### Beyond Condorcet winners: Majority Judgement

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There have been rumours on Wikipedia that Michael Balinksi passed away this Februari 4th. If this is indeed the case, then my heartfelt sympathies go out to those around him. I never knew him personally, but he was an amazing scientist and I’ve used results of his more than once. He was quite a phenomenal speaker.

Today I want to talk about a single-winner voting system that Balinksi introduced with Rida Laraki. It is called majority judgement and it is so brilliant that it almost makes me wonder what voting theorists could have been doing both before and since then.

One big concept in social choice theory is majority rule: if most people think that A is better than B, then A should be the winner. Most multi-candidate voting systems generalize this in various ways, always preserving that if candidate A would beat every other candidate B, C, etc, in a pairwise competition, then A should win the election. If candidate A satisfies this criterium, we call A a Condorcet winner. The leading wisdom in social choice theory was that any good voting rule should let the Condorcet winner win (if it exists).

According to my informal sample from the Effective Altruism community, EA’s favourite voting system seems to be approval voting, which is one of these voting systems that generalizes majority rule to multiple candidates.

The genius of majority judgement is that it moves past the Condorcet winner paradigm and considers a perspective beyond that.

To illustrate, let’s assume we’re running an election among two candidates, the red candidate and the blue candidate, and every voter gets a ballot with for each candidate the options “Amazing”, “Good”, “Mediocre” and “Terrible” to express how good of a president they think the candidate would be. Let us for simplicity assume that our voting population consists of 5 groups, the A’s, B’s, C’s, D’s and E’s, and everyone in a group votes the exact same way. The outcome of the election is in the table below.

 A B C % of population 40 20 40 Red candidate Amazing Mediocre Terrible Blue candidate Good Terrible Amazing

The red candidate wins the Condorcet-style election: 60% of the population prefers the red candidate over the blue candidate. But the blue candidate is clearly better: 80% of the population considers the blue candidate to be “Good” or better, while 60% of the population considers the red candidate to be “Mediocre” or worse.

Majority judgement is a voting rule designed to have the blue candidate win in the above election. The actual algorithm is a bit involved, but the first step is to compare the median vote: if at least 50% of the voters think the blue candidate is “Good” or better and at least 50% of the voters think that the red candidate is “Mediocre” or worse, than the blue candidate will win. If the median opinion is a tie, a more complicated tie-breaking rule is entered.1

I think the concept is very elegant, and I believe that the outcomes really would be better than with a system that elects Condorcet winners. In a talk that Balinksi gave, which I was lucky enough to attend, he pointed out another advantage of the majority judgement rule: it allows voters to express what they think of the candidates. You wouldn’t be asking anyone to “vote for the lesser evil”, everyone can keep their conscience clear. Majority judgement admits a clear way of expressing frustration with both candidates: rank both of them very badly. It also helps that the different options are given in words instead of ranking by numbers, for the latter turns out to entice voters to rate their favourite candidate 10/10 and all others 0/10.

1. The exact rule satisfies a number of optimality criteria and is the only rule to do so. For this post I want to skip the details. []

### Funding opportunity in academic publishing

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I’ve recently listened to a talk by Jean-Sébastien Caux, the founder, implementer and chairman of SciPost. It looks like a charity that might be worth funding.

Academic publishing is a complete disaster. The incentive structure is messed up. Papers must have more than a certain minimum of content*coolness, but simultaneously they shouldn’t be too long. This results in researchers sticking two unrelated papers together and calling it one thing, and cutting single papers up into multiple pieces to get published. If your result is too simple then it will get rejected so people make their papers more difficult on purpose. There is no pipeline for communicating minor improvements and comments on other people’s papers to the outside world.

Peer review might not literally be a farce but it is closer to being so than anyone is really comfortable with. Because it all happens behind closed doors, peer reviews seldom have constructive feedback in them and reviewers will most likely harm their own careers if they spend time and effort into reviewing that could be spent doing research. People submit everything to the best journals first, trying one rung lower when their paper gets rejected. The review process can take years. Reviews are all hidden from the wider public, so the only way to judge a paper’s quality if you’re not in the field is by looking at citation counts and the journal a paper appeared in.

Publishers make a lot of profit selling the research community’s own results back to them. Journals and impact factors are silly inventions that date back to the dark ages before internet existed and serve little to no use in modern times.

Enter SciPost. Imagine a love child of the free software movement, open access academic publishing, and modern discussion platform design. Submission is free. A manuscript is public from the moment one of the editors decides it is probably worth getting reviewed, with all the fancy DOI’s and what-not that you could ask for. Both the content and the platform itself are licenced under free licences. Reviews are public, either with the authors name or anonymously, which turns out to greatly improve review quality. Reviews are citable objects with DOI’s and everything. A number of reviews get invited, but anyone can submit a review if they’d like. People can post comments on reviews. Their funding comes entirely from sponsors. Their average costs per publication are under $400, way less than the article processing fees of most open access journals. They keep themselves to principles of openness way beyond the Fair Open Access Principles. Right now SciPost publishes papers in physics. They want to expand to other disciplines, but money is the major bottleneck. Over the past 3 years they’ve gotten around$250k in total funding, so the marginal gains from additional funds should be pretty good.

