arXiv subjects

Philipp Hennig

Publications and source records attributed to Philipp Hennig.

At least 127 records · Page 7Linked to original sources

arXiv2017

Early Stopping without a Validation Set

Early stopping is a widely used technique to prevent poor generalization performance when training an over-expressive model by means of gradient-based optimization. To find a good point to halt the optimizer, a common practice is to split the dataset into a training and a smaller validation set to obtain an ongoing estimate of the generalization performance. We propose a novel early stopping criterion based on fast-to-compute local statistics of the computed gradients and entirely removes the need for a held-out validation set. Our experiments show that this is a viable approach in the setting of least-squares and logistic regression, as well as neural networks.

cs.LG↗

arXiv2017

Krylov Subspace Recycling for Fast Iterative Least-Squares in Machine Learning

Solving symmetric positive definite linear problems is a fundamental computational task in machine learning. The exact solution, famously, is cubicly expensive in the size of the matrix. To alleviate this problem, several linear-time approximations, such as spectral and inducing-point methods, have been suggested and are now in wide use. These are low-rank approximations that choose the low-rank space a priori and do not refine it over time. While this allows linear cost in the data-set size, it also causes a finite, uncorrected approximation error. Authors from numerical linear algebra have explored ways to iteratively refine such low-rank approximations, at a cost of a small number of matrix-vector multiplications. This idea is particularly interesting in the many situations in machine learning where one has to solve a sequence of related symmetric positive definite linear problems. From the machine learning perspective, such deflation methods can be interpreted as transfer learning of a low-rank approximation across a time-series of numerical tasks. We study the use of such methods for our field. Our empirical results show that, on regression and classification problems of intermediate size, this approach can interpolate between low computational cost and numerical precision.

cs.LG↗

arXiv2017

Fast Bayesian Optimization of Machine Learning Hyperparameters on Large Datasets

Bayesian optimization has become a successful tool for hyperparameter optimization of machine learning algorithms, such as support vector machines or deep neural networks. Despite its success, for large datasets, training and validating a single configuration often takes hours, days, or even weeks, which limits the achievable performance. To accelerate hyperparameter optimization, we propose a generative model for the validation error as a function of training set size, which is learned during the optimization process and allows exploration of preliminary configurations on small subsets, by extrapolating to the full dataset. We construct a Bayesian optimization procedure, dubbed Fabolas, which models loss and training time as a function of dataset size and automatically trades off high information gain about the global optimum against computational cost. Experiments optimizing support vector machines and deep neural networks show that Fabolas often finds high-quality solutions 10 to 100 times faster than other state-of-the-art Bayesian optimization methods or the recently proposed bandit strategy Hyperband.

cs.LG↗

arXiv2017

Virtual vs. Real: Trading Off Simulations and Physical Experiments in Reinforcement Learning with Bayesian Optimization

In practice, the parameters of control policies are often tuned manually. This is time-consuming and frustrating. Reinforcement learning is a promising alternative that aims to automate this process, yet often requires too many experiments to be practical. In this paper, we propose a solution to this problem by exploiting prior knowledge from simulations, which are readily available for most robotic platforms. Specifically, we extend Entropy Search, a Bayesian optimization algorithm that maximizes information gain from each experiment, to the case of multiple information sources. The result is a principled way to automatically combine cheap, but inaccurate information from simulations with expensive and accurate physical experiments in a cost-effective manner. We apply the resulting method to a cart-pole system, which confirms that the algorithm can find good control policies with fewer experiments than standard Bayesian optimization on the physical system only.

cs.RO↗

arXiv2016

Dual Control for Approximate Bayesian Reinforcement Learning

Control of non-episodic, finite-horizon dynamical systems with uncertain dynamics poses a tough and elementary case of the exploration-exploitation trade-off. Bayesian reinforcement learning, reasoning about the effect of actions and future observations, offers a principled solution, but is intractable. We review, then extend an old approximate approach from control theory---where the problem is known as dual control---in the context of modern regression methods, specifically generalized linear regression. Experiments on simulated systems show that this framework offers a useful approximation to the intractable aspects of Bayesian RL, producing structured exploration strategies that differ from standard RL approaches. We provide simple examples for the use of this framework in (approximate) Gaussian process regression and feedforward neural networks for the control of exploration.

