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Register a new userOn Explore-Then-Commit Strategies
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Added by
Emilie Kaufmann
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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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.
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In this paper, we study multi-armed bandit problems in explore-then-commit setting. In our proposed explore-then-commit setting, the goal is to identify the best arm after a pure experimentation (exploration) phase and exploit it once or for a given finite number of times. We identify that although the arm with the highest expected reward is the most desirable objective for infinite exploitations, it is not necessarily the one that is most probable to have the highest reward in a single or finite-time exploitations. Alternatively, we advocate the idea of risk-aversion where the objective is to compete against the arm with the best risk-return trade-off. Then, we propose two algorithms whose objectives are to select the arm that is most probable to reward the most. Using a new notion of finite-time exploitation regret, we find an upper bound for the minimum number of experiments before commitment, to guarantee an upper bound for the regret. As compared to existing risk-averse bandit algorithms, our algorithms do not rely on hyper-parameters, resulting in a more robust behavior in practice, which is verified by the numerical evaluation.
We revisit lower bounds on the regret in the case of multi-armed bandit problems. We obtain non-asymptotic, distribution-dependent bounds and provide straightforward proofs based only on well-known properties of Kullback-Leibler divergences. These bounds show in particular that in an initial phase the regret grows almost linearly, and that the well-known logarithmic growth of the regret only holds in a final phase. The proof techniques come to the essence of the information-theoretic arguments used and they are deprived of all unnecessary complications.
The promise of reinforcement learning is to solve complex sequential decision problems autonomously by specifying a high-level reward function only. However, reinforcement learning algorithms struggle when, as is often the case, simple and intuitive rewards provide sparse and deceptive feedback. Avoiding these pitfalls requires thoroughly exploring the environment, but creating algorithms that can do so remains one of the central challenges of the field. We hypothesise that the main impediment to effective exploration originates from algorithms forgetting how to reach previously visited states (detachment) and from failing to first return to a state before exploring from it (derailment). We introduce Go-Explore, a family of algorithms that addresses these two challenges directly through the simple principles of explicitly remembering promising states and first returning to such states before intentionally exploring. Go-Explore solves all heretofore unsolved Atari games and surpasses the state of the art on all hard-exploration games, with orders of magnitude improvements on the grand challenges Montezumas Revenge and Pitfall. We also demonstrate the practical potential of Go-Explore on a sparse-reward pick-and-place robotics task. Additionally, we show that adding a goal-conditioned policy can further improve Go-Explores exploration efficiency and enable it to handle stochasticity throughout training. The substantial performance gains from Go-Explore suggest that the simple principles of remembering states, returning to them, and exploring from them are a powerful and general approach to exploration, an insight that may prove critical to the creation of truly intelligent learning agents.
We consider the framework of stochastic multi-armed bandit problems and study the possibilities and limitations of forecasters that perform an on-line exploration of the arms. These forecasters are assessed in terms of their simple regret, a regret notion that captures the fact that exploration is only constrained by the number of available rounds (not necessarily known in advance), in contrast to the case when the cumulative regret is considered and when exploitation needs to be performed at the same time. We believe that this performance criterion is suited to situations when the cost of pulling an arm is expressed in terms of resources rather than rewards. We discuss the links between the simple and the cumulative regret. One of the main results in the case of a finite number of arms is a general lower bound on the simple regret of a forecaster in terms of its cumulative regret: the smaller the latter, the larger the former. Keeping this result in mind, we then exhibit upper bounds on the simple regret of some forecasters. The paper ends with a study devoted to continuous-armed bandit problems; we show that the simple regret can be minimized with respect to a family of probability distributions if and only if the cumulative regret can be minimized for it. Based on this equivalence, we are able to prove that the separable metric spaces are exactly the metric spaces on which these regrets can be minimized with respect to the family of all probability distributions with continuous mean-payoff functions.
We consider the problem of learning an unknown function $f_{star}$ on the $d$-dimensional sphere with respect to the square loss, given i.i.d. samples ${(y_i,{boldsymbol x}_i)}_{ile n}$ where ${boldsymbol x}_i$ is a feature vector uniformly distributed on the sphere and $y_i=f_{star}({boldsymbol x}_i)+varepsilon_i$. We study two popular classes of models that can be regarded as linearizations of two-layers neural networks around a random initialization: the random features model of Rahimi-Recht (RF); the neural tangent kernel model of Jacot-Gabriel-Hongler (NT). Both these approaches can also be regarded as randomized approximations of kernel ridge regression (with respect to different kernels), and enjoy universal approximation properties when the number of neurons $N$ diverges, for a fixed dimension $d$. We consider two specific regimes: the approximation-limited regime, in which $n=infty$ while $d$ and $N$ are large but finite; and the sample size-limited regime in which $N=infty$ while $d$ and $n$ are large but finite. In the first regime we prove that if $d^{ell + delta} le Nle d^{ell+1-delta}$ for small $delta > 0$, then RF, effectively fits a degree-$ell$ polynomial in the raw features, and NT, fits a degree-$(ell+1)$ polynomial. In the second regime, both RF and NT reduce to kernel methods with rotationally invariant kernels. We prove that, if the number of samples is $d^{ell + delta} le n le d^{ell +1-delta}$, then kernel methods can fit at most a a degree-$ell$ polynomial in the raw features. This lower bound is achieved by kernel ridge regression. Optimal prediction error is achieved for vanishing ridge regularization.
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