### 6. Astronomical waste, astronomical schmaste

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[Part of this badly written blog post has been superseded by a slightly better written forum post over on the EA forum. I might clean up the other parts in the future as well, and if so I’ll publish them at the EA forum as well.]

Previously: [1] [2] [3] [4] [5][latest].

Epistemic status: there is nothing wrong with writing your bottom line first. The purpose of this article is to get my initial thoughts on AI risk down before I start reading more about the topic, because I fear that I might unknowingly grant AI risk proponents that the implicit assumptions they’re making are true. As I procrastinated a lot on writing this post, there have been an number of articles put out that I did not read. I do not intend this document to be a conclusive argument against ai risk so much as an attempt to justify why it might be reasonable to think ai risk is not real.

Is this text too long? Click here for the summary of the argument.

In this post, I want to tackle the astronomical waste argument as used to justify AI-related existential risk prevention as an EA cause area. I will first describe the argument that people make. After that, I will discuss a number of meta-heuristics to be skeptical of it. Lastly, I want to take the astronomical waste argument face-on and describe why it is so absurdly unlikely for AI risk to be simultaneously real and preventable that the expected value of working on AIS is still not very good.

## Astronomical waste

The astronomical waste argument as most people tell it basically goes like this: the potential good that could be gotten if happy beings colonized the entire universe would be huge, so even if there is a tiny risk of space-colonization not happening, that costs a lot of value in expectation. Moreover, if we can decrease the risk by just a tiny bit, the expected utility generated is still big, so it might be a very cost-effective way to do good.

As many wise people have said before me, “Shut up and calculate.” I will be giving rough estimates without researching them a great lot, because these quantities are not that well-known to humanity either. For the duration of this post, I will be a speciesist and all-around awful person because that simplifies the estimates. Bostrom roughly estimates that colonizing the Virgo supercluster would yield $10^{38}$ human lives per century. The Virgo SC is one of about 10 million superclusters in the observable universe and we have roughly $10^{9}$ centuries left before entropy runs out, making a total of roughly $2^{180}$ potential human lives left in the universe.

I will try to argue that donating $\5000\approx\2^{13}$ to an AI risk charity today will counterfactually produce less than one life saved in expectation. To make that happen, we collect 180 bits of unlikeliness for the hypothesis that donating that sum of money to AI Safety organizations saves a lives.

You need to collect less bits if your counterfactual cause area is more cost-effective than malaria prevention. Possibly $\log_2(5000/0.20) \approx 14$ bits fewer with a charity like ALLFED.

#### On meta-uncertainty

Some of my LessWrong-reading friends would argue that it is impossible to have credence $2^{-200}$ in anything because my own thinking is fallible and I’ll make mistakes in my reasoning with probability much higher than that. I reject that assertion: if I flip 200 coins then my expected credence for most series of outcomes should inevitably be close to $2^{-200}$, because all $2^{200}$ events are mutually exclusive and their probabilities must sum up to at most $1$.

## Discounting the future (30 bits)

Inhabiting the observable universe might take a really long, and in all this time there is some probability of going extinct for reasons other than AI risk. Hence we should discount the total spoils of the universe by a decent fraction. 30 bits.

## Biases (20 bits)

I expect many EA’s to be wrong in their utility calculation, so I think I should propose mechanisms that cause so many EA’s to be wrong. Two such mechanisms are described in previous entries in this series [2] (9 bits) [3] (1 bits) and I want to describe a third one here.

When we describe how much utility could fit in the universe, our reference class for numbers is “how many X fits in the universe”, where X ranges over things like {atoms, stars, planets}. These numbers are huge, typically expressed as $10^n$ for $n \in \mathbb{N}$.

When we describe how likely certain events are, the tempting reference class is “statements of probability”, typically expressed as $ab.cdefghij... \%$. Writing things this way, it seems absurd to have your number start with more than 10 zeros.

The combination of these vastly different scales together with anchoring being a thing, makes that we should expect people to over-estimate the probability of unlikely effects and hence the expected utility of prevention measures.

I expect myself to be subject to these biases still, so I think it is appropriate to count a number of bits to counteract this bias. 20 bits.

## Counterfactual actions (-1 bit)

Nothing is effective in and of itself, effectiveness is relative to a counterfactual action. For this blog post, the counterfactuals will be working on algorithmic fairness and/or digital rights campaigning/legislation, and mainstream machine learning research and engineering. -1 bit.

