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Register a new userUnifying the stochastic and the adversarial Bandits with Knapsack
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Anshuka Rangi
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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This paper investigates the adversarial Bandits with Knapsack (BwK) online learning problem, where a player repeatedly chooses to perform an action, pays the corresponding cost, and receives a reward associated with the action. The player is constrained by the maximum budget $B$ that can be spent to perform actions, and the rewards and the costs of the actions are assigned by an adversary. This problem has only been studied in the restricted setting where the reward of an action is greater than the cost of the action, while we provide a solution in the general setting. Namely, we propose EXP3.BwK, a novel algorithm that achieves order optimal regret. We also propose EXP3++.BwK, which is order optimal in the adversarial BwK setup, and incurs an almost optimal expected regret with an additional factor of $log(B)$ in the stochastic BwK setup. Finally, we investigate the case of having large costs for the actions (i.e., they are comparable to the budget size $B$), and show that for the adversarial setting, achievable regret bounds can be significantly worse, compared to the case of having costs bounded by a constant, which is a common assumption within the BwK literature.
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We study adversarial scaling, a multi-armed bandit model where rewards have a stochastic and an adversarial component. Our model captures display advertising where the click-through-rate can be decomposed to a (fixed across time) arm-quality component and a non-stochastic user-relevance component (fixed across arms). Despite the relative stochasticity of our model, we demonstrate two settings where most bandit algorithms suffer. On the positive side, we show that two algorithms, one from the action elimination and one from the mirror descent family are adaptive enough to be robust to adversarial scaling. Our results shed light on the robustness of adaptive parameter selection in stochastic bandits, which may be of independent interest.
Consider a player that in each round $t$ out of $T$ rounds chooses an action and observes the incurred cost after a delay of $d_{t}$ rounds. The cost functions and the delay sequence are chosen by an adversary. We show that even if the players algorithms lose their no regret property due to too large delays, the expected discounted ergodic distribution of play converges to the set of coarse correlated equilibrium (CCE) if the algorithms have no discounted-regret. For a zero-sum game, we show that no discounted-regret is sufficient for the discounted ergodic average of play to converge to the set of Nash equilibria. We prove that the FKM algorithm with $n$ dimensions achieves a regret of $Oleft(nT^{frac{3}{4}}+sqrt{n}T^{frac{1}{3}}D^{frac{1}{3}}right)$ and the EXP3 algorithm with $K$ arms achieves a regret of $Oleft(sqrt{ln Kleft(KT+Dright)}right)$ even when $D=sum_{t=1}^{T}d_{t}$ and $T$ are unknown. These bounds use a novel doubling trick that provably retains the regret bound for when $D$ and $T$ are known. Using these bounds, we show that EXP3 and FKM have no discounted-regret even for $d_{t}=Oleft(tlog tright)$. Therefore, the CCE of a finite or convex unknown game can be approximated even when only delayed bandit feedback is available via simulation.
We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). We show that in adversarial regimes with a $(Delta,C,T)$ self-bounding constraint the algorithm achieves $mathcal{O}left(left(sum_{i eq i^*} frac{1}{Delta_i}right)log_+left(frac{(K-1)T}{left(sum_{i eq i^*} frac{1}{Delta_i}right)^2}right)+sqrt{Cleft(sum_{i eq i^*}frac{1}{Delta_i}right)log_+left(frac{(K-1)T}{Csum_{i eq i^*}frac{1}{Delta_i}}right)}right)$ regret bound, where $T$ is the time horizon, $K$ is the number of arms, $Delta_i$ are the suboptimality gaps, $i^*$ is the best arm, $C$ is the corruption magnitude, and $log_+(x) = maxleft(1,log xright)$. The regime includes stochastic bandits, stochastically constrained adversarial bandits, and stochastic bandits with adversarial corruptions as special cases. Additionally, we provide a general analysis, which allows to achieve the same kind of improvement for generalizations of Tsallis-INF to other settings beyond multiarmed bandits.
We propose an algorithm for stochastic and adversarial multiarmed bandits with switching costs, where the algorithm pays a price $lambda$ every time it switches the arm being played. Our algorithm is based on adaptation of the Tsallis-INF algorithm of Zimmert and Seldin (2021) and requires no prior knowledge of the regime or time horizon. In the oblivious adversarial setting it achieves the minimax optimal regret bound of $Obig((lambda K)^{1/3}T^{2/3} + sqrt{KT}big)$, where $T$ is the time horizon and $K$ is the number of arms. In the stochastically constrained adversarial regime, which includes the stochastic regime as a special case, it achieves a regret bound of $Oleft(big((lambda K)^{2/3} T^{1/3} + ln Tbig)sum_{i eq i^*} Delta_i^{-1}right)$, where $Delta_i$ are the suboptimality gaps and $i^*$ is a unique optimal arm. In the special case of $lambda = 0$ (no switching costs), both bounds are minimax optimal within constants. We also explore variants of the problem, where switching cost is allowed to change over time. We provide experimental evaluation showing competitiveness of our algorithm with the relevant baselines in the stochastic, stochastically constrained adversarial, and adversarial regimes with fixed switching cost.
We introduce a new model of stochastic bandits with adversarial corruptions which aims to capture settings where most of the input follows a stochastic pattern but some fraction of it can be adversarially changed to trick the algorithm, e.g., click fraud, fake reviews and email spam. The goal of this model is to encourage the design of bandit algorithms that (i) work well in mixed adversarial and stochastic models, and (ii) whose performance deteriorates gracefully as we move from fully stochastic to fully adversarial models. In our model, the rewards for all arms are initially drawn from a distribution and are then altered by an adaptive adversary. We provide a simple algorithm whose performance gracefully degrades with the total corruption the adversary injected in the data, measured by the sum across rounds of the biggest alteration the adversary made in the data in that round; this total corruption is denoted by $C$. Our algorithm provides a guarantee that retains the optimal guarantee (up to a logarithmic term) if the input is stochastic and whose performance degrades linearly to the amount of corruption $C$, while crucially being agnostic to it. We also provide a lower bound showing that this linear degradation is necessary if the algorithm achieves optimal performance in the stochastic setting (the lower bound works even for a known amount of corruption, a special case in which our algorithm achieves optimal performance without the extra logarithm).
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