What do you mean by a product topology here? The product topology being used for a stochastic processes? That requires a topology on the state space in the first place. Right now I have not specified any topologies.
Regarding the stochastic aspect, I have thought about that, but so far I have not yet seen a benefit by including it because any stochastic approach can somehow be seen as just a deterministic approach on the level of distributions. I.e. if a Model M is actually a random variable, and a task T is also a random variable, then the important thing, which is the function M×T⟶R+ and which would now be a random object, can be replaced by a function P(M)×P(T)⟶P(R+). I.e. we map distributions of models and tests to distributions of scores.
Nevertheless on a bit of a different note, consider the following.
I described a task as something which a model can generate an answer to which is then somehow scored. If instead we consider the score of a model on a task to represent the expected value of correct answers given a large amount of tries, then we can say that
T(r⋅M)=r⋅T(M)
i.e. we get a new axiom! This states that tasks are no just any functions, but 1-homogeneous functions. But tasks are certainly not linear, as cooperation of a model with itself may bring no improvement on non-parallelizable tasks.
To clarify, suppose that the agents are chatbots. Then given a sequence of previous messages M, it outputs a probability distribution over the next message that the agent wants to say. For example, if the task is rock-paper-scissors, it would output a probability distribution with three possible outputs, “rock”, “paper”, and “scissors”, each with 1⁄3 probability.
Under this structure, there is a product topology indexed over the set is sequences of messages.
If you only want to use the structure defined in the post, another topology would be the finest topology that makes your two operations continuous.
What do you mean by a product topology here? The product topology being used for a stochastic processes? That requires a topology on the state space in the first place. Right now I have not specified any topologies.
Regarding the stochastic aspect, I have thought about that, but so far I have not yet seen a benefit by including it because any stochastic approach can somehow be seen as just a deterministic approach on the level of distributions. I.e. if a Model M is actually a random variable, and a task T is also a random variable, then the important thing, which is the function M×T⟶R+ and which would now be a random object, can be replaced by a function P(M)×P(T)⟶P(R+). I.e. we map distributions of models and tests to distributions of scores.
Nevertheless on a bit of a different note, consider the following.
I described a task as something which a model can generate an answer to which is then somehow scored. If instead we consider the score of a model on a task to represent the expected value of correct answers given a large amount of tries, then we can say that
T(r⋅M)=r⋅T(M)i.e. we get a new axiom! This states that tasks are no just any functions, but 1-homogeneous functions. But tasks are certainly not linear, as cooperation of a model with itself may bring no improvement on non-parallelizable tasks.
To clarify, suppose that the agents are chatbots. Then given a sequence of previous messages M, it outputs a probability distribution over the next message that the agent wants to say. For example, if the task is rock-paper-scissors, it would output a probability distribution with three possible outputs, “rock”, “paper”, and “scissors”, each with 1⁄3 probability.
Under this structure, there is a product topology indexed over the set is sequences of messages.
If you only want to use the structure defined in the post, another topology would be the finest topology that makes your two operations continuous.