## When is AI risky? (tl;dr)

This is a rough sketch of my argument. AI safety can only be an effective cause area if

1. The future of the non-extinct universe would be good.
2. The probability of an AI-related extinction event is big.
3. It is possible to find ways to decrease that probability.
4. It is feasible to impose those risk mitigation measures everywhere.
5. The AI risk problem won’t be solved by regular commercial and/or academic AI research anyway.
6. A single AI-related extinction event could affect any lifeform in the universe ever.
7. Without AI first causing a relatively minor (at most country-level) accident first.
8. Presently possible AI safety research should be an effective way of decreasing that probability.

I upper bounded the quantity in 1 by $2^{200}$ good lifes. Properties 2 and 3 are necessary for AI Safety work to be useful. Property 5 is necessary for AI safety work to have meaningful counterfactual impact. Property 6 is necessary because otherwise other happy life forms might fill the universe instead, and the stakes here on earth are nowhere near $2^{200}$. If property 7 does not hold, it might mean that people will abandon the AI project, and it would be too easy to debug risky AI’s. Property 8 is in contrast to AI safety work only really be possible after major progress from now has been made in AI capabilities research, and is hence a statement about the present day.

The basic premise of the argument is that there is an inherent tension between properties 2 up to 6 being true at once. AI risk should be big enough for properties 2 and 6 to hold, but small enough for 3 and 5 to hold. I think that this is a pretty narrow window to hit, and which would mean that AI safety is very unlikely to be an effective cause area, or at least it is not so for its potential of saving the universe from becoming paperclips. I am also highly skeptical of both 7 and 8, even assuming that 2 up to 6 hold.

## AI is fake (8 bits)

I think it is likely that we won’t be making a what we now think of as “artificial intelligence”, because current conceptions of AI are inherently mystical. Future humans might one day make something that present-day humans would recognize as AI, but the future humans won’t think of it like that. They won’t have made computers think, they would have demystified thinking to the point where they understand what it is. They won’t mystify computers, they will demystify humans. Note that this is a belief about the state of the world, while [2] is about how we think about the world. Hence, I think both deserve to earn bits separately. 5 bits.

I am not sure that intelligence is a meaningful concept outside principal component analysis. PCA is a statistical technique that gives a largest component of variation in a population independently of whether that axis of variation has an underlying cause. In particular, that might mean that superhuman intelligence cannot exist. That does not preclude thinking at superhuman speeds from existing but would still impy serious bounds on how intelligent an AI can be. 1 bit.

No matter the above, all reasonably possible computation is restricted to polynomial-time solvable problems, fixed-parameter tractable problems and whatever magic modern ILP-, MINLP-, TSP- and SAT-solvers use. This gives real upper bounds on what even the most perfect imaginable AI could do. The strength of AI would lie in enabling fast and flexible communication and automation, not in solving hard computational problems. I hereby accuse many AI-enthousiasts of forgetting this fact, and will penalize their AI-risk fantasies for it. 2 bits.

## AI x-risk is fake (31 bits)

The risks of using optimization algorithms are well-documented and practitioners have a lot of experience in how to handle such software reponsibly. This practical experience literally dates back to the invention of optimization in what is by far my favourite anecdote I’ve ever heard. Optimization practitioners are more responbile than you’d think, and with modern considerations of fairness and adversarial input they’ll only get more responsible over time. If there are things that must be paid attention to for algorithms to give good outcomes, practitioners will know about them. 3 bits.

People have been using computers to run ever more elaborate optimization algorithms pretty much since the introduction of the computer. ILP-solvers might be among the most sophisticated pieces of software in existence. And they don’t have any problems with reward hacking. Hence, reward hacking is probably only a fringe concern. 3 bits.

Debugging is a long and arduous process, both for developing software and for designing the input for the software (both the testing input and the real-world inputs). That means that the software will be run on many different inputs and computers before going in production, each an independent trial. So, if software has a tendency to give catastrophically wrong answers, it will probably already do so in an early stage of development. Such bugs probably won’t survive into production, so any accidents are purely virtual or at most on small scales. 5 bits.

Even if AI would go wrong in a bad way, it has to go really really wrong for it to be an existential thread. Like, one thing that is not ab existential thread is if an AI decided to release poison gas from every possible place in the US. That might kill everyone there, but even the poison gas factories could run indefinitely, the rest of the world could just nuke all of North America long before the whole global atmosphere is poisonous. 10 bits.

Moreover, for the cosmic endowment to be at risk, an AI catastrophy should impact every lifeform that would ever come to exist in the lightcone. That is a lot of ground to cover in a lot of detail. 10 bits.

## AI x-risk is inevitable (28 bits)

Okay, let’s condition on all the above things going wrong anyway. Is AI-induced x-risk inevitable in such a world? Probably.

• There should be a way of preventing the catastrophies. 5 bits.
• Humans should be able to discover the necessary knowledge. 3 bits.
• These countermeasures have to be universally implemented. 10 bits.
• Even against bad actors and anti-natalist terrorists. 10 bits.

