November 2019

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2Vladimir Slepnev19d (11/?)
Superdense coding.
Alice is told two classical bits of information and sends a qubit to Bob, who
can then recover the two classical bits. Again, it relies on Alice and Bob
sharing a prepared pair beforehand. It's the opposite of quantum teleportation,
where Alice sends two classical bits and Bob can recover a qubit.
First let's talk about bases. This is the usual basis: |00>, |01>, |10>, |11>.
This is the Bell basis: (|00> + |11>)/√2, (|00> - |11>)/√2, (|10> + |01>)/√2,
(|10> - |01>)/√2. Check for yourself that each two vectors are orthogonal to
each other. To "measure a state in a different basis" means to apply a rotation
from one basis into another, then measure. For example, if you have a state and
you know that it's one of the Bell basis states, you can figure out which one,
by rotating into the usual basis and measuring.
One cool thing about the Bell basis is that you can change any basis vector into
any other basis vector by operations on the first qubit only. For example,
rotating the first qubit by 90 degrees turns (|00> + |11>)/√2 into (|10> -
|01>)√2. Flipping the first qubit gives (|10> + |01>)√2, and so on.
Now superdense coding is easy. Alice and Bob start by sharing two halves of a
Bell state. Then depending on which two classical bits Alice needs to send, she
either leaves the state alone or rotates into one of the other three, by
operating only on her qubit. Then she sends her qubit to Bob, who now has both
qubits and can rotate them into the usual basis and measure them, recovering the
two classical bits.

1[comment deleted]16d (13/?)
Global phase.
Consider a quantum system that's even smaller than a qubit. Something with only
one possible measurement outcome, instead of two. Let's call it "quantum nit",
or "qunit", by analogy with "unit" in typed programming, which means a type with
one element.
A qunit has only one possible mixed state, with the density matrix ((1)). But
its pure states are more complicated: a one-qunit pure state is a number whose
norm is 1, so either 1 or -1 (or more if we allow complex numbers). The same is
true for an N-qunit pure state - it's also a single number whose norm is 1! The
reason is that the tensor product of N 1-dimensional spaces is 1-dimensional,
not N-dimensional. So the joint state of all qunits in the world is a single
number, which is called "global phase".
Now imagine we have a system with a bunch of qubits, and take its tensor product
with a qunit. We can move a scalar factor from one side of a tensor product to
the other, and that shouldn't change anything. So global phase can't affect any
measurement, it's just an extra degree of freedom in the theory.

116d (12/?)
The uncertainty principle, or something like it.
When you measure a qubit cos(φ)|0> + sin(φ)|1>, the result has variance
(1-cos(4φ))/4. (I'm skipping over the trig calculations here and below.) If you
have a choice between either measuring the qubit, or rotating it by an angle ψ
and then measuring it, then the sum of variances of the two operations is at
least (1-|cos(2ψ)|)/2. In particular, if ψ=π/4, the sum of variances is at least
1/2.
So no matter what state a qubit is in, and how precisely you know its state,
there are two possible measurements such that the sum of variances of their
results is at least 1/2. The reason is that the bases corresponding to these
measurements are at an angle to each other, not aligned. Apparently in the real
world, an object's "position" and "momentum" are two such measurements, so
there's a limit on their joint precision.
You can also carry out one measurement and then the other, but that doesn't help
- after the first measurement you have variance in the second. Moreover, after
the second you have fresh variance in the first. This lets you get an infinite
stream of fair coinflips from a single qubit: start with |0>, rotate by π/4,
measure, rotate by π/4, measure...

