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I have a regression problem with 5-6k variables. I divide my data into 3 non-overlapping sets: training, validation, and testing. I train using only the training set, and generate a lot of different linear regression models by choosing a different set of 200 variables for each model (I try about 100k such subsets). I score a model as $\min(R^2_{\text{training data}}, R^2_{\text{validation data}})$. Using this criterion, I end up choosing a model. It turns out that the model chosen has very similar $R^2$ on the training and the validation data. However, when I try this model on the testing data, it has much lower $R^2$. So it seems I am somehow overfitting on both the training and the validation data. Any ideas on how can I get a more robust model?

I tried increasing the training data size, but that didn't help. I am thinking of perhaps shrinking the size of each subset.

I have tried using regularization. However, the models I obtain using the lasso or the elastic net have much lower $R^2$ on the training set as well as the validation set, as compared to the model I obtain by doing the subset selection approach. Therefore, I don't consider these models, since I assume that if Model A performs better than Model B on both the training set as well as the validation set, Model A is clearly better than Model B. I would be very curious if you disagree with this.

On a related note, do you think $R^2$ is a bad criteria for choosing my models?

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While this sounds somewhat like overfitting, I think it's actually more likely that you've got some kind of "bug" in your code or your process. I would start by verifying that your test set isn't somehow systematically different from the training/validation set. Suppose your data is sorted by date (or whatever). If you used the first 50% for training, the next 25% for validation, and the rest for testing, you may have accidentally stratified your data in a way that makes the training data somewhat representative of the validation data, but less so for the testing data. This is fairly easy to do by accident.

You should also ensure you're not "double-dipping" in the validation data somehow, which sometimes happens accidentally.

Alternately, CV's own @Frank Harrell has reported that a single train/test split is often too variable to provide useful information on a system's performance (maybe he can weigh in with a citation or some data). You might consider doing something like cross-validation or bootstrapping, which would let you measure both the mean and variance of your accuracy measure.

Unlike Mikera, I don't think the problem is your scoring mechanism. That said, I can't imagine a situation where your $R^2_{training} < R^2_{validation}$, so I'd suggest scoring using the validation data alone.

More generally, I think $R^2$ or something like it is a reasonable choice for measuring the performance of a continuous-output model, assuming you're aware of its potential caveats. Depending on exactly what you're doing, you may also want to look at the maximum or worst-case error too. If you are somehow discretizing your output (logistic regression, some external thresholds), then looking at precision/recall/AUC might be a better idea.

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  • $\begingroup$ Thank you for your reply Matt. I do indeed have data sorted by date. I divide it into 3 parts and use the first part for training, the next part for validation, and the last part for testing. For my application, the testing data will always be chronologically after the training and validation set, though I can intermingle training and validation set however I like (including doing cross validation). I will give cross validation a try. Although, I don't expect it to do that well because my $R^2$ on both the training and validation set are pretty close. <Rest continued in next comment> $\endgroup$ – user10 Oct 1 '13 at 14:02
  • $\begingroup$ I will also check the variance in $R^2$ for different days. If the variance is high, I would expect cross validation to be helpful. If not, I would expect it to give similar results to just having a validation set like I already have. Thanks again! $\endgroup$ – user10 Oct 1 '13 at 14:04
  • $\begingroup$ The ordering per se isn't the problem; it's that the training set might be more representative of the validation set than the test set. For example, imagine you're predicting a store's sales. If your training set contains June, July, and August it will probably do a good job of predicting September's sales too (the validation set). However, it might completely fall apart when tested on November and December's sales: people are buying holiday gifts and winter clothes instead of shorts and sunscreen, etc. $\endgroup$ – Matt Krause Oct 2 '13 at 5:56
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You are overfitting because you are using min(training r-square,validation r-square) data to produce a score, which is in turn being used to drive model selection. Because your training r-square is likely to be equal or lower (you just ran a regression on it, after all), this is roughly equivalent to doing model selection on the r-square of the training data.

This has the effect of fitting too tightly to the training data, and ignoring the validation data.

If you used just validation r-square then you should get a better result.

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  • $\begingroup$ But isn't that the point of having a validation data set? Shouldn't I just pick a model which has low error on the validation (i.e. out of sample) set? $\endgroup$ – user10 Sep 30 '13 at 1:13
  • $\begingroup$ Sorry I slightly misread your question. I have amended the answer. $\endgroup$ – mikera Sep 30 '13 at 1:21
  • $\begingroup$ Since I ran a regression, my training $R^2$ should generally be greater than the validation $R^2$, and since I use the min, isn't it equivalent to doing model selection using the validation set? It just so happens that since I use min, I get a good $R^2$ for both training and validation, which unfortunately doesn't translate to the test set. Thanks! $\endgroup$ – user10 Sep 30 '13 at 6:03

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