## AI becomes safe anyway (15 bits)

Let’s split up the AI safety problem into two distinct subproblems. I don’t know the division in enough detail to give a definition, so I’ll describe them by association. The two categories roughly map onto the distinction from [4], and also roughly onto what LW-sphere folks call the control problem and the alignment problem.

 Capitalist’s AI problem Social democrat’s AI problem x/s-risk Cyberpunk dystopia risk Must be solved tomake money using AI Must be solved to havealgorithms producesocial good Making AI optimal Making algorithms fair Solving required forfurthering a singleentity’s values Solving required forfurthering sentientbeings’ collectivevalues. Only real if certainimplausibleassumptions are true Only real if hedonisticutilitarianism is false,or if bad actors hatehedonistic utility. Prevent the light conefrom becoming paperclips Fully Automated LuxuryGay Space Communism Specific to AGI Applies to all algorithms Fear of Skynet Fear of Moloch Beating back unknowninvaders from mindspace Beating back unthinkinglyoptimistic programmers Have AI do what we want Know what we wantalgorithms to do What AIS-focussed EAscare about What the rest of theworld cares about

I’m calling 20 bits on the capitalist’s problem getting solved by capitalists, and 15 bits on the social democrat’s problem getting solved by the rest of humanity. We’re interested in the minimum of the two. 15 bits.

## Working on AIS right now is ineffective (15 bits)

There are two separate ways of being inefficient to account for. AIS research might be ineffective right now no matter what because we lack the knowledge to do useful research, or AIS work might in general be less effective than work on for example FAT algorithms.

The first idea is justified from the viewpoint that making AI will mostly involve demystifying the nature of intelligence, versus obtaining the mystical skill of producing intelligence. Moreover it is reasonable to think given that current algorithms are not intelligent. 5 bits.1

The second idea concerns whether AI safety will be mostly a policy issue or a research issue. If it is mostly policy with just a bit of technical research, it will be more effective to practice getting algorithms regulated in the first place. We can gain practice, knowledge and reputation for example by working on FATML, and I think it likely that this is a better approach at the current moment in time. 5 bits.

Then the last concern is that AIS research is just AI capabilities research by another name. It might not be exactly true, but the fit is close. 5 bits.

## Research is expensive (13 bits)

Let’s get a sense of scale here. You might be familiar with these illustrations. If not, check them out. That tiny bit is the contribution of 4 year researcher-years. One researcher-year costs at least $50,000. The list of projects getting an ERC Starting Grant of 1.5 million euros. Compared to ambitious projects like “make AI safe”, the ERC recipient’s ambitions are tiny and highly specialised. What’s more, these are grant applications, so they are necessarily an excaggeration of what will actually happen with the money. It is not a stretch to estimate that it would cost at least$50 million to make AI safe (conditional on all of the above being such that AIS work is necessary). So a donation of \$5000 would be at most 0.0001 of the budget. 13 bits.

## Hard to estimate issues (10 bits)

I’ll aggregate these because I don’t trust myself to put individual values on each of them.

• Will the universe really get filled by conscious beings?
• Will they be happy?
• Is it better to lead a happy life than to not exist in the first place?
• Is there a moral difference between there being two identical happy universes A and B versus them being identical right up to the point where A’s contents get turned to paperclips but B continues to be happy? And how does anthropic bias factor in to this?
• Has any being ever had a net-positive life?

## Sub-1-bit issues (5 bits)

I listed all objections where I was at least 50% confident in them being obstacles. But there are probably quite a number of potential issues that I haven’t thought of because I don’t expect them to be issues with enough probability. I estimate their collective impact to count for something. 5 bits.

## Conclusion

It turns out I only managed to collect 174 bits, not the 180 bits I aimed for. I see this as weak evidence for AIS being better than malaria prevention but not better than something like ALLFED. Of course, we should keep in mind that all the numbers are made up.

Maybe you disagree with how many bits I handed out in various places, maybe you think I double-counted some bits, or maybe you think that counting bits is inherently fraught and inconsistent. I’d love to hear your thoughts via email at beth@bethzero.com, via Reddit at u/beth-zerowidthspace or at the EA forum at beth​.

1. Note that this argument is different from the previous one under the “AI is fake” heading. The previous argument is about the nature of intelligence and whether it permits AI risk existing versus not existing, this argument is about our capability to resolve AI risk now versus later. []

### 5. Are neural networks intelligent?

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I am skeptical of AI Safety (AIS) as an effective cause area, at least in the way AIS is talked about by people in the effective altruism community. However, it is also the cause area that my skills and knowledge are the best fit for contributing, so it seems worthwhile for me to think my opposition to it through.﻿

Previously: [1] [2] [3] [4][latest].

Epistemic status: this argument has more flaws than I can count. Please don’t take it seriously. [See the post-script]

Let’s answer this abstract philosophical question using high-dimensional geometry.

