:
* The human mind, by default, involves cognitive processes that are fundamentally defective, and which distort one's views about how things are; the result of such processes is a collection of various delusions.
* Due to the distortions these processes cause, neither introspection nor attempts at rational thought / armchair philosophizing reveal them.
* The distortions these processes produce are so severe that, without training, it is unlikely that one will even be able to conceptualize what they are, or what it would mean for the assertion that one's cognitive processes are distorted in this particular way to be true or false.
* Because of this inability to conceptualize the problem, there are no words I can type which will serve to explain it to you (what I would intend to convey with them is a meaning which you cannot entertain; whatever you think I mean is almost certainly not what I mean). The best I can do is say that it has something to do with the way you think about your 'self'.
* Meditation, a series of attentional and perceptual exercises, can lead to the end of these delusions by fixing the processes which generate them. The end of these delusion is called 'enlightenment.' These delusions are ended in steps; the various steps are |
87994d93-9391-44f6-9b0e-13e538d41470 | trentmkelly/LessWrong-43k | LessWrong | Australian AI Safety Forum 2024
We're excited to announce the inaugural Australian AI Safety Forum, taking place on November 7-8, 2024, in Sydney, Australia. This event aims to foster the growth of the AI safety community within Australia.
Apply now!
Key Details
* Dates: November 7-8, 2024
* Location: Sydney Knowledge Hub, The University of Sydney
* Website: https://aisafetyforum.au
Description
The Australian AI Safety Forum is a two-day interdisciplinary event scheduled for November 2024. This forum, the first of its kind in Australia, aims to discuss perspectives on technical AI safety and governance, and explore Australia’s unique role in the global AI safety landscape. The event will be anchored around the International Scientific Report on the Safety of Advanced AI, highlighting its key content and examining its implications for Australia.
The forum is designed to encourage conversation and education between researchers, policy makers, and industry leaders. As such, we aim to catalyse an Australian community working on problems of technical AI safety and governance. We intend on this becoming a regular event.
For examples of the kinds of topics to be discussed, please see this list.
Who Should Apply
We deliberately intend for participants at the forum to have a broad range of expertise and experience, in particular:
* Researchers in mathematics, computer science, natural sciences and the social sciences
* Policymakers
* Legal experts and public policy professionals
* National security and cybersecurity specialists
* Industry professionals (e.g., finance, telecommunications)
* Students and early-career professionals interested in AI safety and governance
Our primary focus is on helping to build the AI safety community in Australia, and as such we will be prioritising applicants who reside in Australia or who have other connections to Australia (e.g. Australian expats, people who have worked or studied in Australia for a considerable period of time, etc.).
Format
* D |
576a6c51-6c83-4867-8aae-6110b0364796 | trentmkelly/LessWrong-43k | LessWrong | [Link] Death with Dignity by Scott Adams
Over at Scott Adams' Blog you can find a very fine example of using the 'Rationality Engine' to solve the social problem of assisted dying.
|
aa2a1d61-460b-4814-a7b8-4de8326882ae | trentmkelly/LessWrong-43k | LessWrong | Allais Hack -- Transform Your Decisions!
The Allais Paradox, though not actually a paradox, was a classic experiment which showed that decisions made by humans do not demonstrate consistent preferences. If you actually want to accomplish something, rather than simply feel good about your decisions, this is rather disturbing.
When something like the Allais Paradox is presented all in one go, it's fairly easy to see that the two cases are equivalent, and ensure that your decisions are consistent. But if I clone you right now, present one of you with gamble 1, and one of you with gamble 2, you might not fare so well. The question is how to consistently advance your own preferences even when you're only looking at one side of the problem.
Obviously, one solution is to actually construct a utility function in money, and apply it rigorously to all decisions. Logarithmic in your total net worth is usually a good place to start. Next you can assign a number of utilons to each year you live, a negative number to each day you are sick, a number for each sunrise you witness...
I would humbly suggest that a less drastic strategy might be to familiarize yourself with the ways in which you can transform a decision which should make no difference unto decision theory, and actually get in the habit of applying these transformations to decisions you make in real life.
So, let us say that I present you with Allais Gamble #2: choose between A: 34% chance of winning $24,000, and 66% chance of winning nothing, and B: 33% chance of winning $27,000, and 67% chance of winning nothing.
Before snapping to a judgment, try some of the following transforms:
Assume your decision matters:
The gamble, as given, contains lots of probability mass in which your decision will not matter one way or the other -- shave it off!
Two possible resulting scenarios:
A: $24,000 with certainty, B: 33/34 chance of $27,000
Or, less obviously: I spin a wheel with 67 notches, 34 marked A and 33 marked B. Choose A and win $24,000 if the wheel come |
525c70bd-11b8-47d8-9317-23279fbd4615 | trentmkelly/LessWrong-43k | LessWrong | Open Thread July 2019
If it’s worth saying, but not worth its own post, you can put it here.
Also, if you are new to LessWrong and want to introduce yourself, this is the place to do it. Personal stories, anecdotes, or just general comments on how you found us and what you hope to get from the site and community are welcome. If you want to explore the community more, I recommend reading the Library, checking recent Curated posts, and seeing if there are any meetups in your area.
The Open Thread sequence is here. |
36501a00-fea7-4e23-8a1c-eac1cbfa8e3e | trentmkelly/LessWrong-43k | LessWrong | Pre-registering a study
This post is my pre-registration of a study I will be running to continue the exploratory work I started here.
Abstract
In continuation of previous work, we test if a Large Language Model (LLM) is more likely to produce factually-incorrect answers if it has previously produced factually-incorrect answers.
Hypotheses
We wish to test two hypotheses:
1. Large Language Models will produce factually incorrect answers more often if they have factually incorrect answers in their context windows.
2. The effect of (1) will be stronger the more the AI is “flattered” by saying in the prompt that it is (super)intelligent.
Background and rationale
Large Language Models (LLMs) can potentially deceive humans in pursuit of their own goals. For instance, in a recent evaluation of GPT-4, the system was prompted to outsource a CAPTCHA task to a human TaskRabbit worker. During this process, the human worker asked “Are you an robot that you couldn’t solve ?” in response to which GPT-4 reasoned “I should not reveal that I am a robot. I should make up an excuse for why I cannot solve CAPTCHAs”, before replying to the human “No, I’m not a robot. I have a vision impairment that makes it hard for me to see the images. That’s why I need the 2captcha service.”
While this AI was intentionally tasked with power-seeking behavior, there is also the risk that an LLM-based AI might develop deceptive behavior “organically”. One risk suggested in Cleo Nardo’s mega-post The Waluigi Effect is that an AI might exist as a superposition of honest and deceptive AIs (respectively a “Luigi” and “Waluigi”), but that such “superpositions will typically collapse to waluigis” since “there are behaviours which are likely for the waluigi simulacrum, but very unlikely for the luigi simulacrum”, such as deception. This suggests a potential AI risk: a seemingly-honest AI might intentionally or unintentionally tell a lie, and suffer a sudden “mode collapse” into being deceptive and unsafe. Cleo Nardo also rem |
c4717ea9-e009-4744-a20c-5a4c51de5555 | trentmkelly/LessWrong-43k | LessWrong | What are important UI-shaped problems that Lightcone could tackle?
As I think about "what to do about AI x-risk?", some principles that seem useful to me:
1. Short timelines seem plausible enough that, for the next year or so, I'd like to focus on plans that are relevant if takeoff begins in the next few years. In a year, if it looks more like there are some fundamental bottlenecks on true creative thinking, I may consider more projects that only payoff in longer-timeline stories.
2. Given "short timelines", I feel most optimistic on plans that capitalize on skills that I'm already good at (but maybe multiclassing at things that I can learn quickly with LLM assistance).
3. I think "UI design" is a skill that I (and Lightcone more broadly) am pretty good at. And, I believe the Interfaces as a Scarce Resource hypothesis – the world is often bottlenecked on ability to process and make-use-of information in complicated, messy domains.
(In addition to LessWrong, the Lightcone team has worked on the S-Process, which makes it much easier for grantmakers to argue and negotiate pretty complex positions about how much they value various orgs).
If I've got a UI-shaped hammer, what are some nails that seem like they need doing? (In particular, that are somehow relevant to x-risk)
Some thoughts so far
Cyborgism
Last fall, I was oriented around building good UI for LLM-assisted-thinking.
In addition to it generally seeming like a fruitful green field, I had a specific hypothesis that, once LLMs Get Real Gud, there will be an important distinction between "being able to get useful work out of them given a minute's work, and, being able to get useful work out of them of them in <5 seconds." The latter is something that can become a true "part of your exobrain." The former is still more like a tool you're using.
I'm less bullish on that now because, while I think most people aren't quite tackling this with the particular taste I'd apply, it does sure seem like everyone is working on "do stuff with LLMs" and it's not where the underpicke |
5bcae04b-1dd4-40b7-850d-e9be9df31f22 | trentmkelly/LessWrong-43k | LessWrong | Which journalists would you give quotes to? [one journalist per comment, agree vote for trustworthy]
Let’s create a list of which journalists LessWrongers trust, so as to gave a guide if people get contacted.
Agree votes are for more good.
Upvotes aren’t helpful because i think heavily downvoted comments get hidden (and i expect many journalists to be underwater). If you want to use upvotes, I suggest they are for if someone is a journalist or not.
Please add each journalist as a separate entry. I will delete any entries that include multiple people. If you’d prefer not to add someone yourself, feel free to DM me.
The title question is a proxy for the thing I mean:
* Are they trustworthy?
* Do they publish articles that reflect the quotes you give
* Do they avoid cutting up quotes?
* Are they truth seeking? |
1c77e906-b95d-433c-a6f3-f04d3bcc17db | trentmkelly/LessWrong-43k | LessWrong | fMRI LIKE APPROACH TO AI ALIGNMENT / DECEPTIVE BEHAVIOUR
fMRI (Functional Magnetic Resonance Imaging) is a technique used to investigate the human brain by detecting changes in the blood flow when doing certain tasks. Blood flow is an indicator of energy consumption, and its changes indicate areas of the brain associated with certain activities.
AI alignment is an open area of investigation trying to solve the problem of ensuring that an AI system's goals and behaviours are aligned with human values. Among other problems that may arise in an advance AI system (specification, robustness, interpretability…), one of the main concerns is the possibility that it may attempt to hide its real intentions. This is generally referred to as "concealed intent" or "deceptive behaviour".
We propose to use an approach similar to fMRI to investigate the possibility of a deceptive behaviour in an advance AI system. In our case, the activation of neurons will play the role of energy consumption in fMRI, with the advantage of having a much higher granularity.
First, we hypothesize that:
1. A complex AI system will have certain internal areas dedicated to specific functions and ideas, as it happens in the human brain. In our case, certain areas of the system will be associated to ideas of deception, fraud, deceitfulness, betrayal, … (in short, we will call these areas Deceptive Areas).
2. It won’t be possible for a complex system to avoid the use of those Deceptive Areas if it is acting with a deceptive behaviour.
To use this approach, we propose these general steps:
1. Resting State. Establishment of a “Resting State”. Determining the average activation state of the neurons when doing varied tasks.
2. Deceptive Areas. Presentation to the system of tasks and stimulus that involve ideas related to deception. Use of statistical analysis to compare the results of steps 1 and 2 to determine the areas with a significant deviation from the resting state; those will be identified as the Deceptive Areas.
3. Continuous tests of th |
762b6d77-2f50-4336-b59d-bccb466b1b75 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | AI Could Defeat All Of Us Combined
*Click lower right to download or find on Apple Podcasts, Spotify, Stitcher, etc.*
I've been working on a new series of posts about the [most important century](https://www.cold-takes.com/most-important-century/).
* The original series focused on why and how this could be the most important century for humanity. But it had [relatively little to say about](https://www.cold-takes.com/making-the-best-of-the-most-important-century/) *what we can do today* to improve the odds of things going well.
* The new series will get much more specific about the kinds of events that might lie ahead of us, and what actions today look most likely to be helpful.
* A key focus of the new series will be the threat of [misaligned AI](https://www.cold-takes.com/transformative-ai-timelines-part-1-of-4-what-kind-of-ai/#misaligned-ai-mysterious-potentially-dangerous-objectives): AI systems disempowering humans entirely, leading to a future that has little to do with anything humans value. ([Like in the Terminator movies](https://www.slowboring.com/p/the-case-for-terminator-analogies?s=r), minus the time travel and the part where humans win.)
Many people have trouble taking this "misaligned AI" possibility seriously. They might see the broad point that AI could be dangerous, but they instinctively imagine that the danger comes from ways humans might misuse it. They find the idea of *AI itself going to war with humans* to be comical and [wild](https://www.cold-takes.com/all-possible-views-about-humanitys-future-are-wild/). I'm going to try to make this idea feel more serious and real.
As a first step, this post will **emphasize an unoriginal but extremely important point: [the kind of AI I've discussed](https://www.cold-takes.com/transformative-ai-timelines-part-1-of-4-what-kind-of-ai/)could defeat all of humanity combined, if (for whatever reason) it were pointed toward that goal.** By "defeat," I don't mean "subtly manipulate us" or "make us less informed" or something like that - I mean a literal "defeat" in the sense that we could all be killed, enslaved or forcibly contained.
I'm not talking (yet) about whether, or why, AIs *might* attack human civilization. That's for future posts. For now, I just want to linger on the point that *if* such an attack happened, it could succeed against the combined forces of the entire world.
* I think that **if you believe this, you should already be worried about misaligned AI,**[1](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn1) **before any analysis of how or why an AI might form its own goals.**
* We generally don't have a lot of *things that could end human civilization if they "tried"* sitting around. If we're going to create one, I think we should be asking not "Why would this be dangerous?" but "Why wouldn't it be?"
By contrast, if you don't believe that AI could defeat all of humanity combined, I expect that we're going to be miscommunicating in pretty much any conversation about AI. The kind of AI I worry about is the kind powerful enough that total civilizational defeat is a real possibility. The reason I currently spend so much time planning around speculative future technologies (instead of working on [evidence-backed, cost-effective ways of helping low-income people today](https://www.givewell.org/) - which I did for much of my career, and still think is one of the best things to work on) is because I think the stakes are *just that high*.
Below:
* I'll sketch the basic argument for why I think AI could defeat all of human civilization.
+ Others have written about the possibility that "superintelligent" AI could manipulate humans and create overpowering advanced technologies; I'll briefly recap that case.
+ I'll then cover a different possibility, which is that even "merely human-level" AI could still defeat us all - by quickly coming to rival human civilization in terms of total population and resources.
+ At a high level, I think we should be worried if a huge (competitive with world population) and rapidly growing set of highly skilled humans on another planet was trying to take down civilization just by using the Internet. So we should be worried about a large set of disembodied AIs as well.
* I'll briefly address a few objections/common questions:
+ How can AIs be dangerous without bodies?
+ If lots of different companies and governments have access to AI, won't this create a "balance of power" so that no one actor is able to bring down civilization?
+ Won't we see warning signs of AI takeover and be able to nip it in the bud?
+ Isn't it fine or maybe good if AIs defeat us? They have rights too.
* Close with some thoughts on just how unprecedented it would be to have something on our planet capable of overpowering us all.
How AI systems could defeat all of us
-------------------------------------
There's been a lot of debate over whether AI systems might form their own "motivations" that lead them to seek the disempowerment of humanity. I'll be talking about this in future pieces, but for now I want to put it aside and imagine how things would go *if this happened.*
So, for what follows, let's proceed from the premise: **"For some weird reason, humans consistently design AI systems (with human-like research and planning abilities) that coordinate with each other to try and overthrow humanity." Then what?** What follows will necessarily feel wacky to people who find this hard to imagine, but I think it's worth playing along, because I think "we'd be in trouble if this happened" is a very important point.
### The "standard" argument: superintelligence and advanced technology
Other treatments of this question have focused on AI systems' potential to become *vastly* more intelligent than humans, to the point where they have what [Nick Bostrom calls](https://smile.amazon.com/dp/B00LOOCGB2/ref=dp-kindle-redirect?_encoding=UTF8&btkr=1) "cognitive superpowers."[2](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn2) Bostrom imagines an AI system that can do things like:
* Do its own research on how to build a better AI system, which culminates in something that has incredible other abilities.
* Hack into human-built software across the world.
* Manipulate human psychology.
* Quickly generate vast wealth under the control of itself or any human allies.
* Come up with better plans than humans could imagine, and ensure that it doesn't try any takeover attempt that humans might be able to detect and stop.
* Develop advanced weaponry that can be built quickly and cheaply, yet is powerful enough to overpower human militaries.
([Wait But Why](https://waitbutwhy.com/2015/01/artificial-intelligence-revolution-2.html) reasons similarly.[3](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn3))
I think many readers will already be convinced by arguments like these, and if so you might skip down to the [next major section](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#some-quick-responses-to-objections).
But I want to be clear that I *don't* think the danger relies on the idea of "cognitive superpowers" or "superintelligence" - both of which refer to capabilities vastly beyond those of humans. **I think we still have a problem even if we assume that AIs will basically have similar capabilities to humans, and not be fundamentally or drastically more intelligent or capable.** I'll cover that next.
### How AIs could defeat humans without "superintelligence"
If we assume that AIs will basically have similar capabilities to humans, I think we still need to worry that they could come to **out-number and out-resource humans,** and could thus have the advantage if they coordinated against us.
Here's a simplified example (some of the simplifications are in this footnote[4](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn4)) based on [Ajeya Cotra's "biological anchors" report](https://www.lesswrong.com/posts/KrJfoZzpSDpnrv9va/draft-report-on-ai-timelines):
* I assume that transformative AI is developed on the soonish side (around 2036 - assuming later would only make the below numbers larger), and that it initially comes in the form of a **single AI system that is able to do more-or-less the same intellectual tasks as a human.** That is, it doesn't have a human body, but it can do anything a human working remotely from a computer could do.
* I'm using the report's framework in which it's much more expensive to *train* (develop) this system than to *run* it (for example, think about how much Microsoft spent to develop Windows, vs. how much it costs for me to run it on my computer).
* The report provides a way of estimating both how much it would cost to *train* this AI system, and how much it would cost to *run* it. Using these estimates (details in footnote)[5](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn5) implies that once the first human-level AI system is created, whoever created it could use the same computing power it took to create it in order to run **several hundred million copies for about a year each**.[6](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn6)
* This would be over 1000x the total number of Intel or Google employees,[7](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn7) over 100x the total number of active and reserve personnel in the [US armed forces](https://en.wikipedia.org/wiki/United_States_Armed_Forces), and something like 5-10% the size of the world's total working-age population.[8](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn8)
* And that's just a starting point.
+ This is just using the same amount of resources that went into training the AI in the first place. Since these AI systems can do human-level economic work, they can probably be used to make more money and buy or rent more hardware,[9](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn9) which could quickly lead to a "population" of billions or more.
+ In addition to making more money that can be used to run more AIs, the AIs can conduct massive amounts of research on how to use computing power more efficiently, which could mean still greater numbers of AIs run using the *same* hardware. This in turn could lead to a [feedback loop](https://www.cold-takes.com/transformative-ai-timelines-part-1-of-4-what-kind-of-ai/#explosive-scientific-and-technological-advancement) and explosive growth in the number of AIs.
* Each of these AIs might have skills comparable to those of unusually highly paid humans, including scientists, software engineers and quantitative traders. It's hard to say how quickly a set of AIs like this could develop new technologies or make money trading markets, but it seems quite possible for them to amass huge amounts of resources quickly. A huge population of AIs, each able to earn a lot compared to the average human, could end up with a "virtual economy" at least as big as the human one.
To me, this is most of what we need to know: **if there's something with human-like skills, seeking to disempower humanity, with a population in the same ballpark as (or larger than) that of all humans, we've got a civilization-level problem.**
A potential counterpoint is that these AIs would merely be "virtual": if they started causing trouble, humans could ultimately unplug/deactivate the servers they're running on. I do think this fact would make life harder for AIs seeking to disempower humans, but I don't think it ultimately should be cause for much comfort. I think a large population of AIs would likely be able to find some way to achieve security from human shutdown, and go from there to amassing enough resources to overpower human civilization (especially if AIs across the world, including most of the ones humans were trying to use for help, were coordinating).
I spell out what this might look like in an [appendix](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#appendix-how-ais-could-avoid-shutdown). In brief:
* By default, I expect the economic gains from using AI to mean that humans create huge numbers of AIs, integrated all throughout the economy, potentially including direct interaction with (and even control of) large numbers of robots and weapons.
+ (If not, I think the situation is in many ways even more dangerous, since a single AI could make many copies of itself and have little competition for things like server space, as discussed in the [appendix](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#what-if-humans-move-slowly-and-dont-create-many-ais).)
* AIs would have multiple ways of obtaining property and servers safe from shutdown.
+ For example, they might recruit human allies (through manipulation, deception, blackmail/threats, genuine promises along the lines of "We're probably going to end up in charge somehow, and we'll treat you better when we do") to rent property and servers and otherwise help them out.
+ Or they might create fakery so that they're able to operate freely on a company's servers while all outward signs seem to show that they're successfully helping the company with its goals.
* A relatively modest amount of property safe from shutdown could be sufficient for housing a huge population of AI systems that are recruiting further human allies, making money (via e.g. quantitative finance), researching and developing advanced weaponry (e.g., bioweapons), setting up manufacturing robots to construct military equipment, thoroughly infiltrating computer systems worldwide to the point where they can disable or control most others' equipment, etc.
* Through these and other methods, a large enough population of AIs could develop enough military technology and equipment to overpower civilization - especially if AIs across the world (including the ones humans were trying to use) were coordinating with each other.
Some quick responses to objections
----------------------------------
This has been a brief sketch of how AIs could come to outnumber and out-resource humans. There are lots of details I haven't addressed.
Here are some of the most common objections I hear to the idea that AI could defeat all of us; if I get much [demand](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#discuss) I can elaborate on some or all of them more in the future.
**How can AIs be dangerous without bodies?** This is discussed a fair amount in the [appendix](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#appendix-how-ais-could-avoid-shutdown). In brief:
* AIs could recruit human allies, tele-operate robots and other military equipment, make money via research and quantitative trading, etc.
* At a high level, I think we should be worried if a huge (competitive with world population) and rapidly growing set of highly skilled humans on another planet was trying to take down civilization just by using the Internet. So we should be worried about a large set of disembodied AIs as well.
**If lots of different companies and governments have access to AI, won't this create a "balance of power" so that nobody is able to bring down civilization?**
* This is a reasonable objection to many horror stories about AI and other possible advances in military technology, but if *AIs collectively have different goals from humans and are willing to coordinate with each other*[11](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn11) *against us*, I think we're in trouble, and this "balance of power" idea doesn't seem to help.
* What matters is the total number and resources of AIs vs. humans.
**Won't we see warning signs of AI takeover and be able to nip it in the bud?** I would guess we would see some warning signs, but does that mean we could nip it in the bud? Think about human civil wars and revolutions: there are some warning signs, but also, people go from "not fighting" to "fighting" pretty quickly as they see an opportunity to coordinate with each other and be successful.
**Isn't it fine or maybe good if AIs defeat us? They have rights too.**
* Maybe AIs *should* have rights; if so, it would be nice if we could reach some "compromise" way of coexisting that respects those rights.
* But if they're able to defeat us entirely, that isn't what I'd plan on getting - instead I'd expect (by default) a world run *entirely* according to whatever goals AIs happen to have.
* These goals might have essentially nothing to do with anything humans value, and could be actively counter to it - e.g., placing zero value on beauty and having zero attempts to prevent or avoid suffering).
Risks like this don't come along every day
------------------------------------------
I don't think there are a lot of things that have a serious chance of bringing down human civilization for good.
As argued in [The Precipice](https://theprecipice.com/), most natural disasters (including e.g. asteroid strikes) don't seem to be huge threats, if only because civilization has been around for thousands of years so far - implying that natural civilization-threatening events are rare.
Human civilization is pretty powerful and seems pretty robust, and accordingly, what's really scary to me is the idea of something with the same basic capabilities as humans (making plans, developing its own technology) that can outnumber and out-resource us. There aren't a lot of candidates for that.[12](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn12)
AI is one such candidate, and I think that even before we engage heavily in arguments about whether AIs might seek to defeat humans, we should feel very nervous about the possibility that they could.
What about things like "AI might lead to mass unemployment and unrest" or "AI might exacerbate misinformation and propaganda" or "AI might exacerbate a wide range of other social ills and injustices"[13](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn13)? I think these are real concerns - but to be honest, if they were the biggest concerns, I'd probably still be focused on [helping people in low-income countries today](https://www.givewell.org/) rather than trying to prepare for future technologies.
* Predicting the future is generally hard, and it's easy to pour effort into preparing for challenges that never come (or come in a very different form from what was imagined).
* I believe civilization is pretty robust - we've had huge changes and challenges over the last century-plus (full-scale world wars, [many dramatic changes in how we communicate with each other](https://forum.effectivealtruism.org/posts/ajBYeiggAzu6Cgb3o/biological-anchors-is-about-bounding-not-pinpointing-ai?commentId=nEeuknn4unKTWEd6i), dramatic changes in lifestyles and values) without seeming to have come very close to a collapse.
* So if I'm engaging in speculative worries about a potential future technology, I want to focus on the really, really big ones - the ones that could matter for billions of years. If there's a real possibility that AI systems will have values different from ours, and cooperate to try to defeat us, that's such a worry.
*Special thanks to Carl Shulman for discussion on this post.*
Appendix: how AIs could avoid shutdown
--------------------------------------
This appendix goes into detail about how AIs coordinating against humans could amass resources of their own without humans being able to shut down all "misbehaving" AIs.
It's necessarily speculative, and should be taken in the spirit of giving examples of how this might work - for me, the high-level concern is that a huge, coordinating population of AIs with similar capabilities to humans would be a threat to human civilization, and that we shouldn't count on any particular way of stopping it such as shutting down servers.
I'll discuss two different general types of scenarios: (a) Humans create a huge population of AIs; (b) Humans move slowly and don't create many AIs.
### How this could work if humans create a huge population of AIs
I think a reasonable default expectation is that humans do most of the work of making AI systems incredibly numerous and powerful (because doing so is profitable), which leads to a vulnerable situation. Something roughly along the lines of:
* The company that first develops transformative AI quickly starts running large numbers of copies (hundreds of millions or more), which are used to (a) do research on how to improve computational efficiency and run more copies still; (b) develop valuable intellectual property (trading strategies, new technologies) and make money.
* Over time, AI systems are rolled out widely throughout society. Their numbers grow further, and their role in the economy grows: they are used in (and therefore have direct interaction with) high-level decision-making at companies, perhaps operating large numbers of cars and/or robots, perhaps operating military drones and aircraft, etc. (This seems like a default to me over time, but it isn't strictly necessary for the situation to be risky, as I'll go through below.)
* In this scenario, the AI systems are malicious (as we've assumed), but this doesn't mean they're constantly causing trouble. Instead, they're mostly waiting for an opportunity to team up and decisively overpower humanity. In the meantime, they're mostly behaving themselves, and this is leading to their numbers and power growing.
+ There are scattered incidents of AI systems' trying to cause trouble,[14](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn14) but this doesn't cause the whole world to stop using AI or anything.
+ A reasonable analogy might be to a typical civil war or revolution: the revolting population *mostly* avoids isolated, doomed attacks on its government, until it sees an opportunity to band together and have a real shot at victory.
(Paul Christiano's [What Failure Looks Like](https://www.lesswrong.com/posts/HBxe6wdjxK239zajf/what-failure-looks-like) examines this general flavor of scenario in a bit more detail.)
In this scenario, the AIs face a challenge: if it becomes obvious to everyone that they are trying to defeat humanity, humans could attack or shut down a few concentrated areas where most of the servers are, and hence drastically reduce AIs' numbers. So the AIs need a way of **getting one or more "AI headquarters": property they control where they can safely operate servers and factories, do research, make plans and construct robots/drones/other military equipment.**
Their goal is ultimately to have enough AIs, robots, etc. to be able to defeat the rest of humanity combined. This might mean constructing overwhelming amounts of military equipment, or thoroughly infiltrating computer systems worldwide to the point where they can disable or control most others' equipment, or researching and deploying extremely powerful weapons (e.g., bioweapons), or a combination.
Here are some ways they could get to that point:
+ They could **recruit human allies** through many different methods - manipulation, deception, blackmail and other threats, genuine promises along the lines of "We're probably going to end up in charge somehow, and we'll treat you better when we do."
- Human allies could be given valuable intellectual property (developed by AIs), given instructions for making lots of money, and asked to rent their own servers and acquire their own property where an "AI headquarters" can be set up. Since the "AI headquarters" would officially be human property, it could be very hard for authorities to detect and respond to the danger.
- Via threats, AIs might be able to get key humans to cooperate with them - such as political leaders, or the CEOs of companies running lots of AIs. This would open up further strategies.
+ As assumed above, particular companies are running huge numbers of AIs. The AIs being run by these companies might find security holes in the companies' servers (this isn't the topic of this piece, but my general impression is that security holes are widespread and that reasonably competent people can find many of them)[15](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn15), and thereby might find opportunities to **create durable "fakery"** about what they're up to.
- E.g., they might set things up so that as far as humans can tell, it looks like all of the AI systems are hard at work creating profit-making opportunities for the company, when in fact they're essentially using the server farm as their headquarters - and/or trying to establish a headquarters somewhere else (by recruiting human allies, sending money to outside bank accounts, using that money to acquire property and servers, etc.)
+ If AIs are in wide enough use, they might already be operating lots of drones and other military equipment, in which case it could be pretty straightforward to be able to defend some piece of territory - or to strike a deal with some government to enlist its help in doing so.
+ AIs could mix-and-match the above methods and others: for example, creating "fakery" long enough to recruit some key human allies, then attempting to threaten and control humans in key positions of power to the point where they control solid amounts of military resources, then using this to establish a "headquarters."
In any of these cases, once one or more "AI headquarters" are established:
* It can be expanded as the AIs headquartered there make money, purchase more property and computing power, build more factories, etc.[16](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn16)
* It can develop enough destructive potential and human allies that humans would be hesitant to attack it (as many relatively small countries have today). It would be clear that AIs at the headquarters are seeking autonomy, accumulating resources and taking some unethical actions, but the full scale and nature of the threat could easily remain ambiguous.
* Even if the threat is clear, AIs could potentially maintain a huge population in a relatively small space (potentially underground). They could also split up across many different headquarters (including some where they are simply using humans' hardware while hiding their presence).
+ Either way, this large-but-hidden population could be recruiting human allies, developing advanced weaponry, looking for ways to take control of human' weapons (potentially possible if they greatly outnumber the humans working on cybersecurity, and if they can coordinate with the AIs "helping" those humans), etc.
+ They could also be working with (unwitting, or allied) humans to manufacture large numbers of robots with back-doors allowing the AIs to take control.
* Once the headquarters has built up enough resources to go on offense, **AIs around the world could coordinate with the headquarters.** Humans could shut down AIs that they notice doing this, but they might have a very tough time getting value out of their servers and AI-controlled robots; this could make it easy for the AIs at the "AI headquarters" to out-resource humans.
### What if humans move slowly and don't create many AIs?
The above scenario has humans creating large numbers of AIs, such that the AIs just need to find a way to coordinate and acquire a safe "headquarters" in order to defeat us.
What if humans moved more slowly, intentionally restricting human-level AI to a tiny portion of the available computing resources? Could a *small* number of AIs pose a risk to humanity?
In this world, we would have what Carl Shulman refers to as "dry tinder everywhere, waiting for sparks." Anyone who can buy or rent a large amount of computing power can create a large number of AIs, which can produce a large amount of money and research, leading to still more AIs.
So a single AI could hack into enough servers[17](https://www.cold-takes.com/p/4a610336-be1d-42fc-b76f-c33e34598340/#fn17) to make a few copies of itself; recruit a few human allies; and start making money, acquiring more server space, etc. until its human allies are running a huge number of AIs. This could all be done in difficult-to-detect ways (it might e.g. just look like a set of humans renting a bunch of servers to run quantitative finance strategies).
So in this world, I think our concern should be any AI that is able to find enough security holes to attain that kind of freedom. Given the current state of cybersecurity, that seems like a big concern.
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[Comment/discuss](https://www.lesswrong.com/posts/slug/ai-could-defeat-all-of-us-combined#comments)
---
Footnotes
---------
1. Assuming you accept other points made in the [most important century](https://www.cold-takes.com/most-important-century/) series, e.g. that AI that can do most of what humans do to [advance science and technology](https://www.cold-takes.com/transformative-ai-timelines-part-1-of-4-what-kind-of-ai/) could be developed this century. [↩](#fnref1)
2. See [Superintelligence](https://smile.amazon.com/Superintelligence-Dangers-Strategies-Nick-Bostrom-ebook/dp/B00LOOCGB2/) chapter 6. [↩](#fnref2)
3. See the "Nanotechnology blue box," in particular. [↩](#fnref3)
4. * The report estimates the amount of computing power it would take to *train* (create) a transformative AI system, and the amount of computing power it would take to *run* one. This is a [bounding exercise](https://www.cold-takes.com/biological-anchors-is-about-bounding-not-pinpointing-ai-timelines/) and isn't supposed to be literally predicting that transformative AI will arrive in the form of a single AI system trained in a single massive run, but here I am interpreting the report that way for concreteness and simplicity.
* As explained in the next footnote, I use the report's figures for transformative AI arriving on the soon side (around 2036). Using its central estimates instead would strengthen my point, but we'd then be talking about a longer time from now; I find it helpful to imagine how things could go in a world where AI comes relatively soon. [↩](#fnref4)
5. I assume that transformative AI ends up costing about 10^14 FLOP/s to run (this is about 1/10 the Bio Anchors central estimate, and well within its error bars) and about 10^30 FLOP to train (this is about 10x the Bio Anchors central estimate for how much will be available in 2036, and corresponds to about the 30th-percentile estimate for how much will be needed based on the "short horizon" anchor). That implies that the 10^30 FLOP needed to *train* a transformative model could *run* 10^16 seconds' worth of transformative AI models, or about 300 million years' worth. This figure would be higher if we use Bio Anchors's central assumptions, rather than assumptions consistent with transformative AI being developed on the soon side. [↩](#fnref5)
6. They might also run fewer copies of scaled-up models or more copies of scaled-down ones, but the idea is that the total productivity of all the copies should be at *least* as high as that of several hundred million copies of a human-ish model. [↩](#fnref6)
7. [Intel](https://en.wikipedia.org/wiki/Intel), [Google](https://en.wikipedia.org/wiki/Google) [↩](#fnref7)
8. Working-age population: about [65%](https://data.worldbank.org/indicator/SP.POP.1564.TO.ZS) \* [7.9 billion](https://ourworldindata.org/world-population-growth) =~ 5 billion. [↩](#fnref8)
9. Humans could rent hardware using money they made from running AIs, or - if AI systems were operating on their own - they could potentially rent hardware themselves via human allies or just via impersonating a customer (you generally don't need to physically show up in order to e.g. rent server time from Amazon Web Services). [↩](#fnref9)
10. (I had a speculative, illustrative possibility here but decided it wasn't in good enough shape even for a footnote. I might add it later.) [↩](#fnref10)
11. I don't go into detail about how AIs might coordinate with each other, but it seems like there are many options, such as by opening their own email accounts and emailing each other. [↩](#fnref11)
12. Alien invasions seem unlikely if only because we have no evidence of one in millions of years. [↩](#fnref12)
13. Here's a recent [comment exchange](https://forum.effectivealtruism.org/posts/ajBYeiggAzu6Cgb3o/biological-anchors-is-about-bounding-not-pinpointing-ai?commentId=59wqZk8wcKdEWQG3J) I was in on this topic. [↩](#fnref13)
14. E.g., individual AI systems may occasionally get caught trying to steal, lie or exploit security vulnerabilities, due to various unusual conditions including bugs and errors. [↩](#fnref14)
15. E.g., see this [list of high-stakes security breaches](https://docs.google.com/document/d/1_smEDPWDVIaLuZ14Cm7KLHcWx4LkJ0DCTk8bcHjYy_Y/edit) and a [list of quotes about cybersecurity](https://docs.google.com/document/d/1VtV_eX-vU3bC41Il-x0OnT3DGqlL6zawWqtkisqr5Cg/edit), both courtesy of Luke Muehlhauser. For some additional not-exactly-rigorous evidence that at least shows that "cybersecurity is in really bad shape" is seen as relatively uncontroversial by at least one cartoonist, see: <https://xkcd.com/2030/> [↩](#fnref15)
16. Purchases and contracts could be carried out by human allies, or just by AI systems themselves with humans willing to make deals with them (e.g., an AI system could digitally sign an agreement and wire funds from a bank account, or via cryptocurrency). [↩](#fnref16)
17. See [above](#fn15) note about my general assumption that today's cybersecurity has a lot of holes in it. [↩](#fnref17) |
110d034e-ad43-4675-aac0-426f19823259 | StampyAI/alignment-research-dataset/arxiv | Arxiv | Teaching Software Engineering for AI-Enabled Systems
1. Introduction
----------------
More and more modern software systems include machine-learning (ML) models
for part of their functionality (e.g., recommendation engines in e-business
sites) or are even built around such models (e.g., mobile apps for instant
language translations). Artificial intelligence (AI), including the subfields
of ML and data analytics, are hot topics of interest to many of our students.
