954,423 research outputs found
The Vapnik-Chervonenkis Dimension: Information versus Complexity in Learning
When feasible, learning is a very attractive alternative to explicit programming. This is particularly true in areas where the problems do not lend themselves to systematic programming, such as pattern recognition in natural environments. The feasibility of learning an unknown function from examples depends on two questions:
1. Do the examples convey enough information to determine the function?
2. Is there a speedy way of constructing the function from the examples?
These questions contrast the roles of information and complexity in learning. While the two roles share some ground, they are conceptually and technically different. In the common language of learning, the information question is that of generalization and the complexity question is that of scaling. The work of Vapnik and Chervonenkis (1971) provides the key tools for dealing with the information issue. In this review, we develop the main ideas of this framework and discuss how complexity fits in
Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization
Relative to the large literature on upper bounds on complexity of convex
optimization, lesser attention has been paid to the fundamental hardness of
these problems. Given the extensive use of convex optimization in machine
learning and statistics, gaining an understanding of these complexity-theoretic
issues is important. In this paper, we study the complexity of stochastic
convex optimization in an oracle model of computation. We improve upon known
results and obtain tight minimax complexity estimates for various function
classes
Passive Learning with Target Risk
In this paper we consider learning in passive setting but with a slight
modification. We assume that the target expected loss, also referred to as
target risk, is provided in advance for learner as prior knowledge. Unlike most
studies in the learning theory that only incorporate the prior knowledge into
the generalization bounds, we are able to explicitly utilize the target risk in
the learning process. Our analysis reveals a surprising result on the sample
complexity of learning: by exploiting the target risk in the learning
algorithm, we show that when the loss function is both strongly convex and
smooth, the sample complexity reduces to \O(\log (\frac{1}{\epsilon})), an
exponential improvement compared to the sample complexity
\O(\frac{1}{\epsilon}) for learning with strongly convex loss functions.
Furthermore, our proof is constructive and is based on a computationally
efficient stochastic optimization algorithm for such settings which demonstrate
that the proposed algorithm is practically useful
Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions
We investigate three related and important problems connected to machine
learning: approximating a submodular function everywhere, learning a submodular
function (in a PAC-like setting [53]), and constrained minimization of
submodular functions. We show that the complexity of all three problems depends
on the 'curvature' of the submodular function, and provide lower and upper
bounds that refine and improve previous results [3, 16, 18, 52]. Our proof
techniques are fairly generic. We either use a black-box transformation of the
function (for approximation and learning), or a transformation of algorithms to
use an appropriate surrogate function (for minimization). Curiously, curvature
has been known to influence approximations for submodular maximization [7, 55],
but its effect on minimization, approximation and learning has hitherto been
open. We complete this picture, and also support our theoretical claims by
empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201
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