arXiv subjects

Philipp Hennig

Publications and source records attributed to Philipp Hennig.

At least 109 records · Page 6Linked to original sources

arXiv2020

BackPACK: Packing more into backprop

Automatic differentiation frameworks are optimized for exactly one thing: computing the average mini-batch gradient. Yet, other quantities such as the variance of the mini-batch gradients or many approximations to the Hessian can, in theory, be computed efficiently, and at the same time as the gradient. While these quantities are of great interest to researchers and practitioners, current deep-learning software does not support their automatic calculation. Manually implementing them is burdensome, inefficient if done naively, and the resulting code is rarely shared. This hampers progress in deep learning, and unnecessarily narrows research to focus on gradient descent and its variants; it also complicates replication studies and comparisons between newly developed methods that require those quantities, to the point of impossibility. To address this problem, we introduce BackPACK, an efficient framework built on top of PyTorch, that extends the backpropagation algorithm to extract additional information from first- and second-order derivatives. Its capabilities are illustrated by benchmark reports for computing additional quantities on deep neural networks, and an example application by testing several recent curvature approximations for optimization.

cs.LG↗

arXiv2019

Conjugate Gradients for Kernel Machines

Regularized least-squares (kernel-ridge / Gaussian process) regression is a fundamental algorithm of statistics and machine learning. Because generic algorithms for the exact solution have cubic complexity in the number of datapoints, large datasets require to resort to approximations. In this work, the computation of the least-squares prediction is itself treated as a probabilistic inference problem. We propose a structured Gaussian regression model on the kernel function that uses projections of the kernel matrix to obtain a low-rank approximation of the kernel and the matrix. A central result is an enhanced way to use the method of conjugate gradients for the specific setting of least-squares regression as encountered in machine learning. Our method improves the approximation of the kernel ridge regressor / Gaussian process posterior mean over vanilla conjugate gradients and, allows computation of the posterior variance and the log marginal likelihood (evidence) without further overhead.

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arXiv2019

Convergence Guarantees for Adaptive Bayesian Quadrature Methods

Adaptive Bayesian quadrature (ABQ) is a powerful approach to numerical integration that empirically compares favorably with Monte Carlo integration on problems of medium dimensionality (where non-adaptive quadrature is not competitive). Its key ingredient is an acquisition function that changes as a function of previously collected values of the integrand. While this adaptivity appears to be empirically powerful, it complicates analysis. Consequently, there are no theoretical guarantees so far for this class of methods. In this work, for a broad class of adaptive Bayesian quadrature methods, we prove consistency, deriving non-tight but informative convergence rates. To do so we introduce a new concept we call weak adaptivity. Our results identify a large and flexible class of adaptive Bayesian quadrature rules as consistent, within which practitioners can develop empirically efficient methods.

stat.ML↗

arXiv2019

Uncertainty Estimates for Ordinal Embeddings

To investigate objects without a describable notion of distance, one can gather ordinal information by asking triplet comparisons of the form "Is object $x$ closer to $y$ or is $x$ closer to $z$?" In order to learn from such data, the objects are typically embedded in a Euclidean space while satisfying as many triplet comparisons as possible. In this paper, we introduce empirical uncertainty estimates for standard embedding algorithms when few noisy triplets are available, using a bootstrap and a Bayesian approach. In particular, simulations show that these estimates are well calibrated and can serve to select embedding parameters or to quantify uncertainty in scientific applications.

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arXiv2019

Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective

We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement sequence to consist of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP---which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a non-linear Bayesian filtering problem and all widely-used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers that are formulated in terms of generating synthetic measurements of the gradient field come out as specific approximations. Based on the non-linear Bayesian filtering problem posed in this paper, we develop novel Gaussian solvers for which we establish favourable stability properties. Additionally, non-Gaussian approximations to the filtering problem are derived by the particle filter approach. The resulting solvers are compared with other probabilistic solvers in illustrative experiments.

stat.ME↗

arXiv2019

DeepOBS: A Deep Learning Optimizer Benchmark Suite

Because the choice and tuning of the optimizer affects the speed, and ultimately the performance of deep learning, there is significant past and recent research in this area. Yet, perhaps surprisingly, there is no generally agreed-upon protocol for the quantitative and reproducible evaluation of optimization strategies for deep learning. We suggest routines and benchmarks for stochastic optimization, with special focus on the unique aspects of deep learning, such as stochasticity, tunability and generalization. As the primary contribution, we present DeepOBS, a Python package of deep learning optimization benchmarks. The package addresses key challenges in the quantitative assessment of stochastic optimizers, and automates most steps of benchmarking. The library includes a wide and extensible set of ready-to-use realistic optimization problems, such as training Residual Networks for image classification on ImageNet or character-level language prediction models, as well as popular classics like MNIST and CIFAR-10. The package also provides realistic baseline results for the most popular optimizers on these test problems, ensuring a fair comparison to the competition when benchmarking new optimizers, and without having to run costly experiments. It comes with output back-ends that directly produce LaTeX code for inclusion in academic publications. It supports TensorFlow and is available open source.

