All of Joseph Van Name's Comments + Replies

Since AI interpretability is a big issue for AI safety, let's completely interpret the results of evolutionary computation. 

Disclaimer: This interpretation of the results of AI does not generalize to interpreting deep neural networks. This is a result for interpreting a solution to a very specific problem that is far less complicated than deep learning, and by interpreting, I mean that we iterate a mathematical operation hundreds of times to get an object that is simpler than our original object, so don't get your hopes up too much.

A basis matroid is ... (read more)

I have originally developed a machine learning notion which I call an LSRDR ($L_{2,d}$

-spectral radius dimensionality reduction), and LSRDRs (and similar machine learning models) behave mathematically and they have a high level of interpretability which should be good for AI safety. Here, I am giving an example of how LSRDRs behave mathematically and how one can get the most out of interpreting an LSRDR.

Suppose that $n$ is a natural number. Let $N$ denote the quantum channel that takes an $n$ qubit quantum state and selects one of those ... (read more)

In this note, I will continue to demonstrate not only the ways in which LSRDRs ($L_{2,d}$-spectral radius dimensionality reduction) are mathematical but also how one can get the most out of LSRDRs. LSRDRs are one of the types of machine learning that I have been working on, and LSRDRs have characteristics that tell us that LSRDRs are often inherently interpretable which should be good for AI safety.

Suppose that $N$ is the quantum channel that maps a $n$ qubit state to a $n$ qubit state where we select one of the 6 qubits at random and se... (read more)

I personally like my machine learning algorithms to behave mathematically especially when I give them mathematical data. For example, a fitness function with apparently one local maximum value is a mathematical fitness function. It is even more mathematical if one can prove mathematical theorems about such a fitness function or if one can completely describe the local maxima of such a fitness function. It seems like fitness functions that satisfy these mathematical properties are more interpretable than the fitness functions which do not, so people should ... (read more)

Here is an example of what might happen. Suppose that for each $u_j$, we select a orthonormal basis $e_{j,1},\dots ,e_{j,s}$ of unit vectors for $V$. Let $R=\left\{\right(u_j,e_{j,k}):1\le j\le n,1\le k\le s\}$. Then

Then for each quantum channel $E$, by the concavity of the logarithm function (which is the arithmetic-geometric mean inequality), we have 

$L(R,E)={\sum }_{j=1}^n{\sum }_{k=1}^n-\log \left(E\right(u_j{u}_j^{\ast })e_{j,k},e_{j,k}⟩)$

$\le {\sum }_{j=1}^n-\log ({\sum }_{k=1}^n⟨E(u_j{u}_j^{\ast })e_{j,k},e_{j,k}⟩)$

$={\sum }_{j=1}^n-\log \left(\text{Tr}\right(E\left)\right)$. Here, equality is reached if and only if 

$E\left(u_j{u}_j^{\ast }\right)e_{j,k},e_{j,k}⟩=E\left(u_j{u}_j^{\ast }\right)e_{j,l},e_{j,l}⟩$ for each $j,k,l$, but this equality... (read more)

The notion of the linear regression is an interesting machine learning algorithm in the sense that it can be studied mathematically, but the notion of a linear regression is a quite limited machine learning algorithm as most relations are non-linear. In particular, the linear regression does not give us any notion of any uncertainty in the output.

One way to extend the notion of the linear regression to encapsulate uncertainty in the outputs is to regress a function not to a linear transformation mapping vectors to vectors, but to regress the function to a ... (read more)

There are some cases where we have a complete description for the local optima for an optimization problem. This is a case of such an optimization problem. 

Such optimization problems are useful for AI safety since a loss/fitness function where we have a complete description of all local or global optima is a highly interpretable loss/fitness function, and so one should consider using these loss/fitness functions to construct AI algorithms.

