In this post I walk through a recent paper about multi-task learning and fill in some mathematical details. Implementation and experiments will follow in a later post.

\[\newcommand{\vx}{\mathbf{x}} \newcommand{\vw}{\mathbf{w}}\]

Multi-task Learning

Multi-task learning is generally defined as solving two or more ML problems in an integrated way that results in synergies and better performance on each of the independent tasks. For example, object recognition and depth perception in artificial vision. Knowing their distance from an observer helps telling a mouse from an elephant or a rook (a chess piece) from a medieval tower, for sure. There are many reasons why this may be helpful and there are even more open questions, but the approach has taken off and is widely adopted at least in AI research circles. A nice overview is available.

One common variant of multi-task learning consists of co-training a single model to perform several tasks at once. In statistical parlance, this would be multivariate (multiple) regression. Since training is achieved in most cases by optimizing a loss function, varying the free parameters of the model, one is left with the problem of formulating a unified loss function that captures the performance on all tasks simultaneously. Unfortunately, this is not so simple. In the image processing example, the output of object recognition is typically a label, whereas the output of depth perception is a depth map (pixel-by-pixel depth prediction). The output is in completely different units and the performance on each task is measured in ways that are not comparable. How much depth error is equivalent to one mislabeled example? Maybe one way would be to let the application be our guide, and convert everything into a “dollar value”. But even if the different tasks consist of predictions in the same units, there are statistical issues to consider. The tasks can differ in their complexity and signal to noise ratio. Imagine predicting multiple stock prices, or the weather at different locations. This can have a negative effect on the balancing act between bias and variance that needs to be achieved for optimal performance. There is a risk to overfit the noisier tasks in an attempt to reduce the error while underfitting the less noisy ones.

The Paper

This problem and a possible solution, with an application to image processing, are the subject of a recent and interesting paper: Multi-Task Learning Using Uncertainty to Weigh Losses for Scene Geometry and Semantics. If I have a problem with that paper, is that they use the word homoscedastic (constant variance) all over the place when their main contribution is to recognize and take into account the heteroscedasticity (variable variance) of multitask learning. If it’s lucky to have constant variance within a task, it would be downright miraculous to have it across tasks. Therefore it is advisable to model the heterogeneity of different tasks with the addition of per-task variance parameter.

Another slight disappointment is that this paper doesn’t explain in full detail how it goes about estimating those per-task variances. Given a well-behaved loss function, which they derive in detail, it’s easy to imagine that they use SGD or equivalent to simultaneously fit the weights of a neural network and the per-task variances. But how this is accomplished is not clear, nor an implementation is provided — email to the first author has not been returned yet. One would think that in this day and age of open science and reproducibility crisis we should do better than that. While trying to fill in the missing details, I derived a closed form solution to the problem of per-task variance fitting, which I present below. In a follow up post, I will provide a commented example and Keras implementation of the loss function thus derived.

Mathematical Derivation of the Loss function

We largely follow the notation in the aforementioned paper. Let \(i\) index the training set and \(j\) the dependent variables (the “tasks” in multi-task learning). Under the assumptions that the such variables, with realizations \(y_{ij}\), are independent, conditional to the prediction returned by a model \(f\), with adjustable parameters \(\vw\), on input \(\vx_i\); and that the error is normally distributed and zero-mean, with variance \(\sigma_j^2\), which depends only on \(j\) (the task), we can write the log-likelihood function as follows:

\[\sum_{ij}\log\left(\frac{1}{\sqrt{2\pi}\sigma_j} \exp\left(-\frac{\left(y_{ji}-f_j(\vx_i;\vw)\right)^2}{2\sigma_j^2}\right)\right)\]

By the basic properties of the \(\log\) function this can be rewritten as:

\[\sum_{ij}\left(-\log(\sqrt{2\pi}) -\frac{1}{2}\log\sigma_j^2 - \frac{\left(y_{ji}-f_j(\vx_i;\vw)\right)^2}{2\sigma_j^2}\right)\]