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arXiv2019

Active Probabilistic Inference on Matrices for Pre-Conditioning in Stochastic Optimization

Pre-conditioning is a well-known concept that can significantly improve the convergence of optimization algorithms. For noise-free problems, where good pre-conditioners are not known a priori, iterative linear algebra methods offer one way to efficiently construct them. For the stochastic optimization problems that dominate contemporary machine learning, however, this approach is not readily available. We propose an iterative algorithm inspired by classic iterative linear solvers that uses a probabilistic model to actively infer a pre-conditioner in situations where Hessian-projections can only be constructed with strong Gaussian noise. The algorithm is empirically demonstrated to efficiently construct effective pre-conditioners for stochastic gradient descent and its variants. Experiments on problems of comparably low dimensionality show improved convergence. In very high-dimensional problems, such as those encountered in deep learning, the pre-conditioner effectively becomes an automatic learning-rate adaptation scheme, which we also empirically show to work well.

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arXiv2019

Fast and Robust Shortest Paths on Manifolds Learned from Data

We propose a fast, simple and robust algorithm for computing shortest paths and distances on Riemannian manifolds learned from data. This amounts to solving a system of ordinary differential equations (ODEs) subject to boundary conditions. Here standard solvers perform poorly because they require well-behaved Jacobians of the ODE, and usually, manifolds learned from data imply unstable and ill-conditioned Jacobians. Instead, we propose a fixed-point iteration scheme for solving the ODE that avoids Jacobians. This enhances the stability of the solver, while reduces the computational cost. In experiments involving both Riemannian metric learning and deep generative models we demonstrate significant improvements in speed and stability over both general-purpose state-of-the-art solvers as well as over specialized solvers.

stat.ML↗

arXiv2018

Active Uncertainty Calibration in Bayesian ODE Solvers

There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al. introduced a sampling-based class of methods that are 'well-calibrated' in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al. pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the trade-off between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations.

math.NA↗

arXiv2018

Probabilistic Linear Solvers: A Unifying View

Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the matrix inverse. These approaches have typically focused on replicating the behavior of the conjugate gradient method as a prototypical iterative method. In this work surprisingly general conditions for equivalence of these disparate methods are presented. We also describe connections between probabilistic linear solvers and projection methods for linear systems, providing a probabilistic interpretation of a far more general class of iterative methods. In particular, this provides such an interpretation of the generalised minimum residual method. A probabilistic view of preconditioning is also introduced. These developments unify the literature on probabilistic linear solvers, and provide foundational connections to the literature on iterative solvers for linear systems.

stat.CO↗

arXiv2018

Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences

This paper is an attempt to bridge the conceptual gaps between researchers working on the two widely used approaches based on positive definite kernels: Bayesian learning or inference using Gaussian processes on the one side, and frequentist kernel methods based on reproducing kernel Hilbert spaces on the other. It is widely known in machine learning that these two formalisms are closely related; for instance, the estimator of kernel ridge regression is identical to the posterior mean of Gaussian process regression. However, they have been studied and developed almost independently by two essentially separate communities, and this makes it difficult to seamlessly transfer results between them. Our aim is to overcome this potential difficulty. To this end, we review several old and new results and concepts from either side, and juxtapose algorithmic quantities from each framework to highlight close similarities. We also provide discussions on subtle philosophical and theoretical differences between the two approaches.

stat.ML↗

arXiv2018

Exact Sampling from Determinantal Point Processes

Determinantal point processes (DPPs) are an important concept in random matrix theory and combinatorics. They have also recently attracted interest in the study of numerical methods for machine learning, as they offer an elegant "missing link" between independent Monte Carlo sampling and deterministic evaluation on regular grids, applicable to a general set of spaces. This is helpful whenever an algorithm explores to reduce uncertainty, such as in active learning, Bayesian optimization, reinforcement learning, and marginalization in graphical models. To draw samples from a DPP in practice, existing literature focuses on approximate schemes of low cost, or comparably inefficient exact algorithms like rejection sampling. We point out that, for many settings of relevance to machine learning, it is also possible to draw exact samples from DPPs on continuous domains. We start from an intuitive example on the real line, which is then generalized to multivariate real vector spaces. We also compare to previously studied approximations, showing that exact sampling, despite higher cost, can be preferable where precision is needed.

