3 research outputs found
On Landscape of Lagrangian Functions and Stochastic Search for Constrained Nonconvex Optimization
We study constrained nonconvex optimization problems in machine learning,
signal processing, and stochastic control. It is well-known that these problems
can be rewritten to a minimax problem in a Lagrangian form. However, due to the
lack of convexity, their landscape is not well understood and how to find the
stable equilibria of the Lagrangian function is still unknown. To bridge the
gap, we study the landscape of the Lagrangian function. Further, we define a
special class of Lagrangian functions. They enjoy two properties: 1.Equilibria
are either stable or unstable (Formal definition in Section 2); 2.Stable
equilibria correspond to the global optima of the original problem. We show
that a generalized eigenvalue (GEV) problem, including canonical correlation
analysis and other problems, belongs to the class. Specifically, we
characterize its stable and unstable equilibria by leveraging an invariant
group and symmetric property (more details in Section 3). Motivated by these
neat geometric structures, we propose a simple, efficient, and stochastic
primal-dual algorithm solving the online GEV problem. Theoretically, we provide
sufficient conditions, based on which we establish an asymptotic convergence
rate and obtain the first sample complexity result for the online GEV problem
by diffusion approximations, which are widely used in applied probability and
stochastic control. Numerical results are provided to support our theory.Comment: 29 pages, 2 figure
The Landscape of Matrix Factorization Revisited
We revisit the landscape of the simple matrix factorization problem. For
low-rank matrix factorization, prior work has shown that there exist infinitely
many critical points all of which are either global minima or strict saddles.
At a strict saddle the minimum eigenvalue of the Hessian is negative. Of
interest is whether this minimum eigenvalue is uniformly bounded below zero
over all strict saddles. To answer this we consider orbits of critical points
under the general linear group. For each orbit we identify a representative
point, called a canonical point. If a canonical point is a strict saddle, so is
every point on its orbit. We derive an expression for the minimum eigenvalue of
the Hessian at each canonical strict saddle and use this to show that the
minimum eigenvalue of the Hessian over the set of strict saddles is not
uniformly bounded below zero. We also show that a known invariance property of
gradient flow ensures the solution of gradient flow only encounters critical
points on an invariant manifold determined by the initial
condition. We show that, in contrast to the general situation, the minimum
eigenvalue of strict saddles in is uniformly bounded below
zero. We obtain an expression for this bound in terms of the singular values of
the matrix being factorized. This bound depends on the size of the nonzero
singular values and on the separation between distinct nonzero singular values
of the matrix.Comment: 19 page
Towards Automatic Evaluation of Dialog Systems: A Model-Free Off-Policy Evaluation Approach
Reliable automatic evaluation of dialogue systems under an interactive
environment has long been overdue. An ideal environment for evaluating dialog
systems, also known as the Turing test, needs to involve human interaction,
which is usually not affordable for large-scale experiments. Though researchers
have attempted to use metrics (e.g., perplexity, BLEU) in language generation
tasks or some model-based reinforcement learning methods (e.g., self-play
evaluation) for automatic evaluation, these methods only show a very weak
correlation with the actual human evaluation in practice. To bridge such a gap,
we propose a new framework named ENIGMA for estimating human evaluation scores
based on recent advances of off-policy evaluation in reinforcement learning.
ENIGMA only requires a handful of pre-collected experience data, and therefore
does not involve human interaction with the target policy during the
evaluation, making automatic evaluations feasible. More importantly, ENIGMA is
model-free and agnostic to the behavior policies for collecting the experience
data (see details in Section 2), which significantly alleviates the technical
difficulties of modeling complex dialogue environments and human behaviors. Our
experiments show that ENIGMA significantly outperforms existing methods in
terms of correlation with human evaluation scores