# DavidHolmes

Karma: 97
• Hi Zack,

Can you clarify something? In the picture you draw, there is a codimension-1 linear subspace separating the parameter space into two halves, with all red points to one side, and all blue points to the other. Projecting onto any 1-dimensional subspace orthogonal to this (there is a unique one through the origin) will thus yield a variable’ which cleanly separates the two points into the red and blue categories. So in the illustrated example, it looks just like a problem of bad coordinate choice.

On the other hand, one can easily have much more pathological situations; for examples, the red points could all lie inside a certain sphere, and the blue points outside it. Then no choice of linear coordinates will illustrate this, and one has to use more advanced analysis techniques to pick up on it (e.g. persistent homology).

So, to my vague question: do you have only the first situation in mind, or are you also considering the general case, but made the illustrated example extra-simple?

Perhaps this is clarified by your numerical example, I’m afraid I’ve not checked.

• Thanks for the reply, Zack.

The reason this objection doesn’t make the post completely useless...

Sorry, I hope I didn’t suggest I thought that! You make a good point about some variables being more natural in given applications. I think it’s good to keep in mind that sometimes it’s just a matter of coordinate choice, and other times the points may be separated but not in a linear way.

• Hi Stuart,

I’m working my way through your Research Agenda v0.9’ post, and am therefore going through various older posts to understand things. I wonder if I could ask some questions about the definition you propose here?

First, that be contained in for some seems not so relevant; can I just assume X, Y and Z are some manifolds ( for some )? And we are given some partial order on X, so that we can refer to being a better world’?

Then, as I understand it, your definition says the following:

Fix X, and Z. Let Y be a manifold and , . Given a local homomorphism , we say that is partially preferred to if for all , we have .

I’m not sure which inequalities should be strict, but this seems non-essential for now. On the other hand, the dependence of this definition on the choice of Y seems somewhat subtle and interesting. I will try to illustrate this in what follows.

First, let us make a new definition. Fix X, , and Z as before. Let , a two-element set equipped with the discrete topology, and let be an immersion of -manifolds. We say that is weakly partially preferred to if for all , we have .

First, it is clear that partial preference implies weak partial preference. More formally:

Claim 1: Fix X, and Z. Suppose we have a manifold Y, points , , and a local homomorphism such that is partially preferred to . Setting with the subspace topology from (i.e. discrete), and taking to be the restriction of from to , we have that is weakly partially preferred to .

Proof: obvious. $\qed$

However, the converse can fail if Z is not contractible. First, let’s prove that the concepts are equivalent for Z contractible:

Claim 2: Fix X, and Z, and assume that Z is contractible. Suppose we have a two-element set and a map making weakly partially preferred to . Then there exist a manifold Y, an injection , and a local homeomorphism whose restriction to is , making partially preferred to .

Proof: Let’s assume for simplicity of notation that X is equidimensional, say of dimension , and write for the dimension of Z. Let Y be the disjoint union of two open balls of dimension , with the inclusion of the centres of the balls. Then take an -neighbourhood of Z in X; it is diffeomorphic to since the normal bundle to Z in X is trivialisable (c.f. https://​​math.stackexchange.com/​​questions/​​857784/​​product-neighborhood-theorem-with-boundary). $\qed$

If we want examples where weak partial preference and partial preference don’t coincide, we should look for an example where Z is not contractible, and its normal bundle in X is not contractible.

Example 3: Let X be the disjoint union of two moebius bands, and let Z be a circle. Note that including Z along the centre of either band gives a submanifold whose tubular neighbourhood is not a product. Assume that is such that one component of X is preferred to the other (and is indifferent within each connected component). Then take , and to be the inclusion of the two circles along the centres of the two moebius bands, such that ends up in the preferred band. This yields a situation where is weakly partially preferred to , but the conclusion of Claim 2 fails, i.e. this cannot be extended to a partial preference for over .

What conclusion should we draw from this? To me, it suggests that the notion of partial preference is not yet quite as one would want. In the setting of Example 3, where X consists of two moebius strips, one of which is preferred to the other, then landing in the preferred strip should be preferred to landing in the un-preferred strip?! And yet the local homeomorphism from a product’ condition gets in the way. This example is obviously quite artificial, and maybe analogous things cannot occur in reality. But I’m not so happy with this as an answer, since our approaches to AI safety should be (so far as possible) robust against the flaws in our understanding of physics.

