->
In our cat recognition example, the “ideal” error rate—that is, one achievable by an “optimal” classifier—is nearly 0%. A human looking at a picture would be able to recognize if it contains a cat almost all the time; thus, we can hope for a machine that would do just as well. ->
Other problems are harder. For example, suppose that you are building a speech recognition system, and find that 14% of the audio clips have so much background noise or are so unintelligible that even a human cannot recognize what was said. In this case, even the most “optimal” speech recognition system might have error around 14%. ->
Suppose that on this speech recognition problem, your algorithm achieves: ->
-
Training error = 15% ->
-
Dev error = 30% ->
The training set performance is already close to the optimal error rate of 14%. Thus, there is not much room for improvement in terms of bias or in terms of training set performance. However, this algorithm is not generalizing well to the dev set; thus there is ample room for improvement in the errors due to variance. ->
This example is similar to the third example from the previous chapter, which also had a training error of 15% and dev error of 30%. If the optimal error rate is ~0%, then a training error of 15% leaves much room for improvement. This suggests bias-reducing changes might be fruitful. But if the optimal error rate is 14%, then the same training set performance tells us that there’s little room for improvement in the classifier’s bias. ->
For problems where the optimal error rate is far from zero, here’s a more detailed breakdown of an algorithm’s error. Continuing with our speech recognition example above, the total dev set error of 30% can be broken down as follows (a similar analysis can be applied to the test set error): ->
-
Optimal error rate (“unavoidable bias”): 14%. Suppose we decide that, even with the best possible speech system in the world, we would still suffer 14% error. We can think of this as the “unavoidable” part of a learning algorithm’s bias. ->
-
Avoidable bias: 1%. This is calculated as the difference between the training error and the optimal error rate. [8] ->
-
Variance: 15%. The difference between the dev error and the training error. ->
To relate this to our earlier definitions, Bias and Avoidable Bias are related as follows:[9] ->
Bias = Optimal error rate (“unavoidable bias”) + Avoidable bias ->
The “avoidable bias” reflects how much worse your algorithm performs on the training set than the “optimal classifier.” ->
The concept of variance remains the same as before. In theory, we can always reduce variance to nearly zero by training on a massive training set. Thus, all variance is “avoidable” with a sufficiently large dataset, so there is no such thing as “unavoidable variance.” ->
Consider one more example, where the optimal error rate is 14%, and we have: ->
-
Training error = 15% ->
-
Dev error = 16% ->
Whereas in the previous chapter we called this a high bias classifier, now we would say that error from avoidable bias is 1%, and the error from variance is about 1%. Thus, the algorithm is already doing well, with little room for improvement. It is only 2% worse than the optimal error rate. ->
We see from these examples that knowing the optimal error rate is helpful for guiding our next steps. In statistics, the optimal error rate is also called Bayes error rate, or Bayes rate. ->
How do we know what the optimal error rate is? For tasks that humans are reasonably good at, such as recognizing pictures or transcribing audio clips, you can ask a human to provide labels then measure the accuracy of the human labels relative to your training set. This would give an estimate of the optimal error rate. If you are working on a problem that even humans have a hard time solving (e.g., predicting what movie to recommend, or what ad to show to a user) it can be hard to estimate the optimal error rate. ->
In the section “Comparing to Human-Level Performance (Chapters 33 to 35), I will discuss in more detail the process of comparing a learning algorithm’s performance to human-level performance. ->
In the last few chapters, you learned how to estimate avoidable/unavoidable bias and variance by looking at training and dev set error rates. The next chapter will discuss how you can use insights from such an analysis to prioritize techniques that reduce bias vs. techniques that reduce variance. There are very different techniques that you should apply depending on whether your project’s current problem is high (avoidable) bias or high variance. Read on! ->
FOOTNOTE:
[8] If this number is negative, you are doing better on the training set than the optimal error rate. This means you are overfitting on the training set, and the algorithm has over-memorized the training set. You should focus on variance reduction methods rather than on further bias reduction methods. ->
[9] These definitions are chosen to convey insight on how to improve your learning algorithm. These definitions are different than how statisticians define Bias and Variance. Technically, what I define here as “Bias” should be called “Error we attribute to bias”; and “Avoidable bias” should be “error we attribute to the learning algorithm’s bias that is over the optimal error rate.” ->