I’ll assume for simplicity that there is a single property called intelligence and the only variation is in how much you have of it. So no verbal intelligence vs visiual intelligence, no being better at math and than at languages, the only variation is in how much intelligence we have. Let us call this direction of variation $g$, scaled to have $\|g\| = 1$, and pretend that it is roughly the thing you get from a singular value decomposition/principal component analysis of human’s intelligence test results.

A typical neural net has many neurons. For example, VGG-19 has ~143 million parameters. Now suppose that we train a VGG-19 net to classify images. This is an optimization problem in $\mathbb{R}^{143 \text{ million}}$, and let’s call the optimal parameter setting $x$. By definition, the trained net has an intelligence of exactly the inner product $g^{\mathsf{T}}x$.1 2

The trained net is intelligent in exactly the extend that intelligence helps you recognize images. If you can recognize images more efficiently by not being intelligent, then the trained net will not be intelligent. But exactly how helpful would intelligence be in recognizing images? I’d guess that a positive amount of intelligence would be better than a negative amount, but other than that I have no clue.

As a good subjective Bayesian, I’ll hence consider the vector $\omega$ of goodness-at-recognizing-images to be chosen uniformly from the unit sphere, conditional on having non-negative intelligence, i.e., uniformly chosen from $\{\omega\in\mathbb{S}^{143\text{ million} - 1} : g^{\mathsf{T}}\omega \geq 0\}$. For this distribution, what is the expected intelligence $\mathbb{E}[g^{\mathsf{T}}x]$? Well, we know, we know that $x$ maximizes $\omega$, so if the set of allowed parameters is nice we would get $g^{\mathsf{T}}x \approx g^{\mathsf{T}}\omega \cdot \|x\|$,3 where $\|x\|$ is how good the net is at recognizing images. We can calculate this expectation and find that, up to a constant factor, $$\mathbb{E}[g^{\mathsf{T}}\omega] \approx \frac{2}{\sqrt{2e\pi(143\text{ million}-1)}}.$$

So the trained VGG-19 neural net is roughly $10^{-5}$ times as intelligent as it is good at recognizing images. Hence, it is probably not very smart.

1. Note that the projection of g into this 143 million-dimensional space might be much shorter than g itself is, that depends on the architecture of the neural net. If this projection is very short, then every parameter setting of the net is very unintelligent. By the same argument that I’m making in the rest of the post, we should expect the projection to be short, but let’s assume that the projection is long for now. []
2. I’m assuming for simplicity that everything is convex. []
3. I have to point out that this is by far the most unrealistic claim in this post. It is true if $x$ is constrained to lie in a ball, but in other cases it might be arbitrarily far off. It might be true for the phenomenon I describe in the first footnote. []

### Gaussian tail bounds

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This is all basic stuff, but at the moment it is too hard find through Google and sometimes I just want to know the answer without having to derive it on the spot. Hence, this post. I hope I’m not ruining anyone’s course’s hand-in exercises.

The standard normal distribution $N(0,1)$ has probability density function $\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. There is no way to integrate this function symbolically in a nice way, but we do at times want to (upper) bound expressions of the form $$\frac{1}{\sqrt{2\pi}}\int_x^\infty e^{-t^2/2} \mathrm{d}t.$$ How can we do this?

One way is to follow this approach. Since $t\geq x$ everywhere, we can upper bound $$\frac{1}{\sqrt{2\pi}}\int_x^\infty e^{-t^2/2} \mathrm{d}t \leq \frac{1}{\sqrt{2\pi}}\int_x^\infty \frac{t}{x} e^{-t^2/2} \mathrm{d}t = \frac{1}{x\sqrt{2\pi}}e^{-x^2/2}.$$

There is another tail bound which is a bit weaker for large $x$, but I like the proof better. We’ll give a tail bound by looking at the moment-generating function $\lambda \mapsto \mathbb{E}[e^{\lambda X}]$, where $X \sim N(0,1)$ is our normally distributed random variable. We can explicitly calculate this expectation and find $$\mathbb{E}[e^{\lambda X}] = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty e^{\lambda x - x^2/2}\mathrm{d}x = \frac{1}{\sqrt{2\pi}}e^{\lambda^2/2}\int_{-\infty}^\infty e^{-(x-\lambda)^2/2}\mathrm{d}x.$$ The last thing is just the entire Gaussian integral shifted a bit and hence $\mathbb{E}[e^{\lambda X}] = e^{\lambda^2/2}$ Now we use Chernoff’s bound (an easy corrollary of Markov’s inequality) to find $\mathbb{P}[X \geq t] \leq \mathbb{E}[e^{\lambda X}]e^{-\lambda t},$ which we can now minimize over the choice of $\lambda$, setting $\lambda=t$, and we conclude that $\mathbb{P}[X \geq t] \leq e^{-t^2/2}$.

