arXiv subjects

Tor Lattimore

Publications and source records attributed to Tor Lattimore.

At least 73 records · Page 4Linked to original sources

arXiv2017

Online Learning with Gated Linear Networks

This paper describes a family of probabilistic architectures designed for online learning under the logarithmic loss. Rather than relying on non-linear transfer functions, our method gains representational power by the use of data conditioning. We state under general conditions a learnable capacity theorem that shows this approach can in principle learn any bounded Borel-measurable function on a compact subset of euclidean space; the result is stronger than many universality results for connectionist architectures because we provide both the model and the learning procedure for which convergence is guaranteed.

cs.LG↗

arXiv2017

A Scale Free Algorithm for Stochastic Bandits with Bounded Kurtosis

Existing strategies for finite-armed stochastic bandits mostly depend on a parameter of scale that must be known in advance. Sometimes this is in the form of a bound on the payoffs, or the knowledge of a variance or subgaussian parameter. The notable exceptions are the analysis of Gaussian bandits with unknown mean and variance by Cowan and Katehakis [2015] and of uniform distributions with unknown support [Cowan and Katehakis, 2015]. The results derived in these specialised cases are generalised here to the non-parametric setup, where the learner knows only a bound on the kurtosis of the noise, which is a scale free measure of the extremity of outliers.

stat.ML↗

arXiv2017

Refined Lower Bounds for Adversarial Bandits

We provide new lower bounds on the regret that must be suffered by adversarial bandit algorithms. The new results show that recent upper bounds that either (a) hold with high-probability or (b) depend on the total lossof the best arm or (c) depend on the quadratic variation of the losses, are close to tight. Besides this we prove two impossibility results. First, the existence of a single arm that is optimal in every round cannot improve the regret in the worst case. Second, the regret cannot scale with the effective range of the losses. In contrast, both results are possible in the full-information setting.

math.ST↗

arXiv2017

Following the Leader and Fast Rates in Linear Prediction: Curved Constraint Sets and Other Regularities

The follow the leader (FTL) algorithm, perhaps the simplest of all online learning algorithms, is known to perform well when the loss functions it is used on are convex and positively curved. In this paper we ask whether there are other "lucky" settings when FTL achieves sublinear, "small" regret. In particular, we study the fundamental problem of linear prediction over a non-empty convex, compact domain. Amongst other results, we prove that the curvature of the boundary of the domain can act as if the losses were curved: In this case, we prove that as long as the mean of the loss vectors have positive lengths bounded away from zero, FTL enjoys a logarithmic growth rate of regret, while, e.g., for polytope domains and stochastic data it enjoys finite expected regret. Building on a previously known meta-algorithm, we also get an algorithm that simultaneously enjoys the worst-case guarantees and the bound available for FTL.

cs.LG↗

arXiv2016

On Explore-Then-Commit Strategies

We study the problem of minimising regret in two-armed bandit problems with Gaussian rewards. Our objective is to use this simple setting to illustrate that strategies based on an exploration phase (up to a stopping time) followed by exploitation are necessarily suboptimal. The results hold regardless of whether or not the difference in means between the two arms is known. Besides the main message, we also refine existing deviation inequalities, which allow us to design fully sequential strategies with finite-time regret guarantees that are (a) asymptotically optimal as the horizon grows and (b) order-optimal in the minimax sense. Furthermore we provide empirical evidence that the theory also holds in practice and discuss extensions to non-gaussian and multiple-armed case.

math.ST↗

arXiv2016

The End of Optimism? An Asymptotic Analysis of Finite-Armed Linear Bandits

Stochastic linear bandits are a natural and simple generalisation of finite-armed bandits with numerous practical applications. Current approaches focus on generalising existing techniques for finite-armed bandits, notably the optimism principle and Thompson sampling. While prior work has mostly been in the worst-case setting, we analyse the asymptotic instance-dependent regret and show matching upper and lower bounds on what is achievable. Surprisingly, our results show that no algorithm based on optimism or Thompson sampling will ever achieve the optimal rate, and indeed, can be arbitrarily far from optimal, even in very simple cases. This is a disturbing result because these techniques are standard tools that are widely used for sequential optimisation. For example, for generalised linear bandits and reinforcement learning.

stat.ML↗

arXiv2016

Free Lunch for Optimisation under the Universal Distribution

Function optimisation is a major challenge in computer science. The No Free Lunch theorems state that if all functions with the same histogram are assumed to be equally probable then no algorithm outperforms any other in expectation. We argue against the uniform assumption and suggest a universal prior exists for which there is a free lunch, but where no particular class of functions is favoured over another. We also prove upper and lower bounds on the size of the free lunch.

math.OC↗

arXiv2016

Causal Bandits: Learning Good Interventions via Causal Inference

We study the problem of using causal models to improve the rate at which good interventions can be learned online in a stochastic environment. Our formalism combines multi-arm bandits and causal inference to model a novel type of bandit feedback that is not exploited by existing approaches. We propose a new algorithm that exploits the causal feedback and prove a bound on its simple regret that is strictly better (in all quantities) than algorithms that do not use the additional causal information.

stat.ML↗

arXiv2016

Thompson Sampling is Asymptotically Optimal in General Environments

We discuss a variant of Thompson sampling for nonparametric reinforcement learning in a countable classes of general stochastic environments. These environments can be non-Markov, non-ergodic, and partially observable. We show that Thompson sampling learns the environment class in the sense that (1) asymptotically its value converges to the optimal value in mean and (2) given a recoverability assumption regret is sublinear.