Theorem: Suppose that $U$ is a real,complex, or quaternionic $n\times n$-matrix that minimizes the quantity&n... (read more)

I do not care to share much more of my reasoning because I have shared enough and also because there is a reason that I have vowed to no longer discuss except possibly with lots of obfuscation. This discussion that we are having is just convincing me more that the entities here are not the entities I want to have around me at all. It does not do much good to say that the community here is acting well or to question my judgment about this community. It will do good for the people here to act better so that I will naturally have a positive judgment about this community.

You are judging my reasoning without knowing all that went into my reasoning. That is not good.

I will work with whatever data I have, and I will make a value judgment based on the information that I have. The fact that Karma relies on very small amounts of information is a testament to a fault of Karma, and that is further evidence of how the people on this site do not want to deal with mathematics.  And the information that I have indicates that there are many people here who are likely to fall for more scams like FTX. Not all of the people here are so bad, but I am making a judgment based on the general atmosphere here. If you do not like my judgment, then the best thing would be to try to do better. If this site has made a mediocre impression on me, then I am not at fault for the mediocrity here.

Let's see whether the notions that I have talked about are sensible mathematical notions for machine learning.

Tensor product-Sometimes data in a neural network has tensor structure. In this case, the weight matrices should be tensor products or tensor sums. Regarding the structure of the data works well with convolutional neural networks, and it should also work well for data with tensor structure to it.

Trace-The trace of a matrix measures how much the matrix maps vectors onto themselves since

$\text{Tr}\left(A\right)=c\cdot E(⟨Av,v⟩)$ where $v$ follows the multivariat... (read more)

Talking about whining and my loss of status is a good way to get me to dislike the LW community and consider them to be anti-intellectuals who fall for garbage like FTX. Do you honestly think the people here should try to interpret large sections of LLMs while simultaneously being afraid of quaternions?

It is better to comment on threads where we are interacting in a more positive manner.

I thought apologizing and recognizing inadequacies was a core rationalist skill. And I thought rationalists were supposed to like mathematics. The lack of mathematical appr... (read more)

I usually think of the field of complex numbers algebraically, but one can also think of the real numbers, complex numbers, and quaternions geometrically. The real numbers are good with dealing with 1 dimensional space, and the complex numbers are good for dealing with 2 dimensional space geometrically. While the division ring of quaternions is a 4 dimensional algebra over the field of real numbers, the quaternions are best used for dealing with 3 dimensional space geometrically. 

For example, if $U,V$ are open subsets of some Euclidean space, ... (read more)

We can use the $L_2-$spectral radius similarity to measure more complicated similarities between data sets.

Suppose that $A_1,\dots ,A_r$ are $m\times m$-real matrices and $B_1,\dots ,B_r$ are $n\times n$-real matrices. Let $\rho \left(A\right)$ denote the spectral radius of $A$ and let $A\otimes B$ denote the tensor product of $A$ with $B$. Define the $L_2$-spectral radius by setting ${\rho }_2(A_1,\dots ,A_r)=\rho (A_1\otimes A_1+\cdots +A_r\otimes A_r{)}^{1/2}$, Define the $L_2$-spectral radius similarity between $A_1,\dots ,A_r$ and $B_1,\dots ,B_r$ as

$\parallel (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r){\parallel }_2=\frac{\rho (A_1\otimes }{}$... (read more)

I am curious about your statement that all large neural networks are isomorphic or nearly isomorphic and therefore have identical loss values. This should not be too hard to test.

Let $A,B,C$ be training data sets. Let $M,N$ be neural networks. First train $M$ on $A$ and $N$ on $B$. Then slowly switch the training sets, so that we eventually train both $M$ and $N$ on just $C$. After fully training $M$ and $N$, one should be able to train an isomorphism between the networks $M$ ... (read more)

I have made a few minor and mostly cosmetic edits to the post about the dimensionality reduction of tensors that produces so many trace free matrices and also to the post about using LSRDRs to solve a combinatorial graph theory problem.