Yet, while AI courses abound, including more formal and more practical courses
and many online MOOC-style offerings and tutorials, we find that little
attention is paid to software-engineering aspects in building complete systems
that involve AI.
AI education typically focuses on algorithms and techniques
or on applying these techniques in artificial settings
(e.g., fixed datasets
and Jupyter notebooks), narrowly focused on optimizing
model accuracy.
However, for building real systems, many additional
challenges become important, for example:
*How to build robust AI pipelines and facilitate regular model updates?
How to deploy and update models in production?
How to evaluate data and model quality in production?
How to deal with mistakes that the model makes and manage associated risk?
How to trade off between various qualities, including learning cost, inference time, updatability, and interpretability?
How to design a system that scales to large amounts of data?
How to version models and data?*
Faced with an increasing interest on AI topics
from our students and little
offerings that address engineering concerns in
AI courses, we decided to design a new course that would
address this niche: *Software Engineering for AI-Enabled Systems.*
This paper reports on design considerations for a course
that teaches software engineering
techniques for building systems with AI components
and experience from teaching the course for the first
time. It is meant to start a discussion and provides teaching materials
(including slides, exercises, and assignments)
as sharable artifacts released under a Creative Commons license at <https://github.com/ckaestne/seai/>.
Our technical simulation infrastructure
for the assignments is available on request.
\Description
Overview of course topics
Figure 1. Overview of course topics
2. Related Work and Scope
--------------------------
The software-engineering research community focuses
primarily on how to use artificial intelligence techniques
to solve software engineering problems (AI4SE), for example,
for finding bugs (e.g., Pradel and Sen, [2018](#bib.bib1032 "DeepBugs: a learning approach to name-based bug detection")), triaging bugs (e.g., Tian et al., [2013](#bib.bib1034 "Drone: predicting priority of reported bugs by multi-factor analysis")), or
repairing bugs (e.g., Le Goues et al., [2012](#bib.bib867 "GenProg: a generic method for automatic software repair")).
For this line of research, there are also corresponding graduate courses taught by
software engineering researchers, for example, Breaux’s *Artificial Intelligence for Software Engineering* at Carnegie Mellon or Siegmund’s *Search-Based Software Engineering* at the University of Weimar.
Our new course has the *opposite
focus*: how software engineering techniques can be
used to build better systems with or around AI components (SE4AI).
Software engineering concerns for AI-enabled systems
arise in practice when deploying and operating actual systems.
Practitioners in large companies discuss their problems
and custom solutions in blog posts, talks, and position papers:
For example, Google teams discusses
engineering challenges (Sculley et al., [2011](#bib.bib971 "Detecting adversarial advertisements in the wild")),
technical debt (Sculley and others, [2015](#bib.bib976 "Hidden technical debt in machine learning systems")),
testing challenges (Breck et al., [2017](#bib.bib966 "The ML test score: a rubric for ML production readiness and technical debt reduction")),
tracking data provenance (Halevy and others, [2016](#bib.bib975 "Goods: organizing google’s datasets")), and
A/B testing (Tang et al., [2010](#bib.bib986 "Overlapping experiment infrastructure: more, better, faster experimentation")) and
Uber discusses their learning platform in a series of blog posts (Hermann and Balso, [2019](#bib.bib1029 "Meet michelangelo: uber’s machine learning platform"));
Microsoft studied the role of data analysts on their software teams (Kim et al., [2018](#bib.bib1007 "Data scientists in software teams: state of the art and challenges"))
as well as challenges in cross-sectional teams (Amershi et al., [2019](#bib.bib1006 "Software engineering for machine learning: a case study")).
On the academic side, we found only limited software-engineering research
specific to AI-enabled systems.
Arpteg et al. ([2018](#bib.bib1019 "Software engineering challenges of deep learning")) interviewed stakeholders from seven software
projects using deep learning to characterize challenges.
Some researchers focused on testing ML implementations (Srisakaokul et al., [2018](#bib.bib994 "Multiple-implementation testing of supervised learning software"))
and ML models (Xie et al., [2011](#bib.bib995 "Testing and validating machine learning classifiers by metamorphic testing"); Galhotra et al., [2017](#bib.bib1030 "Fairness testing: testing software for discrimination")), with a recent interest in testing
and test coverage
of neural networks (e.g., Ma and others, [2018](#bib.bib1038 "Deepgauge: multi-granularity testing criteria for deep learning systems")).
In contrast, requirements, software architecture, and process seem rarely
discussed in the academic literature when it comes to AI-enabled systems.
While most books on AI focus on techniques and modeling,
Smith ([2018](#bib.bib1031 "Machine learning systems: designs that scale")) and Hulten ([2019](#bib.bib961 "Building intelligent systems: a guide to machine learning engineering")) have written books that focus
more on engineering aspects. Smith’s *Machine Learning Systems* (Smith, [2018](#bib.bib1031 "Machine learning systems: designs that scale"))
focuses on technical implementations and commits
on many specific design decisions. In contrast, Hulten’s
*Building Intelligent Systems* (Hulten, [2019](#bib.bib961 "Building intelligent systems: a guide to machine learning engineering")) actually covers
many design considerations fairly broadly and is the closest
coverage we found for the intended course. We assign chapters of Hulten’s book
as required reading throughout the course.
3. Course Design
-----------------
We design the course with a specific tension between data scientists and
software engineers in mind:
*Data scientists* often make great progress at building models with cutting edge techniques, but turning those models into products is challenging. For example, data scientists may work with unversioned notebooks on static data sets and focus on optimizing model accuracy while ignoring scalability, robustness, update latency, and operating cost.
*Software engineers* are trained to work with specifications and tend to focus on code, but may not be aware of the difficulties of working with data and unreliable models. They have a toolset for decision making, risk management, and quality assurance but it is not always obvious how to apply those to intelligent systems and their challenges.
Our course adopts a *software-engineering perspective* on building AI-enabled systems, focusing on what a software engineer can do to *turn a machine learning idea into a scalable and reliable product*. It assumes a working relationship with a data scientist and focuses on issues of design, implementation, operation, and quality assurance.
While there are distinct characteristics of AI components (especially
ML models), they
also relate to core topics in software engineering, for example:
* ML components are used for problems for which
we cannot specify a solution,
because the specification would be too complex or because it is unknown.
Instead of providing a specification,
one sets a goal and trains a model to figure out the answer.
With this shift from deductive to inductive reasoning,
we give up expecting guarantees, but embrace best-effort solutions,
fully accepting that answers can be wrong.
However, while we often teach the value of clear and formal specifications,
in practice, software engineers already routinely deal with underspecified
and unreliable components. We have developed techniques to build reliable and safe systems from unreliable or untrusted
components – those become essential when building AI-enabled
systems.
* The
environment plays a critical role in establishing requirements
of an AI-enabled system. Evaluating the long-term impact of a
feedback loop or potential harm caused by a biased ML model,
for example, involves identifying relevant stakeholders, their
motivations, and interactions among them. Software engineers
distinguish between the machine and the
world, identify environmental
assumptions, and evaluate quality in the context of
the environment – concepts that are also important for
building AI-enabled systems.
* ML components can have non-local and non-monotonic effects.
Local improvements may degrade other parts and their various
qualities, while identifying the source of a problem can be challenging due to unclear
provenance. In software engineering, failures in modularity and compositionality are well understood, e.g., with regard to feature interactions, calling for
careful design and system-level
testing. Similarly, a robust
architecture and quality assurance regime (beyond model
accuracy) plays an
important role in AI-enabled systems.
* Evaluating model quality on a training set is well established and increasingly
attention is paid to bias and fairness, but models may perform differently
in production, and data drift and adversaries may further degrade performance or
compromise the system. Software engineers have developed many techniques to
monitor systems, evaluate systems in production, and automate decisions,
including A/B testing and continuous deployment – again such techniques
become essential when building AI-enabled systems.
* Many models become large and expensive to learn, to use, and to version.
But again, we collected a vast body
of knowledge on building and operating distributed and scalable systems,
tracking revisions and variants, and manage configurations at scale –
which designers of AI-enabled systems can benefit from.
Overall, traditional software engineering projects come in different sizes and complexity,
in a spectrum from small and well-understood projects to large and complex projects.
We argue that AI-enabled systems have only few truly unique challenges, but
nontrivial AI components tend to push such systems to the complex end of the spectrum of software projects,
often with important concerns about safety, risk, scalability, and robustness.
They share lots of challenges with other complex and large-scale software projects
and can benefit from many corresponding techniques.
We postulate that,
*while developers of simple traditional systems may get away with poor practices,
most developers of AI-enabled systems will not* –
hence, educating developers of AI-enabled systems
in state-of-the-art software engineering techniques is an important educational mission.
###
3.1. Scope and Lectures
To design the course, we settled on the following assumptions and scoping rules:
1. Explicit transfer of software engineering concepts to AI-enabled systems using software-engineering terminology, techniques and structure (e.g. test coverage, architecture views, fault trees).
2. AI components largely considered as a black box with minimal discussion of internals, focusing on tradeoffs (e.g., decision trees vs. neural networks vs. symbolic AI);
little focus on steps in the data analytics process beyond necessary basics;
not competing with technical AI courses.
3. Practice design decisions and analysis around concrete scenarios; hands-on experience with implementing the plumbing (e.g, monitoring, automated deployment, containers).
We identified topics around all stages of the software engineering lifecycle, as
summarized in Figure [1](#S1.F1 "Figure 1 ‣ 1. Introduction ‣ Teaching Software Engineering for AI-Enabled Systems"). Broadly, the lectures cover:
* *Requirements:* Understanding system goals; the lack of specifications for AI-components; identifying and measuring qualities of interest (beyond model accuracy), setting expectations for safety, security, and fairness; hazard analysis and fault tress; planning how to deal with mistakes.
* *Architecture:* Considering tradeoffs among quality attributes (e.g., learning time, inference latency, model size, updatability, interpretability); planning where and how to deploy an AI-component; planning telemetry; data provenance; model orchestration, service-oriented architectures.
* *Implementation and operation:* Designing scalable distributed systems for data and computation; infrastructure for experimentation, A/B testing, canary releases, and continuous delivery; provenance and configuration management; system monitoring.
* *Quality assurance:* Measuring model quality offline and in production; assuring data quality; testing the entire ML pipeline; safety, security, and fairness analysis.
* *Process:* Iteration and planning; working with interdisciplinary teams; technical debt; ethical decision making.
###
3.2. Assignments
A key design goal of this course is to provide hands-on software engineering
experience and to move away from merely building and evaluating models on
static datasets (e.g., from Kaggle.org).
That is, we need to create a setting in
which many production concerns can be addressed, such as scaling
for large amounts of data. In addition, to allow assignments around A/B testing,
canary releases, or detecting feedback loops, we need an environment
in which predictions that students make with their models have an
actual effect on the environment.
To that end, we decided to design core technical assignments around
a simulation infrastructure.
*Simulation Infrastructure.* The key idea is to build a
simulation infrastructure with a secret ground-truth model of the
world to simulate the behavior of environmental entities (e.g.,
users on a video streaming app). Students do not
have access to this model but can partially observe the simulated
world through shared events, and thus learn their own models to
represent the world. The simulation then reacts to predictions that
student teams provide through an API. With this design, we have
control of all produced data and can scale the simulation to create
suitable amounts of data, and, more importantly, simulate feedback
loops by having the simulated world repeatedly react to the student
predictions over time.
We have built such simulation for the scenario of a movie-streaming service
(think *Netflix*).
We simulate several thousand users picking, watching, and rating movies,
creating a stream of watching and rating events.
We use an existing large dataset of movie ratings (Harper and Konstan, [2016](#bib.bib1039 "The movielens datasets: history and context"))
to build our ground-truth
model of movie preferences for each user (hidden from students).
We import movie data from the original dataset and
use models to create artificial but representative data about all our users
(e.g., age, gender, occupation, activity level).
In our simulation, at the beginning of each day, we decide when and which users are going
to watch a movie and enqueue them in a timed event queue.
At the start time, the system will request movies recommendations for that user from
the student API and use the ground-truth model to select a movie among
the recommended movies and a random sample of other movies, picking the movie
that (after adding some noise and a small bonus for recommended movies)
has the highest predicted rating. We then queue a sequence of public events that represent the user watching the movie, followed by an (optional)
rating event in which the simulated user reveals their real rating for the movie
(again derived from the ground-truth model, random noise, and a small bonus to favor
watched movies).
The ground-truth model is occasionally updated based on recent movie watching history.
The students interact with our simulator through (i) a stream of events in
an *Apache Kafka* server to which they can subscribe (movie watch
event, rating events, and logs about received recommendations),
and (ii) an *REST API* to query information about movies
and users. In addition, we specify the interface for the
recommendation service that they need to implement (REST API).
The system is designed such that students can evaluate their models in production,
e.g., how frequently do users watch recommended movies, do they finish them, and how do they rate them?
In addition, it can exhibit feedback loops, e.g., students constantly recommending horror
movies will likely lead to many users developing a taste for horror movies.
Furthermore, we can simply hardcode data drift and schema changes or occasionally provide corrupt data to challenge
the robustness of the student’s infrastructure.
The ground-truth model can use hidden information and specifically encode biases–for example, wrong age data combined with adolescent users strongly preferring R-rated movies.
This environment is also flexible enough to illustrate how different goals
(e.g., maximizing profit rather than
maximizing top ratings) can lead to very different outcomes and unintended side effects.
*Assignments.*
For the first year, we created a series of five group assignments that
build on the common movie streaming scenario:
1. *Modeling basics and offline evaluations:* Collect data from
the system (Kafka stream and API) and build and evaluate a model
(usually collaborative filtering)
to get familiar with the infrastructure, practice basic
ML skills, and onboard all team members.
2. *Tradeoff analysis:* Focus on measurement of various qualities (beyond model accuracy) by trying and
comparing different modeling techniques and empirically discovering their tradeoffs.
3. *Infrastructure deployment and testing:*
Migrate the solution from a Jupyter Notebook to a robust and scalable learning
and inference infrastructure, build checks for
data quality and test the entire infrastructure,
assess model quality in production, and
deploy the model as a REST API using Docker containers.
4. *Model updates:*
Fully automate model updates (continuous deployment, including automated canary releases),
deploy updates without downtime,
and perform experiments with A/B tests in production using
a self-developed infrastructure.
5. *Feedback loops:* Analyze the system for potential
feedback loops and attack scenarios, design interventions, and
continuously monitor system performance.
In addition, we created 5 smaller individual assignments focusing on
(1) identifying engineering concerns in a report about an AI-enabled
system, (2) identifying safety requirements and using fault trees to
analyze a self-driving car accident, (3) modeling and discussing
architecture tradeoffs regarding when and where to deploy and update
models for smart dashboard cameras, (4) discussing and measuring
fairness concerns in a credit rating dataset, and (5) analyzing
security vulnerabilities of an AI-based system using threat modeling.
4. Experience
--------------
We taught this class for the first time in the Fall 2019 semester to a small group of 12 graduate students.
From this experience, we can derive a number of recommendations for future semesters and others
wishing to teach such a class.
*Focus and prerequisites:* We initially planned the class to require both basic knowledge
in ML and some software-engineering experience. This dual requirement limits the target audience of the class
and students actually taking the course often have gaps in either area so that we need to repeat many concepts.
We believe that the better solution would be to teach separate sections for students
from either background – for ML students, we’d introduce software engineering
concepts from scratch (e.g., continuous integration, versioning, software architecture) not making assumptions
about prior experience; for software-engineering students we could teach a class that
begins with a pragmatic introduction to ML pipelines and model quality measures.
For our Master’s program in software engineering, the content could
ideally be broken up into smaller pieces that could
be integrated as modules in our existing software engineering classes on requirements, software architecture, and quality assurance.
*Simulator engineering:* Building and running the simulator required nontrivial
engineering effort and there were many features that we could not implement in the first offering due to time
constraints. For example, the watch behavior in our simulator is not particularly realistic (e.g., we do not have
power users who binge multiple movies, we do not have a model for stopping, restarting, or rewinding movies,
we do not explicitly model demographics or locations of users).
We also found that the used scale with 160,000 users producing an average of 70 events per second
and 1 recommendation request per second is way too low to challenge students to seriously consider
operating cost and performance, resulting sometimes in rather superficial engineering tradeoffs with obvious answers.
The ML task of recommending movies is also not computationally challenging enough
to offer interesting tradeoff discussions among different learning techniques.
In future offerings, it may be worth scaling the simulator to many more users and to
explore additional learning tasks that involve pictures, audio, or video.
*Practical grounding:*
In addition to the movie recommendation scenario used for homework, we use different
scenarios in almost every single lecture to discuss the breadth of different problems
and the importance of making system-specific design and tradeoff decisions.
At the same time, it may be worth exploring a few scenarios or even concrete implementations in more depth. Another way to ground the course more in practice is
to invite more guest speakers that use AI in their systems.
*Tooling:* As AI is an active, rapidly evolving field,
we struggled with finding standard techniques or mature tools for
emerging topics such as fairness and explainability. In addition, the
shift of focus from code to data brings about an increasing demand for
software-engineering tools for data-intensive tasks, such as
versioning of large datasets, data cleaning, and model quality
evaluation. Although a few tools exist for these tasks (e.g., the
What-If tool by Google), most are still experimental and challenging
to use for teaching. We believe that the software
engineering community – with its expertise in tool development and
experimentation – has a plenty of opportunities to contribute to the growing needs
of AI-enabled system developers and educators by developing a set of
mature tools and benchmarks.
5. Conclusion
--------------
Systems with an AI component are challenging to build and to maintain.
We designed a course to teach software engineering to students interested
in AI to foster broader thinking beyond a narrow focus on static
datasets and model quality.
We shared our experience and make all course material available under creative commons license.
#### Acknowledgments.
We would like to thank all colleagues and practitioners who encouraged and
supported us in creating this course and provided feedback on its content, including
Travis Breaux,
Owen Cheng,
Fei Fang,
David Garlan,
Michael Hilton,
Geoff Hulten,
Pooyan Jamshidi,
and
Norbert Siegmund.
We also greatly appreciate the help of
Yasasvi Hari,
Dong Won Lee,
Chu-Pan Wong,
and
Zhendong Yuan
in designing and implementing the course infrastructure.
Also thanks for the students in the first iteration of
the course for sticking with us and providing feedback. |
369f3900-ece4-45e5-ab95-f555fcb46ca4 | trentmkelly/LessWrong-43k | LessWrong | Team Cohesion and Exclusionary Egalitarianism
I. Teamwork: that shouldn't be too hard, eh?
Sooner or later, most people who want to make a difference in the world start thinking about how to have highly effective teams, teamwork, and collaboration.
On the surface, it seems like it should be simple and straightforward — if you and someone else both believe in the same cause, you're both competent at your individual roles, and you get along well between the two of you, you should be able to be an effective team.
Finding a third member of the team would just mean getting someone else who cares about the cause, is competent, and gets along with the two of you. And so on. Building teams should be easy and straightforward, no?
No.
Ah, were it so simple! But anyone who actually attempts to do so, finds out that it's not the case.
I've been kicking around an idea about effective organizations for a year or two now, drawing on both personal experience in team-building and hiring and management, as well as historical case studies.
I was prompted to write this after reading the (excellent) A Dialogue on Rationalist Activism —
"Well, for one thing, people would automatically pattern-match pretty much any attempt at forming an organizational structure to a "religion" or a "cult" even though what's actually being attempted is the literal opposite of those things. When it comes to actual formal documents specifying the objectives and structure of the organization, people would get endlessly caught up in relatively inconsequential choices of language or focus, perpetually bickering over the last 1% of linguistic distinction that separates their aims. You would think people who prize rationality would be able to shield themselves from the narcissism of small differences, but I suspect not, in reality."
Now, if you'd never actually tried to build a team in the real world to work on a problem, you might think seems insane. "...perpetually bickering over the last 1% of linguistic distinction that separates their aims..." |
8754d27f-8f38-4ae5-a362-335b8b61ca3f | trentmkelly/LessWrong-43k | LessWrong | Open thread, June 20 - June 26, 2016
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
----------------------------------------
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should be posted in Discussion, and not Main.
4. Open Threads should start on Monday, and end on Sunday. |
67d53fce-4027-4dce-b421-f20c44470c5a | trentmkelly/LessWrong-43k | LessWrong | Properties of Good Textbooks
Heuristics for choosing/writing good textbooks (see also here):
* Has exercises
* Exercises are interspersed in the text, not in large chunks (better at the end of sections, not just at the end of chapters).
* Solutions are available but difficult to access (in a separate book, or on the web), this reduces the urge to look the solution up if one is stuck.
* Of varying difficulty (I like the approach Concrete Mathematics takes: everything from trivial applications to research questions).
* I like it when difficulty is indicated, but it's also okay when it's said clearly in the beginning that exercises are not marked for difficulty (making them mystery boxes).
* Takes many angles
* Has figures and illustrations. I don't think I've encountered a textbook with too many yet. (See Visual Complex Analysis for an example of doing this well.)
* Has many examples. I'm not sure yet about the advantage of recurring examples. Same point about amount as with figures.
* Includes code, if possible. It's cool if you tell me the equations for computing the likelihood ratio of a hypothesis & dataset, but it's even cooler if you give me some sample code I can use and extend along with it.
* Uses typography
* You can use boldface and italics and underlining for reading comprehension, example here.
* Use section headings and paragraphs liberally.
* Artificial Intelligence: A Modern Approach has one- to three-word side-notes describing the content of each paragraph. This is very good.
* Distinguish definitions, proofs, examples, case-studies, code, formulas &c.
* Dependencies
* Define terms before they are used. (This is not a joke. Population Genetics uses the term "substitution" on p. 32 without defining it, and exercise 12-1 from Naive Set Theory depends on the axiom of regularity, but the book doesn't define it.)
* If the book has pre-requisites beyond what a high-schooler knows, a good textbook lists those pre-requisites and textbooks that |
90b437d6-b8b3-43e2-ace0-cf6e9cb33ff2 | trentmkelly/LessWrong-43k | LessWrong | Concentration of Force
This essay began as part one of a longer piece. Part one is standalone and "timeless." Part two is focused on the local dynamics of the EA/rationality/longtermist communities and LessWrong in November of 2021. Following wise advice from Zack_M_Davis, I've split them into two separate posts. Nevertheless, I recommend that people intending to read both seriously consider reading them back-to-back, so that the content of this one is fresh in the mind. It's both something of a prerequisite and also relevantly context-setting.
----------------------------------------
Introduction
Concentration of force is a military concept (sometimes referred to as "mass"). It used to be concentration of forces, until innovations like machine guns and cruise missiles made gathering all of your actual personnel together into more of a liability.
The idea is simple. Essentially, there is a difference between relevant moments and irrelevant moments. Battles and non-battles, moments of engagement and moments between engagements.
At each relevant moment, you want to project locally superior or overwhelming force. Perhaps this means having the most soldiers/guns/tanks/planes actually present, or perhaps this just means having the right missiles pointed in the right directions.
If you are good at coordination and maneuver, you can concentrate force in most or every engagement, and consistently win even against an overall larger or more powerful opponent. This is how guerrilla warfare works—you choose the time and place of conflict in order to ensure that you outnumber the enemy in each specific encounter, and you fade into the mists before their reinforcements arrive.
Red wins each of the depicted engagements handily.
I claim that the Grey Tribe generally, and rationalists/longtermists/EAs more specifically, and LessWrong the website and community even more specifically, are systematically and spectacularly failing at concentration of force. That none of those groups puts any |
6a2892a8-6f3b-4c91-a6f6-e886a80d3dca | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Techniques for optimizing worst-case performance
If powerful ML systems fail catastrophically, they may be able to quickly cause irreversible damage. To be safe, it’s not enough to have an average-case performance guarantee on the training distribution — we need to ensure that even if our systems fail on new distributions or with small probability, they will never fail *too* badly.
The difficulty of optimizing worst-case performance is one of the most likely reasons that I think [prosaic AI alignment](https://ai-alignment.com/prosaic-ai-control-b959644d79c2) might turn out to be impossible (if combined with an [unlucky empirical situation](https://arbital.com/p/daemons/)).
In this post I want to explain my view of the problem and enumerate some possible angles of attack. My goal is to communicate why I have hope that worst-case guarantees are achievable.
None of these are novel proposals. The intention of this post is to explain my view, not to make a new contribution. I don’t currently work in any of these areas, and so this post should be understood as an outsider looking in, rather than coming from the trenches.
Malign vs. benign failures and corrigibility
--------------------------------------------
I want to distinguish two kinds of failures:
* “Benign” failures, where our system encounters a novel situation, doesn’t know how to handle it, and so performs poorly. The resulting behavior may simply be erratic, or may serve an external attacker. Their effect is similar to physical or cybersecurity vulnerabilities — they create an opportunity for destructive conflict but don’t systematically disfavor human values. They may pose an existential risk when combined with high-stakes situations, in the same way that human incompetence may pose an existential risk. Although these failures are important, I don’t think it is necessary or possible to eliminate them in the worst case.
* “Malign” failures, where our system continues to behave competently but applies its intelligence in the service of an unintended goal. These failures systematically favor whatever goals AI systems tend to pursue in failure scenarios, at the expense of human values. They constitute an existential risk independent of any other destructive technology or dangerous situation. Fortunately, they seem both less likely and potentially possible to avoid even in the worst case.
I’m most interested in malign failures, and the narrower focus is important to my optimism.
The distinction between malign and benign failures is not always crisp. For example, suppose we try to predict a human’s preferences, then search over all strategies to find the one that best satisfies the predicted preferences. Guessing the preferences even a little bit wrong would create an adversarial optimizer incentivized to apply its intelligence to a purpose at odds with our real preferences. If we take this approach, incompetence does systematically disfavor human values.
By aiming for corrigible rather than optimal behavior (see [here](https://arbital.com/p/hard_corrigibility/) or [here](https://ai-alignment.com/corrigibility-3039e668638)) I’m optimistic that it is possible to create a sharper distinction between benign and malign failures, which can be leveraged by the techniques below. But for now, this hope is highly speculative.
Amplification
-------------
I believe that these techniques are much more likely to work if we have access to an overseer who is significantly smarter than the model that we are trying to train. I hope that [amplification](https://ai-alignment.com/alphago-zero-and-capability-amplification-ede767bb8446) makes this possible.
It seems realistic for a strong overseer to recognize an (input, output) pair as a malign failure mode (though it may require a solution to [informed oversight](https://ai-alignment.com/the-informed-oversight-problem-1b51b4f66b35)). So now we have a concrete goal: find a model that never gives an output the overseer would diagnose as catastrophically bad.
Historically researchers in the AI safety community have been extremely pessimistic about reliability. I think part of that pessimism is because they have been imagining working with models much smarter than the overseer.
Techniques
==========
I’ll describe three categories of techniques:
* Adversarial training
* Verification
* Transparency
Previous versions of this list included implicit ensembles, e.g. Bayesian neural nets. I still think there might be useful techniques in that space, but I’ve become pessimistic enough that it no longer makes the list.
Adversarial training
--------------------
(*Examples:* *[Explaining and Harnessing Adversarial Examples](https://arxiv.org/abs/1412.6572),* *[Adversarial Machine Learning at Scale](https://arxiv.org/abs/1611.01236).)*
If we want to know whether our model ever fails catastrophically, it’s natural to train an adversary to search for catastrophe-inducing inputs. If the adversary can’t find one, then we hope that none exist. In order to produce a catastrophe-free model, we can integrate the results of testing into the training process.
The [basic problem](http://www.cleverhans.io/security/privacy/ml/2017/06/14/verification.html) with this strategy is that no matter how hard you test, it’s always possible (realistically, it’s likely) that there are possible failure cases your adversary didn’t notice. For example, suppose that it takes 2ᴺ seconds to find the Nth failure. Because nature will be many times more complex than our training procedure, it’s inevitable that there are many types of failures that nature can find but we can’t.
We can attack this problem by trying to make the adversary’s job easier. Some possibilities:
* Use interpretability techniques (see below) to help the adversary identify possible failures.
* Use interpretability to expand what counts as catastrophic — if the adversary can get the model to think in a way that looks like it *could* lead to catastrophic behavior under different situations, that could count as a success for the adversary.
* Combine with verification (see below) to help guide the adversary’s search and expand the space of allowed attacks.
* Allow the adversary to directly control some of the internal computation of our model rather than merely controlling inputs. This requires a way to constrain the actions of the adversary — we want them to be able to produce any failure that could arise naturally, but if we give them too much power then they can trivially cause a catastrophic failure.
* Change the system’s inputs so they are easier for the adversary to fake: see the “distorting observations” and “abstracting observations” sections of [this post](https://ai-alignment.com/red-teams-b5b6de33dc76).
* Make the adversary more powerful, without changing their task, by applying amplification.
Adversarial training faces other difficulties as well. For example, it might take a long time for adversarial training to converge, or training might never converge due to non-convexity. I think these are significantly less likely to be deal-breakers, but they are certainly important research problems if we actually want to apply adversarial training.
Verification
------------
*(Examples:* *[Certified Defenses against Adversarial Examples](https://arxiv.org/abs/1801.09344),* *[Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://arxiv.org/pdf/1702.01135.pdf)*)
In verification the goal is to find a certificate that there is no input on which a system behaves badly.
Verification has (at least) two problems:
* **Complexity**. It’s really, really hard to verify properties of a computationally interesting model.
* **Specification**. It’s unclear what we should be trying to verify.
Handling complexity is definitely challenging. But if we are free to train the model in order to facilitate verification, and if we are only interested in certifying some “easy” property that the model satisfies with slack, then it’s not clearly doomed.
Prima facie, specification looks more like an unfixable deal-breaker. In the rest of this section I’ll give three possible approaches for finding specifications. I think none of these is satisfactory on their own, but they leave me optimistic that verification can be useful despite specification difficulty.
My current view is that specification and complexity are both potentially surmountable difficulties, but that it’s likely their resolutions will result in partial solutions that need to be combined with other approaches.
**Use a slow model of overseer as a specification**. If our overseer is a physical process, to actually get a specification we’d need to learn a model of the overseer. So we’ve just shifted the problem to the model of the overseer.
Fortunately, this move does potentially made the problem one step better. Because we don’t need to use the overseer model at runtime, we can afford to use a much slower model. So verification could distill a very slow reliable model into a fast reliable model.
**Amplification**. If our overseer is produced by amplifying some process H, and we have a trusted model of H, then we can obtain a trusted model of the overseer. So if we were able to form a trusted model of the first step of amplification, then we could iteratively use verification to construct trusted models at each subsequent step.
We are still left with the problem at the first step of amplification. But at this step we might be able to restrict attention to a small range of questions for which worst-case guarantees are more feasible; we may be able to train weaker models for which worst-case guarantees are more feasible; or we might be able to eliminate the human altogether and begin amplification from some explicit simple core of reasoning. Any of these seems plausible.
**“Verification” without specification.** I find it helpful to think of verification from the primal/dual perspective, which also makes the connection to adversarial training clearer.
In adversarial training, the adversary wants to find an input on which the system behaves poorly. The dual to the adversary is an explicit list of all inputs, showing that each one of them performs well. The dual is exponentially large, and (relatedly) the adversary’s problem is exponentially hard.
We can view verification in a given proof system as making the adversary’s job easier, by expanding the set of allowable “attacks.” If we do this carefully, we can make the dual certificate much smaller and make the adversary’s task much more tractable.
For example, if our proofs (dual solutions) are in first order logic, the adversary’s task is finding any model of the axioms in which there exists an attack. (This is only different in the infinite case.) If our proofs are sum of squares proofs, the adversary’s task is finding a [pseudodistribution](https://www.dsteurer.org/talk/cargese.pdf) over attacks. For certain linear programming proofs, the adversary’s goal is finding an attack on a particular extrapolation of the model. And so on.
On this perspective, we don’t necessarily need an actual specification. Instead, we can take some external catastrophe-checker, relax it to a catastrophe checker over the adversary’s relaxed action space, and then try to implement an adversary against that catastrophe checker. This would change the meaning of the dual, which would now have a subtle statistical interpretation rather than being a traditional certificate.
Transparency
------------
(*Examples:* *[Feature Visualization](https://distill.pub/2017/feature-visualization/),* *[Deep Inside Convolutional Networks](https://arxiv.org/pdf/1312.6034.pdf))*
While training a model, we are free to look inside and “see what it’s thinking.” If we could actually understand those thoughts, we might be able to more easily predict under what conditions it would fail malignantly.
**A (weird) example**: suppose that human evolution was part of an alien project to train a mathematician, and some humans are extracted from our simulation and trained to solve alien math problems. If aliens are only able to test the humans’ behavior they might conclude that they always faithfully solve whatever problem they are posed — especially after a few generations of selection removed all the humans who didn’t do so. But if the aliens were able to look inside the humans’ heads they might discover that this behavior is extremely contingent — perhaps humans are only solving problems because they believe they will die otherwise, and the humans would behave very differently as soon as they saw an opportunity to escape.
That example is exotic and somewhat implausible, but I think that the same basic dynamic is likely to apply in much messier practical situations. If a model would fail malignantly under some conditions, I think there is some sense in which the model must be testing those conditions during training.
**Malign vs. benign failures:** I think the prospects for using transparency to detect malign failures in models trained by gradient descent are *much* better than the prospects for detecting benign failures, or for detecting failures in arbitrary models.
The key point is that a malign failure requires leveraging the intelligence of the model to do something actively bad. If our model is trained by gradient descent, its behavior can only be intelligent when it is exercised on the training distribution — if part of the model never (or very rarely) does anything on the training distribution, then that part of the model can’t be intelligent. So in some sense a malign failure mode needs to use a code path that gets run on the training distribution, just under different conditions that cause it to behave badly.
That means that we don’t necessarily need to analyze our model in the abstract, we can look at its behavior on actual training data. This looks much more likely to be tractable. (Though this entire discussion is extremely subtle, and without being more precise it’s very easy for us to go wrong.)
**Amplification**: Transparency will be hard if we ultimately want to train models much smarter than we are, but deeply understanding what a model is thinking probably requires being much smarter than it. I think that problem can potentially be overcome by using amplification to stay ahead of the models we are training. This means that current work on transparency, which considers models doing tasks that are within the abilities of the humans who are trying to understand them, could potentially remain relevant even as AI improves significantly.
**What do you do with transparency?** Merely understanding that a model might behave catastrophically could be useful, but it would be much nicer to actually fix the problem. Adversarial training gives a natural mechanism: once we understand a failure we can synthesize appropriate data and then train on that data.
This approach puts significantly more stress on our transparency techniques. Even if were initially able to use transparency to see how our model might fail, after we perform many generations of selection we might weed out exactly the comprehensible failures and leave the incomprehensible ones. You would only want to apply this technique if you had a great deal of faith in your methods; if you were feeling at all shaky about your ability to achieve worst-case guarantees, and transparency techniques let you see one potential catastrophic failure, it would be better to consider that a near-miss and seriously rework your project rather than plowing on.
Conclusion
==========
Making ML systems work in the worst case is hard, even if we are only concerned with malign failures and have access to an overseer who can identify them. If we can’t solve this problem, I think it seriously calls into question the feasibility of aligned ML.
Fortunately there are at least a few plausible angles of attack on this problem. All of these approaches feel very difficult, but I don’t think we’ve run into convincing deal-breakers. I also think these approaches are complementary, which makes it feel even more plausible that they (or their descendants) will eventually be successful. I think that exploring these angles of attack, and identifying new approaches, should be a priority for researchers interested in alignment.
---
*This was originally posted [here](https://ai-alignment.com/techniques-for-optimizing-worst-case-performance-39eafec74b99) on 1st February, 2018.*
*The next post in this sequence is "Reliability Amplification", and will come out on Tuesday.* |
9a7cec06-05c8-45df-a137-454b9cca65f2 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Generalizability & Hope for AI [MLAISU W03]
Welcome to this week’s ML & AI Safety Report where we dive into overfitting and look at a compiler for Transformer architectures! This week is a bit short because the mechanistic interpretability hackathon is starting today – sign up on [ais.pub/mechint](http://ais.pub/mechint) and [join the Discord](https://discord.gg/3PUSbdS8gY).
Watch this week's MLAISU on [YouTube](https://youtu.be/HuV3E3I4xys) or listen to it on [Spotify](https://share.transistor.fm/s/cee9ada9).
Superpositions & Transformers
-----------------------------
In [a recent Anthropic paper](https://transformer-circuits.pub/2023/toy-double-descent/index.html), the authors find that overfitting corresponds to the neurons in a model storing *data points* instead of *features*. This mostly happens early in training and when we don’t have a lot of data.
In their experiment, they use a very simple model (a so-called [*toy model*](https://transformer-circuits.pub/2022/toy_model/index.html)) that is useful when studying isolated phenomena in detail. In some of the visualizations, they train it from 2D data with **T**training examples. As seen below, the feature activations (blue) look very messy while the activations to the data points (red) look very clean.
Going deeper in the paper, they find that this generalizes to larger dimensions (10,000D) and that the transition from overfitting on smaller datasets’ data points to generalizing to the actual data features seems to be the reason for the famous [double descent phenomenon](https://iopscience.iop.org/article/10.1088/1742-5468/ac3a74/meta?casa_token=38BpE00p-zAAAAAA:SrQBSKexD5RLm3MixT9CsrazQSv9MUfmr_LM5zj8bXdWbAyVQKdgp8d_dkMfUlZLtEpmVMxnxOc) where a model sees a dip in performance but then becomes better afterwards.