stat.ML↗

arXiv2016

Automatic LQR Tuning Based on Gaussian Process Global Optimization

This paper proposes an automatic controller tuning framework based on linear optimal control combined with Bayesian optimization. With this framework, an initial set of controller gains is automatically improved according to a pre-defined performance objective evaluated from experimental data. The underlying Bayesian optimization algorithm is Entropy Search, which represents the latent objective as a Gaussian process and constructs an explicit belief over the location of the objective minimum. This is used to maximize the information gain from each experimental evaluation. Thus, this framework shall yield improved controllers with fewer evaluations compared to alternative approaches. A seven-degree-of-freedom robot arm balancing an inverted pole is used as the experimental demonstrator. Results of a two- and four-dimensional tuning problems highlight the method's potential for automatic controller tuning on robotic platforms.

cs.RO↗

arXiv2016

Probabilistic Line Searches for Stochastic Optimization

In deterministic optimization, line searches are a standard tool ensuring stability and efficiency. Where only stochastic gradients are available, no direct equivalent has so far been formulated, because uncertain gradients do not allow for a strict sequence of decisions collapsing the search space. We construct a probabilistic line search by combining the structure of existing deterministic methods with notions from Bayesian optimization. Our method retains a Gaussian process surrogate of the univariate optimization objective, and uses a probabilistic belief over the Wolfe conditions to monitor the descent. The algorithm has very low computational cost, and no user-controlled parameters. Experiments show that it effectively removes the need to define a learning rate for stochastic gradient descent.

cs.LG↗

arXiv2015

Batch Bayesian Optimization via Local Penalization

The popularity of Bayesian optimization methods for efficient exploration of parameter spaces has lead to a series of papers applying Gaussian processes as surrogates in the optimization of functions. However, most proposed approaches only allow the exploration of the parameter space to occur sequentially. Often, it is desirable to simultaneously propose batches of parameter values to explore. This is particularly the case when large parallel processing facilities are available. These facilities could be computational or physical facets of the process being optimized. E.g. in biological experiments many experimental set ups allow several samples to be simultaneously processed. Batch methods, however, require modeling of the interaction between the evaluations in the batch, which can be expensive in complex scenarios. We investigate a simple heuristic based on an estimate of the Lipschitz constant that captures the most important aspect of this interaction (i.e. local repulsion) at negligible computational overhead. The resulting algorithm compares well, in running time, with much more elaborate alternatives. The approach assumes that the function of interest, $f$, is a Lipschitz continuous function. A wrap-loop around the acquisition function is used to collect batches of points of certain size minimizing the non-parallelizable computational effort. The speed-up of our method with respect to previous approaches is significant in a set of computationally expensive experiments.

stat.ML↗

arXiv2015

Probabilistic Numerics and Uncertainty in Computations

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data has led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimisers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.

math.NA↗

arXiv2014

Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature

We propose a novel sampling framework for inference in probabilistic models: an active learning approach that converges more quickly (in wall-clock time) than Markov chain Monte Carlo (MCMC) benchmarks. The central challenge in probabilistic inference is numerical integration, to average over ensembles of models or unknown (hyper-)parameters (for example to compute the marginal likelihood or a partition function). MCMC has provided approaches to numerical integration that deliver state-of-the-art inference, but can suffer from sample inefficiency and poor convergence diagnostics. Bayesian quadrature techniques offer a model-based solution to such problems, but their uptake has been hindered by prohibitive computation costs. We introduce a warped model for probabilistic integrands (likelihoods) that are known to be non-negative, permitting a cheap active learning scheme to optimally select sample locations. Our algorithm is demonstrated to offer faster convergence (in seconds) relative to simple Monte Carlo and annealed importance sampling on both synthetic and real-world examples.

stat.ML↗

arXiv2014

Probabilistic ODE Solvers with Runge-Kutta Means

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct this family such that posterior means match the outputs of the Runge-Kutta family exactly, thus inheriting their proven good properties. Remaining degrees of freedom not identified by the match to Runge-Kutta are chosen such that the posterior probability measure fits the observed structure of the ODE. Our results shed light on the structure of Runge-Kutta solvers from a new direction, provide a richer, probabilistic output, have low computational cost, and raise new research questions.