"What's the problem?"-Neural networks are horribly uninterpretable, so it would be nice if we could use more interpretable AI models or at least better interpretability tools. Neural networks seem to include a lot of random information, so it would be good to use AI models that do not include so much random information. Do y... (read more)

I would go further than this. Future architectures will not only be designed for improved performance, but they will be (hopefully) increasingly designed to optimize safety and interpretability as well, so they will likely be much different than the architectures we see today. It seems to me (this is my personal opinion based on my own research for cryptocurrency technologies, so my opinion does not match anyone else's opinion) that non-neural network machine learning models (but which are probably still trained by moving in the direction of a vector field... (read more)

The $L_2$-spectral radius similarity is not transitive. Suppose that $A_1,\dots ,A_r$ are $m\times m$-matrices and $B_1,\dots ,B_r$ are real $n\times n$-matrices. Then define ${\rho }_2(A_1,\dots ,A_r)=\rho (A_1\otimes A_1+\cdots +A_r\otimes A_r{)}^{1/2}$. Then the generalized Cauchy-Schwarz inequality is satisfied:

$\rho (A_1\otimes B_1+\cdots +A_r\otimes B_r)\le {\rho }_2(A_1,\dots ,A_r){\rho }_2(B_1,\dots ,B_r)$.

We therefore define the $L_{2,d}$-spectral radius similarity between $(A_1,\dots ,A_r)$ and $(B_1,\dots ,B_r)$ as $\parallel (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r)\parallel =\frac{\rho (A_1\otimes B_1+\cdots +A_r\otimes B_r)}{{\rho }_2(A_1,\dots ,A_r){\rho }_2(B_1,\dots ,B_r)}$. One should think of the $L_2$-spectral radius similarity as a gene... (read more)

I appreciate your input.  I plan on making more posts like this one with a similar level of technical depth. Since I included a proof with this post, this post contained a bit more mathematics than usual. With that being said, others have stated that I should be aware of the mathematical prerequisites for posts like this, so I will keep the mathematical prerequisites in mind.

Here are some more technical thoughts about this.

1. We would all agree that the problem of machine learning interpretability is a quite difficult problem; I believe that the solution

If you have any questions about the notation or definitions that I have used, you should ask about it in the mathematical posts that I have made and not here. Talking about it here is unhelpful, condescending, and it just shows that you did not even attempt to read my posts. That will not win you any favors with me or with anyone who cares about decency. 

Karma is not only imperfect, but Karma has absolutely no relevance whatsoever because Karma can only be as good as the community here.

P.S. Asking a question about the notation does not even signify an... (read more)

I am pointing out something wrong with the community here. The name of this site is LessWrong. On this site, it is better to acknowledge wrongdoing so that the people here do not fall into traps like FTX again. If you read the article, you would know that it is better to acknowledge wrongdoing or a community weakness than to double down.

I already did that. But it seems like the people here simply do not want to get into much mathematics regardless of how closely related to interpretability it is. 

 P.S. If anyone wants me to apply my techniques to GPT, I would much rather see the embedding spaces as more organized objects. I cannot deal very well with words that are represented as vectors of length 4096 very well. I would rather deal with words that are represented as 64 by 64 matrices (or with some other dimensions). If we want better interpretability, the data needs to be structured in a more organized fashion so that it is easier to apply interpretability tools to the data.

"Lesswrong has a convenient numerical proxy-metric of social status: site karma."-As long as I get massive downvotes for talking correctly about mathematics and using it to create interpretable AI systems, we should all regard karma as a joke. Karma can only be as good as the community here.

Let's compute some inner products and gradients.

Set up: Let $K$ denote either the field of real or the field of complex numbers. Suppose that $d_1,\dots ,d_r$ are positive integers. Let $m_0,\dots ,m_n$ be a sequence of positive integers with $m_0=m_n=1$. Suppose that $X_{i,j}$ is an $m_{i-1}\times m_i$-matrix whenever $1\le j\le d_i$. Then from the matrices $X_{i,j}$, we can define a $d_1\times \cdots \times d_r$-tensor $T\left(\right(X_{i,j}{)}_{i,j})=(X_{1,i_1}\dots X_{n,i_n}{)}_{i_1,\dots ,i_n}$. I have been doing computer experiments where I use this tensor to approximate other tensors by minimizing the... (read more)

So in my research into machine learning algorithms, I have stumbled upon a dimensionality reduction algorithm for tensors, and my computer experiments have so far yielded interesting results. I am not sure that this dimensionality reduction is new, but I plan on generalizing this dimensionality reduction to more complicated constructions that I am pretty sure are new and am confident would work well.

