Improved Second-Order Bounds for Prediction with Expert Advice

This work studies external regret in sequential prediction games with both positive and negative payoffs. External regret measures the difference between the payoff obtained by the forecasting strategy and the payoff of the best action. In this setting, we derive new and sharper regret bounds for the well-known exponentially weighted average forecaster and for a new forecaster with a different multiplicative update rule. Our analysis has two main advantages: first, no preliminary knowledge about the payoff sequence is needed, not even its range; second, our bounds are expressed in terms of sums of squared payoffs, replacing larger first-order quantities appearing in previous bounds. In addition, our most refined bounds have the natural and desirable property of being stable under rescalings and general translations of the payoff sequence

Adaptation to Easy Data in Prediction with Limited Advice

We derive an online learning algorithm with improved regret guarantees for `easy' loss sequences. We consider two types of `easiness': (a) stochastic loss sequences and (b) adversarial loss sequences with small effective range of the losses. While a number of algorithms have been proposed for exploiting small effective range in the full information setting, Gerchinovitz and Lattimore [2016] have shown the impossibility of regret scaling with the effective range of the losses in the bandit setting. We show that just one additional observation per round is sufficient to circumvent the impossibility result. The proposed Second Order Difference Adjustments (SODA) algorithm requires no prior knowledge of the effective range of the losses, $\varepsilon$, and achieves an $O(\varepsilon \sqrt{KT \ln K}) + \tilde{O}(\varepsilon K \sqrt[4]{T})$ expected regret guarantee, where $T$ is the time horizon and $K$ is the number of actions. The scaling with the effective loss range is achieved under significantly weaker assumptions than those made by Cesa-Bianchi and Shamir [2018] in an earlier attempt to circumvent the impossibility result. We also provide a regret lower bound of $\Omega(\varepsilon\sqrt{T K})$, which almost matches the upper bound. In addition, we show that in the stochastic setting SODA achieves an $O\left(\sum_{a:\Delta_a>0} \frac{K^3 \varepsilon^2}{\Delta_a}\right)$ pseudo-regret bound that holds simultaneously with the adversarial regret guarantee. In other words, SODA is safe against an unrestricted oblivious adversary and provides improved regret guarantees for at least two different types of `easiness' simultaneously.Comment: Fixed a mistake in the proof and statement of Theorem

Second-order Quantile Methods for Experts and Combinatorial Games

We aim to design strategies for sequential decision making that adjust to the difficulty of the learning problem. We study this question both in the setting of prediction with expert advice, and for more general combinatorial decision tasks. We are not satisfied with just guaranteeing minimax regret rates, but we want our algorithms to perform significantly better on easy data. Two popular ways to formalize such adaptivity are second-order regret bounds and quantile bounds. The underlying notions of 'easy data', which may be paraphrased as "the learning problem has small variance" and "multiple decisions are useful", are synergetic. But even though there are sophisticated algorithms that exploit one of the two, no existing algorithm is able to adapt to both. In this paper we outline a new method for obtaining such adaptive algorithms, based on a potential function that aggregates a range of learning rates (which are essential tuning parameters). By choosing the right prior we construct efficient algorithms and show that they reap both benefits by proving the first bounds that are both second-order and incorporate quantiles

Adaptive Online Prediction by Following the Perturbed Leader

When applying aggregating strategies to Prediction with Expert Advice, the learning rate must be adaptively tuned. The natural choice of sqrt(complexity/current loss) renders the analysis of Weighted Majority derivatives quite complicated. In particular, for arbitrary weights there have been no results proven so far. The analysis of the alternative "Follow the Perturbed Leader" (FPL) algorithm from Kalai & Vempala (2003) (based on Hannan's algorithm) is easier. We derive loss bounds for adaptive learning rate and both finite expert classes with uniform weights and countable expert classes with arbitrary weights. For the former setup, our loss bounds match the best known results so far, while for the latter our results are new.Comment: 25 page

A Second-order Bound with Excess Losses

We study online aggregation of the predictions of experts, and first show new second-order regret bounds in the standard setting, which are obtained via a version of the Prod algorithm (and also a version of the polynomially weighted average algorithm) with multiple learning rates. These bounds are in terms of excess losses, the differences between the instantaneous losses suffered by the algorithm and the ones of a given expert. We then demonstrate the interest of these bounds in the context of experts that report their confidences as a number in the interval [0,1] using a generic reduction to the standard setting. We conclude by two other applications in the standard setting, which improve the known bounds in case of small excess losses and show a bounded regret against i.i.d. sequences of losses

Optimality of Universal Bayesian Sequence Prediction for General Loss and Alphabet

Various optimality properties of universal sequence predictors based on Bayes-mixtures in general, and Solomonoff's prediction scheme in particular, will be studied. The probability of observing $x_t$ at time $t$, given past observations $x_1...x_{t-1}$ can be computed with the chain rule if the true generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If $\mu$ is unknown, but known to belong to a countable or continuous class \M one can base ones prediction on the Bayes-mixture $\xi$ defined as a $w_\nu$-weighted sum or integral of distributions \nu\in\M. The cumulative expected loss of the Bayes-optimal universal prediction scheme based on $\xi$ is shown to be close to the loss of the Bayes-optimal, but infeasible prediction scheme based on $\mu$. We show that the bounds are tight and that no other predictor can lead to significantly smaller bounds. Furthermore, for various performance measures, we show Pareto-optimality of $\xi$ and give an Occam's razor argument that the choice $w_\nu\sim 2^{-K(\nu)}$ for the weights is optimal, where $K(\nu)$ is the length of the shortest program describing $\nu$. The results are applied to games of chance, defined as a sequence of bets, observations, and rewards. The prediction schemes (and bounds) are compared to the popular predictors based on expert advice. Extensions to infinite alphabets, partial, delayed and probabilistic prediction, classification, and more active systems are briefly discussed.Comment: 34 page

Lipschitz Adaptivity with Multiple Learning Rates in Online Learning

We aim to design adaptive online learning algorithms that take advantage of any special structure that might be present in the learning task at hand, with as little manual tuning by the user as possible. A fundamental obstacle that comes up in the design of such adaptive algorithms is to calibrate a so-called step-size or learning rate hyperparameter depending on variance, gradient norms, etc. A recent technique promises to overcome this difficulty by maintaining multiple learning rates in parallel. This technique has been applied in the MetaGrad algorithm for online convex optimization and the Squint algorithm for prediction with expert advice. However, in both cases the user still has to provide in advance a Lipschitz hyperparameter that bounds the norm of the gradients. Although this hyperparameter is typically not available in advance, tuning it correctly is crucial: if it is set too small, the methods may fail completely; but if it is taken too large, performance deteriorates significantly. In the present work we remove this Lipschitz hyperparameter by designing new versions of MetaGrad and Squint that adapt to its optimal value automatically. We achieve this by dynamically updating the set of active learning rates. For MetaGrad, we further improve the computational efficiency of handling constraints on the domain of prediction, and we remove the need to specify the number of rounds in advance.Comment: 22 pages. To appear in COLT 201