# Defining an objective function for machine learning task of trading

A simplified example. Given:

• asset's price time series
• fixed distances to stop and target.

A function of these inputs has two possible output values: $$1$$ if price is likely to hit the target earlier than stop and $$0$$ otherwise. This function is implemented by means of machine learning, e.g. a neural network.

For reasonably and equally sized target and stop, the feasible win rate will be close to $$\frac{1}{2}$$. Most of times the outcome is not predictable and the right choice is not to open a position. From time to time the odds are marginally better than $$\frac{1}{2}$$; this is when I want the algorithm to output a $$1$$. Cherry picking.

When putting this up as a binary classifier, it appears that the widely used binary cross-entropy loss

1. strongly penalizes confident mis-classifications
2. weakly rewards being rightfully more confident

and is not necessarily appropriate here.

To be trainable by means of stochastic gradient descent, this neural network needs to produce a continuous output $$\in(0;1)$$, and the objective function must have a useful derivative. This is why simply counting winners and losers wouldn't help. The objective function needs to weigh network's confident and less confident decisions differently.

How would you deal with it?

Here is a similar question, probably better written.

• This question seems contradictory in its description. Above the function has two possible output values {0,1} - but the function is a neural network, which typically, as you state below has a continuous output in (0,1). Have you parametrised your neural network to output only {0,1} in the final layer, is that a requirement?
– Attack68
Feb 20, 2019 at 23:21
• @Emma, thanks for tips. It is just a simple ResNet. Feb 21, 2019 at 8:25
• @Attack68, it is actually common to have a continuous output, the training wouldn't be feasible otherwise. For inference, the output is just compared with threshold. Feb 21, 2019 at 8:26
• So your neural network, as normal, outputs a value in (0,1) and weights are trained by comparing continuous output with known {0,1} classes, allowing for the calculation of some loss function and derivatives. Afterwards you implement a thresholding filter to select samples which are more likely to result in 1? Your overall objective is then to minimise the number of misclassifications, am I right? The problem then depends on how many predictions you require, zero gives zero misclassifications but is useless information..
– Attack68
Feb 21, 2019 at 8:55
• @Attack68, you pinpointed it. I expect to balance between trade frequency and win rate by adjusting the threshold. Feb 21, 2019 at 9:33

The code below is written in Wolfram Mathematica.

For example, we have some training data. And we are trying to predict: long (1) or short (0).

SeedRandom[0];
n = 10000;
X = RandomReal[{-1, 1}, {n, 100, 5}];
Y = RandomInteger[{0, 1}, n];

net = NetChain[
{
LongShortTermMemoryLayer[64],
SequenceLastLayer[],
ElementwiseLayer[Ramp],
LinearLayer[2],
SoftmaxLayer[]
},
"Input" -> {100, 5},
"Output" -> NetDecoder[{"Class", {0, 1}}]
]


SeedRandom[0];
netT = NetTrain[
net,
X -> Y,
All,
LossFunction -> CrossEntropyLossLayer["Index"],
BatchSize -> 64, MaxTrainingRounds -> 10, TargetDevice -> "GPU"
]


cm = ClassifierMeasurements[netT["TrainedNet"], X -> Y]


cm["ConfusionMatrixPlot"]


cm["Precision"]


<|0->0.569919,1->0.580981|>

cm["Recall"]


<|0->0.641173,1->0.506965|>

proba = netT["TrainedNet"][X, "Probabilities"];
proba0 = Lookup[proba, 0];
proba1 = Lookup[proba, 1];

PairedHistogram[Pick[proba0, Y, 0], Pick[proba0, Y, 1]]


Table[
{
Select[Pick[proba0, Y, 0], # >= i &] // Length,
Select[Pick[proba0, Y, 1], # >= i &] // Length
} // {i, ##, N[#[[1]]/(#[[1]] + #[[2]])]} &,
{i, 0.5, 0.7, 0.01}
] // MatrixForm


PairedHistogram[Pick[proba1, Y, 1], Pick[proba1, Y, 0]]


Table[
{
Select[Pick[proba1, Y, 1], # >= i &] // Length,
Select[Pick[proba1, Y, 0], # >= i &] // Length
} // {i, ##, N[#[[1]]/(#[[1]] + #[[2]])]} &,
{i, 0.5, 0.7, 0.01}
] // MatrixForm


Calculating the expected P&L

EV[p_] := p*80 - (1 - p)*100 - 4

Plot[
{
Select[Pick[proba0, Y, 0], # >= x &] // Length,
Select[Pick[proba0, Y, 1], # >= x &] // Length
} // EV[#[[1]]/(#[[1]] + #[[2]])]*(#[[1]] + #[[2]]) &,
{x, 0.5, 0.7}
]


Plot[
{
Select[Pick[proba1, Y, 1], # >= x &] // Length,
Select[Pick[proba1, Y, 0], # >= x &] // Length
} // EV[#[[1]]/(#[[1]] + #[[2]])]*(#[[1]] + #[[2]]) &,
{x, 0.5, 0.7}
]


• Impressive! So your message is - "cross entropy loss will do it just fine"? Feb 22, 2019 at 13:14
• @Xpector Exactly! But after training the network, you should manually adjust the optimal threshold for trading. It depends on your EV. In my example: 80 points is the average profit, -100 is the average loss, -4 is the fees. Feb 22, 2019 at 14:23

Whilst reading this I realized that it would be a really good application for meta-labeling. The idea behind meta-labeling is to build a secondary model that determines if the signals {0, 1} from the primary model are correct or not.

By doing this the secondary model outputs a value between 0 and 1 indicating how confident the model is that the primary model is correct or not. This output can then be passed to a bet sizing algorithm which maps the output to a position size. The core idea being that we want to take large positions on trades that are likely to be true and smaller positions on trades when we are unsure.

To give some intuition behind this. Lets take a trend following strategy as an example. Now moving average crossover strategies are known to under perform when the market moves sideways. The choppy nature causes a lot of transaction fees.

The secondary model will pick up that under some volatility conditions and perhaps a low auto correlation, that we are in a side ways trend and thus the primary models signal (a 1 in this case) is likely to be false and so it assigns it a low probability.

More about this technique can be read about in Chapter 3 of Advances in Financial Machine Learning. A toy example of meta-labeling can be found here Meta-Labeling on MNIST Data.