GPFlowSampling in Robotics

[AdrianPrados]

Good afternoon, I am applying GPFlow to a robotic process based on Gaussian Process for manipulation. I have a couple of questions about the algorithm.

Firstly, the first function generated in the examples above is randomly generated with random_fourier. I was wondering if it would be possible to initialise the prior function with information from a previously generated GP solution.

Secondly, I've been testing the codes and I haven't seen how the covariance of the final solution is treated. Are we working with it for simplicity in the examples, or are we just not looking to update the covariance for the new solution?

Thank you very much for your work and I hope it will be useful for me.

[j-wilson]

Hi Adrian,

1. Can you elaborate on what you mean by "GP solution"? If you mean a GP posterior given (noisy) observations of a function, then the answer is "yes". More generally, one can also approach these problems by supposing that the current function $f_{t}$ is correlated to earlier functions $f_{t-i}$ where $i > 0$. It all depends on what you're trying to achieve.

2. This repository implements methods for generating function draws from GPs. GPFlow models implement predict_f methods for computing posterior means and (co)variances though, so that may fit your needs. The model API notebook may be helpful here.

Additional tidbits:

• There is a companion repository, GPFlowPILCO, that may be of interest (esp. for robotics).
• The gpytorch-based implementation of this package exposes additional functionality.
• There is a PR up to extends the gpytorch implementation to support sum and product kernels, which may be useful if you wish to incorporate domain knowledge regarding how a system responds to specific inputs.

[AdrianPrados]

Good morning, and sorry for my earlier explanation. By GP solution, I meant a GP posterior with noisy observations. Our idea is to apply it to Learning from Demonstration, where we have learned the structure of knowledge of some movement primitives (paths in Cartesian or Configuration space of our robot) through several human demonstrations.

From these demonstrations, we generate a GP posterior using the noisy observations, which in this case are our demonstrations. We are now applying the process presented in GPFlow to quickly adapt in case of information changes (such as adding waypoints or modifying initial and final points).

As for the covariances, that’s actually how we’ve been handling them — once we obtain the corrected mean values, we apply our own prediction method to generate the corresponding covariance values. So, as you suggested, we will continue following that approach.

Thank you very much for your quick response and for your work in this field. We hope to achieve some promising results and, of course, we’ll make sure to cite your excellent work, which serves as a foundation for our implementations.

Best regards.

[j-wilson]

By GP solution, I meant a GP posterior with noisy observations. Our idea is to apply it to Learning from Demonstration, where we have learned the structure of knowledge of some movement primitives (paths in Cartesian or Configuration space of our robot) through several human demonstrations.

From these demonstrations, we generate a GP posterior using the noisy observations, which in this case are our demonstrations. We are now applying the process presented in GPFlow to quickly adapt in case of information changes (such as adding waypoints or modifying initial and final points).

That makes a lot of sense; very cool! Sounds a bit like fine-tuning of foundational models. Yes, you can certainly do this. The naive approach is to jointly condition on the $n$ initial and $m$ task-specific observations. This has benefit of being easy to understand/implement, but the downside of incurring $O((n+m)^3)$ time complexity. If this method is fast enough for your use case, then read no further haha.

Another option is to precompute the Cholesky factor of the covariance matrix $k(\mathbf{X}_n, \mathbf{X}_n)$, where $\mathbf{X}_n$ are the inputs associated with the initial domain-knowledge observations. You can then augment said Cholesky factor to incorporate $m$ additional observations using tfp.math.cholesky_concat. If I recall correctly, the time complexity here is $O(mn^2 + m^3)$. Note that this approach also holds when computing (updated) posterior covariance matrices.

Lastly, it is possible to directly update a path $f \vert \mathbf{y}_n$ to account for $m$ additional observations. This is a bit different from the previous approaches in the sense that the "randomness" is the same. Hence, in regions where $f \vert \mathbf{y}_n$ is not correlated with the additional observations, the path would effectively stay the same. I can dig up some backlogged material on this if that's what you need. I'd have to double check, but I think the time complexity here is the same as the previous approach.

[AdrianPrados]

In principle, I believe the $O((n+m)^3)$ complexity would be sufficient for processing speed, although I will still need to check it. The Cholesky precomputation method also seems quite interesting to me and could be a valuable idea I wasn't aware of—I'll look into it further. As for the third option, it also seems like a very good approach to use, although I'm unfamiliar with its implementation. If you have any additional material related to that possible implementation, that would be great. Thank you so much for all your help, and if you agree, we can continue the conversation via email so as not to overload the issue with more specific problems from my implementation. My email is: aprados@ing.uc3m.es