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Suppose the output of my network is $y \in [0, 1]$ for a given input x.

The loss function to be minimized is $f(x) = -\Sigma (w_i y_i + | y_i - y_{i-1}|)$, with real weights $w_i$.

The dependency of one output to the other means I can't calculate derivatives and plug this into a backpropagation algorithm. (I've tried and failed.)

I see two questions:

  • Have I given up on calculus too early? Is it possible to differentiate $f(x)$ for use in backpropagation?
  • If not, what efficient methods can I use to find an optimal solution? Right now I randomly tinker with weights and biases, and accept them if I find a new minimum.
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  • $\begingroup$ A trivial solution to minimize this objective function is to ignore the input and force all outputs to the same value (e.g. force to 0 by setting a large negative bias). Are you sure this is what you want to minimize? Also, if $y_i \in \{0, 1\}$ then $f(y)$ can only take integer values, and isn't differentiable, so gradient-based methods like backprop aren't possible. $\endgroup$
    – user20160
    Commented Jan 25, 2017 at 9:25
  • $\begingroup$ Mayor oops, I didn't have enough coffee yet. I forgot to add weights. And yes y doesn't need to be integer. Thanks for your comment! $\endgroup$
    – keepitwiel
    Commented Jan 25, 2017 at 9:56

2 Answers 2

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Let $\theta$ denote a vector of network parameters to learn (weights/biases) and $g$ denote the function the network implements, so $y_i = g(x_i)$ (implicitly depends on $\theta$). It looks like the gradient of your loss function w.r.t. $\theta$ is:

$$-\sum_{i=1}^n [ w_i \nabla_\theta g(x_i) + \text{sgn}(g(x_i) - g(x_{i-1})) [\nabla_\theta g(x_i) - \nabla_\theta g(x_{i-1})] ]$$

On one hand, this seems pretty clean, because you can easily find the gradient of $g$, evaluate it for the different inputs, and plug in those values. There are 2 tricky parts:

Because of the absolute value in the loss function, its gradient is undefined when two subsequent outputs are identical. Depending on your data, that might be a very unlikely circumstance. Otherwise, I suppose you could call the expression above a subgradient and use it to do subgradient descent, which looks pretty much like regular gradient descent. The sgn function will return 0 where subsequent outputs are identical. Consider that something similar occurs with ReLUs; the activation function is non-differentiable at 0, but the subgradient is typically declared to be 0, 0.5, or 1 at this point, and standard gradient-based learning methods work fine (e.g. see here).

As you mention, standard online training methods (e.g. stochastic gradient descent) might not work so well because of the dependence on sequential outputs. You could try doing batch training, where you sweep through all data points to compute the gradient and update parameters once per sweep.

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  • $\begingroup$ Thanks for your well stated reply, will check out subgradient descent! $\endgroup$
    – keepitwiel
    Commented Jan 25, 2017 at 14:02
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You could try some other optimization scheme, such as particle swarm optimization, or perhaps Bayesian optimization using a library such as Bayesopt.

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