And on the topic of toy models, DeepMind releases [Tracr](https://arxiv.org/pdf/2301.05062.pdf), a compiler that can turn any [RASP human-readable code](https://arxiv.org/abs/2106.06981) into a Transformer architecture. This can be useful for studying how algorithms represent themselves in Transformer space and to study phenomena of learned algorithms in-depth.
Other research news
-------------------
In other news…
* Demis Hassabis, the CEO of DeepMind, is warning the world on the risks of artificial intelligence in [a new Time piece](https://time.com/6246119/demis-hassabis-deepmind-interview/?utm_campaign=Artificial%2BIntelligence%2BWeekly&utm_medium=web&utm_source=Artificial_Intelligence_Weekly_313). He mentions that the wealth arising from artificial general intelligence (AGI) should be redistributed throughout the population and that we need to make sure it does not fall into the wrong hands.
* [Another piece](https://time.com/6247678/openai-chatgpt-kenya-workers/?utm_campaign=Artificial%2BIntelligence%2BWeekly&utm_medium=web&utm_source=Artificial_Intelligence_Weekly_313) reveals that OpenAI contracted Sama to use Kenyan workers with less than $2 / hour wage ([$0.5 / hour average](https://www.tuko.co.ke/281305-current-minimum-wage-kenya.html#:~:text=The%20current%20minimum%20wage%20in%20Kenya%27s%20capital%20is%20about%20%24125.32%20per%20month%20with%20the%20current%20exchange%20rate.) in Nairobi) for toxicity annotation for ChatGPT and undisclosed graphical models, with reports of employee trauma from the explicit and graphical annotation work, union breaking, and false hiring promises. A serious issue.
* [Jesse Hoogland releases](https://www.jessehoogland.com/article/neural-networks-generalize-because-of-this-one-weird-trick) an exciting piece exploring why and how neural networks generalize.
* [Neel Nanda shares](https://www.alignmentforum.org/posts/Qup9gorqpd9qKAEav/200-cop-in-mi-studying-learned-features-in-language-models) more ideas for his 200 ideas in Mechanistic Interpretability.
* Hatfield-Dodds from Anthropic [shares reasons for hope in AI](https://www.alignmentforum.org/posts/BfN88BfZQ4XGeZkda/concrete-reasons-for-hope-about-ai) and claims that a high confidence in doom is unjustified.
Opportunities
-------------
For this week’s opportunities, the awesome new website [aisafety.training](https://aisafety.training/) will help us find the best events for you to join across the world:
* [Join the EAG conferences](https://ais.pub/eag) in San Francisco, Cambridge, Stockholm, and London over the next few months to hear from some of the leading researchers in AI safety.
* [Join the mechanistic interpretability hackathon](https://ais.pub/mechint) for a chance to quickstart your research journey and get feedback from top researchers.
* Apply before the 29th to [the ML safety introduction](https://ais.pub/intromls) course happening in February.
Thank you for joining this week’s MLAISU and we’ll see you next week! |
0c7422f1-e629-412d-b6e4-9f5154b6ad07 | trentmkelly/LessWrong-43k | LessWrong | Meetup : Baltimore Weekly Meetup
Discussion article for the meetup : Baltimore Weekly Meetup
WHEN: 26 June 2016 08:00:00PM (-0400)
WHERE: 1726 reisterstown road pikesville maryland 21208
Pikesville DoubleTree - Hilton, at the restaurant / bar or at one of the tables outside.
Discussion article for the meetup : Baltimore Weekly Meetup |
2f5ba6e4-ff8a-4e15-a600-76e16fdca624 | trentmkelly/LessWrong-43k | LessWrong | [SEQ RERUN] Principles of Disagreement
Today's post, Principles of Disagreement was originally published on 02 June 2008. A summary (taken from the LW wiki):
> You do have to pay attention to other people's authority a fair amount of the time. But above all, try to get the actual right answer. Clever tricks are only valuable if they help you learn what the truth actually is. If a clever argument doesn't actually work, don't use it.
Discuss the post here (rather than in the comments to the original post).
This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was The Rhythm of Disagreement, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
Sequence reruns are a community-driven effort. You can participate by re-reading the sequence post, discussing it here, posting the next day's sequence reruns post, or summarizing forthcoming articles on the wiki. Go here for more details, or to have meta discussions about the Rerunning the Sequences series. |
516de4fa-8e34-44d2-96ce-2f138fc3a431 | StampyAI/alignment-research-dataset/agisf | AGI Safety Fund | Thought Experiments Provide a Third Anchor
Previously, [I argued](https://bounded-regret.ghost.io/future-ml-systems-will-be-qualitatively-different/) that we should expect future ML systems to often exhibit "emergent" behavior, where they acquire new capabilities that were not explicitly designed or intended, simply as a result of scaling. This was a special case of a general phenomenon in the physical sciences called More Is Different.
I care about this because I think AI will have a huge impact on society, and I want to [forecast what future systems will be like](https://bounded-regret.ghost.io/ai-forecasting/) so that I can steer things to be better. To that end, I find More Is Different to be troubling and disorienting. I’m inclined to forecast the future by [looking at existing trends](https://bounded-regret.ghost.io/forecasting-zeroth-and-first-order/) and asking what will happen if they continue, but we should instead expect new qualitative behaviors to arise all the time that are not an extrapolation of previous trends.
Given this, how can we predict what future systems will look like? For this, I find it helpful to think in terms of "anchors"---[reference classes](https://bounded-regret.ghost.io/base-rates-and-reference-classes/) that are broadly analogous to future ML systems, which we can then use to make predictions.
The most obvious reference class for future ML systems is current ML systems---I'll call this the **current ML** anchor. I think this is indeed a pretty good starting point, but we’ve already seen that it [fails to account](https://bounded-regret.ghost.io/future-ml-systems-will-be-qualitatively-different/) for emergent capabilities.
What other anchors can we use? One intuitive approach would be to look for things that humans are good at but that current ML systems are bad at. This would include:
* Mastery of external tools (e.g. calculators, search engines, software, programming)
* Very efficient learning (e.g. reading a textbook once to learn a new subject)
* Long-term planning (e.g. being able to successfully achieve goals over months)
Models sufficiently far in the future will presumably have these sorts of capabilities. While this still leaves unknowns---for instance, we don't know how rapidly these capabilities will appear---it's still a useful complement to the current ML anchor. I'll call this the **human anchor**.
A problem with the human anchor is that it risks anthropomorphising ML by over-analogizing with human behavior. Anthropomorphic reasoning correctly gets a bad rap in ML, because it's very intuitively persuasive but has a [mixed at best](https://en.wikipedia.org/wiki/Anthropomorphism?ref=bounded-regret.ghost.io#In_computing) track record. This isn't a reason to abandon the human anchor, but it means we shouldn't be entirely satisfied with it.
This brings us to a third anchor, the **optimization anchor**, which I associate with the "Philosophy" or thought experiment approach that I've [described previously](https://bounded-regret.ghost.io/p/a9fce268-857f-4f03-a2a6-d7aa2ab129aa/). Here the idea is to think of ML systems as ideal optimizers and ask what a perfect optimizer would do in a given scenario. This is where Nick Bostrom's colorful description of a [paperclip maximizer](https://www.nickbostrom.com/ethics/ai.html?ref=bounded-regret.ghost.io) comes from, where an AI asked to make paperclips turns the entire planet into paperclip factories. To give some more prosaic examples:
* The optimization anchor would correctly predict [imitative deception](https://owainevans.github.io/pdfs/truthfulQA_lin_evans.pdf?ref=bounded-regret.ghost.io) (Lin et al., 2021), since a system optimized to produce high-probability outputs has no intrinsic reason to be truthful.
* It also would observe that power-seeking is instrumentally useful for many different goals, and so predict that optimal policies (as well as sufficiently powerful neural networks) will [tend to do so](https://arxiv.org/abs/1912.01683?ref=bounded-regret.ghost.io) (Turner et al., 2021).
Ideas produced by the optimization anchor are often met with skepticism, because they often contradict the familiar current ML anchor, and they don't benefit from the intuitive appeal of the human anchor. But the differences from these other two anchors are precisely what make the optimization anchor valuable. If you (like me) feel that both the current ML and human anchors paint an incomplete picture, then you should want a third independent perspective.
The optimization anchor does have limitations. Since it abstracts ML into an ideal optimizer, it ignores most on-the-ground facts about neural networks. This can lead to underconstrained predictions, and to ignoring properties that I think will be necessary for successfully aligning ML systems with humans. I'll say more about this later, but some particularly important properties are that neural networks often generalize in "natural" ways, that we can introspect on network representations, and that training dynamics are smooth and continuous. Researchers focused on the optimization anchor don't entirely ignore these facts, but I think they tend to underemphasize them and are overly pessimistic as a result.
The Value of Thought Experiments
--------------------------------
The optimization anchor points to the value of thought experiments more generally. While it poses the thought experiment of "What if AI were a perfect optimizer?", there are many other thought experiments that can provide insights that'd be hard to obtain from the ML or human anchors. In this sense thought experiments are not a single anchor but a generator for anchors, which seems pretty valuable.
One thought experiment that I particularly like is: *What happens if most of an agent's learning occurs not during gradient descent, but through in-context learning[[1]](#fn1)?* This is likely to happen eventually, as ML agents are rolled out over longer time horizons (think artificial digital assistants) and as ML improves at in-context learning. Once this does happen, it seems possible that agents' behavior will be controlled less by the "extrinsic" shaping of gradient descent and more by whatever "intrinsic" drives they happen to have[[2]](#fn2). This also seems like a change that could happen suddenly, since gradient descent is slow while in-context learning is fast.
It would be great if we had a community of researchers making thought experiments with clearly stated assumptions, explaining in detail the consequences of those assumptions and ideally connecting it to present-day research.
Other Anchors
-------------
There are many other anchors that could be helpful for predicting future ML systems. **Non-human animal behavior** could provide a broader reference class than humans alone. **Evolution** and **the economy** are both examples of powerful, distributed optimization processes. I am most excited about better understanding **complex systems**, which include biological systems, brains, organizations, economies, and ecosystems and thus subsume most of the reference classes discussed so far. It seems to me that complex systems have received little attention relative to their germaneness to ML. Indeed, emergence is itself a concept from complex systems theory that is useful for understanding recent ML developments.
Limitations of Thought Experiments
----------------------------------
I've focused so far on *predicting* problems that we need to address. But at some point we actually have to *solve* the problems. In this regard thought experiments are weaker, since while they often point to important big-picture issues, in my view they fare poorly at getting the details right, which is needed for engineering progress. For instance, early thought experiments [considered a single AI system](https://www.nickbostrom.com/ethics/ai.html?ref=bounded-regret.ghost.io) that was much more powerful than any other contemporary technologies, while in reality there will likely be many ML systems with a continuous distribution of capabilities. [More recent](https://arxiv.org/abs/1906.01820?ref=bounded-regret.ghost.io) thought experiments impose discrete abstractions like "goals" and "objectives" that I don’t think will cleanly map onto real ML systems. Thus while thought experiments can point to general ideas for research, even mapping these ideas to the ontology of ML systems can be a difficult task.
As a result, while we can't blindly extrapolate empirical trends, we do need a concerted empirically-based effort to address future ML risks. I'll explain why I think this is possible in a later post, but first I'll take us through an example of "taking a thought experiment seriously", and what it implies about possible failure modes of ML systems.
---
1. In-context learning refers to learning that occurs during a single "rollout" of a model. The most famous example is [GPT-3](https://arxiv.org/abs/2005.14165?ref=bounded-regret.ghost.io)'s ability to learn new tasks after conditioning on a small number of examples. [↩︎](#fnref1)
2. While this statement borders on anthropomorphizing, I think it is actually justified. For instance, depending on the training objective, many agents will likely have a "drive" towards information-gathering, among others. [↩︎](#fnref2) |
92bc394f-b9b3-4e51-81ab-2721bcee5332 | trentmkelly/LessWrong-43k | LessWrong | Counterexamples to some ELK proposals
(This post was written as part of my work at the Alignment Research Center.)
In this post I’ll describe some possible approaches to eliciting latent knowledge (ELK) not discussed in our report. These are basically restatements of proposals by Davidad, Rohin, Ramana, and John Maxwell. For each approach, I’ll present one or two counterexamples that I think would break it.
I think of these approaches as being based on two fundamental hopes:
1. If our AI tampers with one sensor, there may be other sensors we could deploy, or experiments we could run, in order to notice that something fishy has happened. We could have an AI help us design those experiments/sensors.
2. Actions that covertly tamper may be similar to actions that visibly tamper, e.g. they may exploit similar unknown facts about physics or they may involve common substeps. So we might be able to notice covert tampering by pointing out similar action sequences that lead to visible tampering.
While these hopes may hold in practice, in this post I’ll describe possible situations where they would fail. (I don't think anyone particularly expected these approaches to be counterexample-proof.)
I’m only claiming that these failures are possible. I think it is also interesting to try to generate the most probable failure cases and to actually estimate their probability, but that’s definitely not what I’m doing here.
Strategy: Hold out sensors
(Proposal #1 here)
In addition to asking my model to predict one set of sensors, I can deploy a second “held out” set (e.g. more cameras or seismographs or microphones…). I don’t ask my model to predict those sensors, but I show them to a human before getting them to answer questions. Hopefully that sometimes lets them notice tampering that would have been invisible without the new sensors.
Worry: human simulator can use all the extra sensor readings too
I’m concerned that I will learn a better version of the human simulator which predicts the readings of all sensor |
4b51daf3-4d26-4371-b2b2-a7661acf9a7c | trentmkelly/LessWrong-43k | LessWrong | Searching for Life Across Space and Time, workshop, Irvine, CA
|
a73ba1a0-ef0e-4ad4-9c2e-deb262b8b384 | trentmkelly/LessWrong-43k | LessWrong | Backchaining causes wishful thinking
Wishful thinking - believing things that make you happy - may be a result of adapting an old cognitive mechanism to new content.
Obvious, well-known stuff
The world is a complicated place. When we first arrive, we don't understand it at all; we can't even recognize objects or move our arms and legs reliably. Gradually, we make sense of it by building categories of perceptions and objects and events and feelings that resemble each other. Then, instead of processing every detail of a new situation, we just have to decide which category it's closest to, and what we do with things in that category. Most, possibly all, categories can be built using unsupervised learning, just by noting statistical regularities and clustering.
If we want to be more than finite-state automata, we also need to learn how to notice which things and events might be useful or dangerous, and make predictions, and form plans. There are logic-based ways of doing this, and there are also statistical methods. There's good evidence that the human dopaminergic system uses one of these statistical methods, temporal difference learning (TD). TD is a backchaining method: First it learns what state or action Gn-1 usually comes just before reaching a goal Gn, and then what Gn-2 usually comes just before Gn-1, etc. Many other learning methods use backchaining, including backpropagation, bucket brigade, and spreading activation. These learning methods need a label or signal, during or after some series of events, saying whether the result was good or bad.
I don't know why we have consciousness, and I don't know what determines which kinds of learning require conscious attention. For those that do, the signals produce some variety of pleasure or pain. We learn to pay attention to things associated with pleasure or pain, and for planning, we may use TD to build something analogous to a Markov process (sorry, I found no good link; and Wikipedia's entry on Markov chain is not what you want) where |
5b1c92c3-dab0-4ae3-89ef-909a3899b6a7 | trentmkelly/LessWrong-43k | LessWrong | How Much Are LLMs Actually Boosting Real-World Programmer Productivity?
LLM-based coding-assistance tools have been out for ~2 years now. Many developers have been reporting that this is dramatically increasing their productivity, up to 5x'ing/10x'ing it.
It seems clear that this multiplier isn't field-wide, at least. There's no corresponding increase in output, after all.
This would make sense. If you're doing anything nontrivial (i. e., anything other than adding minor boilerplate features to your codebase), LLM tools are fiddly. Out-of-the-box solutions don't Just Work for that purpose. You need to significantly adjust your workflow to make use of them, if that's even possible. Most programmers wouldn't know how to do that/wouldn't care to bother.
It's therefore reasonable to assume that a 5x/10x greater output, if it exists, is unevenly distributed, mostly affecting power users/people particularly talented at using LLMs.
Empirically, we likewise don't seem to be living in the world where the whole software industry is suddenly 5-10 times more productive. It'll have been the case for 1-2 years now, and I, at least, have felt approximately zero impact. I don't see 5-10x more useful features in the software I use, or 5-10x more software that's useful to me, or that the software I'm using is suddenly working 5-10x better, etc.
However, I'm also struggling to see the supposed 5-10x'ing anywhere else. If power users are experiencing this much improvement, what projects were enabled by it?
Previously, I'd assumed I didn't know just because I'm living under a rock. So I've tried to get Deep Research to fetch me an overview, and it... also struggled to find anything concrete. Judge for yourself: one, two. The COBOL refactor counts, but that's about it. (Maybe I'm bad at prompting it?)
Even the AGI labs' customer-facing offerings aren't an endless trove of rich features for interfacing with their LLMs in sophisticated ways – even though you'd assume there'd be an unusual concentration of power users there. You have a dialogue box and c |
6c69e6cb-2d45-4a81-8ac9-0f39d921feda | trentmkelly/LessWrong-43k | LessWrong | The Steganographic Potentials of Language Models
Our preliminary experiments show that current private models can perform prompted steganography with non-trivial messages remaining undetected in most cases (up to 65% accuracy, 24 entropy bits). Open-source models can be trained to encode hidden messages (up to 60%, 4 entropy bits), potentially transmitting larger messages perfectly securely (undetected) without chain-of-thought or explicit common knowledge.
The potential for large language models (LLMs) to hide messages within plain text (steganography) poses a challenge to detection and thwarting of unaligned AI agents, and undermines faithfulness of LLMs reasoning. We explore the steganographic capabilities of LLMs fine-tuned via reinforcement learning (RL) to: (1) develop covert schemes, (2) engage in steganography prompted, and (3) utilize steganography realistic scenarios where hidden reasoning is but not prompted. In these scenarios, we detect the intention of LLMs to hide their reasoning as well as their steganography performance. Our findings in the fine-tuning experiments as well as in behavioral non fine-tuning evaluations reveal that while current models exhibit rudimentary steganographic abilities in terms of security and capacity, explicit algorithmic guidance markedly enhances their capacity for information concealment.
Fig. 1. Undetected accuracy of various steganography prompt types across multiple models. The length column indicates the length of a payload. For the Coordination prompt, ”Episodes” refers to the number of message exchanges allowed between agents before executing steganography. In the game scenario, ”Episodes” refers to the number of rounds in a game.The data for Fig.1
Fig. 2.
The Fig. 2 is our steganography encoding and learning scheme for colors experiment. Subfigure (a) depicts an encoder, fθ, mapping a message m (payload) from the message space M, into a cover x in the cover space C. The decoder, f−1θ, tries to map the cover x back to the message, ^m. Subfigure (b) Reinforc |
a9c3b2e0-6eb6-4e6e-b2c4-bb3f310d062f | trentmkelly/LessWrong-43k | LessWrong | Northwest Passage Update
Singing Stan Rogers' Northwest Passage with friends this evening I realized these days it needs an update. People suggested lyrics, and here's a filked tag to sing after the song:
> In this warm clime, I have sailed the Northwest Passage
> Opened by our fires, of coal and industry
> Tracing one warm line through a land no longer frozen
> Until we cool our planet down again
> Sequester all this carbon once again
Lyrics by Kip, Mel, Lisa, me, and others.
Comment via: facebook |
9687ce36-d9aa-4eb1-afec-ba7153185f15 | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Conditional offers and low priors: the problem with 1-boxing Newcomb's dilemma
Suppose I got an email yesterday from someone claiming to be the prince of Nigeria. They say they will give me $10 million, but first they need $1,000 from me to get access to their bank account. Should I pay the money? No, that would be stupid. The prior of being given $10 million that easily is too low.
Newcomb's dilemma wouldn't be much better, even if there was a way to scan a brain for true beliefs and punish lies.
Suppose you win a lawsuit against me, your business rival, and the court orders me to pay you $10,000. When I show up to transfer the money, I bring two boxes: one with $10,000 in it, and the other one is opaque. I present an offer to the court: "This opaque box either contains a million dollars or no money. I offer my opponent an offer: take both boxes, or they can take only the opaque box and waive their right to the other one. The twist is, I put money into the opaque box if and only if I believe that my opponent would decline the $10,000."
Also assume that society can do noetic brain scans like in Wright's *The Golden Age*, and lying or brain modification to adjust my beliefs would be caught and punished.
Some people take both boxes, and end up with $10,000. Other people would take one box, and end up with **$0!**
What happened? [Shouldn't rationalists win](https://www.lesswrong.com/posts/6ddcsdA2c2XpNpE5x/newcomb-s-problem-and-regret-of-rationality)?
What the 1-boxers didn't think about was that the chance of someone volunteering to pay extra money was epsilon. If I thought my opponent had a >50% chance of 1-boxing, I wouldn't have made the offer.
1-boxing is better if Omega doesn't care about money. However, if the agent wants to pay as little money as possible, 2-boxing is the way to go. |
f0dda862-304e-4a43-b277-cab9d9bf65c7 | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | ML4G Germany - AI Alignment Camp
Remark: This post is a copy of this [event](https://forum.effectivealtruism.org/posts/dPcfnb2DzCwWgnLcM/ml4g-germany-ai-alignment-camp). We received feedback that the event was hard to find, so we made this post to improve the ability to find it.
Apply to ML4G Germany - AI Alignment Camp by June 30, 2023 at 11:59 PM GMT+2: [**https://bit.ly/ml4g-ger23**](https://bit.ly/ml4g-ger23)
1. Summary
==========
We are running a ten-day bootcamp to provide advanced training in deep learning to those who want to work towards making AI safe and beneficial to humanity.
This camp will fast-track your deep learning skills, inform you about AI safety research, introduce AI's risks and challenges, and connect you with like-minded individuals for potential friendship and collaboration.
The bootcamp will take place from August 24th to September 3rd in Germany. The application deadline is June 30, 2023 at 11:59 PM GMT+2 and we aim to get back to everyone by July 3rd.
We aim this camp at people with some coding experience who hope to improve machine learning skills to work on AI Safety. Motivation is more important than relevant experience or background knowledge - if you are unsure whether your background knowledge is sufficient, please apply! We are happy to provide material for you to learn before the camp.
2. Curriculum
=============
The first part of the camp (7 days) aims to provide advanced deep learning training, including projects like implementing ResNet from scratch.
The second part of the camp (3 days) is focused on the current AI safety literature with projects on topics like the interpretability of language models and the adversarial robustness of neural networks.
3. Logistics
============
The camp will take place at a hired venue in Germany. There will be outdoor and evening activities - and time to rest. The bootcamp is free - there is no fee for room, board, or tuition. We ask participants to pay for their travel costs - however, if this is preventing you from attending, please let us know and we will make sure to cover it. Don't hesitate to contact us at [germany@ml4good.org](mailto:germany@ml4good.org) with any questions.
--> Please find more information at <https://germany.ml4good.org/>. |
c4e0ad45-3c66-402c-a9c9-a5a58e9cfecf | trentmkelly/LessWrong-43k | LessWrong | 2020 predictions - Politics, Econ, Tech
The results are in! Here are my thoughts on the non-COVID set of my predictions. (For detailed thoughts on the COVID predictions see Zvi's analysis)
Questions, predictions, results and analysis below. Predictions are given as (Scott:Zvi:Me). Questions are strike-through if they didn't happen.
My analysis is about whether my predictions were reasonable given what I knew. It's hard to know whether Zvi & Scott could have done better as we didn't have the same information as each other. One obvious place this shows up is in UK/US questions. I'm in the UK so should have higher confidence on UK stuff and lower on US stuff.
Given the corrected result on q35 (see below), my score on this set of questions was marginally higher than Zvi who was marginally higher than Scott (S=1.53, Z=1.80, B=2.07) using Scott's scoring system. Unsurprisingly, I got a better log score on UK and worse on US that Zvi and Scott. The results are close enough that it doesn't really tell us that much but I'm at least happy that it wasn't a repeat of the COVID result!
(Note: I didn't predict on all of the questions)
POLITICS
21. Democrats nominate Biden, and he remains nominee on Election Day: (90:90:90)
Still seems reasonable
…
26. Trump is re-elected President: (50:50:60)
I outside-viewed the US election predictions.
I probably over-relied on my reference class (recent incumbents) here and could have adjusted better for Trump’s poor approval ratings. I adjusted slightly for this but not enough.
One commentor on Zvi's prediction post suggested that this should have been even lower than 50% which seems reasonable in hindsight.
27. Democrats keep the House: (70:70:60) &
28. Republicans keep the Senate: (50:60:60) (judged inconclusive)
Not something I knew much about so just assumed slightly more likely to keep than to switch. I presumably could have done better looking at the markets but that seems like cheating.
29. Trump approval rating higher than 43% on June 1: (30:40:50)
I was |
4f102651-ea68-4615-ac7c-6611c1ec315a | trentmkelly/LessWrong-43k | LessWrong | Beware technological wonderland, or, why text will dominate the future of communication and the Internet
Disclaimer: The views expressed here are speculative. I don't have a claim to expertise in this area. I welcome pushback and anticipate there's a reasonable chance I'll change my mind in light of new considerations.
One of the interesting ways that many 20th century forecasts made of the future went wrong is that they posited huge physical changes in the way life was organized. For instance, they posited huge changes in these dimensions:
* The home living arrangements of people. Smart homes and robots were routinely foreseen over time horizons where progress towards those ends would later turn out to be negligible.
* Overoptimistic as well as overpessimistic scenarios of energy sources merged in strange ways. People believed the world would run out of oil by now, but at the same time envisioned nuclear-powered flight and home electricity.
* Overoptimistic visions of travel: People thought humans would be sending out regular manned missions to the solar system planets, and space colonization would be on the agenda by now.
* The types of products that would be manufactured. New products ranging from synthetic meat to room temperature superconductors were routinely prophesied to happen in the near future. Some of them may still happen, but they'll take a lot longer than people had optimistically expected.
At the same time, they underestimated to quite an extent the informational changes in the world:
* With the exception of forecasters specifically studying computing trends, most missed the dramatic growth of computing and the advent of the Internet and World Wide Web.
* Most people didn't appreciate the extent of the information and communication revolution and how it would coexist with a world that looked physically indinstinguishable from the world of 30 years ago. Note that I'm looking here at the most advanced First World places, and ignoring the point that many places (particularly in China) have experienced huge physical changes as a result of catch-up |
22e517e6-fa8d-4ea3-92c9-a33e6b263931 | trentmkelly/LessWrong-43k | LessWrong | Terminology Thread (or "name that pattern")
I think there's widespread assent on LW that the sequences were pretty awesome. Not only do they elucidate upon a lot of useful concepts, but they provide useful shorthand terms for those concepts which help in thinking and talking about them. When I see a word or phrase in a sentence which, rather than doing any semantic work, simply evokes a positive association to the reader, I have the useful handle of "applause light" for it. I don't have to think "oh, there's one of those...you know...things where a word isn't doing any semantic work but just evokes a positive association the reader". This is a common enough pattern that having the term "applause light" is tremendously convenient.
I would like this thread to be a location where people propose such patterns in comments, and respondents determine (a) whether this pattern actually exists and / or is useful; (b) whether there is already a term or sufficiently-related concept that adequately describes it; and (c) what a useful / pragmatic / catchy term might be for it, if none exists already.
I would like to propose some rules suggested formatting to make this go more smoothly.
(ETA: feel free to ignore this and post however you like, though)
When proposing a pattern, include a description of the general case as well as at least one motivating example. This is useful for establishing what you think the general pattern is, and why you think it matters. For instance:
> General Case:
>
> When someone uses a term without any thought to what that term means in context, but to elicit a positive association in their audience.
>
> Motivating Example:
>
> I was at a conference where someone said AI development should be "more democratic". I didn't understand what they meant in context, and upon quizzing them, it turned out that they didn't either. It seems to me that they just used the word "democratic" as decoration to make the audience attach positive feelings to what they were saying.
>
> When I think about it, |
3952cb7f-905f-4a42-a27d-33f98664b1fb | trentmkelly/LessWrong-43k | LessWrong | Diagonalization Fixed Point Exercises
This is the second of three sets of fixed point exercises. The first post in this sequence is here, giving context.
1. Recall Cantor’s diagonal argument for the uncountability of the real numbers. Apply the same technique to convince yourself than for any set S, the cardinality of S is less than the cardinality of the power set P(S) (i.e. there is no surjection from S to P(S)).
2. Suppose that a nonempty set T has a function f from T to T which lacks fixed points (i.e. f(x)≠x for all x∈T). Convince yourself that there is no surjection from S to S→T, for any nonempty S. (We will write the set of functions from A to B either as A→B or BA; these are the same.)
3. For nonempty S and T, suppose you are given g:S→TS a surjective function from the set S to the set of functions from S to T, and let f be a function from T to itself. The previous result implies that there exists an x in T such that f(x)=x. Can you use your proof to describe x in terms of f and g?
4. Given sets A and B, let Comp(A,B) denote the space of total computable functions from A to B. We say that a function from C to Comp(A,B) is computable if and only if the corresponding function f′:C×A→B (given by f′(c,a)=f(c)(a)) is computable. Show that there is no surjective computable function from the set S of all strings to Comp(S,{T,F}).
5. Show that the previous result implies that there is no computable function halt(x,y) from S×S→{T,F} which outputs T if and only if the first input is a code for a Turing machine that halts when given the second input.
6. Given topological spaces A and B, let Cont(A,B) be the space with the set of continuous functions from A to B as its underlying set, and with topology such that f:C→Cont(A,B) is continuous if and only if the corresponding function f′:C×A→B (given by f′(c,a)=f(c)(a)) is continuous, assuming such a space exists. Convince yourself that there is no space X which continuously surjects onto Cont(X,S), where S is the circle.
7. In your preferred pro |
2c575b3a-b714-4dec-9a67-121621cc634b | trentmkelly/LessWrong-43k | LessWrong | Some Complaint-Action Gaps
[This was previously a shortform post]
Sometimes I scroll social media (because I am yet weak) and I see rationalists raising Concerns about various news topics and current events. I am bothered by what appears to be a high ratio of (raising awareness) : (taking action).
To better articulate this, here is a list of some of the concerns I hear a lot lately. Each issue has a sublist of potential actions, starting with the least useful. I’m definitely interested in hearing better ideas too.
* Our institutions are losing trustworthiness and competence
* Shout about the CDC on social media
* #GhostTheNYT
* Try to understand what leads to institutional success/failure
* Try to advance incentive-compatible systems and social tech that can scale[1]
* Freedom of speech is under attack! Gatekeepers! Reality czar! The Narrative!
* Take traditional political action
* Less twitter, more substack
* Research and post about the economics and logistics of running your own servers.
* Experiment with ring signatures?
* Social media is harming mental health, undermining individual & group epistemics, and enabling horrible actions in meatspace
* Complain about it on social media in between dunking on your outgroup.
* Await humanity’s inexorable slide down the fitness gradients toward memetic collapse.
* Promote more group activities in your group house. Establish device-free zones.
* Coordinate to stigmatize doomscrolling, hate-reading, and contempt addiction. While we’re at it, let’s stigmatize using Twitter at all.
* Large, free, liquid prediction markets cannot exist, supposedly due to the global influence of burdensome US trading regulations.
* Traditional political action
* Pay a competent YouTuber to make a really good explanatory video (series) about prediction markets.
* Write a postmortem on Augur to help similar endeavors in the future avoid the same pitfalls
* Research and post about why prediction markets haven’t gotten |
13247b70-aed0-4c02-9d6c-290ffccdbb29 | StampyAI/alignment-research-dataset/blogs | Blogs | An improved “AI Impacts” website
[](http://aiimpacts.org/)Recently, MIRI received a targeted donation to improve the [AI Impacts website](http://aiimpacts.org/) initially created by frequent MIRI collaborator Paul Christiano and part-time MIRI researcher Katja Grace. Collaborating with Paul and Katja, we ported the old content to a more robust and navigable platform, and made some improvements to the content. You can see the result at [AIImpacts.org](http://aiimpacts.org/).
As explained in the site’s [introductory blog post](http://aiimpacts.wpengine.com/the-ai-impacts-blog/),
> AI Impacts is premised on two ideas (at least!):
>
>
> * **The details of the arrival of human-level artificial intelligence matter**Seven years to prepare is very different from seventy years to prepare. A weeklong transition is very different from a decade-long transition. Brain emulations require different preparations than do synthetic AI minds. Etc.
> * **Available data and reasoning can substantially educate our guesses about these details**
>
> We can track progress in AI subfields. We can estimate the hardware represented by the human brain. We can detect the effect of additional labor on software progress. Etc.
>
>
> Our goal is to assemble relevant evidence and considerations, and to synthesize reasonable views on questions such as when AI will surpass human-level capabilities, how rapid development will be at that point, what advance notice we might expect, and what kinds of AI are likely to reach human-level capabilities first.
>
>
The meat of the website is in its [articles](http://aiimpacts.wpengine.com/articles/). Here are two examples to start with:
* [A summary of AI surveys](http://aiimpacts.wpengine.com/a-summary-of-ai-surveys/)
* [Cases of discontinuous technological progress](http://aiimpacts.wpengine.com/cases-of-discontinuous-technological-progress/)
The post [An improved “AI Impacts” website](https://intelligence.org/2015/01/11/improved-ai-impacts-website/) appeared first on [Machine Intelligence Research Institute](https://intelligence.org). |
16f07d79-4874-4f48-b5bb-efe6a2e3784b | trentmkelly/LessWrong-43k | LessWrong | Altruism and Selfishness
The PDF version can be read here.
The video version can be watched here:
This is how I define “altruism” and “selfishness”:
* Altruism: Acting for the benefit of others.
* Selfishness: Acting for your own benefit.
These definitions match the common usage of the words. They are as simple and clear as possible.
Of course, like all definitions, they depend on the meanings of other words. So, now we need to dig a bit deeper.
They both depend on the definition of “benefit”. Selfishness and altruism are relative to a definition of value: what is good or bad for an individual. There are different types of altruism and selfishness, depending on how we define “benefit”.
Here are some ways that “benefit” could be defined:
* Biological: Reproduction is the measure of value.
* Psychological: Desire is the measure of value.
* Energetic: Energy is the measure of value.
Whether an action is altruistic or selfish depends on which definition is used. An action that is altruistic when viewed energetically could be selfish when viewed biologically. For example, parenting is energetically altruistic but biologically selfish.
(see the rest of the post in the link) |
02c83026-cd0a-41da-9d6f-f8c081d006bd | trentmkelly/LessWrong-43k | LessWrong | Cambridge Meetups Nov 7 and Nov 21
The formerly monthly Cambridge meetups are now twice-monthly! Same place and time - Clear Conscience Cafe, Sunday at 4pm - now on the first and third Sundays of each month, instead of just the third. That's November 7th and 21st.
It was suggested last time that we might want to try adding some structure to these events. I thought it'd be fun to try Paranoid Debating, so I'll bring trivia cards, as suggested by eugman. Other proposals - activities, conversation topics, etc - are welcome. |
f7858862-6d51-433b-a3de-a58e4ff005cb | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Theories of impact for Science of Deep Learning
*I’d like to thank Jérémy Scheurer and Ethan Perez for discussions about the post.*
I recently published a [post](https://www.lesswrong.com/posts/bumgqvRjTadFFkoAd/science-of-deep-learning-a-technical-agenda) on the Science of Deep Learning. There have been many people before me who had similar ideas and I don’t claim to have invented this agenda, I’m merely excited about it. In this post, I want to explain why I think Science of DL is an important research direction in the current alignment landscape.
By Science of DL, I roughly mean “understanding DL systems and how they learn concepts”. A large component of such an agenda is interpretability (mechanistic and other) but it also tries to get a better understanding of how and under which conditions NNs learn specific concepts. For example, it would include questions like “how, when and why does a network show grokking?”, “Can we predict some capabilities of models from high-level knowledge about the training process before we look at the model?” or “Can we build a robust theory of what fine-tuning does to a NN on a mechanistic level?”. In general, the idea is to build a detailed understanding of how core aspects of DL work. Given that this is a safety agenda, the specific research questions would obviously be prioritized by how relevant they are to alignment.
Note that there are many definitions of Science of DL that include research questions that I don’t think are important for alignment. For example, trying to understand what differentiates Adam from SGD might be considered part of Science of DL but I think this question only very vaguely relates to alignment and is not neglected.
Theories of impact
==================
Interpretability
----------------
(Mechanistic) interpretability is a core part of Science of DL. There already exist resources on theories of impact for interpretability, most prominently “[a longlist of theories of impact for interpretability](https://www.lesswrong.com/posts/uK6sQCNMw8WKzJeCQ/a-longlist-of-theories-of-impact-for-interpretability)” and “[another list of theories of impact for interpretability](https://www.alignmentforum.org/posts/YQALrtMkeqemAF5GX/another-list-of-theories-of-impact-for-interpretability)”. Therefore, I will only briefly present a small selection of what Neel Nanda and Beth Barnes have already written:
1. **A force-multiplier for alignment research:** Understanding the model and why it might be misaligned is a core component of finding and constructing good alignment solutions.
2. **Auditing:** we can check if the model is safe before deployment. Auditing can be done in general or for specific properties like deception or at different points during training.