## Multi-variate Gaussians

Now $X \in \mathbb{R}^d$ is $N(0,I_d)$ normally distributed, i.e., $X$ is a vector with iid Gaussian $N(0,1)$ entries. What tail bounds do we get on $\|X\|$? We start off with Markov’s inequality again. $$\mathbb{P}[\|X\| > t] = \mathbb{P}[e^{\lambda\|X\|^2} > e^{\lambda t^2}] \leq \frac{\mathbb{E}[e^{\lambda\|X\|^2}]}{e^{\lambda t^2}}.$$

Deriving the moment generating function $\lambda \mapsto \mathbb{E}[e^{\lambda\|X\|^2}]$ of $X^2$ is an elementary calculation. $$\int_{-\infty}^\infty e^{\lambda x^2} \cdot e^{-x^2/2} \mathrm{d}x = \int_{-\infty}^\infty e^{\frac{-x^2}{2(\sqrt{1-2/\lambda})^2}}\mathrm{d}x = \frac{\sqrt{2\pi}}{\sqrt{1-2\lambda}}.$$

The coordinates of $X$ are iid, so $\mathbb{E}[e^{\lambda\|X\|^2}] = \mathbb{E}[e^{\lambda X_1^2}]^d = (1-2\lambda)^{-d/2}.$ The minimizer is at $\lambda=(1-1/t^2)/2$, and we find, requiring $t \geq 1$ for the last inequality,$$\mathbb{P}[\|X\| > t] \leq e^{-d(t^2-2\log t - 1)/2} \leq e^{-d(t-1)^2}.$$

## Operator norm of Gaussian matrices

The operator norm or spectral norm of a $n \times n$ matrix $M$ is defined as $$\|M\| := \max_{x \in \mathbb{R}^n} \frac{\|Mx\|}{\|x\|}.$$

Now if $M$ were a matrix with every entry independently $N(0,1)$, what would the largest singular value of this random Gaussian matrix be? I’ll give an easy tail bound based on a net argument.

An $\eta$-net, $\eta > 0$, on the sphere is a subset $N \subset \mathbb{S}^{d-1}$ such that, for every point $x \in \mathbb{S}^{d-1}$ there is a net element $n \in N$ such that $\|x-n\| \leq \eta$ but every two net elements are at distance at least $\eta$ from each other. A greedy algorithm can construct an $\eta$-net, and any $\eta$-net has size at most $(4/\eta)^d$.1

Now let $N\subset \mathbb{S}^{d-1}$ be a $1/2$-net. By the above, the size of the net is bounded by $|N| \leq 8^d$.

The function $x \mapsto \|Mx\|$ is $\|M\|$-Lipschitz. Hence we can bound $$\|M\| \leq \max_{x\in\mathbb{S}^{d-1}} \min_{\omega \in N} \|M\omega\| + \|M\|\cdot\|x-\omega\| \leq \max_{x\in\mathbb{S}^{d-1}} \min_{\omega \in N} \|M\omega\| + \|M\|/2.$$ So we have now proved that $\|M\| \leq 2\max_{\omega\in N} \|M\omega\|$.

Now, as $M\omega$ is $N(0,I_d)$ normally distributed for any $\omega\in\mathbb{S}^{d-1}$, we can use the union bound over all points of $N$ and conclude that, for all $t \geq 1$, $$\mathbb{P}[\|M\| \geq 2t\sqrt{d}] \leq 8^d e^{-d(t-1)^2/2}.$$

## Maximum of $n$ Gaussians

The distribution of the maximum $\mathbb{P}[\max_{i \leq n} X_i \geq t]$ of $n$ independent identically distributed variables $X_1,\ldots,X_n \sim N(0,1)$ is, up to a constant factor, tight with the union bound $$\mathbb{P}[\max_{i \leq n} X_i \geq t] \leq ne^{-t^2/2}.$$

Hence the expected maximum is $\mathbb{E}[\max_{i \leq n} X_i] = O(\sqrt{\ln n})$.

## Average width of the simplex

Let $x_1,\dots,x_{d+1} \in \mathbb{R}^d$ be the vertices of a regular simplex such that $\|x_i\| = 1$ for all $i \in [d+1]$. If $\omega \in \mathbb{S}^{d-1}$ is chosen uniformly at random, the difference $\max_{i,j\in[d+1]} |\omega^{\mathsf{T}}(x_i-x_j|$ is called the average width of the simplex. We can bound this up to a constant factor using our knowledge of Gaussians. Let $H_t := \{y\in\mathbb{R}^d : \omega^{\mathsf{T}}y = t\}$. The $d-2$-dimensional volume of $H_t\cap \mathbb{S}^{d-1}$ is $(1-t^2)^{(d-1)/2}$ times the volume of $\mathbb{S}^{d-2}$ by Pythatoras’ theorem. Recalling that $(1+1/\lambda)^\lambda \approx e$, you can prove that the distribution of $\omega^{\mathsf{T}}x_i$ is approximately $N(0,1/\sqrt{d-1})$. The Gaussian tail bound now says that the average width of the simplex is $O(\frac{\sqrt{\ln d}}{\sqrt d}).$

1. See e.g., Jiří Matoušek, Lectures on Discrete Geometry (Springer, 2002), page 314. The proof is based on a simple packing argument where balls of radius $\eta/2$ around each net element have to fit disjointly inside the ball of radius $1+\eta/2 \leq 1$ centered at the origin. []

### 4. Who is worried about AI Risk?

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I am skeptical of AI Safety (AIS) as an effective cause area, at least in the way AIS is talked about by people in the effective altruism community. However, it is also the cause area that my skills and knowledge are the best fit for contributing, so it seems worthwhile for me to think my opposition to it through.﻿

Previously: [1] [2] [3][latest].