Apologies for the overly-long comment, and for the imperfect LaTeX (I’ve not used this type of form much before).

• Thanks for pointing me to this updated version :-). This seems a really neat trick for writing down a utility function that is compatible with the given preorder. I thought a bit more about when/​to what extent such a utility function will be unique, in particular if you are given not only the data of a preorder, but also some information on the strengths of the preferences. This ended up a bit too long for a comment, so I wrote a few things in outline here:

https://​​www.lesswrong.com/​​posts/​​7ncFy84ReMFW7TDG6/​​categorial-preferences-and-utility-functions

It may be quite irrelevant to what you’re aiming for here, but I thought it was maybe worth writing down just in case.

• Thanks for the comment Charlie.

If I am indifferent to a gamble with a probability of ice cream, and a probability 0.8 of chocolate cake and 0.2 of going hungry

To check I understand correctly, you mean the agent is indifferent between the gambles (probability of ice cream) and (probability 0.8 of chocolate cake, probability 0.2 of going hungry)?

If I understand correctly, you’re describing a variant of Von Neumann–Morgenstern where instead of giving preferences among all lotteries, you’re specifying a certain collection of special type of pairs of lotteries between which the agent is indifferent, together with a sign to say in which direction’ things become preferred? It seems then likely to me that the data you give can be used to reconstruct preferences between all lotteries...

If one is given information in the form you propose but only for an incomplete' set of special triples (c.f.weak preferences’ above), then one can again ask whether and in how many ways it can be extended to a complete set of preferences. It feels to me as if there is an extra ambiguity coming in with your description, for example if the set of possible outcomes has 6 elements and I am given the value of the Betterness function on two disjoint triples, then to generate a utility function I have to not only choose a translation’ between the two triples, but also a scaling. But maybe this is better/​more realistic!

. By special types’, I mean indifference between pairs of gambles of the form
(probability of A) vs (probability of B and probability of C)
for some , and possible outcomes A, B, C. Then the sign says that I prefer higher probability of B (say).

• Thanks! I like the way your optimisation problem handles non-closed cycles.

I think I’m less comfortable with how it treats disconnected components—as I understand it you just translate each separately to have centre of mass’ at 0. If one wants to get a utility function out at the end one has to make some kind of choice in this situation, and the choice you make is probably the best one, so in that sense it seems very good.

But for example it seems vulnerable to creating ‘virtual copies’ of worlds in order to shift the centre of mass and push connected components one way or the other. That was what started me thinking about including strength of preference—if one adds to your setup a bunch of virtual copies of a world between which one is almost indifferent’ then it seems it will shift the centre of mass, and thus the utility relative to come other chain. Of course, if one is actually indifferent then the ‘virtual copies’ will be collapsed to a single point in your , but if they are just extremely close then it seems it will affect the utility relative to some other chain. I’ll try to explain this more clearly in a comment to your post.

• This seems really neat, but it seems quite sensitive to how one defines the worlds under consideration, and whether one counts slightly different worlds as actually distinct. Let me try to illustrate this with an example.

Suppose we have a consisting of 7 worlds, , with preferences
and no other non-trivial preferences. Then (from the sensible case’), I think we get the following utilities:

.

Suppose now that I create two new copies , of the world which each differ by the position of a single atom, so as to give me (extremely weak!) preferences , so all the non-trivial preferences in the new are now summarised as

Then the resulting utilities are (I think):

.

In particular, before adding in these ‘trivial copies’ we had , and now we get . Is this a problem? It depends on the situation, but to me it suggests that, if using this approach, one needs to be careful in how the worlds are specified, and the ‘fine-grainedness’ needs to be roughly the same everywhere.

• (actually, my formula doubles the numbers you gave)

Are you sure? Suppose we take with , , then , so the values for should be as I gave them. And similarly for , giving values . Or else I have mis-understood your definition?

I’d simply see that as two separate partial preferences

Just to be clear, by “separate partial preference” you mean a separate preorder, on a set of objects which may or may not have some overlap with the objects we considered so far? Then somehow the work is just postponed to the point where we try to combine partial preferences?