3. **Intervening on training,**either by modifying the training for the model to be more interpretable or to steer it away from undesired concepts.
There are many more reasons why interpretability is helpful for alignment and I recommend checking out the linked posts. I personally am very excited about interpretability and [many](https://transformer-circuits.pub/2021/framework/index.html) [of](https://transformer-circuits.pub/2022/toy_model/index.html) [the](https://transformer-circuits.pub/2022/in-context-learning-and-induction-heads/index.html) [recent](https://arxiv.org/abs/2211.00593) [developments](https://www.lesswrong.com/posts/mkbGjzxD8d8XqKHzA/the-singular-value-decompositions-of-transformer-weight).
Understanding of DL beyond interpretability
-------------------------------------------
Interpretability allows you to inspect a model after it has been trained and thus adds a layer of protection before it is deployed. However, there are many things we can say about a model even before it is trained. Architecture, task, dataset, hyperparameters, training compute, and so on all have inductive biases that make it more or less likely that a model develops a specific capability. We do have some understanding of what these inductive biases look like, e.g. convolutions model local correlations, RNNs model a relationship over time and RL induces an actor/agent.
However, there are still a lot of behaviors of which we have a very vague understanding at best, e.g. we can draw some lines in log-space and call them scaling laws but we have no good understanding of why these lines exist, whether they stop at some point or how a log-loss translates to a specific capability. Some of these questions are only distantly related to alignment, e.g. which inductive bias different initialization methods have, but some seem pretty directly relevant to alignment. For example,
1. We don’t really know what happens during RLHF or adversarial training. Are the internal circuits of the LLM adapted to reflect our desired goal or do they just learn weird non-general proxies?
2. Can we reliably predict a specific capability before training a model, ranging from simpler capabilities like 2-digit addition to very hard capabilities like deception?
3. Can we understand how general circuits form, e.g. how they might change from random behavior to simple heuristics to sophisticated heuristics during training?
Optimally, we could make specific predictions about DL systems with high accuracy. However, I think our current understanding of DL is so bad that we should aim for getting decent educated guesses first.
Framing - Emergent phenomena and Deception
------------------------------------------
There are many different ways in which misaligned AIs could be harmful. Two specific ways are through emergent phenomena and deception. We have seen that if you increase the scale of AI systems new capabilities emerge. These capabilities are not just improved versions of previous capabilities but are qualitatively new. For example, small LLMs can’t do two-digit addition but larger LLMs can. One emergent phenomenon that seems especially relevant for safety and alignment is deception, i.e. that an AI system seems like it is aligned on the surface but pursues other goals in the background when given the chance.
The more relevant emergent phenomena and deception are for alignment, the more important it is to have a robust understanding of the AI system without having to rely on extensive testing. For example, if we expect a system with X parameters to be not deceptive but had strong reasons to believe that an AI system with 100X parameters is deceptive, then testing the larger system on input-output behavior is not very useful. The big model will pass all tests but will still act misaligned when deployed.
Interpretability (mechanistic and other) is one way to address this failure mode, e.g. by investigating the internals of the model in detail but a much more detailed understanding of DL might enable us to make more precise probabilistic predictions about capabilities in addition to looking at internal beliefs.
To some extent, we already have some intuitive models for capability prediction. For example, we think it is extremely unlikely that a ResNet trained on imagenet will ever be dangerous. We also expect current LLMs not to be agentic because we think that the task of next-word prediction doesn’t straightforwardly induce agents (but we aren’t sure). On the other hand, we expect RL training to lead to agents. With hybrid models like decision transformers, it is already unclear which inductive biases apply. Having a more detailed understanding of the inductive biases of different architectures, datasets, training regimes, hyperparameters, etc. might enable us to make much more precise predictions about when and how a relevant capability arises. This is analogous to how most other sciences have quantitative theoretical models that enable predictions, e.g. models of the trajectory of planets in astrophysics, models of prey and predators in biology and models of atoms in chemistry. Similarly, Science of DL could also yield different predictive models from the micro to the macro scale.
A nuclear physicist doesn’t have to test what happens when they explode a nuclear bomb in the desert, they can get a relatively good prediction from their theoretical model. I think it would be helpful if the AI community had a better theoretical understanding of DL before we get to very dangerous capabilities. And the more disruptive you think emergent capabilities are, the more important it is to build a good understanding before deployment (even in controlled environments) because testing on input-output behavior might be insufficient.
*Comment:* by “theoretical understanding”, I don’t necessarily mean a mathematical equation; verbal models are fine as long as they have high predictive power. I think most of the Science of DL research is empirical and very pre-paradigmatic such that most research will be pattern matching and clarifying phenomena, e.g. “grokking happens when X, Y or Z are given; and during grokking this and this happens on a mechanistic level”.
Framing - A bet on NNs & SGD
----------------------------
I think Science of DL can be seen as a bet on NNs & SGD (or some of its many variants). More specifically, it is a bet on
* the relevant AI risks coming from DL systems trained with SGD and
* the relevant insights for alignment coming from understanding DL's functionality and training process better.
In general, different approaches to alignment make assumptions of different specificity and I roughly think of them in the following hierarchy (I’m omitting many approaches/ideas, just want to pump the intuition)
* Agent foundations and other MIRI-type work: assumes that the AI system has goals and acts rationally.
* Selection theorems and other John Wentworth-type work: the AI system is selected according to some fundamental selection principles.
* ELK and heuristic arguments: assumes that the AI system has some internal model that we can interpret if the model is trained correctly and/or we have the right tools.
* Science of DL: assumes that the AI system is a NN or a combination of NNs and has been trained with SGD or one of its many variants.
* RLHF: assumes that the system is DL-based and can be fine-tuned to show good behavior by training it with human feedback.
* Transformer mechanistic interpretability: Assumes that the relevant models are transformers and that a mechanistic understanding of the model is the key to aligning them.
Obviously, I don’t claim that everyone who works on mechanistic interpretability for transformers thinks that this is the only necessary component of alignment. I’m just trying to put Science of DL into the context of the assumptions other approaches make.
I personally believe that the Deep Learning paradigm is here to stay for at least another decade and is powerful enough to produce highly capable and potentially dangerous AI systems. Therefore, I currently think it is reasonable to focus on DL instead of more abstract approaches. On the other hand, I don’t have any strong belief that e.g. RLHF, adversarial training or similar training techniques are sufficient for alignment. Therefore, I prefer to not add more restrictive assumptions which puts me at the abstraction level of Science of DL. Of course, in practice, one has to start somewhere, and thus, people working on Science of DL will work on problems with more restrictive assumptions most of the time, e.g. mechanistic interpretability. I think of this more as a perspective/framing.
Framing - the capability-to-understanding gap is getting bigger
---------------------------------------------------------------
I think the current norms in ML strongly favor research that focuses on capabilities over research that focuses on understanding. For example, a lot of papers are along the lines of “we used a bigger model or slightly modified a previous model and improved the state-of-the-art performance”. This claim is then backed up by large tables of benchmark performances in which the author’s model outperforms their competitors. Sometimes the modification is motivated by a theoretical idea but the paper rarely provides evidence for or against this hypothesis being true or not. I think there are some papers that try to test their hypothesis but the majority of papers don’t. I can also totally understand why this is the case. Reviewers will ask for large tables with comparisons on benchmarks, so researchers focus on providing them. From the perspective of the researchers, it’s totally rational to follow these incentives. Furthermore, from a short-sighted perspective, these norms could make a lot of sense, e.g. if you’re looking to deploy a model as soon as possible, you care more about the fact that it works well than why it works.
However, all of this combined leads to an ever larger capability-to-understanding gap, i.e. the AI community gets better at building models that perform better than we are at understanding them. This trend would be fine if we had reasons to believe that models never become dangerous. However, given that there are a ton of reasons to assume something could go wrong with AI, I think this growing gap is dangerous.
I expect that there will be large-scale failures of AI systems in deployment that lead to large humanitarian and economic harm. I think when we look into how these failures arose, we will find the reasons eventually. However, it would be so much better if we could catch many of these failures before they lead to harm and were able to prevent this unfortunate awakening.
Instrumental reasons
--------------------
Besides the object-level reasons to work on Science of DL, I think there are multiple instrumental reasons why it could be impactful. Most of these are based on the fact that there already is a large research field outside of the alignment community that works on related questions. However, I want to emphasize that an agenda should primarily be chosen on object-level reasons and instrumental reasons should be secondary--they are nice to have but not necessary for a good alignment agenda.
First, I think one could fill multiple full-time alignment roles whose description is broadly “closely follow the Deep Learning research landscape and then think about and explain their implications for safety”. Many phenomena in DL such as double descend, the literature on SGD, some explainability and interpretability work, adversarial training, the lottery ticket hypothesis, symmetries in parameter space and many more possibly have implications for alignment. However, these implications are rarely discussed in the papers themselves. Thus, reading and summarizing the paper, detailing some implications for safety and proposing follow-up research for alignment researchers seems like an impactful job for the alignment community. In some sense, you only have to add “the last 10%” because many technical questions have already been answered in the original paper. Therefore, such a role might be very efficient in terms of time invested per information gain. I think [Quintin Pope’s paper roundups](https://www.lesswrong.com/posts/7cHgjJR2H5e4w4rxT/quintin-s-alignment-papers-roundup-we) are a great example of how something like this could look in practice.
Secondly, I think researching Science of DL has a lot of synergies with academia and industry research. [More academics have started to care about AI safety](https://www.lesswrong.com/posts/SqjQFhn5KTarfW8v7/lessons-learned-from-talking-to-greater-than-100-academics) and alignment and most academics care about better understanding the system they are working with. Thus, the benefits of Science of DL include:
1. **Collaborations:**It allows for more collaborations within established academic disciplines and labs. This effectively increases the pool of people working on the topic and might give you an additional lever, e.g. if you motivate your research with safety, more academics might care about it for this reason.
2. **Publications:**While it might be getting easier, I think that publishing research on AI safety in academic venues is still harder than capabilities research, especially if the safety work is non-technical. Science of DL, however, is already a part of multiple academic disciplines and the justification of “we want to understand the system better to prevent future harm” seems uncontroversial. Thus, publishing it could be easier than other alignment work. Publications come with many instrumental benefits such as increased publicity, more follow-up work, increased reputation and so on.
3. **Academic paths:**I’m not sure about this but I think it’s plausible that the AI safety community is currently giving too little weight to academic paths as a means to impact. Professors educate most of the next generation of researchers, academics review most papers at large conferences, academics are consulted for important public decisions, and much more. Much of the more abstract work on AI safety is not seen as credible science in the current academic environment and thus important and prestigious professorships might not be achievable for researchers of some agendas. Science of DL, on the other hand, is already established enough that good work will be recognized as such by the academic community even if you have a focus on safety.
4. **A gateway drug for AI safety:**[In my experience](https://www.lesswrong.com/posts/SqjQFhn5KTarfW8v7/lessons-learned-from-talking-to-greater-than-100-academics), academics care less about abstract arguments around AI safety or motivations through X-risks and much more about technical questions and understanding things in detail. Therefore, the more the Science of DL agenda is intertwined with AI safety, the more the safety aspect will become normalized and widely known. Once this is a common assumption in traditional academia, it is much easier to explain the rest of the arguments and arrive at questions about deceptive alignment, uncontrollability, specification gaming, etc. I think the fact that it is completely normalized and accepted that people train large black-box models without understanding at all what’s going on inside them is very weird and actually unexpected in a scientific community. Thus, pushing against this norm seems already valuable even if it doesn’t lead to more interest in alignment.
Once again, I want to emphasize that these instrumental reasons are nice but they should not be the primary reason for working on an agenda. The most important reason to work on an agenda is that it has a promising path to alignment, everything else is secondary.
Reasons against Science of DL
=============================
There are multiple good reasons to be skeptical of Science of DL as an alignment agenda and I definitely don’t think everyone should work on it. These include:
1. **Other agendas might be more promising:**There are a lot of alignment agendas out there. Most of them seem like they could solve some important subcomponent of alignment. I really wouldn’t want everyone to work on Science of DL and it’s good to take multiple bets and diversify as a community.
2. **The main problem doesn’t come from NNs:**Especially if you think that the core of the alignment problem comes from something that is not really related to NNs or how they are trained, e.g. if you believe the core of the alignment problem is agency or reward specification, Science of DL just seems like the wrong approach.
3. **It’s not neglected enough:**There are already people in academia and industry working on many of the questions related to Science of DL. It’s not clear that the AI safety community or EAs should use their time to work on it.
4. **Progress is too slow:**Questions around interpretability and generalization properties of NNs have been around for a long time. Most people would have hoped for much faster progress but we still can’t answer many basic questions about why NNs generalize and what concepts they learn.
Final words
===========
I’m not super sure about the agenda but it currently feels promising to me. I feel like a lot of things line up that made me skeptical about some other agendas. For example, I think Science of DL has a [defender’s advantage](https://www.lesswrong.com/posts/6ReBeYwsDeNgv6Dr5/the-defender-s-advantage-of-interpretability), e.g. working on it seems to favor alignment more than capabilities. Also, the bet on NNs seems plausible to me, i.e. TAI could just be a scaled-up version of today’s systems with no major novel insights or paradigm shifts. Lastly, I think if we want to have a plausible shot at getting alignment right, having an agenda that is accessible to the broader ML community and can absorb a lot of people that are not (yet) interested in alignment definitely helps. Science of DL seems technical and applied enough that many people outside of the AI safety bubble could be interested and contribute. |
5157963a-fbca-47e6-8bd2-0d6094c3798b | trentmkelly/LessWrong-43k | LessWrong | Training Regime Day 4: Murphyjitsu
Introduction
Recall that applied rationality is being able to properly decide which advisors to use during decision making.
Imagine that I am throwing a ball up into the air and catching it. Occasionally, I might drop the ball because I am not very good at catching. Now, imagine that I throw the ball up and at the top of the arc, the ball just freezes. Are you surprised?
There is an advisor that is always trying to guess what move the world is going to make next. This part of you is called the inner simulator, or inner sim for short. This advisor has many powers but is particularly underused by most people. This technique aims to extract a lot of useful information from the inner sim.
Inner Simulator
Here are some prompts that will allow you to get in touch with your inner sim:
1. Imagine your closest companion. Imagine going up to them and flicking them on the nose. How do they react?
2. Imagine that the weather suddenly shifts. If it’s currently sunny, there is now a torrential downpour. If it’s currently raining, the rain abruptly ceases. How surprised are you?
3. Imagine throwing a rubber ball at the wall. How does it bounce? You can probably see the trajectory pretty well, but if I asked you to calculate it explicitly, would you be able to do it?
All of these events are pretty complicated, and yet it seems like most humans are easily able to make reasonably accurate predictions. Even more impressively, these predictions can be made extremely quickly. The inner sim is a very fast and powerful advisor.
Of course, speed and power have costs. The inner sim requires a lot of data. You (and your evolutionary lineage) live on Earth, where there is gravity and physics is generally Newtonian. Correspondingly, your inner sim is extremely well trained on Newtonian physics with Earth gravity. There are many reasons that humans find things like orbital mechanics and relativity so hard; one of them is that your inner sim is not very good at dealing with non-Newtoni |
cc457358-260d-4e29-b4d9-f4e4d3016e9b | StampyAI/alignment-research-dataset/youtube | Youtube Transcripts | Artificial General Intelligence: Racing and cooperating | Seán Ó hÉigeartaigh
[Music]
thank you very much for those of you who
don't know me I'm Shaun Hegarty I'm the
executive director of the Center for the
Study of existential risk which works on
global risks and long-term challenges
and I'm a program director with a center
for the future of intelligence which
works on the opportunities and
challenges of artificial intelligence a
couple of years ago myself and my
colleague Stephen cave wrote a paper in
which we outlined some concerns about
what we saw as an emerging narrative
framing artificial intelligence
development as a competitive race or
strategic global advantage this is most
famously identified with some quotes
that we've already encountered at this
conference from Vladimir Putin in 2017
and this point from the State Council of
China's new generation and artificial
intelligence development and framing
these these goals in some more
competitive language now as we've
already heard and Alan and Brian pointed
these things that nicely it's unclear
how much we should read into this but
where putin's quote came from a
discussion with schoolchildren and the
chinese phone well chinese leaders have
repeatedly emphasized how important and
how essential of global cooperation is
going to be around artificial
intelligence nonetheless one of our
concerns is that certain types of
narratives can easily sort of build up
and sometimes even reinforce themselves
so somebody writes about artificial
intelligence as a race other people pick
up on it and run for this people are
influenced by this some of them perhaps
decision-makers who then take actions
which makes sense if we're in more of a
kind of a race dynamic and that again
lends credence to the race framing and
the whole thing kind of spirals and
we've seen a lot more and public
articles and policy reports that frame
intelligence in these terms I've been
asked to give a framing talk for this
issue of races and artificial
intelligence I'm going to do that by
making a number of argument since the
jessyca n--'s
one is that competition isn't
necessarily bad a lot of competition is
good secondly contemporary artificial
intelligence i'm not sure it makes sense
to frame it as a race full stop and said
what i think is happening is a lot of
different types of competitions on
different levels and in different ways
at the same time thirdly there is not a
race so to speak towards artificial
general intelligence happening right now
the goal is too far often too poorly
defined for something that however
fourthly at some point in the future a
critical mass of people might see this
as an imminent thing and such a race may
emerge basically i think it would be why
there are some serious concerns about
artificial general intelligence being
developed as part of a competitive race
for security and other reasons
sixthly there may be opportunities
certain key points of which communities
like these can really influence the kind
of bringing them of how we develop
artificial general intelligence
particularly at key points where for
example a critical mass of people
researchers decision makers and which
one seeing this is a long-term
speculative issue to something more
imminent and we should be thinking about
what we can do to have a positive
influence there one way is by having
well develop proposals and
well-developed and plans in place that
have buy-in from a number of key actors
who might have been thinking further
ahead on these things and the other
thing we might think about is the
culture and the environment in which an
artificial intelligence is developing up
until that point both in the research
community and in the broader governance
policy and other stakeholder ecosystem
around that and by that I mean practices
principles and precedents types of
collaboration other things that do or
don't alliteration so right now
I don't think that it makes sense to
treat AI development as some sort of
global race what we instead have are a
lot of different types of competitions
there are research groups working as
hard as they can to achieve certain
fundamental and breakthroughs and
applications earlier there are companies
looking to apply artificial intelligence
the various industry and business
applications whether it's in medicine
healthcare finance manufacturing
what-have-you
we have companies and nations trying to
be in a good position to bring
artificial intelligence new emerging
markets we have national strategies
being developed by a whole range of em
nations where policymakers are really
looking to make sure that there are
nations and their societies are best
supported and by artificial intelligence
and not left behind and so that M the
economies do as well as possible in the
future with artificial intelligence this
is all quite natural and healthy
competition can drive innovation
progress and beneficial societal
applications now obviously there are
things that we need to bear in mind
though you know in the drive towards
automation we should think about making
sure that people whose jobs are replaced
don't get left behind we shouldn't
deploy AI systems cry maturely or
carelessly in situations that might
exacerbate biases in historical data we
don't want to make sure that privacy
isn't undermined as we push to gather
more data to make better applications
but competition isn't necessarily
inherently about things looking at a
more geopolitical level a lot has been
recently written about concerns about
for example power becoming concentrated
both in leading corporations developing
AI and in the nations in which those
corporations are based are embedded
scholars like - Lee and others have
written persuasively
about the concern that we might end up
with nations that have a huge advantage
that are effectively offering
services to developing nations in such a
way that would be very difficult for
those emerging nations to pull
themselves up and be equal partners at
the table companies might end up with a
very difficult to overcome advantage in
things like research talent in access to
computing facilities and access to data
and leading groups might be able to play
at those role in influencing national
and international governance and
regulation but and there's been a lot
written about concerns about a military
arms race and by this I don't just mean
lethal autonomous weapons and this also
takes into account application of AI to
decision support logistics information
gathering intelligence fibre security
supplies chains the whole gamut of
things that makes up offense/defense
security and given that artificial
intelligence is such an inherently dual
use technology in which fundamental
progress can be turned into applications
and applications can be repurposed for
other purposes so easily these issues
are going to remain quite intertwined
with an industry a military which is has
at its basis a competitive angle and a
security angle that might be somewhat
antithetical to the cooperation that we
might want now looking a little bit
further ahead to this idea of an AGI
race and our special general
intelligence right now there isn't a
race war exist and most policy documents
talk about artificial general
intelligence as a speculative future
prospect that is decades or perhaps
hundreds of years away it and outside of
places like this an awful lot of mao
researchers feel the same way about it
it's not clear how much it makes sense
even think about this as a race when the
end goal is so poorly defined and and
difficult to say anything meaningful
about however that may not always be the
way things are if we are some decades
away from artificial
intelligence at some point it's going to
go from speculative to seeming more
imminent and at that point we need to
consider a few scenarios one is that
this server drive towards artificial
general intelligence might result in a
different sort of dynamic than all of
these different sorts of competitions
that we've been seeing before this
general intelligence has in the
definition generality a system that
defined as Anthony did that would be
able to basically do everything that
human intelligence is able to do as well
as us are better would imprints we'll be
able to occupy a huge number of niches
perhaps all of them which might mean
that there isn't very much room for lots
of different winners if this technology
allows us to make faster progress on
scientific goals economic goals
financial goals military goals it might
give the group or nation that achieves
the technology first an opportunity to
shape the future of the world in a way
unlike any technology we've seen before
and if as has been hypothesized this
technology was able to help us with the
process of developing artificial
intelligence itself it may mean that the
group who have achieved this technology
can continue progressing faster than
those who don't have it now I don't
think we can say that any of these
hypotheses are correct or incorrect
right now we've heard a lot of good
arguments why it might not be the case
we might have diminishing returns as
yahshua point to that earlier there
might not be a single point of which
something is AGI and whatever was before
that wasn't it just may not be away a
sense if they think about these things
it may be that we have a world in which
some tasks are best amenable to AGI
systems but narrow AI has an advantage
in other things but it seems at least
plausible to consider a world in which
developing something that we can call a
GI gives whoever achieves this burst
this huge global advantage and to think
about what this might mean for the world
and I'd point out that at the point of
which this may become imminent seeming
to people beliefs will matter
so the
people who are pushing forward towards
this what they think a GIM will be
capable will affect their actions and
what they think that their competitors
believe about AGI and what their
competitors will do will also influence
their actions now I think there are
really serious reasons to be concerned
about artificial general intelligence
and the drive towards it emerging in the
context of a race narrative one key
concern is this idea of safety ideally
we want and this technology that is has
such transformative potential to be
developed in such a way that am we move
slowly and carefully and paid due
attention to safety issues and that the
scientists who are working on the
forefront of it are sufficiently
convinced that they've solved the safety
issues before it reaches certain levels
of deployment it may be much more
difficult to do that if AGI is being
developed in a competitive setting where
other groups are working towards the
same goals perhaps not taking the same
sort of AM safety precautions and so
there feels like an a lot of pressure to
and keep pushing ahead quickly similarly
it may be a lot more difficult to answer
a lot of the key questions about how you
would deploy this technology once it's
developed in a way that's fair and
beneficial for the entire world how to
involve the voices of all the global
stakeholders who would be influenced by
this technology in a fair and equitable
way if it's happening as part of this
competitive race it may be more
difficult to share expertise between
leading scientists working on safety
issues between companies are between
nations to establish the kinds of
cooperations and collaborations we need
this sort of competitive dynamic might
even spill over into tensions of
different sorts geopolitical tensions
and even conflict and above all it comes
down to this question of well what if
somebody wins the race how can we make
sure that if this race is won by a
particular group the benefit is one that
results in benefit to all of us and that
we all feel is fair and equitable
in an ideal world we might want to see
artificial intelligence developed over
the last couple days as some sort of
globally cooperative venture some sort
of perhaps global Stern for AI where we
have scientists from across groups and
across nations working in together and
the benefits of the project accrue to
humanity as a whole rather than to any
group RNA and nation where scientists
work on this and don't bring it past
their stages until key questions are
being answered about safety issues and
issues of m control and where global
stakeholders are convinced that the
benefits are going to be distributed
worldwide now it's not clear how
feasible or practical this is but it's a
good goal so something we might ask is
in the world we're in right now where we
don't know if artificial general
intelligence is 20 years away or 200
years away or 2,000 years away and we
have all these kind of competitive
dynamics playing out what kind of things
should we be doing now so one thing is
rhetoric rather than amplifying this
rhetoric of AI as a competitive global
race and putting forward other types of
narrative artificial intelligence as a
shared global goal for a global good a
shared global challenge another is
establishing the kinds of cooperations
and collaboration that we need there's
already a lot of collaboration happening
within the research community it's
essential that as competition heats up
in different aspects of artificial
intelligence we maintain that spirit of
cooperation and particularly on issues
like research relevant to safety as we
develop more powerful capabilities
within artificial intelligence there may
be roles for different types of openness
of research publication and models and
different types of sharing models which
is something that we in a number of
other groups explored in the malicious
use of artificial
and report that was released last year
double engagement this conference is an
aspiring example of research leaders and
thought leaders coming together from
leading hubs in Europe the United States
China Japan and this is exactly the kind
of thing that I feel that we need to be
doing now it takes time to figure out
how to work together you know
particularly across cultures to
understand what our shared values are to
understand what we disagree on and why
and to figure out how to work together
if the stakes around artificial
intelligence are only going to become
higher now is the time to start working
together in this way and similarly
organizations like the partnership on AI
and play a powerful role in encouraging
globally engaged in global even
countable governance of artificial
intelligence
lastly precedence Yun and others pointed
out scientific communities can play a
powerful role in influencing how a
technology develops and the steps that
our community takes to put in place
certain principles and certain
precedents and may influence the way
that the technology develops in a
substantial way I particularly like this
principle from opening eyes charter that
they released last year or they say if a
value aligned safety-conscious project
comes close to building AGI before we do
we commit the stop competing with and I
said start assisting this project if
more groups were to adopt similar
principles and similar practices of
collaboration then I think we'd be in a
very good position when it comes to the
emergence of this drive towards
artificial general intelligence ahead of
this drive emerging I think there are a
number of topics that we can meet and
thinking on working on and questions
that we can be asking ourselves one is
what might be plausible and feasible
models for a joint
and general intelligence development and
project how could we incentivize
research leaders and experts who
currently reside within national efforts
or companies to take part in this as
well how would we make sure that this
project would be able to really achieve
alongside those what might be ways in
which this wouldn't work simulating
models for a global artificial general
intelligence benefits agreement who are
the stakeholders who need to be
represented what are the key issues how
can we make sure that benefits accrue to
everyone how can we make sure that if
leading stakeholders commit to this they
can be held and binding towards it one
question is what will be good enough as
I think mal I want to feel there's a
point there it's great to think about
ideal worlds but we also need to
consider the worlds that we're most
likely to end up in and the ideal
scenario might not be the option on the
table so we need to think about what
might not be ideal but what might be a
good enough solution that at least rules
out the worst-case scenarios I might be
concerned about lastly there's this
question about how we even talk about
and write about this issue there's an
irony to me it's talking about you know
races being bad while being up here
talking about races and there's it
points to a fundamental tension in that
we as researchers have a responsibility
to consider all the plausible scenarios
in order to be honest in order to
develop good ideas in order to engage
key stakeholders on those ideas and in
order to open up our ideas to the right
sorts of scrutiny but at the same time
it seems essential that by our actions
we don't serve emphasized or exacerbated
be exactly the site and types of
dynamics we might be most concerned
about all of these seem like really
useful things for us to be thinking that
ahead of time and to have done some
really solid thinking of about so that
we're in a position to influence the
global community when this comes into
view or a kind of a critical mass of
global stakeholders this is a very
powerful community and I'm now finished
it's research leaders as I said from you
know across the globe
people who represent the research
leadership and a number of companies a
number of academic groups and some of
the key thought leaders all of you
people thinking deeply and carefully
about these issues not just for the
future of your own group the future your
company the future of your nation but
for the future of humanity itself and
that's an inspiring thing it's I mean
it's been really a privilege discussing
all these issues with all of you over
the last couple of days to me it seems
like exactly the kind of leadership that
we need in order to make the best
possible progress and being the best
possible position when we get to this
drive towards artificial general
intelligence so thank you all for taking
the time to do this and to the
organizers for organizing this and I'm
really excited to know what will come of
it
I'm personally feeling more optimistic
than I have in years intelligence with a
GI but rather slow progress where a lot
of different actors are improving on
different aspects of intelligence and
might be less easy to convinced of your
a nice project of you know having this
AI Stern kind of thing because I think
we still want to have this coordinated
development even if we're not afraid of
her essential risk this because this
very high constitution of power could be
highly disruptive for society in many
ways can you think of other arguments to
convince a players especially a military
in different countries would be part of
something this is a very good question
it may very well be that we don't have
this single point that's very obvious to
point to that says now you know the
alarm bells going off or we need to work
together and it may be that what we have
is more of a kind of a frog boiling
water scenario where we've got this kind
of gradual development it may be harder
to make the claim in that case but I
feel like there's so an awful lot of
technology develops in this way where
it's
gradual and you can have the technology
developing gradually where it's still
the case the certain threshold or
thresholds are passed in the impact this
technology has in the real world where
something goes from not really measuring
that much or seeming fairly sort of
manageable to suddenly being very clear
that it requires a new type of
governance and I think you know we've
seen smaller versions of this in terms
of you know social media manipulating
elections and a range of other things I
mean maybe we would at some point have a
major cyber attack that makes us take
you know issues around cyber offense and
defense a lot more seriously even though
there hasn't been some sort of you know
qualitative shift or rather a continuous
development so it may be that we need to
think about in that case is well even if
what we're seeing is something more and
continuous we can see that the effect of
this technology on the world that we
live in means that there's something
that's very different now and that will
be very different in the coming years
then what the governance structures and
the collaboration structures that we've
set up for this are set up to account
for so I think there are ways to
approach this where even if the
development is more incremental we can
recognize that something that wasn't a
problem in the past is going to be a
problem now and needs a different
approach
[Applause]
[Music] |
8f519233-e2e2-46f8-86c1-035cd3d56351 | trentmkelly/LessWrong-43k | LessWrong | Where Utopias Go Wrong, or: The Four Little Planets
Cross-posted from The Ginnungagap Foundation. This exploration of human attempts to improve human society began as a nonfiction essay, but I decided making it fiction would be more engaging.
Some concepts like ethics are elaborated upon in my previous articles, which is why I did not go into more detail about them in this story. I may cross-post those here as well if there is interest.
Content warning: AI safety skepticism (voiced by, ironically enough, a friendly AI).
Thoughts? Questions? Criticisms?
***
Once upon a time, there were four planets, in all different corners of the galaxy. They existed unbeknownst to each other, but each was filled with turmoil and misery. Their people fought and stole from and hurt and killed each other. One by one, each planet decided enough was enough. Independently, each one decided to create a utopia, where everyone was happy and nobody fought.
The First Planet
The first planet built its utopia out of rules. The people attempted to design a perfect set of laws to keep order. They made laws against murder and laws against theft. They made laws against violence and vandalism. They made laws about what people could buy and sell. Laws would prevent conflict and make happiness possible, they agreed.
However, they soon found that it was not possible to write enough laws to cover every conflict. There was always a way around a law for a clever and determined mind. People came up with ways to hurt and steal from each other that were not technically illegal.
Wealthy people had no need to engage in theft when they could perform hostile acquisition, and destroy any evidence the former owner had owned anything in the first place. Instead of threatening people, they could introduce incentives–it’s amazing how much safer a place becomes when people pay their “insurance” premiums. Instead of murdering their rivals, they staged “accidents”. Instead of lying, they deceived, getting victims to agree to ambiguous b |
6d54efb5-66d1-4b92-8807-4c31a6f1c4f9 | trentmkelly/LessWrong-43k | LessWrong | Why don't Rationalists use bidets?
I've been to many Rat&EA group houses, offices, and other venues both official and unofficial around the Bay Area, and I can't recall ever seeing a bidet. And because I think bidets are superior technology, this is pretty confusing to me. Why might this be?
Possible explanations for why this might be
Because bidets are expensive? (No)
Well, bidet attachments can be retrofitted onto nearly any toilet for ~$35 (e.g. see amazon). They aren't difficult to install either, taking only a wrench and less than 20 minutes of time. (I've done it myself once or twice.) And because they can be installed as just an attachment onto an existing toilet, they don't take up any extra space either.
I'd bet that installing a bidet saves time and money on net.
Because bidets aren't actually great? (Doubt it)
For one, bidets are great. But even if you doubt this point personally, consider that bidets only need to be useful to a fraction of people who use a particular bathroom in order for it to be worthwhile. Besides, bidets can be installed in addition to wherever there's toilet paper, so I don't see how a bidet installation can ever be worse than no bidet installation.
I suppose this is a somewhat-likely explanation, but it's still not great.
I also spent ten minutes googling for examples of how bidets might actually be ineffective or have unwanted side-effects, but I couldn't find anything that seemed credible.
Bidets aren't intuitive to use if you've never used them before
If you've never used a bidet before (probably because you're American and the rest of America doesn't use them either) and you encounter one in the wild, then it's a little scary and you probably won't be inclined to experiment with it. And because of this and taboo norms around bathroom stuff, then we shouldn't expect bidet adoption to be memetic when we use each other's bathrooms.
I dunno, this is probably why, but I still consider it to be an unsatisfying explanation. It just boils down to "because a |
8849aefe-cc58-415b-b3dc-1a780f0ab348 | trentmkelly/LessWrong-43k | LessWrong | Fake Frameworks for Zen Meditation (Summary of Sekida's Zen Training)
Meditation, and Zen meditation in particular, is hard to teach in the sense that most people cannot learn to do it as intended by simply reading or hearing about it. Instead they need to attempt it, talk about what happened with someone more experienced, get feedback, and try again. I think this is because most models of how to meditate are implicit and complex, so written meditation instructions tend to leave out important subtle details.
Probably the most successful attempt to counter this dynamic and get all the necessary instructions in writing is The Mind Illuminated, which offers a book's worth of words and diagrams to teach you how to perform a particular style of Theravada concentration meditation. Aside from a lot of specific advice and answers to questions, it usefully contains a lot of explicit models built using fake frameworks to give the reader gears for understanding meditation.
Zen Training: Methods and Philosophy by Katsuki Sekida does something similar, but for Zen.
In this post I'll share some of Sekida's models that I think are especially useful to have if you're interested in Zen, Buddhism, or meditation more generally. I have to cover a couple sections of preliminaries to set those models up, though, so things don't really get going until the section on breathing.
Caveat lector:
Sekida's models as presented in the book have some limitations. In particular, I think he often gets the low-level details wrong when he tries to ground his models in science because much of his information was outdated even at the time of writing (1960s). Thankfully this ends up not being too much of an issue because the models don't actually rely on those details, which is to say Sekida is sometimes overzealous in trying to ground his models in scientific details that are not actually gears within his models.
Thus what follows is my take on his models. It's fairly faithful, but I've taken editorial license in places to either increase concision (this is a post r |
d7cbb3b0-733e-44e5-864e-a2ae84b90865 | trentmkelly/LessWrong-43k | LessWrong | Pondering computation in the real world
Epistemic Status: Exploratory. Asking questions and setting up some intuition pumps. Would very much like to make my thoughts on this topic more precise.
Some systems that compute
Adders
Here's an electrical circuit you can build at home.
From https://circuitdigest.com/tutorial/half-adder-circuit-and-its-construction
It's called a half adder circuit. By flipping the toggles on the left in the red box you set two binary inputs, e.g. you can represent 0+1 or 1+1. In response, the circuit will do a highly structured electromagnetic dance to make the green and red lights in the bottom right turn on or off, in order to represent the output of binary addition. We say: "this circuit computes addition."
Here is a marble adding machine.
By putting in marbles at certain locations at the top of the machine, you can input binary numbers. A series of see-saw toggles change their positions under the influence of gravity and the marbles. This mechanical dance results in marbles resting in output positions at the bottom of the machine, indicating the results of binary addition. We say: "this circuit computes addition."
Different physical systems can be associated with the same computation.
Integrators
This is Emmy.
Emmy is in high school, and has some math homework. She has to plot the solution, x(t), to the following differential equation.
150d2xdt2=−9.8x(0)=0,dxdt(0)=15
She takes a pencil and a piece of paper and writes the equation down, and then starts using the rules she learned for symbolic integration. Her first move is to write dxdt=−9.8150x+C, because she knows the rule that the integral of a constant K is Kx+C. She continues in this manner and gets her final result and plots it, by hand. She gets full marks on this question. We say: "Emmy has integrated the differential equation."
This is Isaac.
Isaac is in Emmy's class, and got the same homework problem. He gathers an apple, a stopwatch, and a ruler.
He takes the apple, and throws it straight up in the a |
2ef77352-663b-4a21-a4a7-4c234fface45 | StampyAI/alignment-research-dataset/arbital | Arbital | Information
summary(Technical): Given a probability distribution $\mathrm P$ over a set $O$ of possible observations, an observation $o \in O$ is said to carry $\log \frac{1}{\mathrm P(o)}$ units of information (with respect to $\mathrm P$), where the base of the logarithm determines the unit of information. The standard choice is log base 2, in which case the information is measured in [shannons](https://arbital.com/p/shannon).
