8,092 research outputs found
Towards an Intelligent Tutor for Mathematical Proofs
Computer-supported learning is an increasingly important form of study since
it allows for independent learning and individualized instruction. In this
paper, we discuss a novel approach to developing an intelligent tutoring system
for teaching textbook-style mathematical proofs. We characterize the
particularities of the domain and discuss common ITS design models. Our
approach is motivated by phenomena found in a corpus of tutorial dialogs that
were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor
for textbook-style mathematical proofs can be built on top of an adapted
assertion-level proof assistant by reusing representations and proof search
strategies originally developed for automated and interactive theorem proving.
The resulting prototype was successfully evaluated on a corpus of tutorial
dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453
Advanced Probabilistic Couplings for Differential Privacy
Differential privacy is a promising formal approach to data privacy, which
provides a quantitative bound on the privacy cost of an algorithm that operates
on sensitive information. Several tools have been developed for the formal
verification of differentially private algorithms, including program logics and
type systems. However, these tools do not capture fundamental techniques that
have emerged in recent years, and cannot be used for reasoning about
cutting-edge differentially private algorithms. Existing techniques fail to
handle three broad classes of algorithms: 1) algorithms where privacy depends
accuracy guarantees, 2) algorithms that are analyzed with the advanced
composition theorem, which shows slower growth in the privacy cost, 3)
algorithms that interactively accept adaptive inputs.
We address these limitations with a new formalism extending apRHL, a
relational program logic that has been used for proving differential privacy of
non-interactive algorithms, and incorporating aHL, a (non-relational) program
logic for accuracy properties. We illustrate our approach through a single
running example, which exemplifies the three classes of algorithms and explores
new variants of the Sparse Vector technique, a well-studied algorithm from the
privacy literature. We implement our logic in EasyCrypt, and formally verify
privacy. We also introduce a novel coupling technique called \emph{optimal
subset coupling} that may be of independent interest
Adaptive posterior contraction rates for the horseshoe
We investigate the frequentist properties of Bayesian procedures for
estimation based on the horseshoe prior in the sparse multivariate normal means
model. Previous theoretical results assumed that the sparsity level, that is,
the number of signals, was known. We drop this assumption and characterize the
behavior of the maximum marginal likelihood estimator (MMLE) of a key parameter
of the horseshoe prior. We prove that the MMLE is an effective estimator of the
sparsity level, in the sense that it leads to (near) minimax optimal estimation
of the underlying mean vector generating the data. Besides this empirical Bayes
procedure, we consider the hierarchical Bayes method of putting a prior on the
unknown sparsity level as well. We show that both Bayesian techniques lead to
rate-adaptive optimal posterior contraction, which implies that the horseshoe
posterior is a good candidate for generating rate-adaptive credible sets.Comment: arXiv admin note: substantial text overlap with arXiv:1607.0189
Adaptive non-parametric estimation in the presence of dependence
We consider non-parametric estimation problems in the presence of dependent
data, notably non-parametric regression with random design and non-parametric
density estimation. The proposed estimation procedure is based on a dimension
reduction. The minimax optimal rate of convergence of the estimator is derived
assuming a sufficiently weak dependence characterized by fast decreasing mixing
coefficients. We illustrate these results by considering classical smoothness
assumptions. However, the proposed estimator requires an optimal choice of a
dimension parameter depending on certain characteristics of the function of
interest, which are not known in practice. The main issue addressed in our work
is an adaptive choice of this dimension parameter combining model selection and
Lepski's method. It is inspired by the recent work of Goldenshluger and Lepski
(2011). We show that this data-driven estimator can attain the lower risk bound
up to a constant provided a fast decay of the mixing coefficients.Comment: 39 pages, 4 figure
Statistical inference for time-inhomogeneous volatility models
This paper offers a new approach for estimating and forecasting the
volatility of financial time series. No assumption is made about the parametric
form of the processes. On the contrary, we only suppose that the volatility can
be approximated by a constant over some interval. In such a framework, the main
problem consists of filtering this interval of time homogeneity; then the
estimate of the volatility can be simply obtained by local averaging.
We construct a locally adaptive volatility estimate (LAVE) which can perform
this task and investigate it both from the theoretical point of view and
through Monte Carlo simulations. Finally, the LAVE procedure is applied to a
data set of nine exchange rates and a comparison with a standard GARCH model is
also provided. Both models appear to be capable of explaining many of the
features of the data; nevertheless, the new approach seems to be superior to
the GARCH method as far as the out-of-sample results are concerned
Distributed Linear Parameter Estimation: Asymptotically Efficient Adaptive Strategies
The paper considers the problem of distributed adaptive linear parameter
estimation in multi-agent inference networks. Local sensing model information
is only partially available at the agents and inter-agent communication is
assumed to be unpredictable. The paper develops a generic mixed time-scale
stochastic procedure consisting of simultaneous distributed learning and
estimation, in which the agents adaptively assess their relative observation
quality over time and fuse the innovations accordingly. Under rather weak
assumptions on the statistical model and the inter-agent communication, it is
shown that, by properly tuning the consensus potential with respect to the
innovation potential, the asymptotic information rate loss incurred in the
learning process may be made negligible. As such, it is shown that the agent
estimates are asymptotically efficient, in that their asymptotic covariance
coincides with that of a centralized estimator (the inverse of the centralized
Fisher information rate for Gaussian systems) with perfect global model
information and having access to all observations at all times. The proof
techniques are mainly based on convergence arguments for non-Markovian mixed
time scale stochastic approximation procedures. Several approximation results
developed in the process are of independent interest.Comment: Submitted to SIAM Journal on Control and Optimization journal.
Initial Submission: Sept. 2011. Revised: Aug. 201
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