25,443 research outputs found
Private Incremental Regression
Data is continuously generated by modern data sources, and a recent challenge
in machine learning has been to develop techniques that perform well in an
incremental (streaming) setting. In this paper, we investigate the problem of
private machine learning, where as common in practice, the data is not given at
once, but rather arrives incrementally over time.
We introduce the problems of private incremental ERM and private incremental
regression where the general goal is to always maintain a good empirical risk
minimizer for the history observed under differential privacy. Our first
contribution is a generic transformation of private batch ERM mechanisms into
private incremental ERM mechanisms, based on a simple idea of invoking the
private batch ERM procedure at some regular time intervals. We take this
construction as a baseline for comparison. We then provide two mechanisms for
the private incremental regression problem. Our first mechanism is based on
privately constructing a noisy incremental gradient function, which is then
used in a modified projected gradient procedure at every timestep. This
mechanism has an excess empirical risk of , where is the
dimensionality of the data. While from the results of [Bassily et al. 2014]
this bound is tight in the worst-case, we show that certain geometric
properties of the input and constraint set can be used to derive significantly
better results for certain interesting regression problems.Comment: To appear in PODS 201
Incremental online learning in high dimensions
this article, however, is problematic, as it requires a careful selection of initial ridge regression parameters to stabilize the highly rank-deficient full covariance matrix of the input data, and it is easy to create too much bias or too little numerical stabilization initially, which can trap the local distance metric adaptation in local minima.While the LWPR algorithm just computes about a factor 10 times longer for the 20D experiment in comparison to the 2D experiment, RFWR requires a 1000-fold increase of computation time, thus rendering this algorithm unsuitable for high-dimensional regression. In order to compare LWPR's results to other popular regression methods, we evaluated the 2D, 10D, and 20D cross data sets with gaussian process regression (GP) and support vector (SVM) regression in addition to our LWPR method. It should be noted that neither SVM nor GP methods is an incremental method, although they can be considered state-of-the-art for batch regression under relatively small numbers of training data and reasonable input dimensionality. The computational complexity of these methods is prohibitively high for real-time applications. The GP algorithm (Gibbs & MacKay, 1997) used a generic covariance function and optimized over the hyperparameters. The SVM regression was performed using a standard available package (Saunders et al., 1998) and optimized for kernel choices. Figure 6 compares the performance of LWPR and gaussian processes for the above-mentioned data sets using 100, 300, and 500 training data point
Incremental Sparse GP Regression for Continuous-time Trajectory Estimation & Mapping
Recent work on simultaneous trajectory estimation and mapping (STEAM) for
mobile robots has found success by representing the trajectory as a Gaussian
process. Gaussian processes can represent a continuous-time trajectory,
elegantly handle asynchronous and sparse measurements, and allow the robot to
query the trajectory to recover its estimated position at any time of interest.
A major drawback of this approach is that STEAM is formulated as a batch
estimation problem. In this paper we provide the critical extensions necessary
to transform the existing batch algorithm into an extremely efficient
incremental algorithm. In particular, we are able to vastly speed up the
solution time through efficient variable reordering and incremental sparse
updates, which we believe will greatly increase the practicality of Gaussian
process methods for robot mapping and localization. Finally, we demonstrate the
approach and its advantages on both synthetic and real datasets.Comment: 10 pages, 10 figure
Incremental Sparse Bayesian Ordinal Regression
Ordinal Regression (OR) aims to model the ordering information between
different data categories, which is a crucial topic in multi-label learning. An
important class of approaches to OR models the problem as a linear combination
of basis functions that map features to a high dimensional non-linear space.
However, most of the basis function-based algorithms are time consuming. We
propose an incremental sparse Bayesian approach to OR tasks and introduce an
algorithm to sequentially learn the relevant basis functions in the ordinal
scenario. Our method, called Incremental Sparse Bayesian Ordinal Regression
(ISBOR), automatically optimizes the hyper-parameters via the type-II maximum
likelihood method. By exploiting fast marginal likelihood optimization, ISBOR
can avoid big matrix inverses, which is the main bottleneck in applying basis
function-based algorithms to OR tasks on large-scale datasets. We show that
ISBOR can make accurate predictions with parsimonious basis functions while
offering automatic estimates of the prediction uncertainty. Extensive
experiments on synthetic and real word datasets demonstrate the efficiency and
effectiveness of ISBOR compared to other basis function-based OR approaches
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