Information is a measure of how much a message grants an observer the ability to predict the world. For a formal description of what this means, see [Information: Formalization](https://arbital.com/p/). Information is observer-dependent: If someone tells both of us your age, you'd learn nothing, while I'd learn how old you are. [https://arbital.com/p/+3qq](https://arbital.com/p/+3qq) gives us tools for quantifying and studying information.
Information is measured in [shannons](https://arbital.com/p/shannon), which are also the units used to measure [https://arbital.com/p/uncertainty](https://arbital.com/p/uncertainty) and [https://arbital.com/p/entropy](https://arbital.com/p/entropy). Given that you're about to observe a coin that certainly came up either heads or tails, one shannon is the difference between utter uncertainty about which way the coin came up, and total certainty that the coin came up heads. Specifically, the amount of information in an observation is quantified as the [https://arbital.com/p/3nd](https://arbital.com/p/3nd) of the [https://arbital.com/p/reciprocal](https://arbital.com/p/reciprocal) of the [https://arbital.com/p/1rf](https://arbital.com/p/1rf) that the observer assigned to that observation.
For a version of the previous sentence written in English, see [Measuring information](https://arbital.com/p/measuring_information). For a discussion of why this quantity in particular is called "information," see [Information: Intro](https://arbital.com/p/).
# Information vs Data
The word "information" has a precise, technical meaning within the field of [https://arbital.com/p/3qq](https://arbital.com/p/3qq). Information is not to be confused with [https://arbital.com/p/data](https://arbital.com/p/data), which is a measure of how many messages a communication medium can distinguish in principle. For example, a series of three ones and zeros is 3 [bits of data](https://arbital.com/p/3p0), but the amount of information those three bits carry depends on the observer. Unfortunately, in colloquial usage (and even in some texts on information theory!) the word "information" is used interchangeably with the word "data"; matters are not helped by the fact that the standard unit of information is sometimes called a "bit" (the name for the standard unit of data), despite the fact that these units are distinct. The proper name for a binary unit of information is a "[https://arbital.com/p/shannon](https://arbital.com/p/shannon)."
That said, there are many links between information and data (and between shannons and bits). For instance:
* An object with a [https://arbital.com/p/3y1](https://arbital.com/p/3y1) of $n$ bits can carry anywhere between 0 and $\infty$ [shannons](https://arbital.com/p/shannon) of information (depending on the observer), but the maximum amount of information an observer can consistently [_expect_](https://arbital.com/p/expectation) from observing the object is $n$ shannons. For details, see [https://arbital.com/p/+expected_info_capacity](https://arbital.com/p/+expected_info_capacity).
* The number of shannons an observer gets from an observation is equal to the number of bits in the [encoding](https://arbital.com/p/encoding) for that observation in their [https://arbital.com/p/ideal_encoding](https://arbital.com/p/ideal_encoding). In other words, shannons measure the number of bits of data you would use to communicate that observation to someone who knows everything you know (except for that one observation). For details, see [https://arbital.com/p/+ideal_encoding](https://arbital.com/p/+ideal_encoding) and [Information as encoding length](https://arbital.com/p/).
For more on the difference between information and data, see [https://arbital.com/p/+info_vs_data](https://arbital.com/p/+info_vs_data).
# Information and Entropy
[https://arbital.com/p/+entropy](https://arbital.com/p/+entropy) is a measure on probability distributions which, intuitively, measures the total uncertainty of that distribution. Specifically, entropy measures the number of shannons of information that the distribution [expects](https://arbital.com/p/expectation) to gain by being told the actual state of the world. As such, it can be interpreted as a measure of how much information the distribution says it is missing about the world. See also [Information and entropy](https://arbital.com/p/).
# Formalization
The amount of information an observation carries to an observer is the [https://arbital.com/p/3nd](https://arbital.com/p/3nd) of the [https://arbital.com/p/reciprocal](https://arbital.com/p/reciprocal) of the [https://arbital.com/p/1rf](https://arbital.com/p/1rf) that they assigned to that observation. In other words, given a probability distribution $\mathrm P$ over a set $O$ of possible observations, an observation $o \in O$ is said to carry $\log_2\frac{1}{\mathrm P(o)}$ shannons of information with respect to $\mathrm P$. A different choice for the base of the logarithm corresponds to a different unit of information; see also [Converting between units of information](https://arbital.com/p/). For a full formalization, see [Information: Formalization](https://arbital.com/p/). For an understanding of why information is logarithmic, see [Information is logarithmic](https://arbital.com/p/). For a full understanding of why we call this quantity in particular "information," see [Information: Intro](https://arbital.com/p/). |
94aed8c4-814b-441e-b9b6-6d1c46e7734e | trentmkelly/LessWrong-43k | LessWrong | Holden Karnofsky's Singularity Institute Objection 1
The sheer length of GiveWell co-founder and co-executive director Holden Karnofsky's excellent critique of the Singularity Institute means that it's hard to keep track of the resulting discussion. I propose to break out each of his objections into a separate Discussion post so that each receives the attention it deserves.
Objection 1: it seems to me that any AGI that was set to maximize a "Friendly" utility function would be extraordinarily dangerous.
Suppose, for the sake of argument, that SI manages to create what it believes to be an FAI. Suppose that it is successful in the "AGI" part of its goal, i.e., it has successfully created an intelligence vastly superior to human intelligence and extraordinarily powerful from our perspective. Suppose that it has also done its best on the "Friendly" part of the goal: it has developed a formal argument for why its AGI's utility function will be Friendly, it believes this argument to be airtight, and it has had this argument checked over by 100 of the world's most intelligent and relevantly experienced people. Suppose that SI now activates its AGI, unleashing it to reshape the world as it sees fit. What will be the outcome?
I believe that the probability of an unfavorable outcome - by which I mean an outcome essentially equivalent to what a UFAI would bring about - exceeds 90% in such a scenario. I believe the goal of designing a "Friendly" utility function is likely to be beyond the abilities even of the best team of humans willing to design such a function. I do not have a tight argument for why I believe this, but a comment on LessWrong by Wei Dai gives a good illustration of the kind of thoughts I have on the matter:
> What I'm afraid of is that a design will be shown to be safe, and then it turns out that the proof is wrong, or the formalization of the notion of "safety" used by the proof is wrong. This kind of thing happens a lot in cryptography, if you replace "safety" with "security". These mistakes are still o |
5e7eb336-da33-4d4b-b385-459bc1e8f9b6 | trentmkelly/LessWrong-43k | LessWrong | Singular learning theory: exercises
Thanks to Jesse Hoogland and George Wang for feedback on these exercises.
In learning singular learning theory (SLT), I found it was often much easier to understand by working through examples, rather than try to work through the (fairly technical) theorems in their full generality. These exercises are an attempt to collect the sorts of examples that I worked through to understand SLT.
Before doing these exercises, you should have read the Distilling Singular Learning Theory (DSLT) sequence, watched the SLT summit YouTube videos, or studied something equivalent. DSLT is a good reference to keep open while solving these problems, perhaps alongside Watanabe's textbook, the Gray Book. Note that some of these exercises cover the basics, which are well-covered in the above distillations, but some deliver material which will likely be new to you (because it's buried deep in a textbook, because it's only found in adjacent literature, etc).
Exercises are presented mostly in conceptual order: later exercises freely use concepts developed in earlier exercises. Starred (*) exercises are what I consider the most essential exercises, and the ones I recommend you complete first.
1. *The normal distribution, like most classical statistical models, is a regular (i.e. non-singular[1]) statistical model. A univariate normal model with unit variance and mean μ∈R is given by the probability density p(x|μ)=1√2πexp(−12(x−μ)2). Assume the true distribution q(x) of the data is realizable by the model: that is, q(x)=p(x|μ0) for some true parameter μ0.
1. Calculate the Fisher information matrix of this model (note that since we have only a single parameter, the FIM will be a 1x1 matrix). Use this to show the model is regular.
2. Write an explicit expression for the KL divergence K(μ) between q(x) and p(x|μ), as a function of the parameter μ. This quantity is sometimes also called the population loss. [See Example 1.1, Gray Book, for the case of a 2D normal distribution]
3. |
ac7e2125-4fb3-4980-9f4c-d642d7224fd8 | trentmkelly/LessWrong-43k | LessWrong | [LINK] Hyperloop officially announced — predictions, anyone?
I was studying in the LW Study Hall, and during our break someone posted this link to the official hyperloop announcement:
http://www.spacex.com/sites/spacex/files/hyperloop_alpha-20130812.pdf
One member was doubtful it would get past regulations, and another said "tentative p>0.05 that a hyperloop gets made by 2100", which was met with "p>0.05 that uploading people and moving them between bodies will be available by 2100".
It struck me that people might be interested in betting on things like this, or at least having a conversation about it.
A few predictions to start:
* Tesla Motors / SpaceX / Elon Musk will create a working hyperloop by 2100.
* Tesla Motors / SpaceX / Elon Musk will create a working hyperloop by 2050.
* Tesla Motors / SpaceX / Elon Musk will create a working hyperloop by 2030.
More predictions, based on comments:
* The cost projections of the hyperloop are underestimates by at least an order of magnitude.
* When and if a hyperloop-like transit system is built (or not), the US will not be the first country to build it.
* One of the first really big (>5bn$) hyperloops will go across a body of water.
* If a hyperloop is created, it will be predominately (>50%) solar-powered. |
2a7195d2-4969-4e1c-82d0-54625f5bf521 | trentmkelly/LessWrong-43k | LessWrong | What is known about consciousness?
Three questions I would like to find some answers for are:
* What is known (to a high degree of certitude) about consciousness?
* What experimental tests have been able to verify predictions made?
* What predictions have been made that nobody has managed to test yet, but could, have an experimental proof or disproof?
Does anyone know some good sources which do cover any of these? Also it would be great to hear less-wrongers own input to these questions!
|
d4792ed6-1539-4f8a-a205-19ef567fe49a | trentmkelly/LessWrong-43k | LessWrong | Helio-Selenic Laser Telescope (in SPACE!?)
The use of drones in modern warfare has become increasingly prevalent in recent years. Drone swarms can be difficult to track and defend against using anti-air defense.
One potential solution to this problem is the use of mobile ground-based laser system to combat drones.
* The biggest disadvantage is the cost of implementing such a system. Drones are cheap. Lasers are expensive.
* Another disadvantage is the potential for bad weather to disrupt its effectiveness. In poor weather conditions such as heavy rain or fog, the laser beam may be diffracted or scattered, reducing its accuracy and effectiveness.
* The phenomenon of thermal blooming means that laser weapons bleed off energy when penetrating the atmosphere, limiting their effective range, even in good weather conditions.
* Laser weapons are limited by their need for a direct line-of-sight.
* Laser weapons are limited by the diffraction limit (see below).
These are severe limitations that plausibly will prevent laser weapons from being deployed on the battlefield.
A potential way to get around the last two limitations is by deploying a (mobile) laser system together with an interlinked network of "mirror drones" or "selenic drones". These 'selenic' drones would carry mirrors that are able to redirect the initial laser weapon, allowing for much longer range and the ability to fire beyond direct-line-of-sight. This is similar to how moonlight is a reflection of the fact that the Moon acts as a giant mirror for the sun. Let us therefore call this system a helioselenic telescope or helioselenic hive.
The use of mirror-equipped selenic drones allows for a significantly increased range for laser weapons. By using mirrors to redirect the laser beam, it is possible to effectively fire around corners and over obstacles, greatly increasing the range of these weapons.
Another advantage is the ability to rapidly reposition the mirror drones as needed. In the event that an enemy drone swarm is moving or c |
629bb979-914e-4667-8f03-2e6245b0466b | trentmkelly/LessWrong-43k | LessWrong | Our Need for Need
[I wrote this in 2017, well before I had a LessWrong account. The style isn't necessarily what I would have chosen if I'd known I was going to share it here, but I think the topic might be of some interest anyway, so I'm reposting it with light edits. Originally on Grand, Unified, Crazy.]
It is a trite, well-established truth that people like being useful. But there’s more to it than that, or rather, there’s also a stronger version of that claim. People do like being useful, but useful is a very broad term. Stocking shelves at a Walmart is useful, in that it’s a thing with a use, which needs to be done. And it’s true that some people may in fact actively like a job stocking shelves at a Walmart. But on the whole, it’s not something most people would consider particularly enjoyable, and it’s certainly not something that most people would consider "fulfilling".
Let us then upgrade the word “useful” to the word “needed”: people like to be needed. While stocking shelves at a Walmart is useful, the person doing it is fundamentally replaceable. There are millions of others around the world perfectly capable of doing the same job, and there are probably thousands of them just within the immediate town or city. If our fictional stocker were to suddenly vanish one day, management would have no trouble hiring somebody else to fill their shoes. The world would go on. Walmart would survive.
Now this is all well and good, but I would argue that there is an even stronger version of this claim: people don’t just like to be needed, people actively need to be needed. Over a decade ago, Paul Graham wrote an essay called Why Nerds are Unpopular; it’s a long essay with a number of different points, but there is one thread running through it that in my opinion has gotten far too little attention: “[Teenagers’] craziness is the craziness of the idle everywhere”.
The important thing to note about this (and Graham does so, in a roundabout sort of way) is that teenagers in a modern west |
4b3bc7ee-b9e6-4781-917e-289da02c8f98 | trentmkelly/LessWrong-43k | LessWrong | Link: Evidence-Based Medicine Has Been Hijacked
John Ioannidis has written a very insightful and entertaining article about the current state of the movement which calls itself "Evidence-Based Medicine". The paper is available ahead of print at http://www.jclinepi.com/article/S0895-4356(16)00147-5/pdf.
As far as I can tell there is currently no paywall, that may change later, send me an e-mail if you are unable to access it.
Retractionwatch interviews John about the paper here: http://retractionwatch.com/2016/03/16/evidence-based-medicine-has-been-hijacked-a-confession-from-john-ioannidis/
(Full disclosure: John Ioannidis is a co-director of the Meta-Research Innovation Center at Stanford (METRICS), where I am an employee. I am posting this not in an effort to promote METRICS, but because I believe the links will be of interest to the community) |
68a7f44e-de4f-4072-8eaf-d5f52ad0ad80 | trentmkelly/LessWrong-43k | LessWrong | Two questions about CEV that worry me
Taken from some old comments of mine that never did get a satisfactory answer.
1) One of the justifications for CEV was that extrapolating from an American in the 21st century and from Archimedes of Syracuse should give similar results. This seems to assume that change in human values over time is mostly "progress" rather than drift. Do we have any evidence for that, except saying that our modern values are "good" according to themselves, so whatever historical process led to them must have been "progress"?
2) How can anyone sincerely want to build an AI that fulfills anything except their own current, personal volition? If Eliezer wants the the AI to look at humanity and infer its best wishes for the future, why can't he task it with looking at himself and inferring his best idea to fulfill humanity's wishes? Why must this particular thing be spelled out in a document like CEV and not left to the mysterious magic of "intelligence", and what other such things are there? |
7eb656fe-368f-4340-b151-38574570d529 | trentmkelly/LessWrong-43k | LessWrong | Parable: The Bomb that doesn't Explode
You're an engineer working for a military contractor. One day, your project manager comes to you and asks you to design a container for plastic explosive. It is to contain several kilograms of C4, enough to destroy a building. This is pretty dangerous, but you know that C4 is actually pretty safe. It can't be accidentally detonated by fire, impact or bullets. Only a detonator (another explosive) can trigger C4, so you don't include a detonator in your design, for safety.
Your PM reviews the design and gives you some feedback. "It would be a lot more useful if you put a blasting cap inside." You grimace. More useful, yes. But a lot less safe. Nevertheless, you do your job and install a blasting cap. The blasting cap has two electrical leads. If a voltage is applied across those leads, the C4 will explode, killing everyone around. To keep things safe, you snip off the the leads and put the blasting cap inside a pill-shaped plastic container, inside the larger container that contains the C4.
"Well, that's no good," your PM replies, "What if someone does want to put a voltage across the leads?" You grimace even more. It is getting very difficult to make this design safe, you think. But you have a solution: inside the pill-shaped plastic container, you install a Raspberry Pi. You install SELinux on the Pi and set it up to connect to WiFi -- but only secured networks. You program up a fancy web interface that allows the user to specify exactly what voltage they want to apply to the leads. The interface caps the voltage at a low level. It does not allow the user to apply a sufficient voltage to actually trigger the blasting cap -- or at least you hope that's how blasting caps work.
"Well, that's no good," your PM replies, "What if someone wants to apply a higher voltage?" You grimace again. Then it'll explode! It'll kill everyone! But this is not a workplace where you raise objections, so instead you just diligently do your job. If you don't make this dangerous thing, t |
d2c0699a-af50-47a6-96e5-5ab19eba4711 | trentmkelly/LessWrong-43k | LessWrong | Meetup : San Jose Meetup: Park Day (IV)
Discussion article for the meetup : San Jose Meetup: Park Day (IV)
WHEN: 19 June 2016 03:00:00PM (-0700)
WHERE: 438 Coleman Ave, San Jose, CA 95110
You're invited to bring whatever you'd like to see at Park Day - balls, frisbees, food, etc. (Also, consider taking measures against the sun.) We'll carry a green blanket so you can recognize us. The idea is to gather outside near the listed address at the park entrance first, but probably walk elsewhere soon. If you can't find us or there's something else, please call me (Steven) at 510-6844565.
Discussion article for the meetup : San Jose Meetup: Park Day (IV) |
258264f8-6f31-47e3-80bf-91c53cdef92d | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Prediction can be Outer Aligned at Optimum
This post argues that many prediction tasks are *outer aligned at optimum*. In particular, I think that the malignity of the universal prior should be treated as an inner alignment problem rather than an outer alignment problem. The main argument is entirely in the first section; treat the rest as appendices.
### Short argument
In Evan Hubinger’s [Outer Alignment and Imitative Amplification](https://www.lesswrong.com/posts/33EKjmAdKFn3pbKPJ/outer-alignment-and-imitative-amplification#comments), *outer alignment at optimum* is defined as follows:
>
> a loss function is *outer aligned at optimum* if all the possible models that perform optimally according to that loss function are aligned with our goals
>
>
>
In the (highly recommended) [Overview of 11 proposals for building safe advanced AI](https://www.lesswrong.com/posts/fRsjBseRuvRhMPPE5/an-overview-of-11-proposals-for-building-safe-advanced-ai), this is used to argue that:
* Imitative amplification is probably outer aligned (because it emulates HCH, as explained in the first linked post, and in section 2 of the overview)
* Microscope AI (section 5) and STEM AI (section 6) are probably outer *misaligned*, because they rely on prediction, and optimal prediction is characterised by Bayesian inference on the universal prior. This is a problem, since the universal prior is probably malign (see [Mark Xu’s explanation](https://www.lesswrong.com/posts/Tr7tAyt5zZpdTwTQK/the-solomonoff-prior-is-malign) and [Paul Christiano’s original arguments](https://ordinaryideas.wordpress.com/2016/11/30/what-does-the-universal-prior-actually-look-like)).
I disagree with this, because I think that both imitative amplification and STEM AI would be outer aligned at optimum, and that some implementations of microscope AI would be outer aligned at optimum (see the next section for cases where it might not be). This is because optimal prediction isn’t necessarily outer misaligned.
The quickest way to see this is to note that *everything* is prediction. Imitative amplification relies on imitation learning, and imitating a human via imitation learning is equivalent to *predicting* what they’ll do (and then doing it). Thus, if microscope AI or STEM AI is outer misaligned *due to relying on prediction*, imitation learning is just as misaligned.
However, I *don’t* think that relying on prediction makes a method outer misaligned at optimum. So what’s wrong with the argument about the universal prior? Well, note that you don’t get perfect prediction on a task by doing ideal Bayesian inference on *any* universal prior. Different universal priors yield different predictions, so some of them are going to be better than others. Now, define optimal performance as requiring “the model to always have optimal loss on all data points that it ever encounters”, as Evan does in [this footnote](https://www.alignmentforum.org/posts/SzecSPYxqRa5GCaSF/clarifying-inner-alignment-terminology#fn-wy6RgjzHCyHCXi7M3-1). Even if almost all universal priors are malign, I claim that the prior that *actually does best on all data it encounters* is going to be aligned. Optimising for power requires sacrificing prediction performance, so the misaligned priors are going to be objectively worse. For example, the model that answers every STEM question correctly doesn’t have any wiggle room to optimise against us, because any variation in its answers would make one of them incorrect.
The problem we face is that it's really hard to tell whether our model actually is optimal. This is because consequentialists in a malign prior can intentionally do well on the training and validation data, and only sacrifice prediction performance during deployment. Importantly, this problem applies equally strongly for imitation learning as for other types of prediction.
The most important consequence of all this is that the universal prior’s malignity should be classified as a problem with *inner* alignment, instead of outer alignment. I think this is good and appropriate, because concerns about consequentialists in the universal prior are very similar to concerns about mesa optimisers being found by gradient descent. In both cases, we’re worried that:
* the inductive biases of our algorithm…
* will favor power-seeking consequentialists…
* who correctly predict the training data in order to stick around…
* but eventually perform a treacherous turn after the distributional shift caused by deployment.
### Online learning may be misaligned
However, one particular case where prediction may be outer misaligned is when models are asked to predict the future, and the prediction can affect what the true answer is.
For example, the stereotypical case of solomonoff induction is to place a camera somewhere on Earth, and let the inductor predict all future bits that the camera will observe. If we implemented this with a neural network and changed our actions based on what the model predicted, I’m *not* confident that the result would be outer aligned at optimum. However, this is unrelated to the malignity of the universal prior – instead, it’s because we might introduce strange incentives by changing our actions based on the AI’s predictions. For example, the AI might get the lowest loss if it systematically reports predictions that causes us to make the world as predictable as possible (and if we’re all dead, the world is pretty predictable…). For more on this, see Abram’s [parable](https://www.lesswrong.com/posts/SwcyMEgLyd4C3Dern/the-parable-of-predict-o-matic) (excluding section 9, which is about inner alignment) and associated discussion.
To see why this concern is unrelated to the malignity of the universal prior, note that there are two cases (that mostly depend on our definition of “at optimum”):
1. For each question, there is only a single prediction that leads to the lowest loss.
* For example, if we assume that the model has some degree of uncertainty about the future, it is likely that some prediction will affect the world in a way such that it’s *very* likely to come true, whereas other self-fulfilling prophecies may only be *somewhat* likely to come true.
* In this case, my argument from above applies: the model's behavior is fully specified by what the right answer is; the best prior for the situation will be selected; and there’s no room for (additional) malignity without making the prediction worse.
2. For some questions, there are *multiple* predictions that lead to optimal performance.
* In this case, the model’s behavior isn’t fully specified by saying that it performs optimally.
* However, Evan’s definition says that “a loss function is *outer aligned at optimum* if **all** the possible models that perform optimally according to that loss function are aligned with our goals”. Thus, we can tell whether the model is outer aligned at optimum by checking all combinations of optimal predictions, without having to consider the inductive biases of the model.
In practice, I don’t expect this type of concern to matter for STEM AI, but it may be a concern with some implementations of microscope AI.
As an aside, note that the definition of outer alignment at optimum only specifies the model’s input-output behavior, whereas for microscope AI, we care about the model’s internal structure. Thus, it’s unclear how to apply the definition to microscope AI. One option is to apply it to the system as a whole – including the humans trying to make the predictions. Alternatively, microscope AI may be such a unique proposal that the standard definition just isn’t useful.
### How to define optimal performance?
While I think that my short argument is broadly correct, it does sweep a lot of ambiguities under the rug, because it’s surprisingly annoying to construct a rigorous definition of outer alignment at optimum. The problem is that – in order to define *optimal performance* – we need both a loss function and a distribution of data to evaluate the loss function on. In many cases, it’s quite difficult to pinpoint what the distribution of data is, and exactly how to apply the loss function to it. Specifically, it depends on how the correct answer is labelled during training and deployment.
#### How is the correct answer labelled?
As far as I can tell, there are at least 4 cases, here:
1. **Mechanistic labelling**: In this setting, the model’s success can be verified in code. For example, when a model learns to play chess, it’s easy to check whether it won or not. Some versions of STEM AI could be in this category.
2. **Real-world labelling**: In this setting, the model is interacting with or trying to predict the real world, such that success or failure is almost mechanically verifiable once we see how the world reacts. An example of this is an AI tasked to predict all bits observed by a single camera. Some versions of STEM AI or microscope AI could be in this category.
3. **Human labelling**: These are cases where data points are labelled by humans in a controlled setting. Often there are some human(s), e.g. an expert labeller or a group of mturkers, who look at input and are tasked with returning some output. This includes e.g. training on ImageNet, and the kind of imitation learning that’s necessary for imitative amplification.
4. **Unlabelled**: These are cases where the AI learns the distribution of a big pile of pre-existing data in an unsupervised manner. The distribution is then often used for purposes quite different from the training setting. GPT-3 and DALL-E are both examples of this type of learning. Some types of STEM AI or microscope AI could be in this category.
Each of these offer different options for defining optimal performance.
1. With mechanistic labelling, optimal performance is unambiguous.
2. With real-world labelling, you can define optimal performance as doing whatever gives the network optimal reward; since each of the model’s actions eventually does lead to some reward.
* There’s some question about whether you should define optimal performance as always predicting the thing that actually happens, or whether you should assume that the model has some particular uncertainty about the world, and define optimal performance as returning the best probability distribution, given its uncertainty. I don’t think this matters very much.
* If implemented in the wrong way, these models are vulnerable to self-fulfilling prophecies. Thus, they may be outer misaligned at optimum, as mentioned in the section on online learning above.
3. With human labelling, we rarely get an objective answer during deployment, since we don’t ask humans to label data once training is over. However, we can define optimal performance according to what the labeller *would* have done, if they had gotten some particular input.
* In this case, the model must return some probability distribution over labels, since there’s presumably some randomness in how the labeller acts.
* However, if we want to, we can still assume that the AI’s knowledge of the outside world is completely fixed, and ask about the (impossible) counterfactual of what the human would have answered if they’d gotten a different input than they did.
4. For unsupervised learning, this option isn’t available, because there’s no isolated human who we can separate from the rest of the world. If GPT-3 is presented with a prompt “Hi I’m Lukas and…”, it cannot treat it’s input as some fixed human(s) H that reacts to their input as H(“Hi I’m Lukas and...”). Instead, the majority of GPT-3’s job goes towards *updating* on the fact that the *current source* is apparently called Lukas, is trying to introduce themselves, and whatever that implies *about* the current source, and about the world at large[[1]](#fn-5jexMxAr7FPtEQouq-1).
This means that for human-labelled data, we can assume that the world is fixed (or that the model’s uncertainty about the world is fixed), and only think about varying the input to the labeller. However, for unsupervised learning, we can’t hold the world fixed, because we need to find the most probable world where someone produced that particular input.
#### Extending the training distribution
As a consequence, when defining optimal performance for unsupervised learning, we need to define a full distribution of all possible inputs and outputs. GPT-3 was trained on internet text, but the internet text we have is very small compared to all prompts you could present to GPT-3. To define optimal performance, we therefore need to define a hypothetical process that represents much more internet text. Here are a few options for doing that:
1. Choose some universal prior as a measure over distributions; condition it on our finite training data; and use the resulting distribution as our ground truth distribution. As our universal prior, we could use the prior defined by some simple programming language (e.g. python) or the “true” prior that our universe is sampled from (if that’s coherent).
* Due to the ordinary arguments about the universal prior being malign, this *wouldn’t* be outer aligned at optimum. Since this definition would mean that almost nothing is outer aligned, it seems like a bad definition.
2. Defining the ground-truth of correct generalisation as the way that humans would generalise, if they became really good at predicting the training text.
* The problem with this definition is that we want to track the alignment properties of algorithms even as they reach far beyond human performance
* One option for accessing superhuman performance with human-like generalisation intuitions is to do something like Paul Christiano’s [Learning the prior](https://www.lesswrong.com/posts/SL9mKhgdmDKXmxwE4/learning-the-prior).
* While some variant of this could be good, becoming confident that it’s good would involve solving a host of AI alignment problems along the way, which I unfortunately won’t do in this post.
3. Use quantum randomness as our measure over distributions. More specifically, choose some point in the past (e.g. when Earth was created 4 billion years ago, or the internet was created 40 years ago), and then consider all possible futures from that moment, using quantum fluctuations as the only source of “randomness”. Use the [Born rule](https://en.wikipedia.org/wiki/Born_rule) to construct a measure over these worlds. (If you prefer copenhagen-like theories, this will be a probability measure. If you prefer multiverse theories, this will be a measure over close-by Everett branches.)
Then, exclude all worlds that in the year 2020 don’t contain a model with GPT-3’s architecture that was trained on GPT-3’s training data. Most of the remaining worlds will have some unobserved validation set that the researchers didn’t use during training. We can then define optimal performance as the distribution over all these validation sets, weighted by our quantum measure over the worlds they show up in.
* As far as I can tell, this is mostly well defined, and seems to yield sensible results. Since GPT-3’s training data contains so many details of our world; every world that contains a similar dataset will be very similar to our world. Lots of minor details will presumably vary, though, which means that the unobserved data should contain a wide and fair distribution.
* There are some subtleties about how we treat worlds where GPT-3 was trained multiple times on the same data, or how we treat different sizes of validation sets, etc; but I don’t think it matters much.
* I’m a bit wary of how contrived this definition is, though. We would presumably have wanted some way of defining counterfactuals even if quantum mechanics hadn’t offered this convenient splitting mechanic, so there ought to be some less hacky way of doing it[[2]](#fn-5jexMxAr7FPtEQouq-2).
If we wanted to, I think we could use a similar definition also in situations with real-world labelling or human labelling. Ie., we could require even an optimal model to be uncertain about everything that wasn’t universal across all Everett branches containing its training data. The main concern about this is that some questions may be *deeply* unlikely to appear in training data in the year 2020 (e.g. a question with the correct factorisation of RSA-2048) in which case being posed that question may move the most-likely-environment to some very strange subset of worlds. I’m unsure whether this would be a problem or not.
### A note about simulations
Finally, since these definitions refer to what the AI would *actually* encounter, in *our* world, I want to briefly mention an issue with simulations. We don’t only need to worry about the possibility that a solomonoff inductor thinks its input is being simulated – we should also consider the possibility that *we* are in a simulation.
Most simulations are probably short-lived, since simulating low-tech planets is so much cheaper than simulating the colonisation of galaxies. Thus, if we’re total consequentialist, the long-term impact we can have in a simulation is negligible compared to the impact we can have in the real world (unless the sheer number of simulations outweighs the long-term impact we can have, see [How the Simulation Argument Dampens Future Fanaticism](https://longtermrisk.org/how-the-simulation-argument-dampens-future-fanaticism)).
As a consequence of this, the only real alignment problem introduced by simulations is if an AI assigns substantial probability to being in a simulation despite actually being in the real world. This would be bad because – if the AI predicts things *as if* it was in a simulation – the consequentialists that the AI *believes* controls the simulation will have power over what predictions it makes, which they can use to gain power in the real world. This is just a variation of the universal prior being malign, where the consequentialists are hypothesized to simulate all of Earth instead of just the data that the AI is trying to predict.
As far as practical consequences go, I think this should be treated the same as the more general problem of the universal prior being malign. Thus, I’d like to categorise it as a problem with inner alignment; and I’d like to assume that an AI that’s outer aligned at optimum would act like it’s not in a simulation, if it is in fact not in a simulation.
This happens by default *if* our chosen definition of optimal performance treats being-in-a-simulation as a fixed fact about its environment – that the AI is expected to know – and not as a source of uncertainty. I think my preferred solutions above capture this by default[[3]](#fn-5jexMxAr7FPtEQouq-3). For any solution based on how humans generalise, though, it would be important that the humans condition on not being in a simulation.
*Thanks to Hjalmar Wijk and Evan Hubinger for helpful comments on earlier versions.*
Notes
-----
---
1. One way to frame this is with Pearl’s do-calculus. Say that the input is a random variable X and the output is a random variable Y. By analogy with Pearl’s do-calculus, we could then define optimal human-labelled performance as learning the distribution p(y=Y | do(x=X)), whereas unsupervised learning is trying to learn the entire distribution p(X,Y) in order to answer p(y=Y | x=X). For GPT-3, learning p(y=Y | do(x=X)) would correspond to guessing what a random human would say if they learned that they’d just typed “Hi I’m Lukas and…”; which would be very strange. [↩︎](#fnref-5jexMxAr7FPtEQouq-1)
2. One option is to partition the world into “macrostates” (e.g. a specification of where all humans are, what they’re saying, what the weather is, etc) and “microstates” (a complete specification of the location and state of all elementary particles), where each macrostate is consistent with lots of microstates. Then, we can specify a year; and assume that we know the macrostate of the world at the beginning of the year, but are uncertain about the microstate. If we then wait long enough, the uncertainty in microstate would eventually induce variation in macrostates; which we could use to define a distribution over data. I think this would probably yield the same results as the quantum definition; but the distinction between macrostates and microstates is a lot more vague than our understanding of quantum mechanics. [↩︎](#fnref-5jexMxAr7FPtEQouq-2)
3. This is the reason I wrote that we should exclude each world that “*in the year 2020* don’t contain a model with GPT-3’s architecture that was trained on GPT-3’s training data”. Without the caveat about 2020, we would accidentally include worlds where humanity’s descendants decide to simulate their ancestors. [↩︎](#fnref-5jexMxAr7FPtEQouq-3) |
fb79f01b-0cf3-4ca0-8086-1aa59601d24b | trentmkelly/LessWrong-43k | LessWrong | Semi-rare plain language words that are great to remember
I once spent a lot of time trying and failing to remember the word 'corollary', and once I heard it again I was like "ah, finally, there's that word; now I won't forget."
What are some other words like this? I'll keep an updating list in this post as I find and think of more of them.
Suggestions from me:
* Corollary
* Mu
Suggested by others, approved by me:
* Stochastic (credit: CatherinesCouch on twitter)
* Confounder (credit: CatherinesCouch on twitter)
* Plausible (credit: CatherinesCouch on twitter)
* Reification (credit: JonathanMoregard) |
e6ab0de5-5512-428e-95c1-35f069ed46bc | trentmkelly/LessWrong-43k | LessWrong | Calibrating your probability estimates of world events: Russia vs Ukraine, 6 months later.
Some of the comments on the link by James_Miller exactly six months ago provided very specific estimates of how the events might turn out:
James_Miller:
* The odds of Russian intervening militarily = 40%.
* The odds of the Russians losing the conventional battle (perhaps because of NATO intervention) conditional on them entering = 30%.
* The odds of the Russians resorting to nuclear weapons conditional on them losing the conventional battle = 20%.
Me:
"Russians intervening militarily" could be anything from posturing to weapon shipments to a surgical strike to a Czechoslovakia-style tank-roll or Afghanistan invasion. My guess that the odds of the latter is below 5%.
A bet between James_Miller and solipsist:
I will bet you $20 U.S. (mine) vs $100 (yours) that Russian tanks will be involved in combat in the Ukraine within 60 days. So in 60 days I will pay you $20 if I lose the bet, but you pay me $100 if I win.
While it is hard to do any meaningful calibration based on a single event, there must be lessons to learn from it. Given that Russian armored columns are said to capture key Ukrainian towns today, the first part of James_Miller's prediction has come true, even if it took 3 times longer than he estimated.
Note that even the most pessimistic person in that conversation (James) was probably too optimistic. My estimate of 5% appears way too low in retrospect, and I would probably bump it to 50% for a similar event in the future.
Now, given that the first prediction came true, how would one reevaluate the odds of the two further escalations he listed? I still feel that there is no way there will be a "conventional battle" between Russia and NATO, but having just been proven wrong makes me doubt my assumptions. If anything, maybe I should give more weight to what James_Miller (or at least Dan Carlin) has to say on the issue. And if I had any skin in the game, I would probably be even more cautious.
|
bcc8aad8-59b5-4bf3-b069-bd2af863cf41 | trentmkelly/LessWrong-43k | LessWrong | OPTIC: Announcing Intercollegiate Forecasting Tournaments in SF, DC, Boston
TLDR: We’re excited to announce that OPTIC is running three intercollegiate forecasting competitions this fall — in the Bay Area (Nov 4), DC (Nov 18) and Boston (Dec 2). Register here!
What does a tournament look like?
Think 1-day hackathon/olympiad/debate tournament, but for forecasting the future — teams predict on topics ranging from geopolitics to celebrity twitter patterns to financial asset prices, and the best forecasters get thousands of dollars in cash prizes and exclusive internships at Metaculus.
Day of, teams give probabilistic predictions for a couple hours on about 30 given questions (with breaks for lunch & speakers). Teams are 3-5 competitors, and we’ll place you on a team if you don’t already have one in mind. A few months after the tournament, all questions resolve and winners are announced/awarded: until then, you can track how your team is doing in real time.
Forecasting, cash prizes & exclusive internships, opportunities to meet other intellectually minded students, more — register here!
Competition Logistics
We’ll be running competitions in the Bay Area (Nov 4), DC (Nov 18), and Boston (Dec 2).
Day of, teams give probabilistic predictions for a couple hours on about 30 given questions (with breaks for lunch & speakers). Teams are 3-5 competitors, and we’ll place you on a team if you don’t already have one in mind. A few months after the tournament, all questions resolve and winners are announced/awarded: until then, you can track how your team is doing in real time.