112d It seems useful to consider agents that reason in terms of an unobservable
ontology, and may have uncertainty over what this ontology is. In particular, in
Dialogic RL
[https://www.alignmentforum.org/posts/dPmmuaz9szk26BkmD/vanessa-kosoy-s-shortform#Wi65Ahs9abL63gPSe]
, the user's preferences are probably defined w.r.t. an ontology that is
unobservable by the AI (and probably unobservable by the user too) which the AI
has to learn (and the user is probably uncertain about emself). However,
onotlogies are more naturally thought of as objects in a category than as
elements in a set. The formalization of an "ontology" should probably be a POMDP
or a suitable Bayesian network. A POMDP involves an arbitrary set of states, so
it's not an element in a set, and the class of POMDPs can be naturally made into
a category. Therefore, there is need for defining the notion of a probability
measure over a category. Of course we can avoid this by enumerating the states,
considering the set of all possible POMDPs w.r.t. this enumeration and then
requiring the probability measure to be invariant w.r.t. state relabeling.
However, the category theoretic point of view seems more natural, so it might be
worth fleshing out.
Ordinary probably measures are defined on measurable spaces. So, first we need
to define the analogue of "measurable structure" (σ-algebra) for categories. Fix
a category C. Denote Meas the category of measurable spaces. A measurable
structure on C is then specified by providing a Grothendick fibration
[https://ncatlab.org/nlab/show/Grothendieck+fibration] B:MFC→Meas and an
equivalence E:B−1(pt)→C. Here, B−1(pt) stands for the essential fiber
[https://ncatlab.org/nlab/show/essential+fiber] of B over the one point space
pt∈Meas. The intended interpretation of MFC is, the category of families of
objects in C indexed by measurable spaces. The functor B is supposed to extract
the base (index space) of the family. We impose the following conditions on MFC
and B:
Given A∈

October 2019

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620d I've been thinking a lot about 'parallel economies' recently. One of the main
differences between 'slow takeoff' and 'fast takeoff' predictions is whether AI
is integrated into the 'human civilization' economy or constructing a separate
'AI civilization' economy. Maybe it's worth explaining a bit more what I mean by
this: you can think of 'economies' as collections of agents who trade with each
other. Often it will have a hierarchical structure, and where we draw the lines
are sort of arbitrary. Imagine a person who works at a company and participates
in its internal economy, and the company participates in national and global
economies, and the person participates in those economies as well. A better
picture has a very dense graph with lots of nodes and links between groups of
nodes whose heaviness depends on the number of links between nodes in those
groups.
As Adam Smith argues, the ability of an economy to support specialization of
labor depends on its size. If you have an island with a single inhabitant, it
doesn't make sense to fully employ a farmer (since a full-time farmer can
generate much more food than a single person could eat), for a village with 100
inhabitants it doesn't make sense to farm more than would feed a hundred mouths,
and so on. But as you make more and more of a product, investments that have a
small multiplicative payoff become better and better, to the point that a planet
with ten billion people will have massive investment in farming specialization
that make it vastly more efficient per unit than the village farming system. So
for much of history, increased wealth has been driven by this increased
specialization of labor, which was driven by the increased size of the economy
(both through population growth and decreased trade barriers widening the links
between economies until they effectively became one economy).
One reason to think economies will remain integrated is because increased size
benefits all actors in the economy on net; a

522d One challenge for theories of embedded agency over Cartesian theories is that
the 'true dynamics' of optimization (where a function defined over a space
points to a single global maximum, possibly achieved by multiple inputs) are
replaced by the 'approximate dynamics'. But this means that by default we get
the hassles associated with numerical approximations, like when integrating
differential equations. If you tell me that you're doing Euler's Method on a
particular system, I need to know lots about the system and about the particular
hyperparameters you're using to know how well you'll approximate the true
solution. This is the toy version of trying to figure out how a human reasons
through a complicated cognitive task; you would need to know lots of details
about the 'hyperparameters' of their process to replicate their final result.
This makes getting guarantees hard. We might be able to establish what the
'sensible' solution range for a problem is, but establishing what algorithms can
generate sensible solutions under what parameter settings seems much harder.
Imagine trying to express what the set of deep neural network parameters are
that will perform acceptably well on a particular task (first for a particular
architecture, and then across all architectures!).

825d Game theory is widely considered the correct description of rational behavior in
multi-agent scenarios. However, real world agents have to learn, whereas game
theory assumes perfect knowledge, which can be only achieved in the limit at
best. Bridging this gap requires using multi-agent learning theory to justify
game theory, a problem that is mostly open (but some results exist). In
particular, we would like to prove that learning agents converge to game
theoretic solutions such as Nash equilibria (putting superrationality aside: I
think that superrationality should manifest via modifying the game rather than
abandoning the notion of Nash equilibrium).
The simplest setup in (non-cooperative) game theory is normal form games.
Learning happens by accumulating evidence over time, so a normal form game is
not, in itself, a meaningful setting for learning. One way to solve this is
replacing the normal form game by a repeated version. This, however, requires
deciding on a time discount. For sufficiently steep time discounts, the repeated
game is essentially equivalent to the normal form game (from the perspective of
game theory). However, the full-fledged theory of intelligent agents requires
considering shallow time discounts, otherwise there is no notion of long-term
planning. For shallow time discounts, the game theory of a repeated game is very
different from the game theory of the original normal form game. In fact, the
folk theorem asserts that any payoff vector above the maximin of each player is
a possible Nash payoff. So, proving convergence to a Nash equilibrium amounts
(more or less) to proving converges to at least the maximin payoff. This is
possible using incomplete models
[https://www.alignmentforum.org/posts/5bd75cc58225bf0670375575/the-learning-theoretic-ai-alignment-research-agenda]
, but doesn't seem very interesting: to receive the maximin payoff, the agents
only have to learn the rules of the game, they need not learn the reward
functions of the othe

