David Janz

Tom Perneczky, Marc Abeille, David Janz

We prove a variance-sensitive regret bound for Thompson sampling in stochastic generalised linear bandits. The argument assumes a warm-up, after which the regret is controlled through using the Gaussian Poincaré inequality. This bypasses the point at which previous optimism-based analyses break down. Removing the warm-up while retaining the same variance-sensitive scaling remains open, and appears nontrivial.

Jihao Andreas Lin, Javier Antorán, Shreyas Padhy et al. · cambridge

Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to conditioning. We explore stochastic gradient algorithms as a computationally efficient method of approximately solving these linear systems: we develop low-variance optimization objectives for sampling from the posterior and extend these to inducing points. Counterintuitively, stochastic gradient descent often produces accurate predictions, even in cases where it does not converge quickly to the optimum. We explain this through a spectral characterization of the implicit bias from non-convergence. We show that stochastic gradient descent produces predictive distributions close to the true posterior both in regions with sufficient data coverage, and in regions sufficiently far away from the data. Experimentally, stochastic gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks. Its uncertainty estimates match the performance of significantly more expensive baselines on a large-scale Bayesian optimization task.

Javier Antorán, David Janz, James Urquhart Allingham et al.

The linearised Laplace method for estimating model uncertainty has received renewed attention in the Bayesian deep learning community. The method provides reliable error bars and admits a closed-form expression for the model evidence, allowing for scalable selection of model hyperparameters. In this work, we examine the assumptions behind this method, particularly in conjunction with model selection. We show that these interact poorly with some now-standard tools of deep learning--stochastic approximation methods and normalisation layers--and make recommendations for how to better adapt this classic method to the modern setting. We provide theoretical support for our recommendations and validate them empirically on MLPs, classic CNNs, residual networks with and without normalisation layers, generative autoencoders and transformers.

Jihao Andreas Lin, Shreyas Padhy, Javier Antorán et al.

As is well known, both sampling from the posterior and computing the mean of the posterior in Gaussian process regression reduces to solving a large linear system of equations. We study the use of stochastic gradient descent for solving this linear system, and show that when \emph{done right} -- by which we mean using specific insights from the optimisation and kernel communities -- stochastic gradient descent is highly effective. To that end, we introduce a particularly simple \emph{stochastic dual descent} algorithm, explain its design in an intuitive manner and illustrate the design choices through a series of ablation studies. Further experiments demonstrate that our new method is highly competitive. In particular, our evaluations on the UCI regression tasks and on Bayesian optimisation set our approach apart from preconditioned conjugate gradients and variational Gaussian process approximations. Moreover, our method places Gaussian process regression on par with state-of-the-art graph neural networks for molecular binding affinity prediction.

David Janz, Alexander E. Litvak, Csaba Szepesvári

We provide the first useful and rigorous analysis of ensemble sampling for the stochastic linear bandit setting. In particular, we show that, under standard assumptions, for a $d$-dimensional stochastic linear bandit with an interaction horizon $T$, ensemble sampling with an ensemble of size of order $d \log T$ incurs regret at most of the order $(d \log T)^{5/2} \sqrt{T}$. Ours is the first result in any structured setting not to require the size of the ensemble to scale linearly with $T$ -- which defeats the purpose of ensemble sampling -- while obtaining near $\smash{\sqrt{T}}$ order regret. Our result is also the first to allow for infinite action sets.

Javier Antorán, Shreyas Padhy, Riccardo Barbano et al.

