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Multiple resources I referred to mention that MSE is great because it's convex. But I don't get how, especially in the context of neural networks.

Let's say we have the following:

  • $X$: training dataset
  • $Y$: targets
  • $\Theta$: the set of parameters of the model $f_\Theta$ (a neural network model with non-linearities)

Then:

$$\operatorname{MSE}(\Theta) = (f_\Theta(X) - Y)^2$$

Why would this loss function always be convex? Does this depend on $f_\Theta(X)$?

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  • $\begingroup$ Although this is an older question, the other duplicate has an accepted answer, and, overall, the answers there have more upvotes, so they seem to be more useful, so I marked this one as a duplicate of the newer one. $\endgroup$
    – nbro
    Nov 16 '21 at 18:24
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Answer in short: MSE is convex on its input and parameters by itself. But on an arbitrary neural network it is not always convex due to the presence of non-linearities in the form of activation functions. Source for my answer is here.

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Convexity

A function $f(x)$ with $x ∈ Χ$ is convex, if, for any $x_1 ∈ Χ$, $x_2 ∈ Χ$ and for any $0 ≤ λ ≤ 1$, $$f(λ x_1 + (1 − λ) x_2) ≤ λf(x_1) + (1 − λ) f (x_2).$$

It can be proven that such convex $f(x)$ has one global minimum. A unique global minimum eliminates traps created by local minima that can occur in algorithms that attempt to achieve convergence on a global minimum, such as the minimization of an error function.

Although an error function may be 100% reliable in all continuous, linear contexts and many non-linear contexts, it does not mean the convergence on a global minimum for all possible non-linear contexts.

Mean Square Error

Given a function $s(x)$ describing ideal system behavior and a model of the system $a(x, p)$ (where $p$ is the parameter vector, matrix, cube, or hypercube and $1 ≤ n ≤ N$), created rationally or via convergence (as in neural net training), the mean square error (MSE) function can be represented as follows.

$$e(β) := N^{-1} \sum_{n} [a(x_n) − s(x_n)]^2$$

The material you are reading is probably not claiming that $a(x, p)$ or $s(x)$ are convex with respect to $x$, but that $e(β)$ is convex with respect to $a(x, p)$ and $s(x)$ no matter what they are. This later statement can be proven for any continuous $a(x, p)$ and $s(x)$.

Confounding the Convergence Algorithm

If the question is whether a specific $a(x, p)$ and method of achieving an $s(x)$ that approximates the $a(x, p)$ within a reasonable MSE convergence margin can be confounded, the answer is, "Yes." That is why MSE is not the only error model.

Summary

The best way summarize is that $e(β)$ should be defined or chosen from a set of stock convex error models based on the following knowledge.

  • Known properties of the system $s(x)$
  • The definition of the approximation model $a(x, p)$
  • Tensor used to generate the next state in the convergent sequence

The set of stock convex error models certainly includes the MSE model because of its simplicity and computational thrift.

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