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I'm pretty new to ML and most of the supervised techniques I've discovered so far are along the lines of

  • Pass model training set inputs and get it to generate predictions
  • evaluate predictions (predicted 'y') against real results (actual 'y') for the training set
  • Perform some kind of gradient descent to tweak the model parameters to give outputs that are closer to the real results

The precise method of training can vary, but the common theme in the examples I've seen so far is that the outputs of the model are the actual things that we want to get out of the system. For example, we might be trying to predict house prices - in this case, an output of the model would actually be a predicted price, and it's that that I would care about as part of my cost function and as a user of the system.

I'm imagining a scenario where we have a robot that's able to hold and use a palette and paintbrush, and we want it to copy existing paintings. (Strictly theoretical - I'm not trying to forge anything here!).

What I imagine I'd want is to

  • present robot with example input (existing paintings)
  • have the robot run a model where the output is some brush motions
  • evaluate the painting resulting from the brush strokes against the original painting

The difference here is that the output of my model is 'brush strokes', but that output (from the model) isn't ultimately what I care about - it's what's rendered by those brush strokes. I don't mind if the robot makes the same strokes that the original artist made, but I do mind about the similarity of what's rendered by those strokes to the original.

So I am wondering what areas of ML I need to look into to find something that works this way, where our cost or fitness is evaluated based on some transform of the model output, rather than the model output itself. I tried searching for examples of where a transform of the model output was incorporated into the cost function, but I couldn't see that it was a common technique.

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Main Idea:

The main idea is the you could measure how good is the output and the input of your model simply representing it by some mathematical object, for example a vector in a high dimensional space and then using some error function to measure the fitness (how good your model is performing).

A possible formalization

The scenario your describing fits in a more general optimization function, the most general optimization scenario is;

$$ min_{x \in X} f(x) $$

where $X$ is the domain. In the case of image processing and computer vision $x$ is a matrix of pixels, this matrix of pixels can be represented by a vector $x \in \mathbb{R}^{m \times n}$. In the case you describe you could also represent the input (painting) as a vector $y\in \mathbb{R}^{m \times n}$. Then $f$ coul be RSME between some $x$ and $y$, i.e. $f(x) = {\| y - x \|}_2$. Notice that $y$ is given and it does not change.

Now the key question here is how to represent a stroke, or more formally a function the values of $x$. A very basic idea can be represent the stroke by the following tuple $(o, e, t, c)$ where $o$ and $e$ are two dimensional points, the origin and the end of the stroke; $t$ is the thickness of the stroke and $c$ is the color of the stroke. Let $S$ be the space of all the strokes, assuming the thickness and colors are finite, S is finite. The possible pairs of $o$ and $e$ are finite namely all the pairs of possible points in $m \times n$. Now let $S^k$ be the composition of $S$ with itself $k$ times.

Finally the problem can be represented the following way:

$$\min_{s \in S^k}{\| y - t(s, x) \|}$$

Where $t$ is the function that applies $s$, that is a number of different strokes to an empty canvas $x$, by empty canvas I mean a matrix full of zeros.

Summarizing the ideas expose above are a possible formalization of your problem, of course is not the only one. The main issue is that the robot does not "learn" is simply computes the optimal set of strokes to apply.

Other ideas

Another possible approach is to use reinforcement learning, this learning paradigm does not needs labeled training samples, it relies in a reward function $R$ to guide the learning process. Also if defines a set of states and actions the agent can be or do. The states could be the strokes and the actions add another stroke, the reward function can be a variation of the error function. Also I am far from being an expert in reinforcement learning I guess is worth looking into it.

A proven approach

The problem you describe has been demonstrated in real life. See this paper: Feedback-guided Stroke Placement for a Painting Machine. The paper demonstrates a system that describes your scenario, also see this site for more work of the authors or this for the site of the project.

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