There are many people talking about the risks of artificial intelligence. I want to roughly split them into three groups for now, because they worry about very different issues that tend to talk past each other, confusing outsiders.

The LessWrong-aligned view seems most popular in the EA community. Examplified by the paperclip maximizer argument, LW-aligned worriers are concerned that an Artifical General Intelligence (AGI) would accomplish their objective in unforeseen ways, and as a consequence should be treated like you should treat an evil genie, except it’d be worse because it would have less understanding of basic words than philosophers have. The principles that AI should satisfy are listed by the Future of Humanity Institute. [Though I suspect at least some of the signatories to have the FATML-aligned view in mind.] A popular book on this is Superintelligence by Nick Bostrom.

Fairness, Accountability and Transparency in Machine-Learning (FATML) is a subfield of machine learning, concerned with making algorithmic decision making fair, accountable and transparent. Exemplified by Amazon’s recent recruiting debacle, FATML-aligned worries are concerned that modern algorithmic decisionmaking will exacerbate existing social, economic and legal inequalities. The princples that AI should satisfy are listed by The Public Voice, and these Google ML guidelines fit as well. [Though I suspect at least some of the signatories to have the LW-aligned view in mind.] Popular books include Weapons of Math Destruction by Cathy O’Neil, Algorithms of Oppression by Safiya Noble and Automating Inequality by Virginia Eubanks.

Other AI-related worries commonly heard in the media, that I want to separate from the previous two categories because, compared to the above categories, these issues are more about politics and less of a technical problem. Worries include killer drones, people losing their jobs because AI replaced them, and who the self-driving car should run over given the choice.

In the next couple of posts on AI-related topics, I will focus on the first two categories. My aim is to use the FATML-aligned view to compare and contrast the LW-aligned view, hopefully gaining some insight in the process. The reason I separate the views this way, is because I agree with the FATML-aligned worries and disagree with the LW-aligned worries.

### 3. Finding meaning in a perfect life

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[This badly written blog post has been superseded by a slightly better written forum post over on the EA forum.]

I am skeptical of AI Safety (AIS) as an effective cause area, at least in the way AIS is talked about by people in the effective altruism community. However, it is also the cause area that my skills and knowledge are the best fit for contributing, so it seems worthwhile for me to think my opposition to it through.﻿

Previously: [1] [2][latest].

My background makes me prone to overrate how important AI Safety is.

My fields of expertise and enjoyment are mathematics and computer science. These skills are useful for the economy and in high demand. The general public is in awe of mathematics and thinks highly of anyone who can do it well. Computer science is the closest thing we have to literal magic.

Wealth, fun, respect, power. The only thing left to desire is cosmic significance, which is exactly the sales pitch of the astronomical waste argument. It would be nice if AI-related existential risk were real, for my labour to potentially make the difference between a meaningless lifeless universe or a universe filled with happyness. It would give objective significance to my life in a way that only religion would otherwise be able to.

This is fertile ground for motivated reasoning, so it is good to be skeptical of any impulse to think AIS is as good as it is claimed to be in cost-effectiveness estimates.

### 2. How do we talk about AI?

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[This badly written blog post has been superseded by a slightly better written forum post over on the EA forum.]

I am skeptical of AI Safety (AIS) as an effective cause area, at least in the way AIS is talked about by people in the effective altruism community. However, it is also the cause area that my skills and knowledge are the best fit for contributing, so it seems worthwhile for me to think my opposition to it through.﻿

Previously: [1][latest].

All sentences are wrong, but some are useful. I think that a certain emotional salience makes us talk about AI in a way that is more wrong than necessary.

A self-driving car and a pre-driven car are the same thing, but I can feel myself thinking about the two in completely different ways.

Self-driving cars are easy to imagine: they are autonomous and you can trust the car like you trust cab drivers; they can make mistakes but probably have good intent, when they encounters an unfamiliar situation they can think about the correct way to proceed, and if something goes wrong then the car is at fault.

A pre-driven car are hard to imagine: it has to have a bunch of rules coded into it by the manufacturer and you can trust the car like you trust a bridge; it does exactly what it was built to do, but if it was built without proper testing or calculations, things will at some point go wrong. When it does, the company and engineers are at fault.

