Currently, I am privileged to be working with Dr. Davide Iannuzzi, Professor in Experimental Physics, and Dr. Maurits Kaptein, assistant professor in Artificial Intelligence, on real-world applications of “Lock in Feedback” – where “Lock in Feedback” refers to their adaptation of the lock-in a amplifier algorithm used in physics and engineering as a generic tool to find maxima of unknown functions by experimentation, even when the maximum of the functions drifts or when the signal to noise ratio is low:

…our method is based on an approach used in physics and engineering, where, if a physical variable y depends on the value of a well controllable physical variable x, the search for argmax

_{x}f(x) can be solved via what is nowadays considered as standard electronics. This approach relies on the possibility of making the variable x oscillate at a fixed frequency and to look at the response of the dependent variable y at the very same frequency by means of a lock-in amplifier. The method is particularly suitable when y is immersed in a high noise level, where other more direct methods would fail. Furthermore, should the entire curve shift (or, in other words, if argmax_{x}f(x) changes in time, also known as concept drift), the circuit will automatically adjust to the new situation and quickly reveal the new maximum position. This approach is widely used in a very large number of applications, both in industry and research, and is the basis for the Lock-in Feedback (LiF) method …^{1}

For the details on the LiF algorithm I refer you to Davide’s and Maurits’ original paper – in a nutshell, Lock in feedback really just goes through the following steps, again and again:

- Oscillate a controllable independent variable X around a set value at a fixed pace.
- Apply the Lock-in amplifier algorithm.to obtain values of the amplitude if the outcome variable Y at the pace you set at step 1.
- Is the amplitude of this variable zero? Congratulations, you have reached lock-in!

That is, you have found the optimal value of Y at the current value of X.

Still, this optimal value might shift over time, so move to step 1 and repeat the process to make sure we maintain lock-in. - Is the amplitude less than, or greater than zero?

Then move the set value around which we are oscillating our independent variable X up or down on the basis of the outcome.

Now move to step 1 and repeat..

Firstly, an equation that represents *x* oscillating with time* t *(1):

Secondly, an equation that integrates yÏ‰ over a time T = 2Ï€N Ï‰ , where N is a positive integer and T denotes the time needed to integrate N full oscillations (2):

Davide and Maurits then apply these equations in one batch-oriented and one continuous time algorithm. Here I will focus on the latter algorithm, LiF-2, as represented by the following pseudocode:

You can clearly recognize the line of pseudocode that represents equation 1 – this is the line that makes *x* oscillate with time *t*. The other lines busy themselves with integrating yÏ‰ over a time T = 2Ï€N Ï‰, as per equation 2.

It is a relatively small step from pseudocode to conventional programming language code. In other words, we can now put LiF to work in some R and Python simulations! More on those simulations in later posts. For now, to wet your appetite, a straightforward conversion of the previous pseudocode to Python:

To demonstrate how you might implement the previous code fragment, a minimal simulation:

1. Kaptein, M., & Ianuzzi, D. (2015). Lock in Feedback in Sequential Experiment. *arXiv*, 7.

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