Two aspects of statistical learning are useful for trading
1. First the ones mentioned earlier: some statistical methods focused on working on live datasets. It means that you know you are observing only a sample of data and you want to extrapolate. You thus have to deal with in sample and out of sample issues, overfitting and so on... From this viewpoint, data-mining is more focused on dead datasets (ie you can see almost all the data, you have an in sample only problem) than statistical learning.
Because statistical learning is about working on live dataset, the applied maths that deal with them had to focus on a two scales problem:
$$\left\{\begin{array}{lcl}
X_{n+1} &=& F_\theta(X_n,\xi_{n+1})\\
{\hat\theta}_{n+1} &=& L(\pi(X_n),{\hat\theta}_n)
\end{array}\right.$$
where $X$ is the (multidimentional) state space to study (you have in it your explanatory variables and the ones to predict), $F$ contains the dynamics of $X$ which need some parameters $\theta$. The randomness of $X$ comes from the innovation $\xi$, which is i.i.d.
The goal of statistical learning is to build a methodology $L$ ith as inputs a partial observation $\pi$ of $X$ and progressively adjust an estimate $\hat\theta$ of $\theta$, so that we will know all that is needed on $X$.
If you think about using statistical learning to find the parameters of a linear regression, we can model the state space like this:
$$\underbrace{\left( \begin{array}{c}
y_{n+1}\\ x_{n+1}
\end{array}\right)}_{X_{n+1}} = \left[ \begin{array}{ccc}
a & b & 1\\
1 & 0 & 0\\
\end{array}\right] \cdot \underbrace{\left( \begin{array}{c}
x_{n}\\1\\ \epsilon_{n+1}
\end{array}\right)}_{\xi_{n+1}}$$
which thus allows to observe $(y,x)_n$ at any $n$; here $\theta=(a,b)$.
Then you need to find a way to progressively build an estimator of $\theta$ using our observations. Why not a gradient descent on the L2 distance between $y$ and the regression:
$$C(\hat a, \hat b)_n = \sum_{k\leq n} (y_k - (\hat a \, x_k + \hat b))^2$$
So we can build these dynamics:
$${\hat a}_{n+1} = {\hat a}_n - \gamma_{n+1} \,\frac{\partial\, C({\hat a}_n, {\hat b}_n)_{n+1}}{\partial\, {\hat a}_n}$$
and similarly for $\hat b$.
Here $\gamma$ is a weighting scheme.
Usually a nice way to build an estimator is to write properly the criteria to minimize and implement a gradient descent that will produce the learning scheme $L$.
Going back to our original generic problem: we need some applied maths to know when couple dynamical systems in $(X,\hat\theta)$ converge, and we need to know how to build estimating schemes $L$ that converge towards the original $\theta$.
To give you pointers on such mathematical results:
- on the first aspects (convergence of the coupled system), you have papers like Stochastic approximation algorithms with constant step size whose average is cooperative, by Michel Benaïm and Morris W. Hirsch, in Ann. Appl. Probab. Volume 9, Number 1 (1999), 216-241.
- and on the second one (study on the convergence of $\hat\theta$ to $\theta$), you have this kind of papers/results: A Stochastic Approximation Method, by Herbert Robbins and Sutton Monro, in Ann. Math. Statist. Volume 22, Number 3 (1951), 400-407.
Now we can go back to the second aspect of statistical learning that is very interesting for quant traders/strategists:
2. The results used to prove the efficiency of statistical learning methods can be used to prove the efficiency of trading algorithms. To see that it is enough to read again the coupled dynamical system that allows to write statistical learning:
$$\left\{\begin{array}{lcl}
M_{n+1} &=& F_\rho(M_n,\xi_{n+1})\\
{\hat\rho}_{n+1} &=& L(\pi(M_n),{\hat\rho}_n)
\end{array}\right.$$
Now $M$ are market variables, $\rho$ is underlying PnL, $L$ is a trading strategy. Just replace minimizing a criteria by maximizing the PnL.
See for instance Optimal split of orders across liquidity pools: a stochatic algorithm approach by: Gilles Pagès, Sophie Laruelle, Charles-Albert Lehalle, in this paper, authors show who to use this approach to optimally split an order across different dark pools simultaneously learning the capability of the pools to provide liquidity and using the results to trade.
The statistical learning tools can be used to build iterative trading strategies (most of them are iterative) and prove their efficiency.