cs.LG↗

arXiv2016

Regret Analysis of the Finite-Horizon Gittins Index Strategy for Multi-Armed Bandits

I analyse the frequentist regret of the famous Gittins index strategy for multi-armed bandits with Gaussian noise and a finite horizon. Remarkably it turns out that this approach leads to finite-time regret guarantees comparable to those available for the popular UCB algorithm. Along the way I derive finite-time bounds on the Gittins index that are asymptotically exact and may be of independent interest. I also discuss some computational issues and present experimental results suggesting that a particular version of the Gittins index strategy is a modest improvement on existing algorithms with finite-time regret guarantees such as UCB and Thompson sampling.

cs.LG↗

arXiv2016

Regret Analysis of the Anytime Optimally Confident UCB Algorithm

I introduce and analyse an anytime version of the Optimally Confident UCB (OCUCB) algorithm designed for minimising the cumulative regret in finite-armed stochastic bandits with subgaussian noise. The new algorithm is simple, intuitive (in hindsight) and comes with the strongest finite-time regret guarantees for a horizon-free algorithm so far. I also show a finite-time lower bound that nearly matches the upper bound.

cs.LG↗

arXiv2016

Optimally Confident UCB: Improved Regret for Finite-Armed Bandits

I present the first algorithm for stochastic finite-armed bandits that simultaneously enjoys order-optimal problem-dependent regret and worst-case regret. Besides the theoretical results, the new algorithm is simple, efficient and empirically superb. The approach is based on UCB, but with a carefully chosen confidence parameter that optimally balances the risk of failing confidence intervals against the cost of excessive optimism.

cs.LG↗

arXiv2016

Conservative Bandits

We study a novel multi-armed bandit problem that models the challenge faced by a company wishing to explore new strategies to maximize revenue whilst simultaneously maintaining their revenue above a fixed baseline, uniformly over time. While previous work addressed the problem under the weaker requirement of maintaining the revenue constraint only at a given fixed time in the future, the algorithms previously proposed are unsuitable due to their design under the more stringent constraints. We consider both the stochastic and the adversarial settings, where we propose, natural, yet novel strategies and analyze the price for maintaining the constraints. Amongst other things, we prove both high probability and expectation bounds on the regret, while we also consider both the problem of maintaining the constraints with high probability or expectation. For the adversarial setting the price of maintaining the constraint appears to be higher, at least for the algorithm considered. A lower bound is given showing that the algorithm for the stochastic setting is almost optimal. Empirical results obtained in synthetic environments complement our theoretical findings.

stat.ML↗

arXiv2015

The Pareto Regret Frontier for Bandits

Given a multi-armed bandit problem it may be desirable to achieve a smaller-than-usual worst-case regret for some special actions. I show that the price for such unbalanced worst-case regret guarantees is rather high. Specifically, if an algorithm enjoys a worst-case regret of B with respect to some action, then there must exist another action for which the worst-case regret is at least Ω(nK/B), where n is the horizon and K the number of actions. I also give upper bounds in both the stochastic and adversarial settings showing that this result cannot be improved. For the stochastic case the pareto regret frontier is characterised exactly up to constant factors.

cs.LG↗

arXiv2014

Asymptotics of Continuous Bayes for Non-i.i.d. Sources

Clarke and Barron analysed the relative entropy between an i.i.d. source and a Bayesian mixture over a continuous class containing that source. In this paper a comparable result is obtained when the source is permitted to be both non-stationary and dependent. The main theorem shows that Bayesian methods perform well for both compression and sequence prediction even in this most general setting with only mild technical assumptions.

cs.IT↗

arXiv2014

Bounded Regret for Finite-Armed Structured Bandits

We study a new type of K-armed bandit problem where the expected return of one arm may depend on the returns of other arms. We present a new algorithm for this general class of problems and show that under certain circumstances it is possible to achieve finite expected cumulative regret. We also give problem-dependent lower bounds on the cumulative regret showing that at least in special cases the new algorithm is nearly optimal.

cs.LG↗

arXiv2014

Optimal Resource Allocation with Semi-Bandit Feedback

We study a sequential resource allocation problem involving a fixed number of recurring jobs. At each time-step the manager should distribute available resources among the jobs in order to maximise the expected number of completed jobs. Allocating more resources to a given job increases the probability that it completes, but with a cut-off. Specifically, we assume a linear model where the probability increases linearly until it equals one, after which allocating additional resources is wasteful. We assume the difficulty of each job is unknown and present the first algorithm for this problem and prove upper and lower bounds on its regret. Despite its apparent simplicity, the problem has a rich structure: we show that an appropriate optimistic algorithm can improve its learning speed dramatically beyond the results one normally expects for similar problems as the problem becomes resource-laden.

cs.LG↗

arXiv2013

The Sample-Complexity of General Reinforcement Learning

We present a new algorithm for general reinforcement learning where the true environment is known to belong to a finite class of N arbitrary models. The algorithm is shown to be near-optimal for all but O(N log^2 N) time-steps with high probability. Infinite classes are also considered where we show that compactness is a key criterion for determining the existence of uniform sample-complexity bounds. A matching lower bound is given for the finite case.

cs.LG↗