69,349 research outputs found
On universal oracle inequalities related to high-dimensional linear models
This paper deals with recovering an unknown vector from the noisy
data , where is a known -matrix and
is a white Gaussian noise. It is assumed that is large and may be
severely ill-posed. Therefore, in order to estimate , a spectral
regularization method is used, and our goal is to choose its regularization
parameter with the help of the data . For spectral regularization methods
related to the so-called ordered smoothers [see Kneip Ann. Statist. 22 (1994)
835--866], we propose new penalties in the principle of empirical risk
minimization. The heuristical idea behind these penalties is related to
balancing excess risks. Based on this approach, we derive a sharp oracle
inequality controlling the mean square risks of data-driven spectral
regularization methods.Comment: Published in at http://dx.doi.org/10.1214/10-AOS803 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Lower Bound for the Optimization of Finite Sums
This paper presents a lower bound for optimizing a finite sum of
functions, where each function is -smooth and the sum is -strongly
convex. We show that no algorithm can reach an error in minimizing
all functions from this class in fewer than iterations, where is a
surrogate condition number. We then compare this lower bound to upper bounds
for recently developed methods specializing to this setting. When the functions
involved in this sum are not arbitrary, but based on i.i.d. random data, then
we further contrast these complexity results with those for optimal first-order
methods to directly optimize the sum. The conclusion we draw is that a lot of
caution is necessary for an accurate comparison, and identify machine learning
scenarios where the new methods help computationally.Comment: Added an erratum, we are currently working on extending the result to
randomized algorithm
Sparse and Non-Negative BSS for Noisy Data
Non-negative blind source separation (BSS) has raised interest in various
fields of research, as testified by the wide literature on the topic of
non-negative matrix factorization (NMF). In this context, it is fundamental
that the sources to be estimated present some diversity in order to be
efficiently retrieved. Sparsity is known to enhance such contrast between the
sources while producing very robust approaches, especially to noise. In this
paper we introduce a new algorithm in order to tackle the blind separation of
non-negative sparse sources from noisy measurements. We first show that
sparsity and non-negativity constraints have to be carefully applied on the
sought-after solution. In fact, improperly constrained solutions are unlikely
to be stable and are therefore sub-optimal. The proposed algorithm, named nGMCA
(non-negative Generalized Morphological Component Analysis), makes use of
proximal calculus techniques to provide properly constrained solutions. The
performance of nGMCA compared to other state-of-the-art algorithms is
demonstrated by numerical experiments encompassing a wide variety of settings,
with negligible parameter tuning. In particular, nGMCA is shown to provide
robustness to noise and performs well on synthetic mixtures of real NMR
spectra.Comment: 13 pages, 18 figures, to be published in IEEE Transactions on Signal
Processin
On Perfect Completeness for QMA
Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with
one-sided error, has been an open problem for years. This note helps to explain
why the problem is difficult, by using ideas from real analysis to give a
"quantum oracle" relative to which they are different. As a byproduct, we find
that there are facts about quantum complexity classes that are classically
relativizing but not quantumly relativizing, among them such "trivial"
containments as BQP in ZQEXP.Comment: 9 pages. To appear in Quantum Information & Computatio
Discovering Blind Spots in Reinforcement Learning
Agents trained in simulation may make errors in the real world due to
mismatches between training and execution environments. These mistakes can be
dangerous and difficult to discover because the agent cannot predict them a
priori. We propose using oracle feedback to learn a predictive model of these
blind spots to reduce costly errors in real-world applications. We focus on
blind spots in reinforcement learning (RL) that occur due to incomplete state
representation: The agent does not have the appropriate features to represent
the true state of the world and thus cannot distinguish among numerous states.
We formalize the problem of discovering blind spots in RL as a noisy supervised
learning problem with class imbalance. We learn models to predict blind spots
in unseen regions of the state space by combining techniques for label
aggregation, calibration, and supervised learning. The models take into
consideration noise emerging from different forms of oracle feedback, including
demonstrations and corrections. We evaluate our approach on two domains and
show that it achieves higher predictive performance than baseline methods, and
that the learned model can be used to selectively query an oracle at execution
time to prevent errors. We also empirically analyze the biases of various
feedback types and how they influence the discovery of blind spots.Comment: To appear at AAMAS 201
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