EDIT (in reply to your edit): I guess e.g. keeping conditions 1,2,3 the same and instead minimising

where is proportion to the reciprocal of the strength of the preference? Of course there are lots of variants on this!

• Sure, in the end we only really care about what comes top, as that’s the thing we choose. My feeling is that information on (relative) strengths of preferences is often available, and when it is available it seems to make sense to use it (e.g. allowing circumvention of Arrow’s theorem).

In particular, I worry that, when we only have ordinal preferences, the outcome of attempts to combine various preferences will depend heavily on how finely we divide up the world; by using information on strengths of preferences we can mitigate this.

• I expect most people on LW to be okay being asked their Cheerful Price to have sex with someone.

I find this a surprising assertion. It does not apply to me, probably it does apply to you. Ordinarily I would ask if you had any other data points, but I don’t want to take the conversation in this direction...

• So the set of worlds, , is the set of functions from to …

I guess the should be a ? Also, you don’t seem to define ; perhaps ?

• I’m sceptical of your decision to treat tenured and non-tenured faculty alike. As tenured faculty, this has long seemed to me to be perhaps the most important distinction.

More generally, what you write here is not very consistent with my own experience of academia (which is in mathematics and in Europe, though I have friends and collaborators in other countries and fields, so I am not totally clueless about how things work there).

Some points I am not seeing in your post are:

1. For many academics, being able to do their own research and work with brilliant students is their primary motivation. Grants etc are mainly valuable in how they facilitate that. This makes for a confusing situation where ‘losers’ in the original LCS model do the minimum work necessary for their paycheck, whereas ‘losers’ in the academic system (as you seem to be defining them?) do the maximum work that is compatible with their health and personal situation. Not only is this conceptually confusing to me, it also means that all other things being equal, the more losers’ one is in academia the more impressive one’s CV will tend to be. Which is I think the opposite of the situation in the conventional LCS hierarchy?

2. The fact that I ‘perform peer review for nothing at all’ apparently makes me clueless. But this is weird; it does not go on my CV, and I do it because I think it is important to the advancement of science. Surely this makes it a loser’ activity?

3. Acceptance of papers and awarding of grants is decided by people external to your university. This makes a huge difference, and I think you miss it by writing So we might analyze this system at the department level, at the university level, or at the all-academia level, but it doesn’t make much of a difference.’.

Perhaps the above makes it sound as if I view academia as an organisational utopia; this is far from the case! But I do not think this post does a good job of identifying problems. I think a post analysing moral mazes in academia would be interesting, but I’m not convinced that the LCS hierarchy is an appropriate model, and this attempt to apply it does not seem to me to make useful category distinctions.

• Can you elaborate on this a bit? I’m sorry to hear that you had a bad experience during fieldwork, though I’m afraid I’m not certain what you refer to by ‘Active Personal Life’. Can you explain how the experience you relate connects to the LCS hierarchy?

• I really liked the content, but I found some of the style (Sit down!′ etc) really off-putting, which I why I only actually read the post on my 3rd attempt. Obviously you’re welcome to write in whatever style you want, and probably lots of other people really like it, I just thought it might be useful to mention that a non-empty set of people find it off-putting.

• Definitely the antagonistic bits—I enjoyed the casual style! Really just the line ‘ Sit down. Sit down. Shut up. Listen. You don’t know nothing yet’ I found quite off-putting—even though in hindsight you were correct!

• if we gave research grants to smart and personable university graduates and gave them carte blanche to do with the money what they wished that would work just as well as the current system

This thought is not unique to you; see e.g. the French CNRS system. My impression is that it works kind of as you would expect; a lot of them go on to do solid work, some do great work, and a few stop working after a couple of years. Of course we can not really know how things would have turned out if the same people had been given more conventional positions,

• I’m not sure I agree with interstice’s reading of the ‘sharp minima’ paper. As I understand it, they show that a given function can be made into a sharp or flat minimum by finding a suitable point in the parameter space mapping to the function. So if one has a sharp minmum that does not generalise (which I think we will agree exists) then one can make the same function into a flat minimum, which will still not generalise as it is the same function! Sorry I’m 2 years late to the party...