Kernel-based methods exhibit well-documented performance in various nonlinear learning tasks. Most of them rely on a preselected kernel, whose prudent choice presumes task-specific prior informatio...

We propose and analyze two new MCMC sampling algorithms, the Vaidya walk and the John walk, for generating samples from the uniform distribution over a polytope. Both random walks are sampling algorithms derived from interior point methods. The former is based on volumetric-logarithmic barrier introduced by Vaidya whereas the latter uses John's ellipsoids. We show that the Vaidya walk mixes in significantly fewer steps than the logarithmic-barrier based Dikin walk studied in past work. For a polytope in ℝd defined by n ≥ d linear constraints, we show that the mixing time from a warm start is bounded as O(n0.5d1.5), compared to the O(nd) mixing time bound for the Dikin walk. The cost of each step of the Vaidya walk is of the same order as the Dikin walk, and at most twice as large in terms of constant pre-factors. For the John walk, we prove an O(d2.5 log4(n/d) bound on its mixing time and conjecture that an improved variant of it could achieve a mixing time of O(d2 poly-log(n/d)). Additionally, we propose variants of the Vaidya and John walks that mix in polynomial time from a deterministic starting point. The speed-up of the Vaidya walk over the Dikin walk are illustrated in numerical examples.

We consider the problem of estimating a probability distribution that maximizes the entropy while satisfying a finite number of moment constraints, possibly corrupted by noise. Based on duality of convex programming, we present a novel approximation scheme using a smoothed fast gradient method that is equipped with explicit bounds on the approximation error. We further demonstrate how the presented scheme can be used for approximating the chemical master equation through the zero-information moment closure method, and for an approximate dynamic programming approach in the context of constrained Markov decision processes with uncountable state and action spaces.

In this paper we develop a deep learning method for optimal stopping problems which directly learns the optimal stopping rule from Monte Carlo samples. As such, it is broadly applicable in situations where the underlying randomness can efficiently be simulated. We test the approach on three problems: the pricing of a Bermudan max-call option, the pricing of a callable multi barrier reverse convertible and the problem of optimally stopping a fractional Brownian motion. In all three cases it produces very accurate results in high-dimensional situations with short computing times.

In stochastic optimization, the population risk is generally approximated by the empirical risk which is in turn minimized by an iterative algorithm. However, in the large-scale setting, empirical ...

We introduce an agglomerative hierarchical clustering (AHC) framework which is generic, efficient and effective. Our approach embeds a sub-family of Lance-Williams (LW) clusterings and relies on in...

This work brings together two powerful concepts in Gaussian processes: the variational approach to sparse approximation and the spectral representation of Gaussian processes. This gives rise to an ...

We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximat...

We study the pricing problem faced by a firm that sells a large number of products, described via a wide range of features, to customers that arrive over time. Customers independently make purchasi...

We present a new algorithm for domain adaptation improving upon a discrepancy minimization algorithm, (DM), previously shown to outperform a number of algorithms for this problem. Unlike many previ...