Machine Learning (stat.ML)

Abstract: We propose a new algorithm for variance reduction when estimating $f(X_T)$ where $X$ is the solution to some stochastic differential equation and $f$ is a test function. The new estimator is $(f(X^1_T) + f(X^2_T))/2$, where $X^1$ and $X^2$ have same marginal law as $X$ but are pathwise correlated so that to reduce the variance. The optimal correlation function $\rho$ is approximated by a deep neural network and is calibrated along the trajectories of $(X^1, X^2)$ by policy gradient and reinforcement learning techniques. Finding an optimal coupling given marginal laws has links with maximum optimal transport.

Abstract: Model selection in the context of bandit optimization is a challenging problem, as it requires balancing exploration and exploitation not only for action selection, but also for model selection. One natural approach is to rely on online learning algorithms that treat different models as experts. Existing methods, however, scale poorly ($\text{poly}M$) with the number of models $M$ in terms of their regret. Our key insight is that, for model selection in linear bandits, we can emulate full-information feedback to the online learner with a favorable bias-variance trade-off. This allows us to develop ALEXP, which has an exponentially improved ($\log M$) dependence on $M$ for its regret. ALEXP has anytime guarantees on its regret, and neither requires knowledge of the horizon $n$, nor relies on an initial purely exploratory stage. Our approach utilizes a novel time-uniform analysis of the Lasso, establishing a new connection between online learning and high-dimensional statistics.