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I thought I understood what "monotonically increasing" meant, but in the textbook Understanding Machine Learning by Shalev-Schwarz and Ben-David, I've encountered the term "monotonically non-decreasing". What does it mean for a function to be monotonically non-decreasing, and how is this different from just monotonically increasing?

Sam Wilkinson
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  • $f(x) = x$ is monically non-decreasing but not (strictly) monotonically increasing. Every monotonically increasing function is monically non-decreasing. I prefer to use either monically non-decreasing or strictly monotonically increasing to avoid ambiguity. – copper.hat Jul 18 '21 at 18:21
  • @MartinR not really, as the bit that is tripping me up is specifically the "monotonically" bit. Perhaps my question could have been phrased as "how does a 'monotonically non-decreasing' function differ from a simply 'non-decreasing' function?" I mean, what, exactly, is the word "monotonically" adding here? I think I get what it does for "monotonically increasing," but not for "monotonically non-decreasing." – Sam Wilkinson Jul 18 '21 at 19:09

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In most modern math texts, “monotonically increasing” is used to mean non-decreasing, and we use “strictly monotonically increasing” if we mean “really increasing.”

I’d guess that, because of the topic of the book, the authors might expect non-math readers, and want to avoid confusion for those readers who are not accustomed to the mathematical usage.

In particular, it is very odd in natural language to call a constant function “monotonically increasing.”

Thomas Andrews
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  • So, if I'm understanding correctly, it should really just be to emphasise that the function may be constant over some range, and even though this would normally seem to be included in the definition of "monotonically increasing", the use of the phrase "monotonically non-decreasing" eliminates any potential ambiguity?

    That certainly makes sense in the context of the book, but I'm not sure what (if anything) the word "monotonically" contributes to "non-decreasing." How would a "monotonically non-decreasing" function differ from a merely "non-decreasing" one?

     – Sam Wilkinson Jul 18 '21 at 19:05
  • Yeah, monotonically is a somewhat frivolous word here. – Thomas Andrews Jul 18 '21 at 19:47