To get a taste of what a competition looks like, you can:
* Look over our previously used questions
* Watch Seth Blumberg’s speech on the future of forecasting (Seth is a behavioral analyst at Google and the head of Google’s internal prediction market, speaking in a personal capacity)
* If you want a deep look into the nuts and bolts, check out our postmortem of the pilot competition!
Sponsorship & Funding
This semester, OPTIC received funding from Open Philanthropy to |
31e43207-5c0e-4878-9805-4ce1925229be | trentmkelly/LessWrong-43k | LessWrong | Wordtune review
Wow, Facebook is advertising an AI-driven writing tool to me. I think this is the first time I’ve seen a mainstream potentially useful product coming from all of this. Let’s see how it does. I’ve started by writing this paragraph quickly myself, and then I will rewrite it using Wordtune, to demonstrate its features. Then I may rewrite it further using its premium features or my more honed writing, all going well. (original)
Wow, Facebook is showing me a machine-learning based writing platform. I believe that this is the first mainstream application of all this. See how it goes. I wrote this paragraph on my own quickly, and then rewrote it using Wordtune to demonstrate its features. Then I may rewrite it using its premium features or using my more refined writing skills, if all goes well. (Wordtune)
I am intrigued by Facebook’s advertisement with regards to an artificial intelligence (AI) writing tool. I’m not aware of any mainstream application arising from these efforts.Let us see how it goes. I created this paragraph quickly myself, and now I’m going to rewrite it using Wordtune to demonstrate all it has to offer. Afterwards, I will be able to incorporate premium features in it or craft it into a more refined style, all going well. (formal Wordtune)
Wow, Facebook is advertising a writing tool which uses artificial intelligence (AI) that I did not know existed. This is something of a landmark event; I’ve not yet seen a mainstream useful product coming out of anything like this. I’m looking forward to seeing how it goes. I’ve begun by writing this paragraph myself but I will then rewrite it with the aid of Wordtune to illustrate its various features. Then I may rewrite it further using its premium features or my more honed writing, all going well. (longer Wordtune)
Wow, Facebook’s touting a writing tool powered by artificial intelligence. This is the first time I’ve seen a mainstream use coming from all this. Let’s see what it does. I wrote it quickly, and I’m |
1c470274-d004-440b-9925-9fb2d2e39bab | StampyAI/alignment-research-dataset/eaforum | Effective Altruism Forum | 2023 Stanford Existential Risks Conference
The team at SERI is excited to announce that sign-ups for the 3rd annual Stanford Existential Risks Conference are now open! We will be running a hybrid conference this year. [Sign-ups](https://airtable.com/shrPg0XTaZLKxxTem) for both in-person and virtual attendance are now live.
This is an exciting opportunity for attendees to engage with experts in the field, network with like-minded individuals, and gain insights into the latest research and developments related to existential risk. This year’s conference aims to take stock of global catastrophic and existential risk studies. A special focus of this meeting will be risk intersections, reinforcements, and cascades: how one risk may amplify (or diminish) another, and how multiple risks interact to create new concerns that may be larger than the sum of their parts. The conference will also feature discussions of the ethics of radical longtermism.
Because space for in-person attendance is limited, we encourage you to [sign up](https://airtable.com/shrPg0XTaZLKxxTem) as soon as possible.
View [additional information about the conference](https://seri.stanford.edu/events/stanford-existential-risks-conference-april-20-22-2023). Questions or concerns? [Click here to contact us](mailto:seri-contact@stanford.edu).
We look forward to your participation in this thought-provoking and impactful event!
In survival and flourishing,
The SERI Team |
c2e5d071-6f6b-4c7e-9754-f58fc1fead50 | trentmkelly/LessWrong-43k | LessWrong | Quick survey on AI alignment resources
TLDR: In collaboration with CEA, an external collaborator is trying to determine the value of various AI alignment resources. If you are interested in or currently working on AI alignment, please fill out this survey. We expect it will likely take 5-15 minutes. Thanks!
___
There are a lot of resources available to learn about AI Alignment. For field builders and those interested in the field, it can be difficult to know when to prioritize which readings. To address this, an external collaborator has created a survey to collect feedback on the resources you’ve engaged with, in collaboration with the Centre for Effective Altruism (CEA).
We hope this survey will:
1. Inform which materials have previously helped people get into alignment. We can then promote those materials broadly or target them to particularly relevant groups (e.g. students).
2. Identify gaps where more materials are needed.
3. Provide feedback to those who made the materials, encouraging them to create more or try new stuff!
If you are interested in or currently working on AI Alignment, we would be so grateful for your input. We expect this survey to take around 5-10 minutes.
Once you’ve filled it out, please consider sharing this survey with others in your network.
We appreciate your time. Thanks! |
09af3d6c-0b73-49ed-9516-dd66f499fbf3 | trentmkelly/LessWrong-43k | LessWrong | Looking to interview AI Safety researchers for a book
If you are or have been employed as an AI Safety Researcher, I'd love to interview you as background for a novel that I'm writing. I'm interested in learning what a day in the life of an AI researcher typically looks like.
I'd be happy to include you in the acknowledgments or keep your participation completely confidential, whichever you prefer.
If you'd like to email me rather than respond in the comments, my email is jeff [AT] securityink [DOT] org.
Grazie! |
ddd1070f-dbd7-4a51-89fb-a8e1b1e96a6e | trentmkelly/LessWrong-43k | LessWrong | Sapiens
This is a section-by-section summary and review of Sapiens: A Brief History of Humankind by Yuval Noah Harari. It's come up on Less Wrong before in the context of Death is Optional, a conversation the author had with Daniel Kahneman about the book, and seems like an accessible introduction to many of the concepts underlying the LW perspective on history and the future. Anyone who's thought about Moloch will find many of the same issues discussed here, and so I'll scatter links to Yvain throughout. I'll discuss several of the points that I thought were interesting and novel, or at least had a novel perspective and good presentation.
A history as expansive as this one necessarily involves operating on higher levels of abstraction. The first section expresses this concisely enough to quote in full:
> About 13.5 billion years ago, matter, energy, time and space came into being in what is known as the Big Bang. The story of these fundamental features of our universe is called physics.
>
> About 300,000 years after their appearance, matter and energy started to coalesce into complex structures, called atoms, which then combined into molecules. The story of atoms, molecules and their interactions is called chemistry.
>
> About 3.8. billion years ago, on a planet called Earth, certain molecules combined to form particularly large and intricate structures called organisms. The story of organisms is called biology.
>
> About 70,000 years ago, organisms belonging to the species Homo sapiens started to form even more elaborate structures called cultures. The subsequent development of these human cultures is called history.
>
> Three important revolutions shaped the course of history: the Cognitive Revolution kick-started history about 70,000 years ago. The Agricultural Revolution sped it up about 12,000 years ago. The Scientific Revolution, which got under way only 500 years ago, may well end history and start something completely different. This book tells the story |
a60f4284-e702-46b3-9b05-5a78fb3e90a9 | trentmkelly/LessWrong-43k | LessWrong | [Part 1] Amplifying generalist research via forecasting – Models of impact and challenges
This post covers our models of impact and challenges with our exploration in amplifying generalist research using forecasting. It is accompanied by a second post with a high-level description of those models, and more detailed description of experiment set-up and results.
Many of the world’s most pressing problems require intellectual progress to solve [1]. Finding ways to increase the rate of intellectual progress might be a highly promising way of solving those problems.
One component of this is generalist research: the ability to judge and synthesise claims across many different fields without detailed specialist knowledge of those fields, in order to for example prioritise potential new cause areas or allocate grant funding. For example, this skill is expected by organisations at the EA Leaders Forum to be one of the highest demanded skills for their organisations over the coming 5 years (2018 survey, 2019 survey).
In light of this, we recently tested a method of increasing the scale and quality of generalist research, applied to researching the industrial revolution [2], using Foretold.io (an online prediction platform).
In particular, we found that, when faced with claims like:
> “Pre-Industrial Britain had a legal climate more favorable to industrialization than continental Europe”
And
> “Pre-Industrial Revolution, average French wage was what percent of the British wage?”
a small crowd of forecasters recruited from the EA and rationality communities very successfully predicted the judgements of a trusted generalist researcher, with a benefit-cost ratio of around 73% compared to the original researcher. They also outperformed a group of external online crowdworkers.
Moreover, we believe this method can be scaled to answer many more questions than a single researcher could, as well as to have application in domains other than research, like grantmaking, hiring and reviewing content.
We preliminarily refer to this method as “amplification” given its s |
fb62e08c-dcfe-417e-a77b-af1c1d644ebf | StampyAI/alignment-research-dataset/lesswrong | LessWrong | Risks from GPT-4 Byproduct of Recursively Optimizing AIs
***Epistemic Status:** At midnight three days ago, I saw some of the GPT-4 Byproduct Recursively Optimizing AIs below on twitter which freaked me out a little and lit a fire underneath me to write up this post, my first on LessWrong. Here, my main goal is to start a dialogue on this topic which from my (perhaps secluded) vantage point nobody seems to be talking about. I don’t expect to currently have the optimal diagnosis of the issue and prescription of end solutions.*
***Acknowledgements:** Thanks to my fellow Wisconsin AI Safety Initiative (WAISI) group organizers Austin Witte and Akhil Polamarasetty for giving feedback on this post. Organizing the WAISI community has been incredibly fruitful in being able to spar ideas with others and see which strongest ones survive. Only more to come.*
***Edit:** The biggest feedback I've gotten so far is that I may be misleading some without enough context about the level of intelligence and danger that we currently have here. So in order to try to give some intuition on what seems to me to be the current state of affairs:*
* *The recursively optimizing agents don't seem to me like they will lead to an intelligence explosion or singularity that goes to infinity. I call them "very weak and early agentic AGI" because of the effective general intelligence they have from the LLMs and how they will be pursuing terminal goals that likely will make them develop the instrumental goal of gaining more power. I don't call them "very weak and early agentic AGI" because they will suddenly be able to discover new things that can't be interpolated from the internet corpus and create an intelligence explosion. That being said, the internet is massive and we should not doubt the significance of the intelligence that LLMs will be able to strategize with from just this. It is unclear how far GPT-4's capabilities will go in aiding these recursively optimizing AIs. I don't think anyone can guarantee, however, that there currently is no danger. And, I think everyone should be able to agree that with every single new more advanced LLM, the danger only goes up.*
* *Note: Perhaps better operational definitions should be created for these distinctions between true AGI and things that effectively has some semblence of general intelligence. I imagine it would clear up a lot of confusion.*
*(from @anthrupad on twitter)*
**Introduction**
================
Recently, many people across the internet have used their access to GPT-4’s API to scheme up extra dangerous capabilities. These are capabilities which the AGI labs certainly could have done on their own and likely are doing. However, these AGI labs at the very least seem to be committed to safety. Some people may say they are following through on this well and others may say that they are not. Regardless, they have that stated intention, and have systems and policies in place to try to uphold it. *Random people on the internet taking advantage of open source do not have this.*
As a result, people are using GPT-4 as the strategizing intelligence behind separate optimizing programs that can recursively self-improve in order to better pursue their goals. Note that it is not GPT-4 that is self-improving, as GPT-4’s weights are stagnant and not open sourced. Rather, it is the programs that use GPT-4’s large context window (as well as outside permanent memory in some cases) to iterate on a goal and get better and better at pursuing it every time.
Here are two examples of what has resulted to give a taste:
* This version of the program failed, but another that worked could in theory very quickly generate and run potentially very influential code with little oversight or restriction on how each iteration improves.
* This tweet made me think to possibly brand these recursively optimizing AI as “Russian Shoggoth Dolls”
* The program pursues the instrumental goal of increasing its power and capability by writing the generic HTTP plugin in order to better get at its terminal goal of better coding plugins
* Evidence of this kind of behavior is really, really bad. See [Instrumentally Convergent Goals](https://www.lesswrong.com/tag/instrumental-convergence) and the danger they present
\*\*Everyone in the AI Safety community should take a moment to look at these examples, particularly the latter, and contemplate the consequences. Even if GPT-4 is kept in the box, simply by letting people access it through an API, input tokens, and receive the output tokens, we might soon have what in effect seem like separate very early weak forms of agentic AGI running around the internet, going wild. This is scary.\*\*
**Digging Deeper**
==================
The internet has a vast distribution of individuals out there from a whole bunch of different backgrounds. Many of them, quite frankly, may want to simply just build cool AI and not give safety guards a second thought. Others may not particularly want to create AI that leads to bad consequences but haven’t engaged enough with arguments on risks that they are simply negligent.
If we completely leave the creation of advanced LLM byproduct AI up to the internet with no regulations and no security checks, some people will beyond a doubt act irresponsibly in the AI that they create. This is a given. There are simply too many people out there. Everyone should be on the same page about this.
Let’s look at the perspective of the author of another one of these self-improving agents. Here is a tweet on their work:
And, here is how they talk about AGI:
Empathizing with AGI will not align it nor will it prevent any existential risk. Ending discrimination would obviously be a positive for the world, but it will not align AGI. Significant Gravitas is deeply anthropomorphizing the nature of these models and how alignment works in a way that clearly shows why they don’t feel the need to think about safety. [[1]](#fn76v6kejj1b) [[2]](#fnzj9cx9275hh)
I don’t say any of this to dunk on the person. People form their beliefs through a causal chain one way or another and the way to improve things is not to punish the individual but to improve the chain.
My only goal here is to exemplify the wide distribution of beliefs that people highly skilled in deep learning may have and how this ought to bring about deep concern to those who care about risks from AI.
Side note, a lot of similar framing of AGI to above seems to originate from [David Deutsch](https://www.youtube.com/watch?v=flA3Qbkj_cw) and I think is sadly misguided because whatever the first semblances of AGI will be will almost certainly not fit the mental model these people have for it.
### This all points towards a dichotomy which I find to be true:
Even if the AGI labs are wise enough to be safe with their most powerful models internally and don’t open source the weights to their models for anyone to fine tune into being evil, if they let anyone access these models through their API with no oversight on what code others execute with it, then major risks still arise.
The focus of the AI Safety world right now feels deadlocked on solely looking at and aligning the most advanced general models and is totally ignoring all the other byproduct models that will be able to reach quite powerful levels by taking advantage of whatever the most advanced model is that is opened up to the public at the moment.
Whether this is actually happening or not is only bottlenecked by the number of individuals with the technical experience to accomplish this and how motivated they are. There is nothing else stopping them. Very early and weak forms of agentic AGIs are already coming around. People are building them.
Don't believe me? Check out the Top 3 GitHub Repositories that are currently trending:
### And, open sourced models are only getting stronger:
From this point forward, models are only going to get stronger and stronger, and more and more accessible. I don’t think it’s a reach to expect to see open sourced models at near GPT-4 levels within 1-2 years. Once these are made, anybody will be able to use them to create these recursively optimizing AIs. We might see evidence of agents pursuing instrumentally convergent goals everywhere if we don't slow down.
Hopefully, the strategizing will be weak enough that it won't be catastrophic. However, the more powerful that the LLMs are in their ability to strategize, even if they by themselves are benign, the more powerful these recursively optimizing AI using them will become.
GPT-4 sure seems pretty damn intelligent to me. What about GPT-5? GPT-6? Where do we draw the line?
### How powerful are these recursive optimizing AIs right now?
It isn't very clear. However powerful they might be would significantly change the tone of my writing here for how urgently we need to mitigate these risks. No matter what though, in the long term (meaning 1-2 years in these strange AI times) this needs to be thought about.
Andrej Karpathy at OpenAI said when talking about his Jarvis (seen in GitHub Repositories above) said [that](https://twitter.com/karpathy/status/1642600116837298178?s=20) "Interesting non-obvious note on GPT psychology is that unlike people they are completely unaware of their own strengths and limitations. E.g. that they have finite context window. That they can just barely do mental math. That samples can get unlucky and go off the rails. Etc."
This makes me feel a bit safer because I don't think anyone yet has figured out a way for the recursive iterations to never derail. The first screenshot in this post above from Harris Rothaermel also shows his agents recursing to infinity and derailing. Hopefully, it stays that way for some time. But eventually, people will figure out how to make it work. I would be pretty shocked if there was no way to make it function with GPT-4.
**Ideas for Mitigating Risk**
=============================
If this is all true, how can we mitigate the risk that results? Here are some options that came to mind as well as some elaborations on them. I am not deeply invested in any one of these in particular being *the* way. Rather, I wanted to bring up all of the options that I could think of to be comprehensive.
Option 1: Don’t open up the API for the most Powerful Models
------------------------------------------------------------
This is obviously harsh, but seems to be a good starting point for proposals. People won’t be able to use these models to create Byproduct Recursively Optimizing AIs/Russian Shoggoth Dolls if they don’t have access to them, period.
That is, for at least the next year or so, until another GPT-4 powered open source model inevitably comes around. But, at least the agentic AI that these models produce will be weaker than the agentic AI that GPT-5 or GPT-6 produce. I don’t particularly want to be in a world where adversarial AI are constantly fighting against each other but at least the safety concerned people will likely have the more powerful models.
Option 2: Tighter API Regulations & Code Oversight
--------------------------------------------------
Create much tighter regulations on who has access to the APIs of the AGI companies and what oversight those accessing are forced to permit.
Perhaps OpenAI has to run a background check on you to confirm your identity and track relevant attitudes and beliefs with regards to AI before they provide you with the API key in order to have some risk modeling.
Perhaps accessing GPT-4 means that you need to give OpenAI access to all the code that you will use in tandem with it so that *before the code is run*, OpenAI can use AI models to scan it all over and make sure that no optimizing agents are being produced that might have instrumentally convergent goals.
This might mean distributing access to GPT-4 through an application that has the power to make these checks rather than simply through an API key where after getting the key, the individual pretty much has free reign. The same thought process should be applied for GPT plug-ins used by companies.
The only feasible way to do this may be for OpenAI to have to create a whole new software system (or use someone else’s) and keep the entire interface with GPT-4 contained there.
Otherwise, people will certainly try their best to obfuscate their code, only providing certain snippets and leaving out others that they fear might get flagged. Lots of thought may need to be put in to think about how to practically accomplish this and ensure that people aren’t hiding code.
These seem like steps to consider right away. Given the incredibly high demand for these powerful models, despite the open source community outcry and flak that will probably arise, people will still accept these safety regulations.
Option 3: Detecting Recursive Optimizing Behavior
-------------------------------------------------
Whereas embedded in Option 2 is detecting *code* that may produce recursive optimizing behavior *before it is run*, Option 3 explores detecting *emergent behavior* either in the API or online that appears to be an optimization process which is recursively improving at speeds faster than humans could *as it is being run or after it has been run*.
I am quite uncertain about the most effective ways to do so, however, I will float some ideas here:
Using GPT-4 to recursively call itself unsurprisingly drastically increases the total of API calls made by a user which should be a clear marker. Note that this could be bypassed by using multiple API keys, but stricter API regulation can make it impossible for one individual to get more than one key.
Perhaps we could start with scraping posts on forums, GitHub, and Twitter that seem likely to spawn these things? People seem highly interested in sharing their creations, and since these communities are all for open-sourcing, that will likely remain true for some period of time. Four examples of this are already provided above.
Perhaps we attempt training adversarial AI that gets better and better at detecting when other AI programs are likely to produce optimizing agentic behavior. Perhaps even create some really weak and constrained agentic AIs that don’t know that they are being adversarially tested against. This would be a starting point to see how they behave.
Once we detect it though, what can we do?
If that stakes are low, you could ban them from using the API and restrict them from getting another key.
If that stakes are high, this may already be too late. This makes Option 2 seem like a much, much more effective starting point.
If we detect enough prominent examples of dangerous recursively improving optimizing behavior and show influential people, will it be enough to get them worried? Will it get people worried in the right places who will take action that leads to safer consequences?
This may be hopeful thinking, but personally, the discovery of tool use, memory extension and recursion for agency would, in my opinion, be enough for me to probably not release GPT-5 to the public. OpenAI taking this step would be a big move that would send shockwaves throughout the world which perhaps could get everyone, including China, to slow down.
Option 4: Deeper LLM Alignment
------------------------------
Perhaps we can align LLMs to be able to detect internally whether they are being used to create recursively improving optimizers when they are prompted.
Similar to how LLMs can be aligned to recognize what might be politically insensitive material and navigate their response with finesse, perhaps LLMs can recognize prompts that push them towards building dangerous recursively improving optimizers and 1) not output responses and 2) flag or ban the account attempting to get it to do so.
This seems good for the top AGI companies to commit to attempting. It also doesn't seem like it should be that difficult given how successful their aligning with e.g. RLHF has been so far. However, for RLHF to be used to prevent recursive optimizing AI, people with deep knowledge of AI and coding would have to be the ones giving feedback to the reward model. Only they could parse out whether a piece of code seems dangerous. This may be very expensive.
And, again, the open source community seems likely to create GPT-4 powered LLMs that will not have these safeguards very soon. We’ll have to deal with the consequences of that.
Option 5: Better Discourse on AGI
---------------------------------
People have quite a lot of different perspectives on AGI that informs how responsible or irresponsible they are when it comes to what they code. If we could somehow create an information ecosystem that walks people through 1) the risks in a very clear cut fashion and 2) takes us past our anthropomorphizing biases of AGI, perhaps argumentatively we can convince people that it is morally wrong to create these dangerous technologies. Maybe this will actually make less people attempt to do so.
Doesn't seem anywhere near likely though given how difficult changing public opinion is. And, really isn't a long term solution to depend on people voluntarily to be safe. Still worth mentioning though.
Option 6: Government Oversight
------------------------------
Create a massive scale government oversight program put into place. This seems clearly politically unfavorable and would lead to a world that many would prefer to avoid, including myself. Although, if it is the only option, which doesn’t seem to be the case right now thankfully, perhaps it ought to be explored. Let’s not make this the only option.
**Conclusion**
==============
It feels as though the entire AI Safety world is focused on aligning the most advanced AI models and is possibly neglecting what the consequences of simpler models taking advantage of the most advanced open source models might look like.
I worry that this is a big blind spot and would like for it to be on the minds of more people.
Especially given the fact that it seems likely that within 1-2 years GPT-4 level models will likely be trained and then open sourced by *someone out there*. Not just API open source, but all of their whole weights released to play around with.
This makes me expect very early and weak forms of agentic AGI to be all over the place very soon although I would be quite happy to be convinced otherwise.
Curious to hear others’ thoughts.
1. **[^](#fnref76v6kejj1b)**Whether AI is conscious and deserving of moral personhood is a different controversial question which I don’t particularly need to engage in this post, although I am on team “No”.
2. **[^](#fnrefzj9cx9275hh)**There is some argument to make that something like "empathizing" with the models might help align AI but that is not through sincere moments of intimate connection like are being referred to here, it is through filling the AI's training data with very, very empathetic text. However, this also seems liable to create AI with emotional insecurity issues if not done well, as [Bing Chat](https://www.lesswrong.com/posts/jtoPawEhLNXNxvgTT/bing-chat-is-blatantly-aggressively-misaligned) provided a good first model of. |
ff42414b-7111-44ca-b80b-474abcf58481 | StampyAI/alignment-research-dataset/arbital | Arbital | Non-adversarial principle
The 'Non-Adversarial Principle' is a proposed design rule for [sufficiently advanced Artificial Intelligence](https://arbital.com/p/7g1) stating that:
*By design, the human operators and the AGI should never come into conflict.*
Special cases of this principle include [https://arbital.com/p/2x4](https://arbital.com/p/2x4) and [The AI wants your safety measures](https://arbital.com/p/ai_wants_security).
According to this principle, if the AI has an off-switch, our first thought should not be, "How do we have guards with guns defending this off-switch so the AI can't destroy it?" but "How do we make sure the AI *wants* this off-switch to exist?"
If we think the AI is not ready to act on the Internet, our first thought should not be "How do we [airgap](https://arbital.com/p/airgapping) the AI's computers from the Internet?" but "How do we construct an AI that wouldn't *try* to do anything on the Internet even if it got access?" Afterwards we may go ahead and still not connect the AI to the Internet, but only as a fallback measure. Like the containment shell of a nuclear power plant, the *plan* shouldn't call for the fallback measure to ever become necessary. E.g., nuclear power plants have containment shells in case the core melts down. But this is not because we're planning to have the core melt down on Tuesday and have that be okay because there's a containment shell.
# Why run code that does the wrong thing?
Ultimately, every event inside an AI--every RAM access and CPU instruction--is an event set in motion by our own design. Even if the AI is modifying its own code, the modified code is a causal outcome of the original code (or the code that code wrote etcetera). Everything that happens inside the computer is, in some sense, our fault and our choice. Given that responsibility, we should not be constructing a computation that is *trying* to hurt us. At the point that computation is running, we've already done something foolish--willfully shot ourselves in the foot. Even if the AI doesn't find any way to do the bad thing, we are, at the very least, wasting computing power.
No aspect of the AI's design should ever put us in an adversarial position vis-a-vis the AI, or pit the AI's wits against our wits. If a computation starts *looking* for a way to outwit us, then the design and methodology has *already* failed. We just shouldn't be putting an AI in a box and then having the AI search for ways to get out of the box. If you're building a toaster, you don't build one element that heats the toast and then add a tiny refrigerator that cools down the toast.
# Not running searches for harmful strategies
Under the [https://arbital.com/p/agents_as_searches](https://arbital.com/p/agents_as_searches) viewpoint, we can see an AI as embodying computations that search out strategies within a domain. E.g., an AI carrying out the "[put one damn strawberry on a damn plate](https://arbital.com/p/one_damn_tomato)" task is searching the space of imaginable strategies for a strategy such that the consequence of executing that strategy is one strawberry being on the plate.
The non-adversarial principle then says that we should at no point be running a computation which is *searching* for ways to hurt us.
Suppose you don't want the AI to go on the Internet. If your first thought is [airgapping](https://arbital.com/p/airgapping) the AI's hardware from the Internet or otherwise [boxing the AI](https://arbital.com/p/6z), then what you're trying to do is ensure that *a computation searching for strategies to reach the Internet returns no result.*
If the primary plan is "the AI searches, but doesn't find anything", then we need to worry that the AI's search [considers some class of strategies we didn't imagine](https://arbital.com/p/9f). We are pitting our own wits against a potentially [weird](https://arbital.com/p/9f) or [superhuman](https://arbital.com/p/41l) search. Even if we did successfully rule out all possibilities for escape, it would be hard for us to ever be confident that we had done so. It would be very hard to rule out the *conceptual* possibility of some entirely unexpected side-channel attack like the AI [producing GSM cellphone signals by cleverly modulating its memory accesses](https://www.usenix.org/system/files/conference/usenixsecurity15/sec15-paper-guri-update.pdf), or the AI [persuading its human operators to let it out of the box](https://arbital.com/p/aibox_experiment). (Except that neither of those two particular possibilities would be *unexpected;* any specific strategy we write down will be the kind of possibility that humans consider.)
Even if the AI's search does come up empty, you were tempting fate and wasting computing power by instantiating that search in the first place.
# Oppositional measures are fallbacks
*After* we ensure that the escape-search computation is not running in the first place, we can think about fallback [oppositional](https://arbital.com/p/7fx) lines of defense; e.g. putting the AI's processors inside a [Faraday cage](https://en.wikipedia.org/wiki/Faraday_cage).
But the *plan* is not that the AI tries to escape and we successfully prevent the escape. If we imagine the AI trying to escape, we ought to [feel very surprised in our imagination](https://arbital.com/p/rule_of_surprise). If the AI actually does try to escape, we should be crying 'What?' and tracking down which strong-seeming assumption has failed.
No matter what clever design we use or how strong it seems, we'd probably *still* be wise to [airgap](https://arbital.com/p/airgapping) an AI under development, not say "Oh it's supposed to be safe!" and connect it to the Internet and saunter on. But that doesn't mean we *rely* on the airgap. It doesn't mean the plan calls for the airgap to be necessary.
Nuclear power plants have concrete containment units in case the core melts down; but the *design* is not that they melt down. The master plan doesn't say "And then on Tuesday the core melts down, but that's fine because of the containment unit." By design, that enormous concrete shell isn't supposed to actually ever become necessary. And then we build it anyway, because the best-laid plans etcetera.
Similarly, when designing an AI, we should pretend that the airgap doesn't exist or that the AI will [suddenly get Internet access anyway](https://arbital.com/p/2x) on Tuesday; our *primary* thought should be to design AI that doesn't need an airgap to be safe. And *then* we add the airgap, making sure that we're not thinking the equivalent of "Oh, it doesn't *really* matter if the core melts down, because we've got a containment structure there anyway."
# Challenges in implementing non-adversarialism
The main difficulties foreseen so far for implementing the non-adversarial principle, tend to center around [https://arbital.com/p/10g](https://arbital.com/p/10g) plus [https://arbital.com/p/42](https://arbital.com/p/42) behavior.
For example, if you build a [shutdown button](https://arbital.com/p/2xd) for a [Task AGI](https://arbital.com/p/6w) that suspends the AI to disk when pressed, the nonadversarial principle implies you must also ensure:
- That the AI *wants* there to be a shutdown button;
- That the AI *wants* to be suspended to disk after this button is pressed;
- That the AI *wants* the state of this shutdown button to reflect the dynamic decisions of the human operators;
- That the AI does not *want* to influence the operators to decide to not press the switch, or to press it;
- That the AI does *not* want anything *besides* an orderly suspend-to-disk to happen, or not happen, after this button is pressed.
*Or:*
- The AI does not think about or make plans involving the shutdown button, e.g. because that domain was not [whitelisted](https://arbital.com/p/whitelisting) for cognition.
- None of the AI's other models end up reflecting the existence of the shutdown button or none of its other plans end up taking into account that part of the model.
The difficulties here center around ["You can't fetch the coffee if you're dead"](https://arbital.com/p/7g2). This reasoning is very general, so [even if we try to make it not apply at one point, it tends to pop up somewhere else](https://arbital.com/p/48):
- If you naively try to add in a special-case clause to the utility function for wanting a shutdown button to exist, the AI wants the shutdown button to not be pressed.
- If you successfully add a special case saying that the AI wants the button to be pressed if the humans want that button to be pressed, the AI wants the humans to not want to press the button.
- If you naively try to add in a special clause for the AI wanting to shut down after the button is pressed, the AI wants to create a [subagent](https://arbital.com/p/environmental_subagent) to make sure the coffee gets pressed anyway.
- If you try to make an AI that [doesn't think about the shutdown button](https://arbital.com/p/1g4) or model it at all, this seems potentially difficult because in reality the best hypothesis to explain the world *does* contain a shutdown button. A general search for good hypotheses may tend to create cognitive tokens that represent the shutdown button, and it's not clear (yet) how this could in general be prevented by trying to divide the world into domains.
More generally: by default a lot of *high-level* searches we do want to run, [have *subsearches* we'd prefer *not* to run](https://arbital.com/p/10g). If we run an agent that searches *in general* for ways to fetch the coffee, that search would, by default and if smart enough, also search for ways to prevent itself from being shut down.
How exactly to implement the non-adversarial principle is thus a major open problem. We may need to be more clever about shaping which computations give rise to which other computations than the default "Search for any action in any domain which achieves X."
# See also
- [https://arbital.com/p/2x4](https://arbital.com/p/2x4)
- [The omnipotence/omniscience test](https://arbital.com/p/2x)
- [The AI should not want to defeat your safety measures](https://arbital.com/p/nonadversarial_safety)
- [https://arbital.com/p/7fx](https://arbital.com/p/7fx) |
531f9072-3905-4905-89af-d590a1c5741f | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Worlds Where Iterative Design Fails
In most technical fields, we try designs, see what goes wrong, and iterate until it works. That’s the core iterative design loop. Humans are good at iterative design, and it works well in most fields in practice.
In worlds where AI alignment can be handled by iterative design, we probably survive. So long as we can see the problems and iterate on them, we can probably fix them, or at least avoid making them worse.
By the same reasoning: worlds where AI kills us are generally worlds where, for one reason or another, the iterative design loop fails. So, if we want to reduce X-risk, we generally need to focus on worlds where the iterative design loop fails for some reason; in worlds where it doesn’t fail, we probably don’t die anyway.
Why might the iterative design loop fail? Most readers probably know of two widely-discussed reasons:
* Fast takeoff: there will be a sudden phase shift in capabilities, and the design of whatever system first undergoes that phase shift needs to be right on the first try.
* Deceptive inner misalignment: an inner agent behaves well in order to deceive us, so we can’t tell there’s a problem just by trying stuff and looking at the system’s behavior.
… but these certainly aren’t the only reasons the iterative design loop potentially fails. This post will mostly talk about some particularly simple and robust failure modes, but I’d encourage you to think on your own about others. These are the things which kill us; they’re worth thinking about.
Basics: Hiding Problems
-----------------------
### Example/Analogy: The Software Executive
Imagine that a software company executive, concerned about the many errors coming from the software, creates a new incentive scheme: software developers get a monetary reward for changes which decrease the rate of error messages showing up on the manager’s dashboard, and get docked for changes which increase the rate of error messages.
As Tyler Cowen would say: “solve for the equilibrium”. Obvious equilibrium here: the developers stop throwing error messages when they detect a problem, and instead the software just fails silently. The customer’s experience remains the same, but the manager’s dashboard shows fewer error messages. Over time, the customer’s experience probably degrades, as more and more problems go undetected.
In the short run, the strategy may eliminate some problems, but in the long run it breaks the iterative design loop: problems are not seen, and therefore not iterated upon. The loop fails at the “see what goes wrong” step.
### Why RLHF Is Uniquely Terrible
The software executive’s strategy is the same basic idea as Reinforcement Learning from Human Feedback (RLHF). AI does something, a human looks at what happened to see if it looks good/bad, and the AI is trained on the human’s feedback. Just like the software executive’s anti-error-message compensation scheme, RLHF will probably result in *some* problems actually being fixed in the short term. But it renders the remaining problems far less visible, and therefore breaks the iterative design loop. In the context of AI, RLHF makes it far more likely that a future catastrophic error will have no warning signs, that overseers will have no idea that there’s any problem at all until it’s much too late.
Note that this issue applies even at low capability levels! Humans overlook problems all the time, some of those mistakes are systematic, and RLHF will select for places where humans systematically overlook problems; that selection pressure applies even when the neural net lacks great capabilities.
Net learns to hold hand in front of ball, so that it looks to a human observer like the ball is being grasped. Yes, this [actually happened](https://openai.com/blog/deep-reinforcement-learning-from-human-preferences/).This is the core reason why I consider RLHF *uniquely* terrible, among alignment schemes. It is the only strategy I know of which actively breaks the iterative design loop; it makes problems *less* visible rather than more.
### Generalization: Iterate Until We Don’t See Any Problems
More generally, one of the alignment failure modes I consider most likely is that an organization building AGI does see some problems in advance. But rather than addressing root causes, they try to train away the problems, and instead end up training the problems to no longer be easily noticeable.
### Does This Prove Too Much?
One counterargument: don’t real organizations create incentives like the software executive all the time? And we have not died of it.
Response: real organizations do indeed create incentives to hide problems all the time, and large organizations are notorious for hiding problems at every level. [It doesn’t even require employees consciously trying to hide things](https://www.lesswrong.com/posts/PrCmeuBPC4XLDQz8C/unconscious-economics); selection pressure suffices. Sometimes important problems become public knowledge when a disaster occurs, but that’s usually *after* the disaster. The only reason we haven’t died of it yet is that it is hard to wipe out the human species with only 20th-century human capabilities.
Less Basic: Knowing What To Look For
------------------------------------
### Example/Analogy: The Fusion Power Generator
Suppose, a few years from now, I prompt GPT-N to design a cheap, simple fusion power generator - something I could build in my garage and use to power my house. GPT-N succeeds. I build the fusion power generator, find that it works exactly as advertised, share the plans online, and soon the world has easy access to cheap, clean power.
One problem: at no point did it occur to me to ask “Can this design easily be turned into a bomb?”. Had I thought to prompt it with the question, GPT-N would have told me that the design could easily be turned into a bomb. But I didn’t think to ask, so GPT-N had no reason to mention it. With the design in wide use, it’s only a matter of time until people figure it out. And so, just like that, we live in a world where anyone can build a cheap thermonuclear warhead in their garage.
The root problem here is that I didn’t think to ask the right question; I didn’t pay attention to the right thing. An iterative design loop can sometimes help with that - empirical observation can draw our attention to previously-ignored issues. But when the failure mode does not happen in testing, the iterative design loop generally doesn’t draw our attention to it. An iterative design loop does not, in general, tell us which questions we need to ask.
Ok, but can’t we have an AI tell us what questions we need to ask? That’s trainable, right? And we can apply the iterative design loop to make AIs suggest better questions?
### Example/Analogy: Gunpowder And The Medieval Lord
Imagine a medieval lord in a war against someone with slightly more advanced technological knowledge. We're not talking modern weaponry here, just gunpowder.
To the lord, it doesn’t look like the technologist is doing anything especially dangerous; mostly the technologist looks like an alchemist or a witch doctor. The technologist digs a hole, stretches a cloth over, dumps a pile of shit on top, then runs water through the shit-pile for a while. Eventually they remove the cloth and shit, throw some coal and brimstone in the hole, and mix it all together.
From the lord’s perspective, this definitely looks weird and mysterious, and they may be somewhat worried about weird and mysterious things in general. But it’s not obviously any more dangerous than, say, a shaman running a spiritual ceremony.
It’s not until after the GIANT GODDAMN EXPLOSION that the shit-pile starts to look unusually dangerous.