Suppose that $K$ is either the field of real numbers or the field of complex numbers. Suppose that $d_1,\dots ,d_n$ are positive integers and $(m_0$... (read more)

So in my research into machine learning algorithms that I can use to evaluate small block ciphers for cryptocurrency technologies, I have just stumbled upon a dimensionality reduction for tensors in tensor products of inner product spaces that according to my computer experiments exists, is unique, and which reduces a real tensor to another real tensor even when the underlying field is the field of complex numbers. I would not be too surprised if someone else came up with this tensor dimensionality reduction before since it has a rather simple description ... (read more)

Thanks for pointing that out. I have corrected the typo.  I simply used the symbol $r$ for two different quantities, but now the probability is denoted by the symbol $\alpha$.

Every entry in a matrix counts for the $L_2$-spectral radius similarity. Suppose that $A_1,\dots ,A_r,B_1,\dots ,B_r$ are real $n\times n$-matrices. Set $A^{\otimes 2}=A\otimes A$. Define the $L_2$-spectral radius similarity between $(A_1,\dots ,A_r)$ and $(B_1,\dots ,B_r)$ to be the number

$\frac{\rho (A_1\otimes B_1+\cdots +A_r\otimes B_r)}{\rho (A_1^{\otimes 2}+\cdots +A_r^{\otimes 2}{)}^{1/2}\rho (B_1^{\otimes 2}+\cdots +B_r^{\otimes 2}{)}^{1/2}}$. Then the $L_2$-spectral radius similarity is always a real number in the interval $[0,1]$, so one can think of the $L_2$-spectral radius similarity as a generalization of the value $\frac{|⟨u,v⟩|}{\parallel u\parallel \cdot \parallel v\parallel }$ where $u,v$&nbs... (read more)

The problem of unlearning would be solved (or kind of solved) if we just used machine learning models that optimize fitness functions that always converged to the same local optimum regardless of the initial conditions (pseudodeterministic training) or at least has very few local optima. But this means that we will have to use something other than neural networks for this and instead use something that behaves much more mathematically. Here the difficulty is to construct pseudodeterministically trained machine learning models that can perform fancy tasks a... (read more)

I think that all that happened here was the matrices $A_1,\dots ,A_r$ just ended up being diagonal matrices. This means that this is probably an uninteresting observation in this case, but I need to do more tests before commenting any further.

Suppose that $q,r,d$ are natural numbers. Let $1<p<\infty$. Let $z_{i,j}$ be a complex number whenever $1\le i\le q,1\le j\le r$. Let $L:M_d(C{)}^r\setminus \{0\}\to [-\infty ,\infty )$ be the fitness function defined by letting $L(X_1,\dots ,X_r)$$=({\sum }_{i=1}^q\log \left(\rho \right({\sum }_{j=1}^r{z}_{i,j}{X}_j\left)\right)/q)-\log (\parallel {\sum }_{j=1}^r{X}_j{X}_j^{\ast }{\parallel }_p)/2$. Here, $\rho \left(X\right)$ denotes the spectral radius of a matrix $X$ while $\parallel X{\parallel }_p$ denotes the Schatten $p$-norm of $X$.

Now suppose that $(A_1,\dots ,A_r)\in M_d(C{)}^r\setminus \{0\}$ is a tuple that maximizes $L(A_1,\dots ,A_r)$. Let $M:C^r\setminus \left\{0\right\}\to [-\infty ,\infty )$ be the fitness functio... (read more)

I forgot to mention another source of difficulty in getting the energy efficiency of the computation down to Landauer's limit at the CMB temperature.