stat.ML↗

arXiv2014

Probabilistic Interpretation of Linear Solvers

This manuscript proposes a probabilistic framework for algorithms that iteratively solve unconstrained linear problems $Bx = b$ with positive definite $B$ for $x$. The goal is to replace the point estimates returned by existing methods with a Gaussian posterior belief over the elements of the inverse of $B$, which can be used to estimate errors. Recent probabilistic interpretations of the secant family of quasi-Newton optimization algorithms are extended. Combined with properties of the conjugate gradient algorithm, this leads to uncertainty-calibrated methods with very limited cost overhead over conjugate gradients, a self-contained novel interpretation of the quasi-Newton and conjugate gradient algorithms, and a foundation for new nonlinear optimization methods.

math.OC↗

arXiv2014

Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics

We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian manifolds, where non-analytic ordinary differential equations are involved in virtually all computations. The probabilistic formulation permits marginalising the uncertainty of the numerical solution such that statistics are less sensitive to inaccuracies. This leads to new Riemannian algorithms for mean value computations and principal geodesic analysis. Marginalisation also means results can be less precise than point estimates, enabling a noticeable speed-up over the state of the art. Our approach is an argument for a wider point that uncertainty caused by numerical calculations should be tracked throughout the pipeline of machine learning algorithms.

stat.ML↗

arXiv2014

Local Gaussian Regression

Locally weighted regression was created as a nonparametric learning method that is computationally efficient, can learn from very large amounts of data and add data incrementally. An interesting feature of locally weighted regression is that it can work with spatially varying length scales, a beneficial property, for instance, in control problems. However, it does not provide a generative model for function values and requires training and test data to be generated identically, independently. Gaussian (process) regression, on the other hand, provides a fully generative model without significant formal requirements on the distribution of training data, but has much higher computational cost and usually works with one global scale per input dimension. Using a localising function basis and approximate inference techniques, we take Gaussian (process) regression to increasingly localised properties and toward the same computational complexity class as locally weighted regression.

cs.LG↗

arXiv2013

Gaussian Probabilities and Expectation Propagation

While Gaussian probability densities are omnipresent in applied mathematics, Gaussian cumulative probabilities are hard to calculate in any but the univariate case. We study the utility of Expectation Propagation (EP) as an approximate integration method for this problem. For rectangular integration regions, the approximation is highly accurate. We also extend the derivations to the more general case of polyhedral integration regions. However, we find that in this polyhedral case, EP's answer, though often accurate, can be almost arbitrarily wrong. We consider these unexpected results empirically and theoretically, both for the problem of Gaussian probabilities and for EP more generally. These results elucidate an interesting and non-obvious feature of EP not yet studied in detail.

stat.ML↗

arXiv2013

Active Learning of Linear Embeddings for Gaussian Processes

We propose an active learning method for discovering low-dimensional structure in high-dimensional Gaussian process (GP) tasks. Such problems are increasingly frequent and important, but have hitherto presented severe practical difficulties. We further introduce a novel technique for approximately marginalizing GP hyperparameters, yielding marginal predictions robust to hyperparameter mis-specification. Our method offers an efficient means of performing GP regression, quadrature, or Bayesian optimization in high-dimensional spaces.

stat.ML↗

arXiv2013

The Randomized Dependence Coefficient

We introduce the Randomized Dependence Coefficient (RDC), a measure of non-linear dependence between random variables of arbitrary dimension based on the Hirschfeld-Gebelein-Rényi Maximum Correlation Coefficient. RDC is defined in terms of correlation of random non-linear copula projections; it is invariant with respect to marginal distribution transformations, has low computational cost and is easy to implement: just five lines of R code, included at the end of the paper.

stat.ML↗

arXiv2012

Quasi-Newton Methods: A New Direction

Four decades after their invention, quasi-Newton methods are still state of the art in unconstrained numerical optimization. Although not usually interpreted thus, these are learning algorithms that fit a local quadratic approximation to the objective function. We show that many, including the most popular, quasi-Newton methods can be interpreted as approximations of Bayesian linear regression under varying prior assumptions. This new notion elucidates some shortcomings of classical algorithms, and lights the way to a novel nonparametric quasi-Newton method, which is able to make more efficient use of available information at computational cost similar to its predecessors.

math.NA↗