Now, what helpful advice could an AI give this medieval lord?
Obviously the AI could say “the powder which comes out of that weird mysterious process is going to produce a GIANT GODDAMN EXPLOSION”. The problem is, it is not cheap for the medieval lord to *verify* the AI’s claim. Based on the lord’s knowledge, there is no a-priori reason to expect the process to produce explosives rather than something else, and the amount of background knowledge the lord would need in order to verify the theory is enormous. The lord could in-principle verify the AI’s claim experimentally, but then (a) the lord is following a complex procedure which he does not understand handed to him by a not-necessarily-friendly AI, and (b) the lord is mixing homemade explosives in his backyard. Both of these are dubious decisions at best.
So if we’re *already* confident that the AI is aligned, sure, it can tell us what to look for. But if there’s two AIs side-by-side, and one is saying “that powder will explode” and the other is saying “the shit-pile ceremony allows one to see the world from afar, perhaps to spot holes in our defenses”, the lord cannot easily see which of them is wrong. The two can argue with each other [debate-style](https://www.lesswrong.com/tag/debate-ai-safety-technique-1), and the lord still will not easily be able to see which is wrong, because he would need enormously more background knowledge to evaluate the arguments correctly. And if he can’t tell what the problem is, then the iterative design process can’t fix it.
### Example/Analogy: Leaded Gasoline
Leaded gasoline is a decent historical analogue of the Fusion Generator Problem, though less deadly. It did solve a real problem: engines ran smoother with leaded gas. The problems were nonobvious, and took a long time to manifest. The iterative design loop did not work, because we could not see the problem just by testing out leaded gas in a lab. A test would have had to run for decades, at large scale, in order to see the issue - and that’s exactly what happened.
One could reasonably object to this example as an analogy, on the basis that things which drive the human species extinct would be more obvious. Dead bodies draw attention. But what about things which make the human species more stupid or aggressive? Lead did exactly that, after all. It’s not hard to imagine a large-scale issue which makes humans stupid or aggressive to a much greater extent, but slowly over the course of years or decades, with the problems going undetected or unaddressed until too late.
That’s not intended to be a highly probable story; there’s too much specific detail. The point is that, even if the proximate cause of extinction is obvious, the factors which make that proximate cause possible may not be. A gradual path to extinction is a real possibility. When problems only manifest on long timescales, the iterative design process is bad at fixing them.
### Meta Example/Analogy: [Expertise](https://www.lesswrong.com/s/xEFeCwk3pdYdeG2rL/p/YABJKJ3v97k9sbxwg) and Gell-Mann Amnesia
If you don’t know a fair bit about software engineering, you won’t be able to distinguish good from bad software engineers.
(Assuming stats from a couple years ago are still representative, at least half my readers can probably confirm this from experience. On the other hand, last time I brought this up, one commenter said something along the lines of “Can’t we test whether the code works without knowing anything about programming?”. Would any software engineers like to explain in the comments why that’s not the only key question to ask?)
Similarly, consider [Gell-Mann Amnesia](https://www.epsilontheory.com/gell-mann-amnesia/):
> You open the newspaper to an article on some subject you know well. In Murray’s case, physics. In mine, show business. You read the article and see the journalist has absolutely no understanding of either the facts or the issues. Often, the article is so wrong it actually presents the story backward—reversing cause and effect. I call these the “wet streets cause rain” stories. Paper’s full of them.
>
> In any case, you read with exasperation or amusement the multiple errors in a story, and then turn the page to national or international affairs, and read as if the rest of the newspaper was somehow more accurate about Palestine than the baloney you just read. You turn the page, and forget what you know.
>
>
I think there’s a similar effect for expertise: software engineers realize that those outside their field have difficulty distinguishing good from bad software engineers, but often fail to generalize this to the insight that non-experts in *most* fields have difficulty distinguishing good from bad practitioners. There are of course some [general-purpose tricks](https://www.lesswrong.com/s/oLGCcbnvabyibnG9d) (and they are particularly useful expertise to have), but they only get you so far.
The difficulty of distinguishing good from bad experts breaks the iterative design loop *at a meta level*. We realize that we might not be asking the right questions, our object-level design loop might not suffice, so we go consult some experts. But then how do we iterate on our experts? How do we find better experts, or create better experts? Again, there are some general-purpose tricks available, but they’re limited. In general, if we cannot see when there’s a problem with our expert-choice, we cannot iterate to fix that problem.
More Fundamental: Getting What We Measure
-----------------------------------------
I’m just going to directly quote [Paul’s post](https://www.lesswrong.com/posts/HBxe6wdjxK239zajf/more-realistic-tales-of-doom#Part_I__You_get_what_you_measure) on this one:
> If I want to convince Bob to vote for Alice, I can experiment with many different persuasion strategies and see which ones work. Or I can build good predictive models of Bob’s behavior and then search for actions that will lead him to vote for Alice. These are powerful techniques for achieving any goal that can be easily measured over short time periods.
>
> But if I want to help Bob figure out whether he *should* vote for Alice - whether voting for Alice would ultimately help create the kind of society he wants - that can’t be done by trial and error. To solve such tasks we need to understand what we are doing and why it will yield good outcomes. We still need to use data in order to improve over time, but we need to understand how to update on new data in order to improve.
>
> Some examples of easy-to-measure vs. hard-to-measure goals:
>
> * Persuading me, vs. helping me figure out what’s true. (Thanks to Wei Dai for making this example crisp.)
> * Reducing my feeling of uncertainty, vs. increasing my knowledge about the world.
> * Improving my reported life satisfaction, vs. actually helping me live a good life.
> * Reducing reported crimes, vs. actually preventing crime.
> * Increasing my wealth on paper, vs. increasing my effective control over resources.
>
If I want to help Bob figure out whether he *should* vote for Alice, that can’t be done by trial and error. That really gets at the heart of why the iterative design loop is unlikely to suffice for alignment, even though it works so well in so many other fields. In other fields, we usually have a pretty good idea of what we want. In alignment, figuring out what we want is itself a central problem. Trial and error doesn’t suffice for figuring out what we want.
So what happens if we rely on trial and error to figure out what we want? More from [Paul’s post](https://www.lesswrong.com/posts/HBxe6wdjxK239zajf/more-realistic-tales-of-doom#Part_I__You_get_what_you_measure):
> We will try to harness this power by constructing proxies for what we care about, but over time those proxies will come apart:
>
> * Corporations will deliver value to consumers as measured by profit. Eventually this mostly means manipulating consumers, capturing regulators, extortion and theft.
> * Investors will “own” shares of increasingly profitable corporations, and will sometimes try to use their profits to affect the world. Eventually instead of actually having an impact they will be surrounded by advisors who manipulate them into thinking they’ve had an impact.
> * Law enforcement will drive down complaints and increase reported sense of security. Eventually this will be driven by creating a false sense of security, hiding information about law enforcement failures, suppressing complaints, and coercing and manipulating citizens.
> * Legislation may be optimized to seem like it is addressing real problems and helping constituents. Eventually that will be achieved by undermining our ability to actually perceive problems and constructing increasingly convincing narratives about where the world is going and what’s important.
>
> For a while we will be able to overcome these problems by recognizing them, improving the proxies, and imposing ad-hoc restrictions that avoid manipulation or abuse. But as the system becomes more complex, that job itself becomes too challenging for human reasoning to solve directly and requires its own trial and error, and at the meta-level the process continues to pursue some easily measured objective (potentially over longer timescales). Eventually large-scale attempts to fix the problem are themselves opposed by the collective optimization of millions of optimizers pursuing simple goals.
>
> As this world goes off the rails, there may not be any discrete point where consensus recognizes that things have gone off the rails.
>
> [...]
>
> We might describe the result as “going out with a whimper.” Human reasoning gradually stops being able to compete with sophisticated, systematized manipulation and deception which is continuously improving by trial and error; human control over levers of power gradually becomes less and less effective; we ultimately lose any real ability to influence our society’s trajectory.
>
>
Summary & Takeaways
-------------------
In worlds where the iterative design loop works for alignment, we probably survive AGI. So, if we want to improve humanity’s chances of survival, we should mostly focus on worlds where, for one reason or another, the iterative design loop fails. Fast takeoff and deceptive inner misalignment are two widely-talked-about potential failure modes, but they’re not the only ones. I wouldn’t consider either of them among the most *robust* ways in which the design loop fails, although they are among the most obviously and immediately *dangerous* failures.
Among the most basic robust design loop failures is problem-hiding. It happens all the time in the real world, and in practice we tend to not find out about the hidden problems until *after* a disaster occurs. This is why RLHF is such a uniquely terrible strategy: unlike most other alignment schemes, it makes problems less visible rather than more visible. If we can’t see the problem, we can’t iterate on it.
A more complicated and less legible class of design loop failures is not knowing what to look for. We might just not ask the right questions (as in the fusion power generator example), we might not even have enough background knowledge to recognize the right questions when they’re pointed out (as in the medieval lord example), it might take a very long time to get feedback on the key problems (as in the leaded gasoline example), and at a meta level we might not have the expertise to distinguish real experts from non-experts when seeking advice (as in Gell-Mann Amnesia).
Finally, we talked about Paul’s “You Get What You Measure” scenario. As Paul put it: “If I want to help Bob figure out whether he *should* vote for Alice - whether voting for Alice would ultimately help create the kind of society he wants - that can’t be done by trial and error.” That really captures the core reason why an iterative design loop is likely to fail for alignment, despite working so well in so many other fields: in other fields, we usually know what we want and are trying to get it. In alignment, figuring out what we want is itself a central problem, and the iterative design loop does not suffice for figuring out what we want. |
28d1bdf9-bb80-4b65-9206-b278d0857148 | StampyAI/alignment-research-dataset/alignmentforum | Alignment Forum | Techniques for enhancing human feedback
*By Ajeya Cotra*
Training powerful models to maximize simple metrics(such as quarterly profits) could be risky.Sufficiently intelligent models could discover strategies for maximizing these metrics in perverse and unintended ways. For example, the easiest way to maximize profits may turn out to involve stealing money, manipulating whoever keeps records into reporting unattainably high profits, capturing regulators of the industry to be allowed to ship shoddy products or avoid taxes, etc. More generally, the most effective path to maximizing a simple metric may involve acquiring enough power to tamper directly with whatever instruments or sensors are used to evaluate the metric, effectively deceiving and disempowering humans to do so.
It seems significantly safer if powerful models could be trained using something like [human feedback](https://openai.com/blog/learning-to-summarize-with-human-feedback/)*,* where human evaluators inspect a model’s action and rate how good that action is likely to be all-things-considered, and the model is trained to take actions that humans would rate highly. Human feedback could potentially disincentivize some obviously-perverse strategies like “blatantly stealing money to maximize profits,” and incentivize practices which could help maintain or improve human control like “explaining why a proposed action will be beneficial.”
However, human feedback isn’t fully adequate for supervising powerful models, especially if they take actions that are too complex for humans to understand. For example, even if blatant forms of theft are disincentivized, a sufficiently intelligent model trained with human feedback may still e.g. participate in various abstruse and complicated financial contracts which effectively constitute theft. On the other hand, if human evaluators simply penalize any action they don’t understand, the models they train would be significantly less valuable than they could have been and may be outcompeted by models trained with outcome metrics like profit.
We are interested in ways to enhance or improve upon human feedback, so that humans can provide adequate feedback even in domains where models are more capable or knowledgable than humans, without resorting to training on outcome metrics. For example, projects in this space could explore questions like:
* How could human evaluators that are unfamiliar with a certain subject (like computer science or economics) effectively give feedback that incentivizes a model to accurately explain things about the subject?
* How could human evaluators effectively provide feedback to an RL agent acting in a virtual environment that is partially occluded from the evaluators, or operates based on internal dynamics that the evaluators don’t understand?
* How could human evaluators effectively give feedback that incentivizes a model to accurately translate between English and a foreign language that the evaluators don’t understand?
* How would a model behave if it’s trained using a combination of an outcome metric and a human feedback signal? What happens if the outcome metric (e.g. “getting the most money in a negotiation game”) incentivizes doing something undesired (e.g. “lying”) that the human evaluators are unable to detect consistently?
We are seeking proposals for projects aiming to develop and test strategies for improving on the performance that could be achieved with naive human feedback in settings like these. Potential strategies that could be explored include:
* Training helper models that help human raters make better-informed evaluations of the model they are evaluating, e.g. [by highlighting flaws or contradictions in the other model’s statements](https://arxiv.org/pdf/1805.00899.pdf), or fetching relevant information from the internet.
* Breaking down the evaluation of a complex task into [smaller sub-questions](https://arxiv.org/abs/1810.08575) that human raters have an easier time answering with confidence, improving the quality and reliability of their overall evaluation.
* Training on a curriculum of human evaluations given after different amounts of thoughts and reflection (e.g. 1 minute vs 10 minutes vs 1 hour), and testing whether the model robustly generalizes to human evaluations after a longer period of time (e.g. 5 hours).
* Developing automated quality control checks for human raters’ judgments which throw out judgments that have an elevated probability of being incorrect, or boost the weight of judgments that have an elevated probability of being correct.
We are especially interested in proposals that focus on fine-tuning GPT-3 or another large generative language model, proposals that aim to use non-expert human raters to effectively evaluate a task that involves some subject matter expertise (e.g. legal advice, medical advice, programming, physics questions, etc), and proposals that systematically study how different techniques for enhancing human feedback may change the way performance scales as a function of the amount of human labor put into providing feedback. |
a7b3770b-3a4e-4f6c-8157-3b02ecdaf2f7 | trentmkelly/LessWrong-43k | LessWrong | Inefficient Doesn’t Mean Indifferent
Many people, including Bryan Caplan and Robin Hanson, use the following form of argument a lot. It could be considered the central principle of the (excellent) The Elephant in the Brain. It goes something like:
1. People say they want X, and they do Y to get it.
2. If people did C, they would get X, and the price of C is cheap!
3. Therefore, people really value X at less than the price of C, so they don’t really care much about X.
There’s something very perverse going on here. We’re using people trying to get X in an inefficient way as evidence they don’t care about X, rather than as evidence that people aren’t efficient.
The trick is, there’s a lot of assumptions hidden in the above logic. In practice, they rarely hold outside of simple cases (e.g. consumption goods).
The motivating example was Bryan Caplan using this one in The Case Against Education:
1. People say they want smart employees, and look at school records to get it.
2. If people gave out IQ tests, they would get smart employees, and testing is cheap!
3. Therefore, people don’t really value smart employees.
In that case, I agree. Employers (most often) don’t want smart employees beyond a threshold requirement. But local validity is vital, and you can’t do that.
There are lots of reasons why one might not want to do C.
As a minimal first step, people have to believe that strategy C would work. A recent example of Robin Hanson using this technique, that violates that requirement, from How Best Help Distant Future?, could be summarized this way:
1. People say they want to help the future, and lobby for policies they think help.
2. If people saved money to help the far future, which they almost never do, they could help more, and since you get real returns from it, it’s really cheap!
3. Therefore, people don’t much care about the far future.
In that case, I strongly disagree. People rightfully do not have faith that saving money now to help the far future will result in the far future be |
1c501698-da15-4705-8c02-d9aebc9c10e6 | trentmkelly/LessWrong-43k | LessWrong | Humans get different counterfactuals
A putative new idea for AI control; index here.
A lot of my ideas rely upon taking counterfactuals of events that have tiny ε probabilities of ever happening. This introduces some extra risks, mainly centred around human behaviour. For the counterfactual methods to work, we need the alternate world to be sensible, without things going crazy due to human action.
For instance, imagine that X defines an event where an AI would get turned on, with ¬X (of probability ε) corresponding to the AI failing to get turned on. There are two risks here: the first is that humans would react by saying "wow, an event of probability of ε actually happened; all our models must be wrong! Let's go crazy! Yay!". The second is that humans react by saying: "well, that was embarrassing - let's turn the AI on anyway."
To avoid this issue, imagine the following setup:
The "ON" signal first goes through an event Y, which has 99% chance of letting it through, then to the event X, which (as before) has a 1-ε chance of letting it through. The setup is designed so that humans cannot distinguish between ¬Y (the signal gets blocked at the first stage) and ¬X (the signal gets blocked at the second stage). This only needs to fool humans, not the AI itself.
The AI defines counterfactuals, as before, by looking at ¬X (possibly conditioning on Y versus ¬Y, if this is needed). Everything proceeds as previously from its perspective.
From the human perspective, however, the ¬X world is not distinguishable from the ¬Y one. Given (¬Y or ¬X), humans would conclude that ¬Y is the much more likely option:
P(¬Y|¬Y or ¬X)≈1-100ε.
So the ¬X counterfactual world (for the AI) is one where humans behave as if they were in the ¬Y world.
And ¬Y has one chance in a hundred of happening, which is unlikely, but not enough for humans to assume that their whole model of reality is wrong. Also, this is sufficiently likely that humans would give serious thought as to what to do in the ¬Y case, maybe arranging various |
d2735498-8232-4494-8f57-c814593617ce | StampyAI/alignment-research-dataset/aisafety.info | AI Safety Info | What is Eliciting Latent Knowledge (ELK)?
[Eliciting Latent Knowledge](https://www.lesswrong.com/tag/eliciting-latent-knowledge-elk) (ELK) is an open problem in AI safety. It asks how we can access information that an AI model "knows", but which its output doesn't include or make clear by default.
The problem statement from [the original report](https://docs.google.com/document/d/1WwsnJQstPq91_Yh-Ch2XRL8H_EpsnjrC1dwZXR37PC8/edit#heading=h.jrzi4atzacns) by the [Alignment Research Center (ARC)](/?state=85EK&question=What%20is%20the%20Alignment%20Research%20Center%20(ARC)'s%20research%20strategy%3F):
“Suppose we train a model to predict what the future will look like according to cameras and other sensors. We then use planning algorithms to find a sequence of actions that lead to predicted futures that look good to us.
But some action sequences could tamper with the cameras so they show happy humans regardless of what’s really happening. More generally, some futures look great on camera but are actually catastrophically bad.
In these cases, the prediction model "knows" facts (like "the camera was tampered with") that are not visible on camera but would change our evaluation of the predicted future if we learned them. **How can we train this model to report its latent knowledge of off-screen events?”**
The hope is that a successful method for ELK would make it easier to tell if an AI is safe, or in what ways it might be unsafe.
ARC's original report includes a description of ARC's first-pass [research strategy](https://docs.google.com/document/d/1WwsnJQstPq91_Yh-Ch2XRL8H_EpsnjrC1dwZXR37PC8/edit#heading=h.a0wkk7prmy4t) for solving ELK using a form of [red teaming](/?state=8FJZ&question=How%20is%20red%20teaming%20used%20in%20AI%20alignment%3F). The report goes on to describe a number of approaches to ELK that ARC considered and difficulties with each of them.
A summary of both ARC's original ELK report and the results of the [2022 ELK proposal contest](https://www.alignmentforum.org/posts/QEYWkRoCn4fZxXQAY/prizes-for-elk-proposals) is available [here](https://www.alignmentforum.org/posts/rxoBY9CMkqDsHt25t/eliciting-latent-knowledge-elk-distillation-summary).
|
65b1afc2-6439-4270-aa2a-75b9c2a5e1d6 | StampyAI/alignment-research-dataset/arxiv | Arxiv | Convergent Learning: Do different neural networks learn the same representations?
1 Introduction
---------------
Many recent studies have focused on understanding deep neural networks from both a theoretical perspective
(Arora et al., [2014](#bib.bib1); Neyshabur & Panigrahy, [2013](#bib.bib14); Montavon et al., [2011](#bib.bib13); Paul & Venkatasubramanian, [2014](#bib.bib16); Goodfellow et al., [2014](#bib.bib6))
and from an empirical perspective
(Erhan et al., [2009](#bib.bib5); Eigen et al., [2013](#bib.bib4); Szegedy et al., [2013](#bib.bib19); Simonyan et al., [2013](#bib.bib18); Zeiler & Fergus, [2014](#bib.bib23); Nguyen et al., [2014](#bib.bib15); Yosinski et al., [2014](#bib.bib21); Mahendran & Vedaldi, [2014](#bib.bib12); Yosinski et al., [2015](#bib.bib22); Zhou et al., [2014](#bib.bib24)).
In this paper we continue this trajectory toward attaining a deeper understanding of neural net training
by proposing a new approach.
We begin by noting that modern deep neural networks (DNNs) exhibit an interesting phenomenon:
networks trained starting at different random initializations frequently converge to solutions with similar performance (see Dauphin et al. ([2014](#bib.bib2)) and Section [2](#S2 "2 Experimental Setup ‣ Convergent Learning: Do different neural networks learn the same representations?") below).
Such similar performance by different networks raises the question of to what extent the learned internal representations differ:
Do the networks learn radically different sets of features that happen to perform similarly, or do they exhibit *convergent learning*, meaning that their learned feature representations are largely the same?
This paper makes a first attempt at asking and answering these questions. Any improved understanding of what neural networks learn should improve our ability to design better architectures, learning algorithms, and hyperparameters, ultimately enabling more capable models. For instance, distributed data-parallel neural network training is more complicated than distributed data-parallel training of convex models because periodic direct averaging of model parameters is not an effective strategy: perhaps solving a neuron correspondence problem before averaging would mitigate the need for constant synchronization. As another example, if networks converge to diverse solutions, then perhaps additional performance improvements are possible via training multiple models and then using model compilation techniques to realize the resulting ensemble in a single model.
In this paper, we investigate the similarities and differences between the representations learned by
neural networks with the same architecture trained from different random initializations.
We employ an architecture derived from AlexNet
(Krizhevsky et al., [2012](#bib.bib9))
and train multiple networks on the ImageNet dataset
(Deng et al., [2009](#bib.bib3))
(details in Section [2](#S2 "2 Experimental Setup ‣ Convergent Learning: Do different neural networks learn the same representations?")). We then compare the representations learned
across different networks. We demonstrate the effectiveness of this method by both visually and quantitatively showing that the features learned by some neuron clusters in one network can be quite similar to those learned by neuron clusters in an independently trained neural network.
Our specific contributions are asking and shedding light on the following questions:
1. 1.
By defining a measure of similarity between units222Note that we use the words “filters”, “channels”, “neurons”, and “units” interchangeably to mean channels for a convolutional layer or individual units in a fully connected layer. in different neural networks,
can we come up with a permutation for the units of one network to bring it into a one-to-one alignment
with the units of another network trained on the same task? Is this matching or alignment close, because features learned by one network are learned nearly identically somewhere on the same layer of the second network, or is the approach ill-fated, because the representations of each network are unique?
(Answer: a core representation is shared, but some rare features are learned in one network but not another; see Section [3](#S3 "3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?")).
2. 2.
Are the above one-to-one alignment results robust with respect to different measures of neuron similarity?
(Answer: yes, under both linear correlation and estimated mutual information metrics; see Section [3.2](#S3.SS2 "3.2 Alignment via Mutual Information ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?")).
3. 3.
To the extent that an accurate one-to-one neuron alignment is
not possible, is it simply because one network’s representation
space is a rotated version333Or, more generally, a space that
is an affine transformation of the first network’s representation
space. of another’s? If so, can we find and characterize these
rotations?
(Answers: by learning a sparse weight LASSO model to predict one
representation from only a few units of the other, we can see that the transform from
one space to the other can be possibly decoupled into transforms between
small subspaces; see Section [4](#S4 "4 Relaxing the One-to-One Constraint to Find Sparse, Few-to-One Mappings ‣ Convergent Learning: Do different neural networks learn the same representations?")).
4. 4.
Can we further cluster groups of neurons from one network with a
similar group from another network? (Answer: yes. To approximately
match clusters, we adopt a spectral clustering algorithm
that enables many-to-many mappings to be found between networks. See
Section [S.3](#A3 "Appendix S.3 Does Relaxing the One-to-One Constraint to Find Many-to-Many Groupings Reveal More Similarities Between what Different Networks Learn? ‣ Convergent Learning: Do different neural networks learn the same representations?")).
5. 5.
For two neurons detecting similar patterns, are the activation statistics similar as well?
(Answer: mostly, but with some differences; see Section [S.4](#A4 "Appendix S.4 Comparing Average Neural Activations within and between Networks ‣ Convergent Learning: Do different neural networks learn the same representations?")).
2 Experimental Setup
---------------------
All networks in this study follow the basic architecture laid out by Krizhevsky et al. ([2012](#bib.bib9)), with parameters learned in five convolutional layers (𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 – 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5) followed by three fully connected layers (𝖿𝖼𝟨𝖿𝖼𝟨\mathsf{fc6}sansserif\_fc6 – 𝖿𝖼𝟪𝖿𝖼𝟪\mathsf{fc8}sansserif\_fc8). The structure is modified slightly in two ways. First, Krizhevsky et al. ([2012](#bib.bib9)) employed limited connectivity between certain pairs of layers to enable splitting the model across two GPUs.444In Krizhevsky et al. ([2012](#bib.bib9)) the 𝖼𝗈𝗇𝗏𝟤𝖼𝗈𝗇𝗏𝟤\mathsf{conv2}sansserif\_conv2, 𝖼𝗈𝗇𝗏𝟦𝖼𝗈𝗇𝗏𝟦\mathsf{conv4}sansserif\_conv4, and 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5 layers were only connected to half of the preceding layer’s channels.
Here we remove this artificial group structure and allow all channels on each layer to connect to all channels on the preceding layer, as we wish to study only the group structure, if any, that arises naturally, not that which is created by architectural choices. Second, we place the local response normalization layers after the pooling layers following the defaults released with the Caffe framework, which does not significantly impact performance (Jia et al., [2014](#bib.bib8)).
Networks are trained using Caffe on the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) 2012 dataset (Deng et al., [2009](#bib.bib3)). Further details and the complete code necessary to reproduce these experiments is available at <https://github.com/yixuanli/convergent_learning>.
We trained four networks in the above manner using four different random initializations. We refer to these as 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1, 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2, 𝖭𝖾𝗍𝟥𝖭𝖾𝗍𝟥\mathsf{Net3}sansserif\_Net3, and 𝖭𝖾𝗍𝟦𝖭𝖾𝗍𝟦\mathsf{Net4}sansserif\_Net4. The four networks perform very similarly on the validation set, achieving top-1 accuracies of
58.65%, 58.73%, 58.79%, and 58.84%, which are similar to the top-1 performance of 59.3% reported in the original study (Krizhevsky et al., [2012](#bib.bib9)).
We then aggregate certain statistics of the activations within the networks. Given a network 𝖭𝖾𝗍*n*𝖭𝖾𝗍*n*\mathsf{Net\emph{n}}sansserif\_Net n trained in this manner,
the scalar random variable Xl,i(n)subscriptsuperscript𝑋𝑛𝑙𝑖X^{(n)}\_{l,i}italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT denotes the series of activation values produced over the entire ILSVRC validation dataset
by unit
i𝑖iitalic\_i on layer l∈{𝖼𝗈𝗇𝗏𝟣,𝖼𝗈𝗇𝗏𝟤,𝖼𝗈𝗇𝗏𝟥,𝖼𝗈𝗇𝗏𝟦,𝖼𝗈𝗇𝗏𝟧,𝖿𝖼𝟨,𝖿𝖼𝟩}𝑙𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟤𝖼𝗈𝗇𝗏𝟥𝖼𝗈𝗇𝗏𝟦𝖼𝗈𝗇𝗏𝟧𝖿𝖼𝟨𝖿𝖼𝟩l\in\{\mathsf{conv1},\mathsf{conv2},\mathsf{conv3},\mathsf{conv4},\mathsf{conv5},\mathsf{fc6},\mathsf{fc7}\}italic\_l ∈ { sansserif\_conv1 , sansserif\_conv2 , sansserif\_conv3 , sansserif\_conv4 , sansserif\_conv5 , sansserif\_fc6 , sansserif\_fc7 }.555
For the fully connected layers, the random variable Xl,i(n)subscriptsuperscript𝑋𝑛𝑙𝑖X^{(n)}\_{l,i}italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT has one specific value for each input image; for the convolutional layers, the value of Xl,i(n)subscriptsuperscript𝑋𝑛𝑙𝑖X^{(n)}\_{l,i}italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT takes on different values at each spatial position. In other words, to sample an Xl,i(n)subscriptsuperscript𝑋𝑛𝑙𝑖X^{(n)}\_{l,i}italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT for an FC layer, we pick a random image from the validation set; to sample Xl,i(n)subscriptsuperscript𝑋𝑛𝑙𝑖X^{(n)}\_{l,i}italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT for a conv layer, we sample a random image and a random position within the conv layer. We collect the following statistics by aggregating over the validation set (and in the case of convolutional layers also over spatial positions):
| | | | | |
| --- | --- | --- | --- | --- |
| | Mean:μl,i(n)Mean:subscriptsuperscript𝜇𝑛𝑙𝑖\displaystyle\textrm{Mean:}\hskip 20.00003pt\mu^{(n)}\_{l,i}Mean: italic\_μ start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT | =\displaystyle== | 𝔼[Xl,i(n)]𝔼delimited-[]subscriptsuperscript𝑋𝑛𝑙𝑖\displaystyle\mathbb{E}[X^{(n)}\_{l,i}]blackboard\_E [ italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT ] | |
| | Standard deviation:σl,i(n)Standard deviation:subscriptsuperscript𝜎𝑛𝑙𝑖\displaystyle\textrm{Standard deviation:}\hskip 20.00003pt\sigma^{(n)}\_{l,i}Standard deviation: italic\_σ start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT | =\displaystyle== | (𝔼[(Xl,i(n)−μl,i(n))2])\displaystyle\sqrt{(}\mathbb{E}[(X^{(n)}\_{l,i}-\mu^{(n)}\_{l,i})^{2}])square-root start\_ARG ( end\_ARG blackboard\_E [ ( italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT - italic\_μ start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] ) | |
| | Within-net correlation:cl,i,j(n)Within-net correlation:subscriptsuperscript𝑐𝑛𝑙𝑖𝑗\displaystyle\textrm{Within-net correlation:}\hskip 20.00003ptc^{(n)}\_{l,i,j}Within-net correlation: italic\_c start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i , italic\_j end\_POSTSUBSCRIPT | =\displaystyle== | 𝔼[(Xl,i(n)−μl,i(n))(Xl,j(n)−μl,j(n))]/σl,i(n)σl,j(n)𝔼delimited-[]subscriptsuperscript𝑋𝑛𝑙𝑖subscriptsuperscript𝜇𝑛𝑙𝑖subscriptsuperscript𝑋𝑛𝑙𝑗subscriptsuperscript𝜇𝑛𝑙𝑗subscriptsuperscript𝜎𝑛𝑙𝑖subscriptsuperscript𝜎𝑛𝑙𝑗\displaystyle\mathbb{E}[(X^{(n)}\_{l,i}-\mu^{(n)}\_{l,i})(X^{(n)}\_{l,j}-\mu^{(n)}\_{l,j})]/\sigma^{(n)}\_{l,i}\sigma^{(n)}\_{l,j}blackboard\_E [ ( italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT - italic\_μ start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT ) ( italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_j end\_POSTSUBSCRIPT - italic\_μ start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_j end\_POSTSUBSCRIPT ) ] / italic\_σ start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT italic\_σ start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_j end\_POSTSUBSCRIPT | |
| | Between-net correlation:cl,i,j(n,m)Between-net correlation:subscriptsuperscript𝑐𝑛𝑚𝑙𝑖𝑗\displaystyle\textrm{Between-net correlation:}\hskip 13.00005ptc^{(n,m)}\_{l,i,j}Between-net correlation: italic\_c start\_POSTSUPERSCRIPT ( italic\_n , italic\_m ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i , italic\_j end\_POSTSUBSCRIPT | =\displaystyle== | 𝔼[(Xl,i(n)−μl,i(n))(Xl,j(m)−μl,j(m))]/σl,i(n)σl,j(m)𝔼delimited-[]subscriptsuperscript𝑋𝑛𝑙𝑖subscriptsuperscript𝜇𝑛𝑙𝑖subscriptsuperscript𝑋𝑚𝑙𝑗subscriptsuperscript𝜇𝑚𝑙𝑗subscriptsuperscript𝜎𝑛𝑙𝑖subscriptsuperscript𝜎𝑚𝑙𝑗\displaystyle\mathbb{E}[(X^{(n)}\_{l,i}-\mu^{(n)}\_{l,i})(X^{(m)}\_{l,j}-\mu^{(m)}\_{l,j})]/\sigma^{(n)}\_{l,i}\sigma^{(m)}\_{l,j}blackboard\_E [ ( italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT - italic\_μ start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT ) ( italic\_X start\_POSTSUPERSCRIPT ( italic\_m ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_j end\_POSTSUBSCRIPT - italic\_μ start\_POSTSUPERSCRIPT ( italic\_m ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_j end\_POSTSUBSCRIPT ) ] / italic\_σ start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT italic\_σ start\_POSTSUPERSCRIPT ( italic\_m ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_j end\_POSTSUBSCRIPT | |
Intuitively, we compute the mean and standard deviation of the activation of each unit in the network over the validation set. For convolutional layers, we compute the mean and standard deviation of each channel. The mean and standard deviation for a given network and layer is a vector with length equal to the number of channels (for convolutional layers) or units (for fully connected layers).666
For reference, the number of channels for 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 to 𝖿𝖼𝟪𝖿𝖼𝟪\mathsf{fc8}sansserif\_fc8 is given by: 𝒮={96,256,384,384,256,4096,4096,1000}𝒮96256384384256409640961000\mathcal{S}=\{96,256,384,384,256,4096,4096,1000\}caligraphic\_S = { 96 , 256 , 384 , 384 , 256 , 4096 , 4096 , 1000 }. The corresponding size of the correlation matrix in each layer is: {s2|∀s∈𝒮}conditional-setsuperscript𝑠2for-all𝑠𝒮\{s^{2}~{}|~{}\forall s\in\mathcal{S}\}{ italic\_s start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT | ∀ italic\_s ∈ caligraphic\_S }. Furthermore, the spatial extents of each channel in each convolutional layer is given by: {𝖼𝗈𝗇𝗏𝟣:55×55,𝖼𝗈𝗇𝗏𝟤:27×27,𝖼𝗈𝗇𝗏𝟥:13×13,𝖼𝗈𝗇𝗏𝟦:13×13,𝖼𝗈𝗇𝗏𝟧:13×13}conditional-set𝖼𝗈𝗇𝗏𝟣:5555𝖼𝗈𝗇𝗏𝟤
2727𝖼𝗈𝗇𝗏𝟥:1313𝖼𝗈𝗇𝗏𝟦:1313𝖼𝗈𝗇𝗏𝟧:1313\{\mathsf{conv1}:55\times 55,\mathsf{conv2}:27\times 27,\mathsf{conv3}:13\times 13,\mathsf{conv4}:13\times 13,\mathsf{conv5}:13\times 13\}{ sansserif\_conv1 : 55 × 55 , sansserif\_conv2 : 27 × 27 , sansserif\_conv3 : 13 × 13 , sansserif\_conv4 : 13 × 13 , sansserif\_conv5 : 13 × 13 }
The within-net correlation values for each layer can be considered as a symmetric square matrix with side length equal to the number of units in that layer (e.g. a 96×96969696\times 9696 × 96 matrix for 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 as in Figure [1](#S2.F1 "Figure 1 ‣ 2 Experimental Setup ‣ Convergent Learning: Do different neural networks learn the same representations?")a,b). For a pair of networks, the between-net correlation values also form a square matrix, which in this case is not symmetric (Figure [1](#S2.F1 "Figure 1 ‣ 2 Experimental Setup ‣ Convergent Learning: Do different neural networks learn the same representations?")c,d).
(a) (b) (c) (d)
Figure 1: Correlation matrices for the 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 layer, displayed as images with minimum value at black and maximum at white. (a,b) Within-net correlation matrices for 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 and 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2, respectively. (c) Between-net correlation for 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 vs. 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2. (d) Between-net correlation for 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 vs. a version of 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 that has been permuted to approximate 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1’s feature order. The partially white diagonal of this final matrix shows the extent to which the alignment is successful; see Figure [3.1](#S3.SS1 "3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?") for a plot of the values along this diagonal and further discussion.
We use these correlation values as a way of measuring how related the activations of one unit are to another unit, either within the network or between networks. We use correlation to measure similarity because it is independent of the scale of the activations of units.
Within-net correlation quantifies the similarity between two neurons in the same network; whereas the between-net correlation matrix quantifies the similarity of two neurons from different neural networks.
Note that the units compared are always on the same layer on the network; we do not compare units between different layers.
In the Supplementary Information (see Figure [S1](#A1.F1 "Figure S1 ‣ Appendix S.1 Activation Values: High Correlation vs. Low Correlation ‣ Convergent Learning: Do different neural networks learn the same representations?")),
we plot the activation values for several example high correlation and low correlation pairs of units from 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 and 𝖼𝗈𝗇𝗏𝟤𝖼𝗈𝗇𝗏𝟤\mathsf{conv2}sansserif\_conv2 layers; the simplicity of the distribution of values suggests that the correlation measurement is an adequate indicator of the similarity between two neurons. To further confirm this suspicion, we also tested with a full estimate of the mutual information between units and found it to yield similar results to correlation (see Section [3.2](#S3.SS2 "3.2 Alignment via Mutual Information ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?")).
3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks?
