171 research outputs found
A refinement of Bennett's inequality with applications to portfolio optimization
A refinement of Bennett's inequality is introduced which is strictly tighter
than the classical bound. The new bound establishes the convergence of the
average of independent random variables to its expected value. It also
carefully exploits information about the potentially heterogeneous mean,
variance, and ceiling of each random variable. The bound is strictly sharper in
the homogeneous setting and very often significantly sharper in the
heterogeneous setting. The improved convergence rates are obtained by
leveraging Lambert's W function. We apply the new bound in a portfolio
optimization setting to allocate a budget across investments with heterogeneous
returns
Frank-Wolfe Algorithms for Saddle Point Problems
We extend the Frank-Wolfe (FW) optimization algorithm to solve constrained
smooth convex-concave saddle point (SP) problems. Remarkably, the method only
requires access to linear minimization oracles. Leveraging recent advances in
FW optimization, we provide the first proof of convergence of a FW-type saddle
point solver over polytopes, thereby partially answering a 30 year-old
conjecture. We also survey other convergence results and highlight gaps in the
theoretical underpinnings of FW-style algorithms. Motivating applications
without known efficient alternatives are explored through structured prediction
with combinatorial penalties as well as games over matching polytopes involving
an exponential number of constraints.Comment: Appears in: Proceedings of the 20th International Conference on
Artificial Intelligence and Statistics (AISTATS 2017). 39 page
A PAC-Bayesian bound for Lifelong Learning
Transfer learning has received a lot of attention in the machine learning
community over the last years, and several effective algorithms have been
developed. However, relatively little is known about their theoretical
properties, especially in the setting of lifelong learning, where the goal is
to transfer information to tasks for which no data have been observed so far.
In this work we study lifelong learning from a theoretical perspective. Our
main result is a PAC-Bayesian generalization bound that offers a unified view
on existing paradigms for transfer learning, such as the transfer of parameters
or the transfer of low-dimensional representations. We also use the bound to
derive two principled lifelong learning algorithms, and we show that these
yield results comparable with existing methods.Comment: to appear at ICML 201
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