I don’t follow the setup in your “Toy Model” section. As I understand it, for Function 1, considered for the example of a neural net, X is the whole of the training data, and Ψ is the set of weights resulting from training on that data.
But then I don’t understand Function 2. Firstly, you’re reusing “f” here. For consistency with the subsequent text, it seems that Function 1 should be called “g”. Then, I don’t see what is meant by the relationship Ω between the variables. (I assume the A...Z here are the same as for Function 1, i.e. the weights of the NN.) Once the network has been trained, the weights have fixed values. Ω1 can just output the value of the first weight, ignoring the values of all the others. The idea of these functions only makes sense if we are considering an ensemble of weight sets, derived from an ensemble of training data. But nothing like this appears in the presentation of Function 1 — nor, for that matter, in the training of NNs. In a world where a single training run of the latest LLM takes nine-figure sums of money running seven-figure quantities of GPUs for months on end, there can never be any such ensemble. Am I totally misunderstanding something here?
I have no idea what the description of Function 3 means.
Sorry about mixing up the f and g notation I’m not particularly used to it.
Ω by creating some functions of A...Z means that in principle you can input arbitrary values for B,C...Z and get an expected value for A, so it is not necessarily any specific value. Ψ aren’t the weights of a neural net, they’re the classified outputs, detecting certain patterns in specific signals. Like how the original perceptron could identify a square from a picture of a square vs a circle, ect. So all function 1 is doing is cutting up the continuum of signals its receiving into discrete chunks.
Function three is some arbitrary sorting function.
Ok, so X is the input to an already-trained NN, Ψ is the resulting output, and g (Function 1) is the input-output function. f (Function 2) is a set of functions, one for each output variable, estimating the value of that variable from all the others. I’m not clear on the distinction between f and Ω and I wonder if the notation is confused here.
I’m still unclear on Function 3. What are the “input functions” that it orders?
Sorry if these questions seem like obsessing over details, but this is how I go about understanding any piece of obscure mathematics: going through it symbol by symbol asking “what exactly does this mean?”
Whether the neural net in function 1 is already trained is up to interpretation, it could include the process of training, all that matters is that there are classified outputs at the end of the process.
Ω is the set of functions, whereas function 2 is the function which makes those functions.
“Input functions” isn’t a thing, function 3 is ordering input and function pairs.
Sorry if these questions seem like obsessing over details, but this is how I go about understanding any piece of obscure mathematics: going through it symbol by symbol asking “what exactly does this mean?”
No worries, I’m sorry if my writing isn’t as intelligible as I’d like. I’m glad someone is taking the time to understand it at all.
I am still finding this obscure. Taking the particular case of an already-trained neural net, X is its input and Ψ is the corresponding output: Ψ = g(X).
Ω is a set of functions, one for each output variable, each of which tries to predict the value of that variable from the others. f produces Ω from… what? You write Ω = f(Ψ), but if Ψ is given then again Ω1 can just return the first value in Ψ without having to estimate anything. I am unclear on what is varying in these definitions, what is fixed, and what the domains and ranges of these functions are.
Function 3, unnamed so I’ll call it h, defines a total ordering on pairs (X,Ω)? That is, pairs where X is a tuple of input values and Ω is (following your verbal description rather than the symbols) a world model. Or maybe a world model given the particular X? What is the purpose of this ordering? From the overall context it’s intended to be a preference ordering, but I would expect it to be a preference ordering over states of the world and independent of X, which is merely the data that guves rise to an estimated world state.
In other words, one of the major points of this post is that you can’t reason across or order world states, nobody does or can, you only ever reason across symbolic representations of world states which are created via some particular process.
Recall that the neural net in function 1 is a classifier, not making predictions about the relationships between the variables. All function 1 does is take a domain of a large set of input signals and correlate them to a smaller range of internal variables. Technically you don’t need a neural net to do this, you can hard code it in various ways, but I like thinking of the original Perceptron diagram in my head for this.
The point of using this toy model rather than just assuming an ordering over world states is to show that any modeling of world state is produced by particular functions using real world data. This encoding itself is what generates the epistemic problems, because the encoding of a semantic meaning to a particular signifier always creates some uncertainty when using that encoding as a reference point. In the toy model, X can be arbitrarily complex phenomenal experience, it encompasses every observable state of the world, so even for phantom values of X all we’re doing is extrapolating to the experience we’d expect in a given situation. By creating a function which gives us the relationship between different values of X, we can make a plan on how to achieve a specific value of X that we want, so if you take current and expected conditions of X, the input, and a function Ω, you can have a plan to achieve a goal. The ordering function therefore orders these possible plans from best to worst based on whatever arbitrary criteria.