Recall that the Stefan Boltzmann equation states that the power being emitted from an object by thermal radiation is equal to $P=A\cdot ϵ\cdot \sigma \cdot T^4$. Here, $P$ stands for power, $A$ is the surface area of the object, $ϵ$ is the emissivity of the object ($ϵ$ is a real number with $0\le ϵ\le 1$),$T$ is the temperature, and $\sigma$ is the Stefan-Boltzmann constant. Here, $\sigma \approx 5.67\cdot {10}^{-8}\cdot W$... (read more)

This post uses the highly questionable assumption that we will be able to produce a Dyson sphere that can maintain a temperature at the level of the cosmic microwave background before we will be able to use energy efficient reversible computation to perform operations that cost much less than $k\cdot T$ energy. And this post also makes the assumption that we will achieve computation at the level of about $k\cdot T\cdot \ln \left(2\right)$ per bit deletion before we will be able to achieve reversible computation. And it gets difficult to overcome thermal noise at an en... (read more)

Let \(X,Y\) be topological spaces. Then a function \(f:X\rightarrow Y\) is continuous if and only if whenever \((x_d)_{d\in D}\) is a net that converges to the point \(x\), the net \((f(x_d))_{d\in D}\) also converges to the point \(f(x)\). This is not very hard to prove. This means that we do not have to discuss as to whether continuity should be defined in terms of open sets instead of limits because both notions apply to all topological spaces. If anything, one should define continuity in terms of closed... (read more)

I have heard of filters and ultrafilters, but I have never heard of anyone calling any sort of filter a hyperfilter. Perhaps it is because the ultrafilters are used to make fields of hyperreal numbers, so we can blame this on the terminology. Similarly, the uniform spaces where the hyperspace is complete are called supercomplete instead of hypercomplete.

But the reason why we need to use a filter instead of a collection of sets is that we need to obtain an equivalence relation.

Suppose that $I$ is an index set and $X_i$ is a set with $|X^{}$... (read more)

Yes. We have 2=[(2,2,2,...)]. But we can compare 2 with (1,3,1,3,1,3,...) since (1,3,1,3,1,3,1,3,...)=1 (this happens when the set of all even natural numbers is in your ultrafilter) or (1,3,1,3,1,3,1,3,...)=3 (this happens when the set of all odd natural numbers is in your ultrafilter). Your partially ordered set is actually a linear ordering because whenever we have two sequences $(a_n{)}_n,(b_n{)}_n$, one of the sets

$\{n:a_n>b_n\},\{n:a_n<b_n\},\{n:a_n=b_n\}$ is in your ultrafilter (you can think of an ultrafilter as a thing that selects one block ... (read more)

I trained a (plain) neural network on a couple of occasions to predict the output of the function $x_1\oplus \cdots \oplus x_5$ where $x_1,\dots ,x_5$ are bits and $\oplus$ denotes the XOR operation. The neural network was hopelessly confused despite the fact that neural networks usually do not have any trouble memorizing large quantities of random information. This time the neural network could not even memorize the truth table for XOR. While the operation $(x_1,\dots ,x_5)\mapsto x_1\oplus \cdots \oplus x_5$ is linear over the field $F_2$, it is quite non-linear over $R$. The inabil... (read more)

Neural networks with ReLU activation are the things you obtain when you combine two kinds of linearity, namely the standard linearity that we all should be familiar with and tropical linearity.

Give $R$ two operations $\otimes ,\oplus$ defined by setting $x\oplus y=max(x,y),x\otimes y=x+y$.  Then the operations $\oplus ,\otimes$ are associative, commutative, and they satisfy the distributivity property $x\otimes (y\oplus z)=(x\otimes y)\oplus (x\otimes z)$. We shall call the operations $\oplus ,\otimes$ tropical operations on $R$.