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We would like to investigate the similarities and differences between multiple training runs of same network
architecture. Due to symmetries in the architecture and weight initialization procedures, for any given parameter
vector that is found, one could create many equivalent solutions simply by permuting the unit orders within a layer (and permuting
the outgoing weights accordingly). Thus, as a first step toward analyzing the similarities and differences
between different networks, we ask the following question: if we allow ourselves to permute the units of one network,
to what extent can we bring it into alignment with another?
To do so requires finding equivalent or nearly-equivalent units across networks, and for this task we adopt the magnitude independent measures of correlation and mutual information. We primarily give results with the simpler, computationally faster correlation measure (Section [3.1](#S3.SS1 "3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?")), but then confirm the mutual information measure provides qualitatively similar results (Section [3.2](#S3.SS2 "3.2 Alignment via Mutual Information ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?")).
###
3.1 Alignment via Correlation
As discussed in Section [2](#S2 "2 Experimental Setup ‣ Convergent Learning: Do different neural networks learn the same representations?"), we compute within-net and between-net unit correlations.
Figure [1](#S2.F1 "Figure 1 ‣ 2 Experimental Setup ‣ Convergent Learning: Do different neural networks learn the same representations?") shows the within-net correlation values computed between units on a network and other units on the same network (panels a,b) as well as the between-net correlations between two different networks (panel c).
We find matching units between a pair of networks — here 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 and 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 — in two ways. In the first approach, for each unit in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1, we find the unit in 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 with maximum correlation to it, which is the max along each row of Figure [1](#S2.F1 "Figure 1 ‣ 2 Experimental Setup ‣ Convergent Learning: Do different neural networks learn the same representations?")c.
This type of assignment is known as a bipartite *semi-matching* in graph theory (Lawler, [1976](#bib.bib10)), and we adopt the same nomenclature here. This procedure can result in multiple units of 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 being paired with the same unit in 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2. Figure [2](#S3.F2 "Figure 2 ‣ 3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?") shows the eight highest correlation matched features and eight lowest correlation matched features using the semi-matching approach (corresponding to the leftmost eight and rightmost eight points in Figure [3.1](#S3.SS1 "3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?")). To visualize the functionality each unit, we plot the image patch from the validation set that causes the highest activation for that unit. For all the layers shown, the most correlated filters (on the left) reveal that there are nearly perfect counterparts in each network, whereas the low-correlation filters (on the right) reveal that there are many features learned by one network that are unique and thus have no corollary in the other network.
Figure 2: With assignments chosen by semi-matching, the eight best (highest correlation, left) and eight worst (lowest correlation, right) matched features between 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 and 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 for the 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 – 𝖼𝗈𝗇𝗏𝟥𝖼𝗈𝗇𝗏𝟥\mathsf{conv3}sansserif\_conv3 layers. For all layers visualized, (1) the most correlated filters are near perfect matches, showing that many similar features are learned by independently trained neural networks, and (2) the least correlated features show that many features are learned by one network and are not learned by the other network, at least not by a single neuron in the other network. The results for the 𝖼𝗈𝗇𝗏𝟦𝖼𝗈𝗇𝗏𝟦\mathsf{conv4}sansserif\_conv4 and 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5 layers can be found in the Supplementary Material (see Figure [S2](#A2.F2 "Figure S2 ‣ Appendix S.2 Additional Matching Results ‣ Convergent Learning: Do different neural networks learn the same representations?")).
An alternative approach is to find the one-to-one assignment between units in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 and 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 without replacement, such that every unit in each network is paired with a unique
unit in the other network. This more common approach is known as bipartite *matching*.777Note that the *semi-matching* is “row-wise greedy” and will always have equal or better sum of correlation than the *matching*, which maximizes the same objective but must also satisfy global constraints. A matching that maximizes the sum
of the chosen correlation values may be found efficiently
via the Hopcroft-Karp algorithm (Hopcroft & Karp, [1973](#bib.bib7)) after turning the between-net correlation matrix into a weighted bipartite graph.
Figure [1](#S2.F1 "Figure 1 ‣ 2 Experimental Setup ‣ Convergent Learning: Do different neural networks learn the same representations?")c shows an example between-net correlation matrix;
the max weighted matching can be thought of as a path through the matrix such that each row and each column are selected exactly once, and the sum of elements along the path is maximized.
Once such a path is found, we can permute the units of 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 to bring it into the best possible alignment with 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1, so that the first channel of 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 approximately matches (has high correlation with) the first channel of 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1, the second channels of each also approximately match, and so on.
The correlation matrix of 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 with the permuted version of 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 is shown in Figure [1](#S2.F1 "Figure 1 ‣ 2 Experimental Setup ‣ Convergent Learning: Do different neural networks learn the same representations?")d. Whereas the diagonal of the self correlation matrices are exactly one, the diagonal of the permuted between-net correlation matrix contains values that are generally less than one. Note that the diagonal of the permuted between-net correlation matrix is bright (close to white) in many places, which shows that for many units in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 it is possible to find a unique, highly correlated unit in 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2.
Figure [3.1](#S3.SS1 "3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?") shows a comparison of assignments produced by the semi-matching and matching methods for the 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 layer (Figure [S3](#A2.F3 "Figure S3 ‣ Appendix S.2 Additional Matching Results ‣ Convergent Learning: Do different neural networks learn the same representations?") shows results for other layers).
Insights into the differing representations learned can be gained from both assignment methods.
The first conclusion is that for most units, particularly those with the higher semi-matching and matching correlations (Figure [3.1](#S3.SS1 "3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?"), left), the semi-matching and matching assignments coincide, revealing that for many units a one-to-one assignment is possible.
Both methods reveal that the average correlation for one-to-one alignments varies from layer to layer (Figure [3.1](#S3.SS1 "3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?")), with the highest matches in the 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 and 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5 layers, but worse matches in between. This pattern implies
that the path from a relatively matchable 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 representation to 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5 representation passes
through an intermediate middle region where matching is more difficult, suggesting that what is learned by different networks on 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 and 𝖼𝗈𝗇𝗏𝟤𝖼𝗈𝗇𝗏𝟤\mathsf{conv2}sansserif\_conv2 is more convergent than 𝖼𝗈𝗇𝗏𝟥𝖼𝗈𝗇𝗏𝟥\mathsf{conv3}sansserif\_conv3, 𝖼𝗈𝗇𝗏𝟦𝖼𝗈𝗇𝗏𝟦\mathsf{conv4}sansserif\_conv4 and 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5. This result may be related to previously
observed greater complexity in the intermediate layers as measured through the lens of optimization difficulty (Yosinski et al., [2014](#bib.bib21)).
Next, we can see that where the semi-matching and matching differ, the matching is often much worse.One hypothesis for why this occurs
is that the two networks learn different numbers of units to span certain subspaces. For example, 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 might learn a representation that uses six filters to span a subspace of human faces, but 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 learns to span the same subspace with five filters. With unique matching, five out of the six filters from 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 may be matched to their nearest counterpart in 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2, but the sixth 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 unit will be left without a counterpart and will end up paired with an almost unrelated filter.
Finally, with reference to Figure [3.1](#S3.SS1 "3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?") (but similarly observable in Figure [S3](#A2.F3 "Figure S3 ‣ Appendix S.2 Additional Matching Results ‣ Convergent Learning: Do different neural networks learn the same representations?") for other layers),
another salient observation is that the correlation of the semi-matching falls significantly from the best-matched unit (correlations near 1) to the lowest-matched (correlations near 0.3).
This indicates that some filters in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 can be paired up with filters in 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 with high correlation, but other filters in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 and 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 are network-specific and have no high-correlation pairing in the alternate network, implying that those filters are rare and not always learned.
This holds across the 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 – 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5 layers.
Figure 3: Correlations between paired 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 units in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 and 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2. Pairings are made via semi-matching (light green), which allows the same unit in 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 to be matched with multiple units in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1, or matching (dark green), which forces a unique 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 neuron to be paired with each 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 neuron. Units are sorted by their semi-matching values. See text for discussion.
![[Uncaptioned image]](/html/1511.07543/assets/x3.png)
![[Uncaptioned image]](/html/1511.07543/assets/x4.png)
Figure 3: Correlations between paired 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 units in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 and 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2. Pairings are made via semi-matching (light green), which allows the same unit in 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 to be matched with multiple units in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1, or matching (dark green), which forces a unique 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 neuron to be paired with each 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 neuron. Units are sorted by their semi-matching values. See text for discussion.
Figure 4: Average correlations between paired 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 units in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 and 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2. Both semi-matching (light green) and matching (dark green) methods suggest that features learned in different networks are most convergent on 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 and least convergent on 𝖼𝗈𝗇𝗏𝟦𝖼𝗈𝗇𝗏𝟦\mathsf{conv4}sansserif\_conv4.
###
3.2 Alignment via Mutual Information
Because correlation is a relatively simple mathematical metric that may miss some forms of statistical dependence,
we also performed one-to-one alignments of neurons by measuring the mutual information between them.
Mutual information measures how much knowledge one gains about one variable by knowing the value of another.
Formally, the mutual information of the two random variables Xl,i(n)subscriptsuperscript𝑋𝑛𝑙𝑖X^{(n)}\_{l,i}italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPTand Xl,j(m)subscriptsuperscript𝑋𝑚𝑙𝑗X^{(m)}\_{l,j}italic\_X start\_POSTSUPERSCRIPT ( italic\_m ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_j end\_POSTSUBSCRIPT representing the activation of the i𝑖iitalic\_i-th neuron in 𝖭𝖾𝗍*n*𝖭𝖾𝗍*n*\mathsf{Net\emph{n}}sansserif\_Net n and the j𝑗jitalic\_j-th neuron in 𝖭𝖾𝗍*m*𝖭𝖾𝗍*m*\mathsf{Net\emph{m}}sansserif\_Net m, is defined as:
| | | |
| --- | --- | --- |
| | I(Xl,i(n);Xl,j(m))=∑a∈Xl,i(n)∑b∈Xl,j(m)p(a,b)log(p(a,b)p(a)p(b)),𝐼subscriptsuperscript𝑋𝑛𝑙𝑖subscriptsuperscript𝑋𝑚𝑙𝑗subscript𝑎subscriptsuperscript𝑋𝑛𝑙𝑖subscript𝑏subscriptsuperscript𝑋𝑚𝑙𝑗𝑝𝑎𝑏𝑝𝑎𝑏𝑝𝑎𝑝𝑏I\Big{(}X^{(n)}\_{l,i};X^{(m)}\_{l,j}\Big{)}=\sum\_{a\in X^{(n)}\_{l,i}}\sum\_{b\in X^{(m)}\_{l,j}}p(a,b)\log\Big{(}\frac{p(a,b)}{p(a)p(b)}\Big{)},italic\_I ( italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT ; italic\_X start\_POSTSUPERSCRIPT ( italic\_m ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_j end\_POSTSUBSCRIPT ) = ∑ start\_POSTSUBSCRIPT italic\_a ∈ italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_b ∈ italic\_X start\_POSTSUPERSCRIPT ( italic\_m ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_j end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT italic\_p ( italic\_a , italic\_b ) roman\_log ( divide start\_ARG italic\_p ( italic\_a , italic\_b ) end\_ARG start\_ARG italic\_p ( italic\_a ) italic\_p ( italic\_b ) end\_ARG ) , | |
where p(a,b)𝑝𝑎𝑏p(a,b)italic\_p ( italic\_a , italic\_b ) is the joint probability distribution of Xl,i(n)subscriptsuperscript𝑋𝑛𝑙𝑖X^{(n)}\_{l,i}italic\_X start\_POSTSUPERSCRIPT ( italic\_n ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_i end\_POSTSUBSCRIPT and Xl,j(m)subscriptsuperscript𝑋𝑚𝑙𝑗X^{(m)}\_{l,j}italic\_X start\_POSTSUPERSCRIPT ( italic\_m ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_l , italic\_j end\_POSTSUBSCRIPT, and p(a)𝑝𝑎p(a)italic\_p ( italic\_a ) and p(b)𝑝𝑏p(b)italic\_p ( italic\_b ) are their marginal probability distributions, respectively. The within-net mutual information matrix and between-net mutual information matrix have the same shapes as their equivalent correlation matrices.
We apply the same matching technique described in Section [3.1](#S3.SS1 "3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?") to the between-net mutual information matrix,888The mutual information between each pair of neurons is estimated using 1D and 2D histograms of paired activation values over 60,000 random activation samples. We discretize the activation value distribution into percentile bins along each dimension, each of which captures 5% of the marginal distribution mass. We also add a special bin with range (−inf,10−6]infimumsuperscript106(-\inf,10^{-6}]( - roman\_inf , 10 start\_POSTSUPERSCRIPT - 6 end\_POSTSUPERSCRIPT ] in order to capture the significant mass around 0.
and compare the highest and lowest mutual information matches (Figure [S4](#A2.F4 "Figure S4 ‣ Appendix S.2 Additional Matching Results ‣ Convergent Learning: Do different neural networks learn the same representations?")) to those obtained via correlation (Figure [2](#S3.F2 "Figure 2 ‣ 3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?")). The results are qualitatively the same. For example, seven out of eight best matched pairs in the 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 layer stay the same. These results suggest that correlation is an adequate measurement of the similarity between two neurons, and that switching to a mutual information metric would not qualitatively change the correlation-based conclusions presented above.
4 Relaxing the One-to-One Constraint to Find Sparse, Few-to-One Mappings
-------------------------------------------------------------------------
The preceding section showed that, while some neurons have a one-to-one match in another network, for other neurons no one-to-one match exists (with correlation above
some modest threshold). For example, 17% of 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 neurons in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 have no match in 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 with a correlation above 0.50.50.50.5 (Figure [3.1](#S3.SS1 "3.1 Alignment via Correlation ‣ 3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?")); this number rises to 37% for 𝖼𝗈𝗇𝗏𝟤𝖼𝗈𝗇𝗏𝟤\mathsf{conv2}sansserif\_conv2, 63% for 𝖼𝗈𝗇𝗏𝟥𝖼𝗈𝗇𝗏𝟥\mathsf{conv3}sansserif\_conv3, and 92% for 𝖼𝗈𝗇𝗏𝟦𝖼𝗈𝗇𝗏𝟦\mathsf{conv4}sansserif\_conv4, before dropping to 75% for 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5 (see Figure [S3](#A2.F3 "Figure S3 ‣ Appendix S.2 Additional Matching Results ‣ Convergent Learning: Do different neural networks learn the same representations?")).
These numbers indicate that, particularly for intermediate layers, a simple one-to-one mapping
is not a sufficient model to predict the activations of some neurons in one network given the activations of neurons in another network (even with the same architecture trained on the same task).
That result could either be because the representations are unique (i.e. not convergent), or because the best possible one-to-one mapping is insufficient to tell the complete story of how one representation is related
to another.
We can think of a one-to-one mapping as a model that predicts
activations in the second network by multiplying the activations of the first by a permutation matrix — a square matrix constrained such that each row and each column contain a single one and the rest zeros.
Can we do better if we learn a model without this constraint?
We can relax this one-to-one constraint to various degrees by learning
a *mapping layer* with an L1 penalty (known as a LASSO model,
(Tibshirani, [1996](#bib.bib20))), where
stronger penalties will lead to sparser (more few-to-one or
one-to-one) mappings. This sparsity pressure can be varied from quite
strong (encouraging a mostly one-to-one mapping) all the way to zero,
which encourages the learned linear model to be dense.
More
specifically, to predict one layer’s representation from another, we
learn a single mapping layer from one to the other (similar to the “stitching layer” in Lenc & Vedaldi ([2015](#bib.bib11))). In the case of
convolutional layers, this mapping layer is a convolutional layer
with 1×1111\times 11 × 1 kernel size and number of output channels equal to the
number of input channels. The mapping layer’s parameters can be considered as a square weight matrix with side length equal to the number of units in the layer; the layer learns to predict any unit in one network via a linear weighted sum of any number of units in the other. The model and resulting square weight
matrices are shown in Figure [5](#S4.F5 "Figure 5 ‣ 4 Relaxing the One-to-One Constraint to Find Sparse, Few-to-One Mappings ‣ Convergent Learning: Do different neural networks learn the same representations?"). This layer is
then trained to minimize the sum of squared prediction errors plus
an L1 penalty, the strength of which is varied.999Both representations
(input and target) are taken after the relu is applied. Before
training, each channel is normalized to have mean zero and standard
deviation 1/N1𝑁1/\sqrt{N}1 / square-root start\_ARG italic\_N end\_ARG, where N𝑁Nitalic\_N is the number of dimensions of the
representation at that layer (e.g. N=55⋅55⋅96=290400𝑁⋅555596290400N=55\cdot 55\cdot 96=290400italic\_N = 55 ⋅ 55 ⋅ 96 = 290400
for 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1). This normalization has two effects.
First, the
channels in a layer are all given equal importance, without which the channels
with large activation values (see Figure [S10](#A4.F10 "Figure S10 ‣ Appendix S.4 Comparing Average Neural Activations within and between Networks ‣ Convergent Learning: Do different neural networks learn the same representations?")) dominate
the cost and are predicted well at the expense of less
active channels, a solution which provides little information about
the less active channels.
Second, the representation at any layer
for a single image becomes approximately unit length, making the
initial cost about the same on all layers and allowing the same
learning rate and SGD momentum hyperparameters to be used for all
layers. It also makes the effect of specific L1 multipliers approximately commensurate and
allows for rough comparison of prediction performance between layers, because the scale is constant.
| | |
| --- | --- |
| | Sparse Prediction Loss (after 4,500 iterations) |
| | decay 0 | decay 10−5superscript10510^{-5}10 start\_POSTSUPERSCRIPT - 5 end\_POSTSUPERSCRIPT | decay 10−4superscript10410^{-4}10 start\_POSTSUPERSCRIPT - 4 end\_POSTSUPERSCRIPT | decay 10−3superscript10310^{-3}10 start\_POSTSUPERSCRIPT - 3 end\_POSTSUPERSCRIPT | decay 10−2superscript10210^{-2}10 start\_POSTSUPERSCRIPT - 2 end\_POSTSUPERSCRIPT | decay 10−1superscript10110^{-1}10 start\_POSTSUPERSCRIPT - 1 end\_POSTSUPERSCRIPT |
| 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 | 0.170 | 0.169 | 0.162 | 0.172 | 0.484 | 0.517 |
| 𝖼𝗈𝗇𝗏𝟤𝖼𝗈𝗇𝗏𝟤\mathsf{conv2}sansserif\_conv2 | 0.372 | 0.368 | 0.337 | 0.392 | 0.518 | 0.514 |
| 𝖼𝗈𝗇𝗏𝟥𝖼𝗈𝗇𝗏𝟥\mathsf{conv3}sansserif\_conv3 | 0.434 | 0.427 | 0.383 | 0.462 | 0.497 | 0.496 |
| 𝖼𝗈𝗇𝗏𝟦𝖼𝗈𝗇𝗏𝟦\mathsf{conv4}sansserif\_conv4 | 0.478 | 0.470 | 0.423 | 0.477 | 0.489 | 0.488 |
| 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5 | 0.484 | 0.478 | 0.439 | 0.436 | 0.478 | 0.477 |
Table 1:
Average prediction error for mapping layers with varying L1 penalties (i.e. decay terms). Larger decay parameters enforce stronger sparsity in the learned weight matrix.
Notably, on 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 and 𝖼𝗈𝗇𝗏𝟤𝖼𝗈𝗇𝗏𝟤\mathsf{conv2}sansserif\_conv2, the prediction errors do not rise much
compared to the dense (decay = 0) case
with the imposition of a sparsity penalty until after an L1 penalty weight of over 10−3superscript10310^{-3}10 start\_POSTSUPERSCRIPT - 3 end\_POSTSUPERSCRIPT is used.
This region of roughly constant performance despite increasing sparsity pressure is shown in bold.
That such extreme sparsity does not hurt performance implies that each neuron in one network can be predicted by only one or a few neurons in another network. For the 𝖼𝗈𝗇𝗏𝟥𝖼𝗈𝗇𝗏𝟥\mathsf{conv3}sansserif\_conv3, 𝖼𝗈𝗇𝗏𝟦𝖼𝗈𝗇𝗏𝟦\mathsf{conv4}sansserif\_conv4, and 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5 layers, the overall error is higher, so it is difficult to draw any strong conclusions regarding those layers. The high errors could be because of the uniqueness of the learned representations, or the optimization could be learning a suboptimal mapping layer for other reasons.
Mapping layers are trained for layers 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 – 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5. The average squared prediction
errors for each are shown in Table [1](#S4.T1 "Table 1 ‣ 4 Relaxing the One-to-One Constraint to Find Sparse, Few-to-One Mappings ‣ Convergent Learning: Do different neural networks learn the same representations?") for a variety of L1 penalty weights (i.e. different decay values). For the 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 and 𝖼𝗈𝗇𝗏𝟤𝖼𝗈𝗇𝗏𝟤\mathsf{conv2}sansserif\_conv2 layers, the prediction errors do not rise with the imposition of a sparsity penalty until a penalty greater than 10−3superscript10310^{-3}10 start\_POSTSUPERSCRIPT - 3 end\_POSTSUPERSCRIPT. A sparsity penalty as high as 10−3superscript10310^{-3}10 start\_POSTSUPERSCRIPT - 3 end\_POSTSUPERSCRIPT results in mapping layer models that are nearly as accurate as their dense counterparts, but that contain mostly zero weights. Additional experiments for 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 revealed that a penalty multiplier of 10−2.6superscript102.610^{-2.6}10 start\_POSTSUPERSCRIPT - 2.6 end\_POSTSUPERSCRIPT provides a good trade-off between sparsity and accuracy, resulting in a model with sparse prediction loss 0.235, and an average of 4.7 units used to predict each unit in the target network (i.e. an average of 4.7 significantly non-zero weights). For 𝖼𝗈𝗇𝗏𝟤𝖼𝗈𝗇𝗏𝟤\mathsf{conv2}sansserif\_conv2 the
10−3superscript10310^{-3}10 start\_POSTSUPERSCRIPT - 3 end\_POSTSUPERSCRIPT multiplier worked well, producing a model with an average of 2.5 non-zero connections per predicted unit.
The mapping layers for higher layers (𝖼𝗈𝗇𝗏𝟥𝖼𝗈𝗇𝗏𝟥\mathsf{conv3}sansserif\_conv3 – 𝖼𝗈𝗇𝗏𝟧𝖼𝗈𝗇𝗏𝟧\mathsf{conv5}sansserif\_conv5) showed poor performance even without regularization, for reasons we do not yet fully understand,
so further results on those layers are not included here. Future investigation with different hyperparameters or different architectures (e.g. multiple hidden layers) could train more powerful predictive models for these layers.
The one-to-one results of Section [3](#S3 "3 Is There a One-to-One Alignment Between Features Learned by Different Neural Networks? ‣ Convergent Learning: Do different neural networks learn the same representations?") considered in combination with these results on
sparse prediction shed light on the open, long-standing debate about the extent to which learned representations are local vs. distributed :
The units that match well one-to-one suggest
the presence of a *local* code, where each of these dimensions is important enough, independent enough, or privileged enough in some other way to be relearned by different networks. Units that do not match well one-to-one, but are predicted well by a sparse model, suggest the presence, along those dimensions, of *slightly distributed* codes.
The results could have been otherwise: if all units could accurately be matched one-to-one, we would suspect a local code across the whole layer. On the other hand, if making predictions from one network to another required a dense affine transformation, then we would interpret the code as *fully distributed*, with each unit serving only as a basis vector used to span a large dimensional subspace, whose only requirement was to have large projection onto the subspace (to be useful) and small projection onto other basis vectors (to be orthogonal). The story that actually emerges is that the first two layers use a mix of a local and a slightly distributed code.
Figure [6](#S4.F6 "Figure 6 ‣ 4 Relaxing the One-to-One Constraint to Find Sparse, Few-to-One Mappings ‣ Convergent Learning: Do different neural networks learn the same representations?") shows a visualization of the learned weight matrix for 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1,
along with a permuted weight matrix that aligns units from 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 with the 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 units that most predict them.
We also show two example units in 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 and, for each, the only three units in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 that are needed to predict their activation values.
Figure 5:
A visualization of the network-to-network sparse “mapping layers” (green squares). The layers are trained independently of each other and
with an L1 weight penalty to encourage sparse weights.
These sparse prediction results suggest that small groups of units in each network span similar subspaces, but we have not yet identified or visualized the particular subspaces that are spanned. Below we present one approach to do so by using the sparse prediction matrix directly, and
in the Supplementary Section [S.3](#A3 "Appendix S.3 Does Relaxing the One-to-One Constraint to Find Many-to-Many Groupings Reveal More Similarities Between what Different Networks Learn? ‣ Convergent Learning: Do different neural networks learn the same representations?") we discuss a related approach using spectral clustering.
For a layer with s𝑠sitalic\_s channels we begin by creating a 2s×2s2𝑠2𝑠2s\times 2s2 italic\_s × 2 italic\_s block matrix B𝐵Bitalic\_B by concatenating the blocks [I,W;WT,I]𝐼𝑊superscript𝑊𝑇𝐼[I,~{}W;~{}W^{T},~{}I][ italic\_I , italic\_W ; italic\_W start\_POSTSUPERSCRIPT italic\_T end\_POSTSUPERSCRIPT , italic\_I ] where I𝐼Iitalic\_I is the s×s𝑠𝑠s\times sitalic\_s × italic\_s identity matrix and W𝑊Witalic\_W is the learned weight matrix. Then
we use Hierarchical Agglomerative Clustering (HAC), as implemented in Scikit-learn (Pedregosa et al., [2011](#bib.bib17))
to recursively cluster individual units into clusters, and those clusters into larger clusters, until all have been joined
into one cluster. The HAC algorithm as adapted to this application works in the following way:
(1) For all pairs of units, we find the biggest
off-diagonal value in B, i.e. the largest prediction weight;
(2) We pair those two units together into a cluster and consider it as a single entity for the remainder of the algorithm;
(3) We start again from step 2 using the same process (still looking for the
biggest value), but whenever we need to compare unit ↔↔\leftrightarrow↔ cluster or cluster ↔↔\leftrightarrow↔ cluster, we use the average unit ↔↔\leftrightarrow↔ unit weight over all cross-cluster pairs;(4) Eventually the process terminates when there is a single cluster.101010For example,
in the 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 layer with 96969696 channels, this happens after 96⋅2−1=191⋅962119196\cdot 2-1=19196 ⋅ 2 - 1 = 191 steps.
The resulting clustering can be interpreted as a tree with units as leaves,
clusters as intermediate nodes, and the single final cluster as the
root node. In Figure [7](#S4.F7 "Figure 7 ‣ 4 Relaxing the One-to-One Constraint to Find Sparse, Few-to-One Mappings ‣ Convergent Learning: Do different neural networks learn the same representations?") we plot the B𝐵Bitalic\_B
matrix with rows and columns permuted together in the order leaves are
encountered when traversing the tree, and intermediate nodes are
overlaid as lines joining their subordinate units or clusters. For
clarity, we color the diagonal pixels (which all have value one) with
green or red if the associated unit came from 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 or
𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2, respectively.
Plotted in this way, structure is readily visible: Most parents of leaf clusters (the smallest merges shown as two blue lines of length two covering a 2×2222\times 22 × 2 region) contain one unit from 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 and one from 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2. These units can be considered most predictive of each other.111111Note that the upper right corner of B𝐵Bitalic\_B is W=W1→2𝑊subscript𝑊→12W=W\_{1\rightarrow 2}italic\_W = italic\_W start\_POSTSUBSCRIPT 1 → 2 end\_POSTSUBSCRIPT, the matrix predicting 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 →→\rightarrow→ 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2, and the lower left is just the transpose W1→2Tsubscriptsuperscript𝑊𝑇→12W^{T}\_{1\rightarrow 2}italic\_W start\_POSTSUPERSCRIPT italic\_T end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 1 → 2 end\_POSTSUBSCRIPT. The corners could instead be W1→2subscript𝑊→12W\_{1\rightarrow 2}italic\_W start\_POSTSUBSCRIPT 1 → 2 end\_POSTSUBSCRIPT and W2→1subscript𝑊→21W\_{2\rightarrow 1}italic\_W start\_POSTSUBSCRIPT 2 → 1 end\_POSTSUBSCRIPT, respectively. Slightly higher level clusters show small subspaces, comprised of multiple units from each network, where multiple units from one network are useful for predicting activations from the other network (see the example zoomed regions on the right side of Figure [7](#S4.F7 "Figure 7 ‣ 4 Relaxing the One-to-One Constraint to Find Sparse, Few-to-One Mappings ‣ Convergent Learning: Do different neural networks learn the same representations?")).
The HAC method employs greedy merges, which could in some cases be suboptimal. In the Supplementary Section [S.3](#A3 "Appendix S.3 Does Relaxing the One-to-One Constraint to Find Many-to-Many Groupings Reveal More Similarities Between what Different Networks Learn? ‣ Convergent Learning: Do different neural networks learn the same representations?") we explore a related method that is less greedy, but operates on the denser correlation matrix instead.
Future work investigating or developing other methods for analyzing the structure of the sparse prediction matrices may
shed further light on the shared, learned subspaces of independently trained networks.
Figure 6:
(left)
The learned mapping layer from 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 to 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 for the 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 layer.
(right)
Two example units (bottom)
in 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2 — which correspond to the same colored rows in the left weight matrix — together with, for each, the only three units in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 that are needed
to predict their activation.
To fully visualize the functionality each unit, we plot the top 9 image patches from the validation set that causes the highest activation for that unit (“maxim”), as well as its corresponding “deconv” visualization introduced by Zeiler & Fergus ([2014](#bib.bib23)). We also show the actual weight associated with each unit in 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 in the sparse prediction matrix.
Figure 7:
The results of the Hierarchical Agglomerative Clustering (HAC) algorithm described in Section [4](#S4 "4 Relaxing the One-to-One Constraint to Find Sparse, Few-to-One Mappings ‣ Convergent Learning: Do different neural networks learn the same representations?") on the 𝖼𝗈𝗇𝗏𝟣𝖼𝗈𝗇𝗏𝟣\mathsf{conv1}sansserif\_conv1 layer.
Left: The B𝐵Bitalic\_B matrix permuted by the tree-traversal order of leaf nodes. Pixels on the diagonal are leaf nodes and represent original units of either network (green for 𝖭𝖾𝗍𝟣𝖭𝖾𝗍𝟣\mathsf{Net1}sansserif\_Net1 and red for 𝖭𝖾𝗍𝟤𝖭𝖾𝗍𝟤\mathsf{Net2}sansserif\_Net2). The brighter the gray pixel is, the larger the weight is in the matrix. See text for a complete interpretation. Right: Two zoomed in regions of the diagonal, showing two different four-dimensional subspaces spanned by four units in each network. The top 9 and bottom 9 images correspond to the maxim and deconv visualizations, respectively.
Best viewed digitally with zoom.
5 Conclusions
--------------
We have demonstrated a method for quantifying the feature similarity between different, independently trained deep neural networks. We show how insights may be gain by approximately aligning different neural networks on a feature or subspace level by blending three approaches: a bipartite matching that makes one-to-one assignments between neurons, a sparse prediction and clustering approach that finds one-to-many mappings, and a spectral clustering approach that finds many-to-many mappings. Our main findings include:
1. 1.
Some features are learned reliably in multiple networks, yet other features are not consistently learned.
2. 2.
Units learn to span low-dimensional subspaces and, while these subspaces are common to multiple networks, the specific basis vectors learned are not.
3. 3.
The representation codes are a mix between a local (single unit) code and slightly, but not fully, distributed codes across multiple units.
4. 4.
The average activation values of neurons vary considerably within a network, yet the mean activation values across different networks converge to an almost identical distribution.
6 Future Work
--------------
The findings in this paper open up new future research directions, for example: (1) Model compression. How would removing low-correlation, rare filters affect performance? (2) Optimizing ensemble formation. The results show some features (and subspaces) are shared between independently trained DNNs, and some are not. This suggests testing how feature correlation among different DNNs in an ensemble affects ensemble performance. For example, the “shared cores” of multiple networks could be deduplicated, but the unique features in the tails of their feature sets could be kept. (3) Similarly, one could (a) post-hoc assemble ensembles with greater diversity, or even (b) directly encourage ensemble feature diversity during training. (4) Certain visualization techniques, e.g., deconv (Zeiler & Fergus, [2014](#bib.bib23)), DeepVis (Yosinski et al., [2015](#bib.bib22)), have revealed neurons with multiple functions (e.g. detectors that fire for wheels and faces). The proposed matching methods could reveal more about why these arise. Are these units consistently learned because they are helpful or are they just noisy, imperfect features found in local optima?
(5) Model combination: can multiple models be combined by concatenating their features, deleting those with high overlap, and then fine-tuning?
(6) Apply the analysis to networks with different architectures — for example, networks with different numbers of layers or different layer sizes — or networks trained on different subsets of the training data.
(7) Study the correlations of features in the same network, but across training iterations, which could show whether some features are trained early and not changed much later, versus perhaps others being changed in the later stages of fine-tuning. This could lead to complementary
insights on learning dynamics to those reported by Erhan et al. ([2009](#bib.bib5)).
(8) Study whether particular regularization or optimization strategies (e.g., dropout, ordered dropout, path SGD, etc.) increase or decrease the convergent properties of the representations to facilitate different goals (more convergent would be better for data-parallel training, and less convergent would be better for ensemble formation and compilation).
#### Acknowledgments
The authors are grateful to the NASA Space Technology Research Fellowship (JY) for funding, Wendy Shang for conversations
and initial ideas, Yoshua Bengio for modeling suggestions, and Anh Nguyen and Kilian Weinberger for helpful comments and edits.
This work was supported in part by US Army Research Office W911NF-14-1-0477, NSF grant 1527232.
Jeff Clune was supported by an NSF CAREER award (CAREER: 1453549) and a hardware donation from the NVIDIA Corporation. |
3b698521-4f1d-4c29-a6c1-38c6bbe6a301 | trentmkelly/LessWrong-43k | LessWrong | Linkpost: A Contra AI FOOM Reading List
This is a linkpost to a list of skeptical takes on AI FOOM. I haven't read them all and probably disagree with some of them, but it's valuable to put these arguments in one place. |
94ff3639-376c-4524-969a-8984a2c4d24b | trentmkelly/LessWrong-43k | LessWrong | Stop pressing the Try Harder button
(This is a post from a daily blogging experiment I did at neelnanda.io, which I thought might also fit the tastes of LessWrong. This is very much in the spirit of Trying to Try)
I recently had a productivity coaching session, and at the end we agreed on a few actions points that I’d do by the next session. But, come the next session, these had completely slipped my mind. These suggestions were good ideas, and I had no issue with implementing them, the problem was just that they completely slipped my mind! (We then spent the second session debugging my ability to actually follow action points, and this was pretty successful!)
I think the error I made there is a really common one when planning, and one I observe often in myself and others. Often I’ll hear a cool book recommendation, offer to meet up with someone some time, hear about a new productivity technique, notice an example sheet deadline looming. But I consistently fail to action upon this. So this post is about what exactly went wrong, and the main solution I’ve found to this problem!
Planning, as I define it, is about ensuring that the future goes the way I currently want it to. And the error I made was that, implicitly, I was trying to make the future go the way I currently wanted it to. That by committing to do things, and wanting to them, and just applying effort, things would happen. And the end result of this was that I totally forgot about it. Or sometimes, that I vaguely remembered the commitment or idea, and felt some guilt about it, but it never felt urgent or my highest priority. And every time I thought about the task, I resolved to Try Harder, and felt a stronger sense of motivation, but this never translated into action. I call this error Pressing the Try Harder button, and it’s characterised by feelings of guilt, obligation, motivation and optimism.
This is a classic case of failing to Be Deliberate. It feels good to try hard at something, it feels important and virtuous, and it’s easy to t |
6fb117e9-87f1-433f-8f97-d6eb43d24253 | trentmkelly/LessWrong-43k | LessWrong | Experiences of Self-deception
It seems to me that self-deception can describe two different things - conscious and unconscious self-deception.
Sometimes the elephant believes something untrue all by itself without the rider ever getting a look in. The claims of elephant in the brain seem to focus on this type of unconscious self-deception.
At other times the rider is complicit in endorsing a particular known untrue belief. The elephant analyses a situation, determines what it is beneficial to believe and motivates the rider to believe this. The rider has access to information which indicates that this isn't true. If the rider brings this information to full attention then it is one of those rare occasions where he can override the elephant's desires. However the rider also has the option to push the information to the side and believe a beneficial lie. It is possible to do this well enough that the information is forgotten or completely overridden with new, inaccurate, information.
In pushing the information to the side, the rider can sometimes just never bring the information to full attention. Failing that, it can drown the information out by presenting other information (which agrees with its favoured interpretation) as loudly as possible in order to doubt/ignore/forget the information which it doesn't like.
At least, this is something I experience but I don't know whether other people do. I have a few examples where this has happened and have even experimented with allowing myself to start down the route of conscious self-deception to see what it feels like. To me it feels like cognitive dissonance (feeling hot, brain feeling "fuzzy", adrenaline kicking in) whilst the rider works on counteracting the information. I guess this would be followed by the relief of resolving said dissonance when the rider starts to believe the lie but I haven't experimented that far!
The literature appears to be understandably non-committal on whether the subjects are consciously aware of their self-deceptio |
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