Large-scale linear models are ubiquitous throughout machine learning, with contemporary application as surrogate models for neural network uncertainty quantification; that is, the linearised Laplace method. Alas, the computational cost associated with Bayesian linear models constrains this method's application to small networks, small output spaces and small datasets. We address this limitation by introducing a scalable sample-based Bayesian inference method for conjugate Gaussian multi-output linear models, together with a matching method for hyperparameter (regularisation) selection. Furthermore, we use a classic feature normalisation method (the g-prior) to resolve a previously highlighted pathology of the linearised Laplace method. Together, these contributions allow us to perform linearised neural network inference with ResNet-18 on CIFAR100 (11M parameters, 100 outputs x 50k datapoints), with ResNet-50 on Imagenet (50M parameters, 1000 outputs x 1.2M datapoints) and with a U-Net on a high-resolution tomographic reconstruction task (2M parameters, 251k output~dimensions).

David Janz, Shuai Liu, Alex Ayoub et al.

We introduce exploration via linear loss perturbations (EVILL), a randomised exploration method for structured stochastic bandit problems that works by solving for the minimiser of a linearly perturbed regularised negative log-likelihood function. We show that, for the case of generalised linear bandits, EVILL reduces to perturbed history exploration (PHE), a method where exploration is done by training on randomly perturbed rewards. In doing so, we provide a simple and clean explanation of when and why random reward perturbations give rise to good bandit algorithms. We propose data-dependent perturbations not present in previous PHE-type methods that allow EVILL to match the performance of Thompson-sampling-style parameter-perturbation methods, both in theory and in practice. Moreover, we show an example outside generalised linear bandits where PHE leads to inconsistent estimates, and thus linear regret, while EVILL remains performant. Like PHE, EVILL can be implemented in just a few lines of code.

Marc Abeille, David Janz, Ciara Pike-Burke

We provide an approach for the analysis of randomised exploration algorithms like Thompson sampling that does not rely on forced optimism or posterior inflation. With this, we demonstrate that in the $d$-dimensional linear bandit setting, when the action space is smooth and strongly convex, randomised exploration algorithms enjoy an $n$-step regret bound of the order $O(d\sqrt{n} \log(n))$. Notably, this shows for the first time that there exist non-trivial linear bandit settings where Thompson sampling can achieve optimal dimension dependence in the regret.

Alireza Bakhtiari, Alex Ayoub, Samuel Robertson et al.

We establish a lower bound on the eluder dimension of generalised linear model classes, showing that standard eluder dimension-based analysis cannot lead to first-order regret bounds. To address this, we introduce a localisation method for the eluder dimension; our analysis immediately recovers and improves on classic results for Bernoulli bandits, and allows for the first genuine first-order bounds for finite-horizon reinforcement learning tasks with bounded cumulative returns.

Arya Akhavan, David Janz, El-Mahdi El-Mhamdi

We study distributed learning in the context of gradient-free zero-order optimisation and introduce FedZero, a federated zero-order algorithm with sharp theoretical guarantees. Our contributions are threefold. First, in the federated convex setting, we derive high-probability guarantees for regret minimisation achieved by FedZero. Second, in the single-worker regime, corresponding to the classical zero-order framework with two-point feedback, we establish the first high-probability convergence guarantees for convex zero-order optimisation, strengthening previous results that held only in expectation. Third, to establish these guarantees, we develop novel concentration tools: (i) concentration inequalities with explicit constants for Lipschitz functions under the uniform measure on the $\ell_1$-sphere, and (ii) a time-uniform concentration inequality for squared sub-Gamma random variables. These probabilistic results underpin our high-probability guarantees and may also be of independent interest.

David Janz, David R. Burt, Javier González

We consider the problem of optimising functions in the reproducing kernel Hilbert space (RKHS) of a Matérn kernel with smoothness parameter $ν$ over the domain $[0,1]^d$ under noisy bandit feedback. Our contribution, the $π$-GP-UCB algorithm, is the first practical approach with guaranteed sublinear regret for all $ν>1$ and $d \geq 1$. Empirical validation suggests better performance and drastically improved computational scalablity compared with its predecessor, Improved GP-UCB.

David Janz, Jiri Hron, Przemysław Mazur et al.

