# Best loss function for nonlinear regression

I have some nonlinear nonnormal data that I am trying to analyze. The data has been normalized to -1 to 1 and detrended with polynomials of an order of 3. I'm trying to determine if there is a special loss function for nonlinear regression type problems. I did some googling but not much showed up. Right now I'm sticking with MSE.

• The best loss function is the one that best represents the real-world loss incurred from inaccurate predictions in production for a specific application. Aug 24, 2021 at 9:35
• @BernhardBarker, generally a good idea. Sometimes it can be improved upon; training loss does not have to match evaluation loss if we want to achieve the best out-of-sample results; see Dave's and my comments under Stephan Kolassa's answer. Aug 24, 2021 at 10:42

There's no such a thing as a loss function "for" a particular kind of model. You could be using nonlinear regression with different loss functions. There are many loss functions and you can even construct one yourself. The choice depends on the nature of your problem and the data you are dealing with. Recall that minimizing some loss is equivalent to maximizing a likelihood function (e.g. using squared error is an equivalent of assuming Gaussian likelihood function), so it is tightly connected to the assumptions you are making about the distribution of errors.

More formally, if you think of the model as of something like

$$y = f(X) + \varepsilon$$

then the choice of model (e.g. linear regression, nonlinear regression, deep neural network, etc) is related to estimating the expectation $$E[y] = f(X)$$, while the choice of the loss function impacts how do you treat the residuals $$y - f(X) = \varepsilon$$.

For example, choosing squared error over absolute error penalizes outliers more, so it would be preferable if this is what you want to achieve. On another hand, absolute error is less prone to outliers, this can be an advantage in another scenario.

The most common choice is defaulting to squared error, though it is somehow an arbitrary choice and doesn't have to be the best in all cases.

• the errors are not normal, data is nonlinear, nonnormal...i probably got the time lag wrong...i just know that regression doesn't work well. Aug 23, 2021 at 17:19
• @intuition data cannot be nonlinear, function can be linear or not. Moreover, linear regression can in many cases approximate well such cases. Normality of residuals has nothing to do with the nature of functional relationship. What can you assume about the residuals?
– Tim
Aug 23, 2021 at 18:41
• that they aren't noise is about the only thing i can assume Aug 23, 2021 at 19:53
• @intuition what do you mean by that?
– Tim
Aug 24, 2021 at 5:03
• then the choice of model (e.g. linear regression, nonlinear regression, deep neural network, etc) is related to estimating the expectation: does this apply e.g. to quantile regression? E.g. in your example of absolute loss, one obtains the conditional median. I guess this also relates to Stephan Kolassa's answer. Aug 24, 2021 at 6:32

Other answers (like bdeonovic's and Tim's) discuss "robustness to outliers". I have to admit that while this point of view is extremely common, I do not like it very much.

I find it more helpful to think in terms of which conditional fit (or prediction) we want.

• Use the squared errors if you want conditional expectations as fits or predictions. ("Outliers" are then simply observations that are "far away" from the expectation, and which therefore pull the expectation towards them. If your aim is an expectation fit/prediction, then you should think long and hard about whether you want "robustness to outliers", because "outliers" are a fact of life.)
• Use the absolute errors if you want condititional medians as fits or predictions.
• Use quantile (AKA pinball) losses if you want conditional quantiles as fits or predictions.

I have written a short paper (Kolassa, 2020, IJF) on this, in the context of forecasting - but the idea holds in the precise same way for fits.

Thus, I would recommend you think about what kind of fit/prediction you want, and then tailor your loss function to this.

• I'm not convinced of this. For example, if $X$ has a Laplace distribution, the mean is a viable estimator of $\mathbb{E}[X]$, but the median is considered better. Applying a similar idea to regression, explicitly estimating conditional medians could be a better way of estimating conditional means than explicitly estimating conditional means, could it not?
– Dave
Aug 23, 2021 at 16:39
• @Dave: I would ask "better in what sense"? At Wikipedia, you would read "the median is considered better[by whom?]" And if your goal is to estimate the conditional median, then sure, use absolute errors. My point is that figuring out which functional of the unknown distribution we are interested in is the key question that determines which loss function to use. Aug 23, 2021 at 17:15
• It's the lower variance estimator and also unbiased.
– Dave
Aug 23, 2021 at 17:17
• Dave has a point there. Empirical counterparts of theoretical quantities of interest are not always the best estimators. Better in the sense that empirical median is the MLE of the target parameter, and MLE has certain optimality properties (so it can be considered best in a well defined sense). Aug 24, 2021 at 6:39
• @JeeyCi: I believe this is a misleading way of looking at things. Better to think of this as "the MAE elicits the conditional median". Nov 22, 2023 at 16:04

Most of the alternative loss functions are for making the regression more robust to outliers. I've seen all of the following in various software package implementations, but I haven't looked too hard into the literature comparing them

1. least absolute deviation
2. least median of squares
3. least trimmed squares
4. metric trimming
5. metric winsorizing
6. Huber Loss
7. Tukey's biweight loss
8. soft L1 loss
9. Cauchy loss
10. arctan loss

How are you doing the optimization? Did you code it yourself? Are you using Gaus-Newton or Gradient Descent? May want to consider Levenberg–Marquardt (interpolates between Gaus-Newton and Gradient Descent)

• Using adam as an optimizer, its with a bilstm-rnn model thats been further optimized using GA. Aug 23, 2021 at 17:15