61mo This is preliminary description of what I dubbed Dialogic Reinforcement Learning
(credit for the name goes to tumblr user @di--es---can-ic-ul-ar--es): the
alignment scheme I currently find most promising.
It seems that the natural formal criterion for alignment (or at least the main
criterion) is having a "subjective regret bound": that is, the AI has to
converge (in the long term planning limit, γ→1 limit) to achieving optimal
expected user!utility with respect to the knowledge state of the user. In order
to achieve this, we need to establish a communication protocol between the AI
and the user that will allow transmitting this knowledge state to the AI
(including knowledge about the user's values). Dialogic RL attacks this problem
in the manner which seems the most straightforward and powerful: allowing the AI
to ask the user questions in some highly expressive formal language, which we
will denote F.
F allows making formal statements about a formal model M of the world, as seen
from the AI's perspective. M includes such elements as observations, actions,
rewards and corruption. That is, M reflects (i) the dynamics of the environment
(ii) the values of the user (iii) processes that either manipulate the user, or
damage the ability to obtain reliable information from the user. Here, we can
use different models of values: a traditional "perceptible" reward function, an
instrumental reward function
[https://www.alignmentforum.org/posts/aAzApjEpdYwAxnsAS/reinforcement-learning-with-imperceptible-rewards]
, a semi-instrumental reward functions, dynamically-inconsistent rewards
[https://www.alignmentforum.org/posts/aPwNaiSLjYP4XXZQW/ai-alignment-open-thread-august-2019#C9gRtMRc6qLv7J6k7]
, rewards with Knightian uncertainty etc. Moreover, the setup is
self-referential in the sense that, M also reflects the question-answer
interface and the user's behavior.
A single question can consist, for example, of asking for the probability of
some sentence in F or the expected

220d (10/?)
Quantum teleportation.
It's not a very good name, maybe we should call it "sending qubits over a
classical channel using a quantum one-time pad". The idea is that Alice and Bob
are far apart from each other and have a prepared pair of qubits, and Alice also
has some qubit in unknown state. She can measure it in a clever way and send
some classical bits to Bob, which allows him to recreate the unknown qubit. Both
qubits of the prepared pair are spent in the process.
Let's say the unknown qubit is a|0>+b|1>, and the prepared pair is (|00> +
|11>)/√2. So the whole state is (a|0> + b|1>) ⊗ (|00> + |11>)/√2 = (a|000> +
a|011> + b|100> + b|111>)/√2. Alice can access the first two qubits and Bob can
access the third.
1. Alice flips the second qubit depending on the first, leading to (a|000> +
a|011> + b|110> + b|101>)/√2.
2. Alice applies a combined reflection and rotation to the first qubit, which
is known as Hadamard gate: |0> becomes (|0> + |1>)/√2, while |1> becomes
(|0> - |1>)/√2. So the whole state becomes (a|000> + a|100> + a|011> +
a|111> + b|010> - b|110> + b|001> - b|101>)/2.
3. We can rewrite that as ((|00> ⊗ (a|0> + b|1>)) + (|01> ⊗ (a|1> + b|0>)) +
(|10> ⊗ (a|0> - b|1>)) + (|11> ⊗ (a|1> - b|0>)))/2.
4. Notice something funny? In the above state, measuring the first two qubits
gives you exactly enough information to know which operation to apply to the
third qubit to get a|0> + b|1>. So if Alice measures both her qubits and
sends the two classical bits to Bob, he can reconstruct the unknown qubit on
his end. And if Carol eavesdrops on the message without having access to
Bob's qubit, she won't learn anything, because the four possible messages
have probability 1/4 each, regardless of a and b.
If the unknown qubit was entangled with something else, that also survives the
transfer, though proving that requires a bit more calculation.