Single loss value for gradient descent in neural network optimization

Question

Suppose I train 2 neural networks in stock trading. First network produces enter-trade signals (very sparse). The second network produces exit-trade signals and starts to give signals right after the first gives positive signal.

Question is whether it is mathematically sane to get gradient using just one loss value, which is a function of a squared difference of cumulative return (after 10K trades, for example) from theoretical cumulative return. Doing this trick for both networks, like sharing gradient.

E.g., L <- (-5000 - 20000)^2.

Clarification example of what I meant:

Suppose neural networks have been randomly initialized.

sum_ret_theoretical <- 20000 # theoretical cumulative return

price, input_1, input_2, neural_output_1, neural_output_2
1.000, 0.1, 0.2, 0, NA
1.001, 0.3, 0.1, 0, NA
1.007, 0.9, 0.2, 1, NA

Here first NN gave positive signal

1.01, 0.1, 0.2, NA, 0
1.016, 0.3, 0.1, NA, 0
1.03, 0.9, 0.2, NA, 1

Here second NN gave positive signal

return(i) = 1.03 - 1.007 - 0.0001 = 0.0229 # (trade gained a positive outcome after excluding a constant cost).

Go through all examples [1 in 1:I] until the end of the sample, accumulate returns in vector Ret. Here first training episode has ended.

sum_ret <- sum(Ret)

L <- (sum_ret - sum_ret_theoretical) ^ 2 # get loss function value after first episode ends

calculate neural networks' updates w.r.t. to L derivation (gradients) ## that is the question.

Can I train NN based on extremely sparse loss values (1 per epoch/episode) that is a function of accumulated NN performance. An example of that maybe a NN that is designed to approximate the sum (mean) of the sample of values from variable X, without looking at each x to get loss.

Answer 1

Notes: if you have two NN who train on the same features, you can make a single one with a two row vector prediction. Also instead of having a loss function converging to sum_ret_theoretical, maybe you could tweak it to have a score to maximize.

Case 1: If you do stochastic gradient descent (batch size = 1), the loss function is either undefined or 0.

Case 2: If your batch size > 1, your loss function can be undefined if neural_output_1 = 1 after neural_output_2 = 1.

Therefore I believe you cannot achieve it.

To deal with temporal data, check out recurrent neural networks. To maximize gain, check out (deep) reinforcement learning.

Here, it looks like you're trying to train with data from the future, which bidirectional RNN could handle. But if you're planning to trade, you can't use data from the future.