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Register a new userAdaptivity in Adaptive Submodularity
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Added by
Hossein Esfandiari
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Adaptive sequential decision making is one of the central challenges in machine learning and artificial intelligence. In such problems, the goal is to design an interactive policy that plans for an action to take, from a finite set of $n$ actions, given some partial observations. It has been shown that in many applications such as active learning, robotics, sequential experimental design, and active detection, the utility function satisfies adaptive submodularity, a notion that generalizes the notion of diminishing returns to policies. In this paper, we revisit the power of adaptivity in maximizing an adaptive monotone submodular function. We propose an efficient semi adaptive policy that with $O(log n timeslog k)$ adaptive rounds of observations can achieve an almost tight $1-1/e-epsilon$ approximation guarantee with respect to an optimal policy that carries out $k$ actions in a fully sequential manner. To complement our results, we also show that it is impossible to achieve a constant factor approximation with $o(log n)$ adaptive rounds. We also extend our result to the case of adaptive stochastic minimum cost coverage where the goal is to reach a desired utility $Q$ with the cheapest policy. We first prove the conjecture of the celebrated work of Golovin and Krause by showing that the greedy policy achieves the asymptotically tight logarithmic approximation guarantee without resorting to stronger notions of adaptivity. We then propose a semi adaptive policy that provides the same guarantee in polylogarithmic adaptive rounds through a similar information-parallelism scheme. Our results shrink the adaptivity gap in adaptive submodular maximization by an exponential factor.
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Data that is gathered adaptively --- via bandit algorithms, for example --- exhibits bias. This is true both when gathering simple numeric valued data --- the empirical means kept track of by stochastic bandit algorithms are biased downwards --- and when gathering more complicated data --- running hypothesis tests on complex data gathered via contextual bandit algorithms leads to false discovery. In this paper, we show that this problem is mitigated if the data collection procedure is differentially private. This lets us both bound the bias of simple numeric valued quantities (like the empirical means of stochastic bandit algorithms), and correct the p-values of hypothesis tests run on the adaptively gathered data. Moreover, there exist differentially private bandit algorithms with near optimal regret bounds: we apply existing theorems in the simple stochastic case, and give a new analysis for linear contextual bandits. We complement our theoretical results with experiments validating our theory.
Recent developments in machine-learning algorithms have led to impressive performance increases in many traditional application scenarios of artificial intelligence research. In the area of deep reinforcement learning, deep learning functional architectures are combined with incremental learning schemes for sequential tasks that include interaction-based, but often delayed feedback. Despite their impressive successes, modern machine-learning approaches, including deep reinforcement learning, still perform weakly when compared to flexibly adaptive biological systems in certain naturally occurring scenarios. Such scenarios include transfers to environments different than the ones in which the training took place or environments that dynamically change, both of which are often mastered by biological systems through a capability that we here term fluid adaptivity to contrast it from the much slower adaptivity (crystallized adaptivity) of the prior learning from which the behavior emerged. In this article, we derive and discuss research strategies, based on analyzes of fluid adaptivity in biological systems and its neuronal modeling, that might aid in equipping future artificially intelligent systems with capabilities of fluid adaptivity more similar to those seen in some biologically intelligent systems. A key component of this research strategy is the dynamization of the problem space itself and the implementation of this dynamization by suitably designed flexibly interacting modules.
We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time an adversary chooses an input distribution with density function bounded above by $tfrac{1}{sigma}$ times that of the uniform distribution; nature then samples an input from this distribution. Crucially, our results hold for {em adaptive} adversaries that can choose an input distribution based on the decisions of the algorithm and the realizations of the inputs in the previous time steps. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of adaptive adversaries to the simpler case of oblivious adversaries. We apply this technique to prove strong smoothed guarantees for three problems: -Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of $sigma$-smooth distributions and the hypothesis class has VC dimension $d$. We bound the regret by $tilde{O}big(sqrt{T dln(1/sigma)} + dsqrt{ln(T/sigma)}big)$. This answers open questions of [RST11,Hag18]. -Online discrepancy minimization: We consider the online Komlos problem, where the input is generated from an adaptive sequence of $sigma$-smooth and isotropic distributions on the $ell_2$ unit ball. We bound the $ell_infty$ norm of the discrepancy vector by $tilde{O}big(ln^2!big( frac{nT}{sigma}big) big)$. -Dispersion in online optimization: We consider online optimization of piecewise Lipschitz functions where functions with $ell$ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is $big( {sigma}/{sqrt{Tell}}, tilde Obig(sqrt{Tell} big)big)$-dispersed. This matches the parameters of [BDV18] for oblivious adversaries, up to log factors.
Discrete integration in a high dimensional space of n variables poses fundamental challenges. The WISH algorithm reduces the intractable discrete integration problem into n optimization queries subject to randomized constraints, obtaining a constant approximation guarantee. The optimization queries are expensive, which limits the applicability of WISH. We propose AdaWISH, which is able to obtain the same guarantee but accesses only a small subset of queries of WISH. For example, when the number of function values is bounded by a constant, AdaWISH issues only O(log n) queries. The key idea is to query adaptively, taking advantage of the shape of the weight function being integrated. In general, we prove that AdaWISH has a regret of only O(log n) relative to an idealistic oracle that issues queries at data-dependent optimal points. Experimentally, AdaWISH gives precise estimates for discrete integration problems, of the same quality as that of WISH and better than several competing approaches, on a variety of probabilistic inference benchmarks. At the same time, it saves substantially on the number of optimization queries compared to WISH. On a suite of UAI inference challenge benchmarks, it saves 81.5% of WISH queries while retaining the quality of results.
The federated learning (FL) framework trains a machine learning model using decentralized data stored at edge client devices by periodically aggregating locally trained models. Popular optimization algorithms of FL use vanilla (stochastic) gradient descent for both local updates at clients and global updates at the aggregating server. Recently, adaptive optimization methods such as AdaGrad have been studied for server updates. However, the effect of using adaptive optimization methods for local updates at clients is not yet understood. We show in both theory and practice that while local adaptive methods can accelerate convergence, they can cause a non-vanishing solution bias, where the final converged solution may be different from the stationary point of the global objective function. We propose correction techniques to overcome this inconsistency and complement the local adaptive methods for FL. Extensive experiments on realistic federated training tasks show that the proposed algorithms can achieve faster convergence and higher test accuracy than the baselines without local adaptivity.
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