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Register a new userPareto-Optimal Learning-Augmented Algorithms for Online Conversion Problems
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Added by
Bo Sun
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper leverages machine-learned predictions to design competitive algorithms for online conversion problems with the goal of improving the competitive ratio when predictions are accurate (i.e., consistency), while also guaranteeing a worst-case competitive ratio regardless of the prediction quality (i.e., robustness). We unify the algorithmic design of both integral and fractional conversion problems, which are also known as the 1-max-search and one-way trading problems, into a class of online threshold-based algorithms (OTA). By incorporating predictions into design of OTA, we achieve the Pareto-optimal trade-off of consistency and robustness, i.e., no online algorithm can achieve a better consistency guarantee given for a robustness guarantee. We demonstrate the performance of OTA using numerical experiments on Bitcoin conversion.
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We introduce the online stochastic Convex Programming (CP) problem, a very general version of stochastic online problems which allows arbitrary concave objectives and convex feasibility constraints. Many well-studied problems like online stochastic packing and covering, online stochastic matching with concave returns, etc. form a special case of online stochastic CP. We present fast algorithms for these problems, which achieve near-optimal regret guarantees for both the i.i.d. and the random permutation models of stochastic inputs. When applied to the special case online packing, our ideas yield a simpler and faster primal-dual algorithm for this well studied problem, which achieves the optimal competitive ratio. Our techniques make explicit the connection of primal-dual paradigm and online learning to online stochastic CP.
Multi-Task Learning (MTL) is a well-established paradigm for training deep neural network models for multiple correlated tasks. Often the task objectives conflict, requiring trade-offs between them during model building. In such cases, MTL models can use gradient-based multi-objective optimization (MOO) to find one or more Pareto optimal solutions. A common requirement in MTL applications is to find an {it Exact} Pareto optimal (EPO) solution, which satisfies user preferences with respect to task-specific objective functions. Further, to improve model generalization, various constraints on the weights may need to be enforced during training. Addressing these requirements is challenging because it requires a search direction that allows descent not only towards the Pareto front but also towards the input preference, within the constraints imposed and in a manner that scales to high-dimensional gradients. We design and theoretically analyze such search directions and develop the first scalable algorithm, with theoretical guarantees of convergence, to find an EPO solution, including when box and equality constraints are imposed. Our unique method combines multiple gradient descent with carefully controlled ascent to traverse the Pareto front in a principled manner, making it robust to initialization. This also facilitates systematic exploration of the Pareto front, that we utilize to approximate the Pareto front for multi-criteria decision-making. Empirical results show that our algorithm outperforms competing methods on benchmark MTL datasets and MOO problems.
Motivated by online decision-making in time-varying combinatorial environments, we study the problem of transforming offline algorithms to their online counterparts. We focus on offline combinatorial problems that are amenable to a constant factor approximation using a greedy algorithm that is robust to local errors. For such problems, we provide a general framework that efficiently transforms offline robust greedy algorithms to online ones using Blackwell approachability. We show that the resulting online algorithms have $O(sqrt{T})$ (approximate) regret under the full information setting. We further introduce a bandit extension of Blackwell approachability that we call Bandit Blackwell approachability. We leverage this notion to transform greedy robust offline algorithms into a $O(T^{2/3})$ (approximate) regret in the bandit setting. Demonstrating the flexibility of our framework, we apply our offline-to-online transformation to several problems at the intersection of revenue management, market design, and online optimization, including product ranking optimization in online platforms, reserve price optimization in auctions, and submodular maximization. We show that our transformation, when applied to these applications, leads to new regret bounds or improves the current known bounds.
We study the combinatorial pure exploration problem Best-Set in stochastic multi-armed bandits. In a Best-Set instance, we are given $n$ arms with unknown reward distributions, as well as a family $mathcal{F}$ of feasible subsets over the arms. Our goal is to identify the feasible subset in $mathcal{F}$ with the maximum total mean using as few samples as possible. The problem generalizes the classical best arm identification problem and the top-$k$ arm identification problem, both of which have attracted significant attention in recent years. We provide a novel instance-wise lower bound for the sample complexity of the problem, as well as a nontrivial sampling algorithm, matching the lower bound up to a factor of $ln|mathcal{F}|$. For an important class of combinatorial families, we also provide polynomial time implementation of the sampling algorithm, using the equivalence of separation and optimization for convex program, and approximate Pareto curves in multi-objective optimization. We also show that the $ln|mathcal{F}|$ factor is inevitable in general through a nontrivial lower bound construction. Our results significantly improve several previous results for several important combinatorial constraints, and provide a tighter understanding of the general Best-Set problem. We further introduce an even more general problem, formulated in geometric terms. We are given $n$ Gaussian arms with unknown means and unit variance. Consider the $n$-dimensional Euclidean space $mathbb{R}^n$, and a collection $mathcal{O}$ of disjoint subsets. Our goal is to determine the subset in $mathcal{O}$ that contains the $n$-dimensional vector of the means. The problem generalizes most pure exploration bandit problems studied in the literature. We provide the first nearly optimal sample complexity upper and lower bounds for the problem.
We study the selective learning problem introduced by Qiao and Valiant (2019), in which the learner observes $n$ labeled data points one at a time. At a time of its choosing, the learner selects a window length $w$ and a model $hatell$ from the model class $mathcal{L}$, and then labels the next $w$ data points using $hatell$. The excess risk incurred by the learner is defined as the difference between the average loss of $hatell$ over those $w$ data points and the smallest possible average loss among all models in $mathcal{L}$ over those $w$ data points. We give an improved algorithm, termed the hybrid exponential weights algorithm, that achieves an expected excess risk of $O((loglog|mathcal{L}| + loglog n)/log n)$. This result gives a doubly exponential improvement in the dependence on $|mathcal{L}|$ over the best known bound of $O(sqrt{|mathcal{L}|/log n})$. We complement the positive result with an almost matching lower bound, which suggests the worst-case optimality of the algorithm. We also study a more restrictive family of learning algorithms that are bounded-recall in the sense that when a prediction window of length $w$ is chosen, the learners decision only depends on the most recent $w$ data points. We analyze an exponential weights variant of the ERM algorithm in Qiao and Valiant (2019). This new algorithm achieves an expected excess risk of $O(sqrt{log |mathcal{L}|/log n})$, which is shown to be nearly optimal among all bounded-recall learners. Our analysis builds on a generalized version of the selective mean prediction problem in Drucker (2013); Qiao and Valiant (2019), which may be of independent interest.
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