42 research outputs found
Convergence of Langevin MCMC in KL-divergence
Langevin diffusion is a commonly used tool for sampling from a given
distribution. In this work, we establish that when the target density is
such that is smooth and strongly convex, discrete Langevin
diffusion produces a distribution with in
steps, where is the dimension of the sample
space. We also study the convergence rate when the strong-convexity assumption
is absent. By considering the Langevin diffusion as a gradient flow in the
space of probability distributions, we obtain an elegant analysis that applies
to the stronger property of convergence in KL-divergence and gives a
conceptually simpler proof of the best-known convergence results in weaker
metrics
Efficient improper learning for online logistic regression
We consider the setting of online logistic regression and consider the regret
with respect to the 2-ball of radius B. It is known (see [Hazan et al., 2014])
that any proper algorithm which has logarithmic regret in the number of samples
(denoted n) necessarily suffers an exponential multiplicative constant in B. In
this work, we design an efficient improper algorithm that avoids this
exponential constant while preserving a logarithmic regret. Indeed, [Foster et
al., 2018] showed that the lower bound does not apply to improper algorithms
and proposed a strategy based on exponential weights with prohibitive
computational complexity. Our new algorithm based on regularized empirical risk
minimization with surrogate losses satisfies a regret scaling as O(B log(Bn))
with a per-round time-complexity of order O(d^2)