We can even perform matrix and vector operations by replacing t... (read more)

I think of tensors as homogeneous non-commutative polynomials. But I have found a way of reducing tensors that does not do anything for 2-tensors but which seems to work well for $n$-tensors where $n\ge 3$. We can consider tensors as homogeneous non-commutative polynomials in a couple of different ways depending on whether we have tensors in $V_1\otimes V_2\otimes \cdots \otimes V_n$ or if we have $V\otimes \cdots \otimes V=V^{\otimes n}$. Let $K\in \{R,C\}$. Given a homogeneous non-commutative polynomial $p(x_1,\dots ,x_r)$ over the field $K$, consider the fitness function $L_{p,d,K}:M_d(K{)}^r\to [0,\infty )$&... (read more)

Perhaps it is best to develop AI systems that we can prove theorems about in the first place. AI systems that we can prove theorems about are more likely to be interpretable anyways. Fortunately, there are quite a few theorems about maxima and minima of functions including uniqueness theorems including the following.

Theorem: (maximum principle) If $C\subseteq R^n$ is a compact set, and $f:C\to [-\infty ,\infty )$ is an upper semicontinuous function that is subharmonic on the interior $C^{\circ }$, then ${max}_{x\in C}f\left(x\right)={max}_{x\in \partial C}f\left(x\right)$.

If $U$ is a bounded domain, and&... (read more)

A double exponential model seems very questionable. Is there any theoretical reason why you chose to fit your model with a double exponential? When fitting your model using a double exponential, did you take into consideration fundamental limits of computation? One cannot engineer transistors to be smaller than atoms, and we are approaching the limit to the size of transistors, so one should not expect very much of an increase in the performance of computational hardware. We can add more transistors to a chip by stacking layers (I don't know how this would... (read more)

I am not expecting any worldwide regulation on AI that prohibits people from using or training unaligned systems (I am just expecting a usual level of regulation). I am mainly hoping for spectral techniques to develop to the point where AI groups will want to use these spectral techniques (or some other method) more and more until they are competitive with neural networks at general tasks or at least complement the deficiencies of neural networks. I also hope that these spectral techniques will remain interpretable and aligned.

Right now, there are several ... (read more)

I agree that interpretability research is risky, and that one should carefully consider whether it is worth it to perform this interpretability research. I propose that a safer alternative would be to develop machine learning models that are

1. quite unrelated to the highest performing machine learning models (this is so that capability gains in these safer models do not translate very well to capability gains for the high performing models),
2. as inherently interpretable as one can make these models,
3. more limited in functionality than the highest performing machi

I do not see any evidence that large language models are equipped to understand the structure behind prime numbers. But transformers along with other machine learning tools should be well-equipped to investigate other mathematical structures. In particular, I am thinking about the mathematical structures called Laver-like algebras that I have been researching on and off since about 2015.

I have developed an algorithm that is capable of producing new Laver-like algebras from old ones. From every Laver-like algebra, one can generate a sequence of non-commutat... (read more)

While I believe that this is a sensible proposal, I do not believe there will be too much of a market (in the near future) for it for societal reasons. Our current society unfortunately does not have a sufficient appreciation for mathematics and mathematical theorems for this to have much of a market cap. To see why this is the case, we can compare this proposal to the proposal for cryptocurrency mining algorithms that are designed to advance science. I propose that a scientific cryptocurrency mining algorithm should attract a much larger market capitaliza... (read more)

You claim that Terry Tao says that Benford's law lack's a satisfactory explanation. That is not correct. Terry Tao actually gave an explanation for Benford's law. And even if he didn't, if you are moderately familiar with mathematics, an explanation for Benford's law would be obvious or at least a standard undergraduate exercise. 

Let $X$ be a random variable that is uniformly distributed on the interval $[m\cdot \ln (10),n\cdot \ln (10\left)\right]$ for some $m<n$. Let $Y$ denote the first digit of $e^X$. Then show that $P(Y=k)={\log }_{10}\left(\frac{k+1}{k}\right)$.... (read more)

I made the Latex compile by adding a space. Let me know if there are any problems.

Is the Latex compiling here?