37 research outputs found
Online Learning with Switching Costs and Other Adaptive Adversaries
We study the power of different types of adaptive (nonoblivious) adversaries
in the setting of prediction with expert advice, under both full-information
and bandit feedback. We measure the player's performance using a new notion of
regret, also known as policy regret, which better captures the adversary's
adaptiveness to the player's behavior. In a setting where losses are allowed to
drift, we characterize ---in a nearly complete manner--- the power of adaptive
adversaries with bounded memories and switching costs. In particular, we show
that with switching costs, the attainable rate with bandit feedback is
. Interestingly, this rate is significantly worse
than the rate attainable with switching costs in the
full-information case. Via a novel reduction from experts to bandits, we also
show that a bounded memory adversary can force
regret even in the full information case, proving that switching costs are
easier to control than bounded memory adversaries. Our lower bounds rely on a
new stochastic adversary strategy that generates loss processes with strong
dependencies
On prediction of individual sequences
Sequential randomized prediction of an arbitrary binary sequence is investigated. No assumption is made on the mechanism of generating the bit sequence. The goal of the predictor is to minimize its relative loss, i.e., to make (almost) as few mistakes as the best ``expert'' in a fixed, possibly infinite, set of experts. We point out a surprising connection between this prediction problem and empirical process theory. First, in the special case of static (memoryless) experts, we completely characterize the minimax relative loss in terms of the maximum of an associated Rademacher process. Then we show general upper and lower bounds on the minimax relative loss in terms of the geometry of the class of experts. As main examples, we determine the exact order of magnitude of the minimax relative loss for the class of autoregressive linear predictors and for the class of Markov experts.Universal prediction, prediction with experts, absolute loss, empirical processes, covering numbers, finite-state machines
Delay and Cooperation in Nonstochastic Bandits
We study networks of communicating learning agents that cooperate to solve a
common nonstochastic bandit problem. Agents use an underlying communication
network to get messages about actions selected by other agents, and drop
messages that took more than hops to arrive, where is a delay
parameter. We introduce \textsc{Exp3-Coop}, a cooperative version of the {\sc
Exp3} algorithm and prove that with actions and agents the average
per-agent regret after rounds is at most of order , where is the
independence number of the -th power of the connected communication graph
. We then show that for any connected graph, for the regret
bound is , strictly better than the minimax regret
for noncooperating agents. More informed choices of lead to bounds which
are arbitrarily close to the full information minimax regret
when is dense. When has sparse components, we show that a variant of
\textsc{Exp3-Coop}, allowing agents to choose their parameters according to
their centrality in , strictly improves the regret. Finally, as a by-product
of our analysis, we provide the first characterization of the minimax regret
for bandit learning with delay.Comment: 30 page
Adaptive maximization of social welfare
We consider the problem of repeatedly choosing policies to maximize social
welfare. Welfare is a weighted sum of private utility and public revenue.
Earlier outcomes inform later policies. Utility is not observed, but indirectly
inferred. Response functions are learned through experimentation.
We derive a lower bound on regret, and a matching adversarial upper bound for
a variant of the Exp3 algorithm. Cumulative regret grows at a rate of
. This implies that (i) welfare maximization is harder than the
multi-armed bandit problem (with a rate of for finite policy sets),
and (ii) our algorithm achieves the optimal rate. For the stochastic setting,
if social welfare is concave, we can achieve a rate of (for
continuous policy sets), using a dyadic search algorithm.
We analyze an extension to nonlinear income taxation, and sketch an extension
to commodity taxation. We compare our setting to monopoly pricing (which is
easier), and price setting for bilateral trade (which is harder)
Nonstochastic Bandits with Composite Anonymous Feedback
International audienceWe investigate a nonstochastic bandit setting in which the loss of an action is not immediately charged to the player, but rather spread over at most d consecutive steps in an adversarial way. This implies that the instantaneous loss observed by the player at the end of each round is a sum of as many as d loss components of previously played actions. Hence, unlike the standard bandit setting with delayed feedback, here the player cannot observe the individual delayed losses, but only their sum. Our main contribution is a general reduction transforming a standard bandit algorithm into one that can operate in this harder setting. We also show how the regret of the transformed algorithm can be bounded in terms of the regret of the original algorithm. Our reduction cannot be improved in general: we prove a lower bound on the regret of any bandit algorithm in this setting that matches (up to log factors) the upper bound obtained via our reduction. Finally, we show how our reduction can be extended to more complex bandit settings, such as combinatorial linear bandits and online bandit convex optimization
Correlation Clustering with Adaptive Similarity Queries
International audienceIn correlation clustering, we are given objects together with a binary similarity score between each pair of them. The goal is to partition the objects into clusters so to minimise the disagreements with the scores. In this work we investigate correlation clustering as an active learning problem: each similarity score can be learned by making a query, and the goal is to minimise both the disagreements and the total number of queries. On the one hand, we describe simple active learning algorithms, which provably achieve an almost optimal trade-off while giving cluster recovery guarantees, and we test them on different datasets. On the other hand, we prove information-theoretical bounds on the number of queries necessary to guarantee a prescribed disagreement bound. These results give a rich characterization of the trade-off between queries and clustering error