I don’t follow the setup in your “Toy Model” section. As I understand it, for Function 1, considered for the example of a neural net, X is the whole of the training data, and Ψ is the set of weights resulting from training on that data.
But then I don’t understand Function 2. Firstly, you’re reusing “f” here. For consistency with the subsequent text, it seems that Function 1 should be called “g”. Then, I don’t see what is meant by the relationship Ω between the variables. (I assume the A...Z here are the same as for Function 1, i.e. the weights of the NN.) Once the network has been trained, the weights have fixed values. Ω1 can just output the value of the first weight, ignoring the values of all the others. The idea of these functions only makes sense if we are considering an ensemble of weight sets, derived from an ensemble of training data. But nothing like this appears in the presentation of Function 1 — nor, for that matter, in the training of NNs. In a world where a single training run of the latest LLM takes nine-figure sums of money running seven-figure quantities of GPUs for months on end, there can never be any such ensemble. Am I totally misunderstanding something here?
I have no idea what the description of Function 3 means.
Sorry about mixing up the f and g notation I’m not particularly used to it.
Ω by creating some functions of A...Z means that in principle you can input arbitrary values for B,C...Z and get an expected value for A, so it is not necessarily any specific value. Ψ aren’t the weights of a neural net, they’re the classified outputs, detecting certain patterns in specific signals. Like how the original perceptron could identify a square from a picture of a square vs a circle, ect. So all function 1 is doing is cutting up the continuum of signals its receiving into discrete chunks.
Function three is some arbitrary sorting function.
Ok, so X is the input to an already-trained NN, Ψ is the resulting output, and g (Function 1) is the input-output function. f (Function 2) is a set of functions, one for each output variable, estimating the value of that variable from all the others. I’m not clear on the distinction between f and Ω and I wonder if the notation is confused here.
I’m still unclear on Function 3. What are the “input functions” that it orders?
Sorry if these questions seem like obsessing over details, but this is how I go about understanding any piece of obscure mathematics: going through it symbol by symbol asking “what exactly does this mean?”
Whether the neural net in function 1 is already trained is up to interpretation, it could include the process of training, all that matters is that there are classified outputs at the end of the process.
Ω is the set of functions, whereas function 2 is the function which makes those functions.
“Input functions” isn’t a thing, function 3 is ordering input and function pairs.
No worries, I’m sorry if my writing isn’t as intelligible as I’d like. I’m glad someone is taking the time to understand it at all.
I am still finding this obscure. Taking the particular case of an already-trained neural net, X is its input and Ψ is the corresponding output: Ψ = g(X).
Ω is a set of functions, one for each output variable, each of which tries to predict the value of that variable from the others. f produces Ω from… what? You write Ω = f(Ψ), but if Ψ is given then again Ω1 can just return the first value in Ψ without having to estimate anything. I am unclear on what is varying in these definitions, what is fixed, and what the domains and ranges of these functions are.
Function 3, unnamed so I’ll call it h, defines a total ordering on pairs (X,Ω)? That is, pairs where X is a tuple of input values and Ω is (following your verbal description rather than the symbols) a world model. Or maybe a world model given the particular X? What is the purpose of this ordering? From the overall context it’s intended to be a preference ordering, but I would expect it to be a preference ordering over states of the world and independent of X, which is merely the data that guves rise to an estimated world state.
In other words, one of the major points of this post is that you can’t reason across or order world states, nobody does or can, you only ever reason across symbolic representations of world states which are created via some particular process.
Recall that the neural net in function 1 is a classifier, not making predictions about the relationships between the variables. All function 1 does is take a domain of a large set of input signals and correlate them to a smaller range of internal variables. Technically you don’t need a neural net to do this, you can hard code it in various ways, but I like thinking of the original Perceptron diagram in my head for this.
The point of using this toy model rather than just assuming an ordering over world states is to show that any modeling of world state is produced by particular functions using real world data. This encoding itself is what generates the epistemic problems, because the encoding of a semantic meaning to a particular signifier always creates some uncertainty when using that encoding as a reference point. In the toy model, X can be arbitrarily complex phenomenal experience, it encompasses every observable state of the world, so even for phantom values of X all we’re doing is extrapolating to the experience we’d expect in a given situation. By creating a function which gives us the relationship between different values of X, we can make a plan on how to achieve a specific value of X that we want, so if you take current and expected conditions of X, the input, and a function Ω, you can have a plan to achieve a goal. The ordering function therefore orders these possible plans from best to worst based on whatever arbitrary criteria.