Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSmoothed Analysis with Adaptive Adversaries
440
0
0.0
(
0
)
Added by
Abhishek Shetty
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
Practical and pervasive needs for robustness and privacy in algorithms have inspired the design of online adversarial and differentially private learning algorithms. The primary quantity that characterizes learnability in these settings is the Littlestone dimension of the class of hypotheses [Ben-David et al., 2009, Alon et al., 2019]. This characterization is often interpreted as an impossibility result because classes such as linear thresholds and neural networks have infinite Littlestone dimension. In this paper, we apply the framework of smoothed analysis [Spielman and Teng, 2004], in which adversarially chosen inputs are perturbed slightly by nature. We show that fundamentally stronger regret and error guarantees are possible with smoothed adversaries than with worst-case adversaries. In particular, we obtain regret and privacy error bounds that depend only on the VC dimension and the bracketing number of a hypothesis class, and on the magnitudes of the perturbations.
Greedy decision tree learning heuristics are mainstays of machine learning practice, but theoretical justification for their empirical success remains elusive. In fact, it has long been known that there are simple target functions for which they fail badly (Kearns and Mansour, STOC 1996). Recent work of Brutzkus, Daniely, and Malach (COLT 2020) considered the smoothed analysis model as a possible avenue towards resolving this disconnect. Within the smoothed setting and for targets $f$ that are $k$-juntas, they showed that these heuristics successfully learn $f$ with depth-$k$ decision tree hypotheses. They conjectured that the same guarantee holds more generally for targets that are depth-$k$ decision trees. We provide a counterexample to this conjecture: we construct targets that are depth-$k$ decision trees and show that even in the smoothed setting, these heuristics build trees of depth $2^{Omega(k)}$ before achieving high accuracy. We also show that the guarantees of Brutzkus et al. cannot extend to the agnostic setting: there are targets that are very close to $k$-juntas, for which these heuristics build trees of depth $2^{Omega(k)}$ before achieving high accuracy.
Deep neural networks are vulnerable to small input perturbations known as adversarial attacks. Inspired by the fact that these adversaries are constructed by iteratively minimizing the confidence of a network for the true class label, we propose the anti-adversary layer, aimed at countering this effect. In particular, our layer generates an input perturbation in the opposite direction of the adversarial one, and feeds the classifier a perturbed version of the input. Our approach is training-free and theoretically supported. We verify the effectiveness of our approach by combining our layer with both nominally and robustly trained models, and conduct large scale experiments from black-box to adaptive attacks on CIFAR10, CIFAR100 and ImageNet. Our anti-adversary layer significantly enhances model robustness while coming at no cost on clean accuracy.
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.
Large scale cryptocurrencies require the participation of millions of participants and support economic activity of billions of dollars, which has led to new lines of work in binary Byzantine Agreement (BBA) and consensus. The new work aims to achieve communication-efficiency---given such a large $n$, not everyone can speak during the protocol. Several protocols have achieved consensus with communication-efficiency, even under an adaptive adversary, but they require additional strong assumptions---proof-of-work, memory-erasure, etc. All of these protocols use multicast: every honest replica multicasts messages to all other replicas. Under this model, we provide a new communication-efficient consensus protocol using Verifiable Delay Functions (VDFs) that is secure against adaptive adversaries and does not require the same strong assumptions present in other protocols. A natural question is whether we can extend the synchronous protocols to the partially synchronous setting---in this work, we show that using multicast, we cannot. Furthermore, we cannot achieve always safe communication-efficient protocols (that maintain safety with probability 1) even in the synchronous setting against a static adversary when honest replicas only choose to multicast its messages. Considering these impossibility results, we describe a new communication-efficient BBA protocol in a modified partially synchronous network model which is secure against adaptive adversaries with high probability.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