Posterior sampling for reinforcement learning (PSRL) is an effective method for balancing exploration and exploitation in reinforcement learning. Randomised value functions (RVF) can be viewed as a promising approach to scaling PSRL. However, we show that most contemporary algorithms combining RVF with neural network function approximation do not possess the properties which make PSRL effective, and provably fail in sparse reward problems. Moreover, we find that propagation of uncertainty, a property of PSRL previously thought important for exploration, does not preclude this failure. We use these insights to design Successor Uncertainties (SU), a cheap and easy to implement RVF algorithm that retains key properties of PSRL. SU is highly effective on hard tabular exploration benchmarks. Furthermore, on the Atari 2600 domain, it surpasses human performance on 38 of 49 games tested (achieving a median human normalised score of 2.09), and outperforms its closest RVF competitor, Bootstrapped DQN, on 36 of those.

Alex Kendall, Jeffrey Hawke, David Janz et al.

We demonstrate the first application of deep reinforcement learning to autonomous driving. From randomly initialised parameters, our model is able to learn a policy for lane following in a handful of training episodes using a single monocular image as input. We provide a general and easy to obtain reward: the distance travelled by the vehicle without the safety driver taking control. We use a continuous, model-free deep reinforcement learning algorithm, with all exploration and optimisation performed on-vehicle. This demonstrates a new framework for autonomous driving which moves away from reliance on defined logical rules, mapping, and direct supervision. We discuss the challenges and opportunities to scale this approach to a broader range of autonomous driving tasks.

David Janz, Jos van der Westhuizen, Brooks Paige et al.

Deep generative models have been successfully used to learn representations for high-dimensional discrete spaces by representing discrete objects as sequences and employing powerful sequence-based deep models. Unfortunately, these sequence-based models often produce invalid sequences: sequences which do not represent any underlying discrete structure; invalid sequences hinder the utility of such models. As a step towards solving this problem, we propose to learn a deep recurrent validator model, which can estimate whether a partial sequence can function as the beginning of a full, valid sequence. This validator provides insight as to how individual sequence elements influence the validity of the overall sequence, and can be used to constrain sequence based models to generate valid sequences -- and thus faithfully model discrete objects. Our approach is inspired by reinforcement learning, where an oracle which can evaluate validity of complete sequences provides a sparse reward signal. We demonstrate its effectiveness as a generative model of Python 3 source code for mathematical expressions, and in improving the ability of a variational autoencoder trained on SMILES strings to decode valid molecular structures.

David Janz, Jos van der Westhuizen, José Miguel Hernández-Lobato

Deep learning techniques have been hugely successful for traditional supervised and unsupervised machine learning problems. In large part, these techniques solve continuous optimization problems. Recently however, discrete generative deep learning models have been successfully used to efficiently search high-dimensional discrete spaces. These methods work by representing discrete objects as sequences, for which powerful sequence-based deep models can be employed. Unfortunately, these techniques are significantly hindered by the fact that these generative models often produce invalid sequences. As a step towards solving this problem, we propose to learn a deep recurrent validator model. Given a partial sequence, our model learns the probability of that sequence occurring as the beginning of a full valid sequence. Thus this identifies valid versus invalid sequences and crucially it also provides insight about how individual sequence elements influence the validity of discrete objects. To learn this model we propose an approach inspired by seminal work in Bayesian active learning. On a synthetic dataset, we demonstrate the ability of our model to distinguish valid and invalid sequences. We believe this is a key step toward learning generative models that faithfully produce valid discrete objects.

David Janz, Brooks Paige, Tom Rainforth et al.

Existing methods for structure discovery in time series data construct interpretable, compositional kernels for Gaussian process regression models. While the learned Gaussian process model provides posterior mean and variance estimates, typically the structure is learned via a greedy optimization procedure. This restricts the space of possible solutions and leads to over-confident uncertainty estimates. We introduce a fully Bayesian approach, inferring a full posterior over structures, which more reliably captures the uncertainty of the model.