Marvels and Pitfalls of the Langevin Algorithm in Noisy High-dimensional Inference
This work addresses algorithmic efficiency in high-dimensional statistical inference, but it is incremental as it compares existing methods without introducing a fundamentally new approach.
The study analyzed the Langevin algorithm's performance in noisy high-dimensional inference using the spiked matrix-tensor model, finding that its algorithmic threshold is sub-optimal compared to approximate message passing (AMP). A landscape-annealing protocol was shown to approach AMP's performance by violating Bayes-optimality.
Gradient-descent-based algorithms and their stochastic versions have widespread applications in machine learning and statistical inference. In this work we perform an analytic study of the performances of one of them, the Langevin algorithm, in the context of noisy high-dimensional inference. We employ the Langevin algorithm to sample the posterior probability measure for the spiked matrix-tensor model. The typical behaviour of this algorithm is described by a system of integro-differential equations that we call the Langevin state evolution, whose solution is compared with the one of the state evolution of approximate message passing (AMP). Our results show that, remarkably, the algorithmic threshold of the Langevin algorithm is sub-optimal with respect to the one given by AMP. We conjecture this phenomenon to be due to the residual glassiness present in that region of parameters. Finally we show how a landscape-annealing protocol, that uses the Langevin algorithm but violate the Bayes-optimality condition, can approach the performance of AMP.