Looking for stationary points of this loss w.r.t. the variance \(\sigma_j^2\), considered a variable with slight abuse of notation — the substitution is immaterial to finding a minimum — and, dropping a constant additive term, we have:

\[\frac{\partial}{\partial\sigma_j^2}\sum_i\left( -\frac{1}{2}\log\sigma_j^2 - \frac{\left(y_{ji}-f_j(\vx_i;\vw)\right)^2}{2\sigma_j^2}\right) = 0\]

Applying the linearity of partial derivatives, and calculating the derivatives for each term we have:

\[\sum_i\left(-\frac{1}{2\sigma_j^2}+ \frac{\left(y_{ji}-f_j(\vx_i;\vw)\right)^2}{2\sigma_j^4}\right) = 0\]

The first term is independent of \(i\) so it can be extracted from the summation:

\[-\frac{N}{2\sigma_j^2}+\sum_i \frac{\left(y_{ji}-f_j(\vx_i;\vw)\right)^2}{2\sigma_j^4}= 0\]

where \(N\) is the size of the training set. We can simplify a common \(1/2\sigma_j^2\) factor:

\[-N+\sum_i \frac{\left(y_{ji}-f_j(\vx_i;\vw)\right)^2}{\sigma_j^2} = 0\]

And, finally, solving for \(\sigma_j^2\), we have:

\[\sigma_j^2 = \frac{1}{N}\sum_i\left(y_{ji}-f_j(\vx_i;\vw)\right)^2\]

We can now substitute this into the likelihood, neglecting a constant term:

\[\sum_{ij}\left( -\frac{1}{2}\log\left(\frac{1}{N}\sum_i \left(y_{ji}-f_j(\vx_i;\vw)\right)^2\right)- \frac{\left(y_{ji}-f_j(\vx_i;\vw)\right)^2}{\frac{2}{N}\sum_i \left(y_{ji}-f_j(\vx_i;\vw)\right)^2}\right)\]

We observe the first term is independent of \(i\) and can be thus pulled out of the summation over that variable:

\[\sum_j\left(-\frac{N}{2}\log\left(\frac{1}{N}\sum_i \left(y_{ji}-f_j(\vx_i;\vw)\right)^2\right)- \frac{\sum_i\left(y_{ji}-f_j(\vx_i;\vw)\right)^2}{\frac{2}{N}\sum_i \left(y_{ji}-f_j(\vx_i;\vw)\right)^2}\right)\]

The second term allows some drastic simplification:

\[\sum_j\left(-\frac{N}{2}\log\left(\frac{1}{N}\sum_i \left(y_{ji}-f_j(\vx_i;\vw)\right)^2\right)-\frac{N}{2}\right)\]

Then by dropping constant additive and multiplicative terms:

\[-\sum_j\log\left(\sum_i \left(y_{ji}-f_j(\vx_i;\vw)\right)^2\right)\]

To obtain a loss function, we need to flip the sign and, optionally, exponentiate to return to the original scale of the quadratic loss:

\[\prod_j\left(\sum_i \left(y_{ji}-f_j(\vx_i;\vw)\right)^2\right)\]

Now, discounting the possibility of errors, this is an interesting result: it says that when variances are unknown we can’t average the losses among different tasks, not matter how weighted, which would be possible in the case of known variances; we have instead to switch to a geometric average of the losses. Otherwise stated, it says that the doubling of the loss on one task cancels out the halving of it on another one, which would not be the case when variances are known. It’s quite neat, and I wouldn’t be surprised if it had been observed before in a different context.

This result has a very practical consequence: we do not need to fit the variances during the training of the neural network. Leveraging the closed form solution we can train in the multitask case with no increase in complexity and by simply implementing this new loss function. And that’s exactly what we are going to do in the next post (link will appear in the footer on the right), together with a couple of experiments. Stay tuned.

Update: Thanks to João Paulo Lima for pointing out an error in the math. Luckily it doesn’t affect the final expression. It has now been corrected in the main text.