You can make these substitutions on any sentence in which a computer is ascribed agency. In the best case, “The neural network learned to recognize objects in images” becomes “The fitted model classifies images in close correspondence with the human-given labels”. In reality, that description might be too generous.

It helps to keep in mind the human component. “The YouTube algorithm shows you exactly those videos that make you spend more time on the platform” is accurate in some sense, but it completely glances over the ways in which in the algorithm does not do that. When you listen to music using YouTube’s autoplay, it isn’t hard to notice that suggestions tend to point backwards in time compared to the upload date of the video you’re watching right now, and that, apart from preventing repeats, autoplay is pretty Markovian (that is mathspeak for the algorithm not doing anything clever based on your viewing history, just “this video is best followed by that video”). Both of those properties are clearly a result from the way in which YouTube’s engineers modelled the problem they were trying to solve, I would describe YouTube’s suggestion as “The YouTube autoplay algorithm was made to link you to videos that most people watched and liked after watching the current video”.

When you rewrite AI-related statements, they tend to become more wordy. That is exactly what you would expect, but does make it unwieldy to have accurate conversations. I leave the search for catchy-but-more-accurate buzzwords as an open problem. I am particularly interested in how to translate the term “artificial general intelligence” (AGI).

### Two conflicting concepts

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Sometimes you hear a word or concept that changes how you look at the world. For me, these include speciecism and epistemic injustice.

Speciecism is analogous to racism and sexism, but for species: treating another being differently because they are of another species. Speciecism is about intent; if you eat chickens because they are chickens and not humans, that is speciecist, but if you eat chickens because you concluded from observation that they are incapable of suffering, that is not speciecist.

Epistemic injustice is when someone is wronged in their capacity as a knower. If you unjustly limit somebody’s ability to access or express knowledge, like forbidding them from learning to read or speak, that is an epistemic injustice.

I am an outspoken anti-speciecist and I think we should do what we can to prevent epistemic injustice in all forms. But some animals have learned enough language to meaningfully communicate with humans. Does that mean I should find it reprehensible that there are no schools for animals? I think I should and I think I do, but I feel hesitant to firmly claim the position.

### Typed languages, units.

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I recently picked up programming again. I used to do it a lot before I went to university, but the constant mind-numbing programming assignments quickly put me off of programming. Apart from the occasional quick bug fix for software I use myself, I haven’t done any serious coding for years.

Until recently, when I needed something coded up during my research. I decided to learn Python, and I like it. It is easy to use, the libraries are extensive and user-friendly, and ipython is a useful tool. There is just one thing that draws my ire: the weak type system. Studying math has given me an appreciation for type checking that is even stricter than most languages.

An example: my length in centimeters plus the outside temperature in °C right now equals 180. This calculation makes no sense, because the units don’t match: you can’t add centimeters to degrees Celcius. But then there’s Python, which just lets you do that.

In [1]: length = 170In [2]: temperature = 10In [3]: length + temperatureOut[3]: 180

Most bugs that stem from typos are of this sort. Those bugs are possible because the type system is too weak. If you have two loops, one iterating over i and one over j, basic unit-based type checking would probably flag any instance of i in a place where you should have typed j instead. If you intend to query A[i][j] then it should be possible to let i have row-index type and j have type-index type, making A[j][i] raise a type error.

Another example: Let $A \in \mathbb{R}^{n \times n}, x \in \mathbb{R}^n$, and we’re interested in the quantity $Ax \in \mathbb{R}^n$. If you’re like me and you can’t remember what rows and what columns are, then that doesn’t have to impact your ability to symbolically do linear algebra: the quantities $xA = A^{\mathsf{T}}x$, $Ax^{\mathsf{T}}$ and $A^{-1} x$ don’t “compile”, so any mathematician that reads it will know you typo-ed if you wrote one of those latter expressions. All operations might be matrices acting on vectors, but the matrices $A^{-1}$ and $A^{\mathsf{T}}$ fundamentally take input from different copies of $\mathbb{R}^n$ than the ones that $x$ and $x^{\mathsf{T}}$ live in. That is why matrix operations make sense even if the matrices aren’t square or symmetric: there is only one way to make sense of any operation. Even if you write it wrong in a proof, most people can see what the typo is. But then there’s Python.

In [4]: import numpy as np  In [5]: x = np.array([1,2])  In [6]: A = np.array([[3,4],[5,6]])  In [7]: np.dot(A,x) Out[7]: array([11, 17])  In [8]: np.dot(A,np.transpose(x)) Out[8]: array([11, 17])  In [9]: np.dot(x,A) Out[9]: array([13, 16])

I am like me and I can’t remember what rows and what columns are. I would like the interpreter to tell me the correct way of doing my linear algebra. At least one of the above matrix-vector-products should throw a type error. Considering the history of type systems, it is not surprising that the first languages didn’t introduce unit-based types. Nonetheless, it is a complete mystery to me why modern languages don’t type this way.