cs.LG↗

arXiv2017

Bayesian Filtering for ODEs with Bounded Derivatives

Recently there has been increasing interest in probabilistic solvers for ordinary differential equations (ODEs) that return full probability measures, instead of point estimates, over the solution and can incorporate uncertainty over the ODE at hand, e.g. if the vector field or the initial value is only approximately known or evaluable. The ODE filter proposed in recent work models the solution of the ODE by a Gauss-Markov process which serves as a prior in the sense of Bayesian statistics. While previous work employed a Wiener process prior on the (possibly multiple times) differentiated solution of the ODE and established equivalence of the corresponding solver with classical numerical methods, this paper raises the question whether other priors also yield practically useful solvers. To this end, we discuss a range of possible priors which enable fast filtering and propose a new prior--the Integrated Ornstein Uhlenbeck Process (IOUP)--that complements the existing Integrated Wiener process (IWP) filter by encoding the property that a derivative in time of the solution is bounded in the sense that it tends to drift back to zero. We provide experiments comparing IWP and IOUP filters which support the belief that IWP approximates better divergent ODE's solutions whereas IOUP is a better prior for trajectories with bounded derivatives.

math.NA↗

arXiv2017

On the Design of LQR Kernels for Efficient Controller Learning

Finding optimal feedback controllers for nonlinear dynamic systems from data is hard. Recently, Bayesian optimization (BO) has been proposed as a powerful framework for direct controller tuning from experimental trials. For selecting the next query point and finding the global optimum, BO relies on a probabilistic description of the latent objective function, typically a Gaussian process (GP). As is shown herein, GPs with a common kernel choice can, however, lead to poor learning outcomes on standard quadratic control problems. For a first-order system, we construct two kernels that specifically leverage the structure of the well-known Linear Quadratic Regulator (LQR), yet retain the flexibility of Bayesian nonparametric learning. Simulations of uncertain linear and nonlinear systems demonstrate that the LQR kernels yield superior learning performance.

eess.SY↗

arXiv2017

A probabilistic model for the numerical solution of initial value problems

Like many numerical methods, solvers for initial value problems (IVPs) on ordinary differential equations estimate an analytically intractable quantity, using the results of tractable computations as inputs. This structure is closely connected to the notion of inference on latent variables in statistics. We describe a class of algorithms that formulate the solution to an IVP as inference on a latent path that is a draw from a Gaussian process probability measure (or equivalently, the solution of a linear stochastic differential equation). We then show that certain members of this class are connected precisely to generalized linear methods for ODEs, a number of Runge--Kutta methods, and Nordsieck methods. This probabilistic formulation of classic methods is valuable in two ways: analytically, it highlights implicit prior assumptions favoring certain approximate solutions to the IVP over others, and gives a precise meaning to the old observation that these methods act like filters. Practically, it endows the classic solvers with `docking points' for notions of uncertainty and prior information about the initial value, the value of the ODE itself, and the solution of the problem.

math.NA↗

arXiv2017

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↗

arXiv2017

Probabilistic Active Learning of Functions in Structural Causal Models

We consider the problem of learning the functions computing children from parents in a Structural Causal Model once the underlying causal graph has been identified. This is in some sense the second step after causal discovery. Taking a probabilistic approach to estimating these functions, we derive a natural myopic active learning scheme that identifies the intervention which is optimally informative about all of the unknown functions jointly, given previously observed data. We test the derived algorithms on simple examples, to demonstrate that they produce a structured exploration policy that significantly improves on unstructured base-lines.

stat.ML↗

arXiv2017

Coupling Adaptive Batch Sizes with Learning Rates

Mini-batch stochastic gradient descent and variants thereof have become standard for large-scale empirical risk minimization like the training of neural networks. These methods are usually used with a constant batch size chosen by simple empirical inspection. The batch size significantly influences the behavior of the stochastic optimization algorithm, though, since it determines the variance of the gradient estimates. This variance also changes over the optimization process; when using a constant batch size, stability and convergence is thus often enforced by means of a (manually tuned) decreasing learning rate schedule. We propose a practical method for dynamic batch size adaptation. It estimates the variance of the stochastic gradients and adapts the batch size to decrease the variance proportionally to the value of the objective function, removing the need for the aforementioned learning rate decrease. In contrast to recent related work, our algorithm couples the batch size to the learning rate, directly reflecting the known relationship between the two. On popular image classification benchmarks, our batch size adaptation yields faster optimization convergence, while simultaneously simplifying learning rate tuning. A TensorFlow implementation is available.

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