# Questions tagged [function-approximation]

For questions related to the concept of function approximation. For example, questions that involve the use of a neural network (which is a function approximator) in the context of RL in order to approximate a value function or questions that are related to universal approximation theorems.

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### Recursive Least squares (RLS) for mini batch

For my application I am considering a learning problem where I simulate a bunch of episodes say '$n$' first, and than carry out the recursive least squares update. Similar to $TD(1)$. I know that RLS ...
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### Is there any thumb rule on the cardinality of state space in order to use the parameterized function to estimate value functions?

Value functions for a given MDP can be learned in at least two ways by experience. The first way (tabular calculation) is generally used in the case of state spaces that are small enough. The second ...
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### Can transformers be used to improve regression?

I was recently reading a bit about transformers and I don't understand them very much but I was wondering if anyone knows if any of their techniques such as attention mechanism or anything has been ...
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### Trying to understand why nonlinearity is important for neural networks by analogy

Is the reason why linear activation functions are usually pretty bad at approximating functions the same reason why combinations of hermitian polynomials or combinations of sines and cosines are ...
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### Can models get 100% accuracy on solved games?

I had a question today that I feel it must have an answer already, so I'm shopping around. If we ask a model to learn the binary OR function, we get perfect accuracy with every model (as far as I know)...
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### Why the optimal Bellman operator of a Q-function can be approximated by a single point

I am currently studying reinforcement learning, especially DQN. In DQN, learning proceeds in such a way as to minimize the norm (least-squares, Huber, etc.) of the optimal Bellman equation and the ...
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### How can “any process you can imagine” be thought of as function computation?

I stumbled upon this passage when reading this guide. Universality theorems are a commonplace in computer science, so much so that we sometimes forget how astonishing they are. But it's worth ...
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### Why is it hard to prove the convergence of the deep Q-learning algorithm?

Why is it hard to prove the convergence of the DQN algorithm? We know that the tabular Q-learning algorithm converges to the optimal Q-values, and with a linear approximator convergence is proved. ...
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### Remove drawbacks of Neural Network regressor as compared Polynomial Regressor

As far as my knowledge goes (might be a bit vague and not mathematical), a Neural Network can and should only be able to approximate a bounded function, which is not the case of a Polynomial Regressor....
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### How to find good features for a linear function approximation in RL with large discrete state set?

I've recently read much about feature engineering in continuous (uncountable) feature spaces. Now I am interested what methods exist in the setting of large discrete state spaces. For example consider ...
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### Why does reinforcement learning using a non-linear function approximator diverge when using strongly correlated data as input?

While reading the DQN paper, I found that randomly selecting and learning samples reduced divergence in RL using a non-linear function approximator (e.g a neural network). So, why does Reinforcement ...
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### Neural network architecture with inputs and outputs being an unkown function each

I am trying to set up a neural network architecture that is able to learn the points of one function (blue curves) from the points of an other one (red curves). I think that it could be somehow ...
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### Smallest possible network to approximate the $sin$ function

The main goal is: Find the smallest possible neural network to approximate the $sin$ function. Moreover, I want to find a qualitative reason why this network is the smallest possible network. I have ...
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### Why is the fraction of time spent in state $s$, $\mu(s)$, not in the update rule of the parameters?

I am reading "Reinforcement Learning: An Introduction (2nd edition)" authored by Sutton and Barto. In Section 9, On-policy prediction with approximation, it first gives the mean squared ...
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### How do we derive the expression for average reward setting in continuing tasks?

In the average reward setting we have: $$r(\pi)\doteq \lim_{h\rightarrow\infty}\frac{1}{h}\sum_{t=1}^{h}\mathbb{E}[R_{t}|S_0,A_{0:t-1}\sim\pi]$$ r(\pi)\doteq \lim_{t\rightarrow\infty}\mathbb{E}[R_{t}...
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### Why do all states appear identical under the function approximation in the Short Corridor task?

This is the Short Corridor problem taken from the Sutton & Barto book. Here it's written: The problem is difficult because all the states appear identical under the function approximation But ...
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### Which machine learning models are universal function approximators?

The universal approximation theorem states that a feed-forward neural network with a single hidden layer containing a finite number of neurons can approximate any continuous function (provided some ...
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### Correct dimensionality of parameter vector for solving an MRP with linear function approximation?

I'm in the process of trying to learn more about RL by shadowing a course offered collaboratively by UCL and DeepMind that has been made available to the public. I'm most of the way through the course,...
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### Can supervised learning be recast as reinforcement learning problem?

Let's assume that there is a sequence of pairs $(x_i, y_i), (x_{i+1}, y_{i+1}), \dots$ of observations and corresponding labels. Let's also assume that the $x$ is considered as independent variable ...
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### What are the differences between artificial neural networks and other function approximators?

Modern artificial neural networks use a lot more functions than just the classic sigmoid, to the point I'm having a hard time really seeing what classifies something as a "neural network" over other ...
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### Can we optimize an optimization algorithm?

In this answer to the question Is an optimization algorithm equivalent to a neural network?, the author stated that, in theory, there is some recurrent neural network that implements a given ...
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### Which functions can't neural networks learn efficiently?

There are a lot of papers that show that neural networks can approximate a wide variety of functions. However, I can't find papers that show the limitations of NNs. What are the limitations of ...
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### Equivalence between expected parameter increments in “Off-Policy Temporal-Difference Learning with Function Approximation”

I am having a hard time understanding the proof of theorem 1 presented in the "Off-Policy Temporal-Difference Learning with Function Approximation" paper. Let $\Delta \theta$ and \$\Delta \...
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### What makes multi-layer neural networks able to perform nonlinear operations?

As I know, a single layer neural network can only do linear operations, but multilayered ones can. Also, I recently learned that finite matrices/tensors, which are used in many neural networks, can ...
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Consider some MDP with no terminal state. We can apply bootstrapping methods (like TD(0)) to learn in these cases no problem, but in policy gradient algorithms that have only a simple monte carlo ...
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### Is there a possibility that there is no relationship between some inputs and outputs?

I'm doing machine learning projects. I took a look at many datasets I worked with, mostly there are already famous datasets that everyone uses. Let's say I decided to make my own dataset. Is there a ...