Smoothed Analysis with Adaptive Adversaries (2102.08446v2)

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 $\tilde{O}\big(\sqrt{T d\ln(1/\sigma)} + d\sqrt{\ln(T/\sigma)}\big)$. This answers open questions of [RST11,Hag18]. -Online discrepancy minimization: We consider the online Koml\'os 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 $\big( {\sigma}/{\sqrt{T\ell}}, \tilde O\big(\sqrt{T\ell} \big)\big)$-dispersed. This matches the parameters of [BDV18] for oblivious adversaries, up to log factors.

• The paper presents a novel framework that extends smoothed analysis to adaptive adversaries, yielding improved regret bounds, discrepancy guarantees, and dispersion insights in online problems.
• It leverages a coupling-based reduction technique to transform adaptive challenges into simpler oblivious settings, ensuring robust theoretical and practical performance.
• The results offer actionable insights for online learning, optimization, and discrepancy minimization, enhancing applications in machine learning, network design, and operational management.

Smoothed Analysis with Adaptive Adversaries: Insights and Implications

The paper "Smoothed Analysis with Adaptive Adversaries" by Haghtalab, Roughgarden, and Shetty introduces new algorithmic guarantees for several online problems within the framework of smoothed analysis. This paper focuses on the novel setup where adaptive adversaries are considered, representing a significant advancement over traditional models that predominantly focus on oblivious adversaries. The authors aim to bridge the gap between average-case and worst-case scenario analyses, offering a nuanced perspective on algorithm performance in practical settings.

Core Contributions

The paper's main contributions are structured around three fundamental online problems: online learning, online discrepancy minimization, and dispersion in online optimization. For each of these problems, the authors leverage a general technique that extends smoothed analysis to settings involving adaptive adversaries.

1. Online Learning:
   • The authors address the online prediction problem within the context of adaptively $\sigma$-smooth distributions, focusing on hypothesis classes with known VC dimensions.
   • The paper provides a regret bound of $\tilde{O}(\sqrt{Td\ln(1/\sigma)} + d\ln(T/\sigma))$, showcasing that the adaptive setting results in learnability characterized by the VC dimension rather than Littlestone dimension as in worst-case analysis.
   • The analysis resolves an open question concerning the learnability against adaptive adversaries, aligning closer with agnostic learning models.
2. Online Discrepancy Minimization:
   • The paper examines the online Komlós problem under adaptive circumstances with isotropic $\sigma$-smooth distributions, achieving an $\ell_\infty$ discrepancy bound of $\tilde{O}(\ln^2(nT/\sigma))$.
   • It contrasts with worst-case bounds of $\Theta(\sqrt{T/n})$, highlighting adaptability's analytical complexity.
   • The isotropy here is crucial, as evidenced by the infeasibility of similar bounds under non-isotropic conditions.
3. Dispersion in Online Optimization:
   • The paper expands into online optimization for piecewise Lipschitz functions, showing that sequences chosen by an $\alpha$-smooth adaptive adversary are $$\big(\sigma/\sqrt{T\ell}, \tilde{O}(\sqrt{T \ell})\big)$$-dispersed.
   • These results parallel assumptions for oblivious settings and provide a robust framework in theoretical and algorithmic terms.

Theoretical Implications

The paper advances theoretical understanding by demonstrating that certain online problems, which appear robust against oblivious adversaries, can be extended to adaptive settings through a coupling-based reduction technique. This technique allows the reduction of adaptive adversary settings to much simpler oblivious ones, maintaining the integrity of algorithmic guarantees even in the face of adversaries capable of intricate input correlations.

Adaptive Adversarial Models:

The implications of extending smoothed analysis to adaptive adversaries are profound. Adaptive adversaries, by definition, can condition their inputs based on the algorithm's historical behavior and state, presenting challenges traditional proof techniques fail to adequately address. This research not only provides an innovative framework for handling such adversaries but also sets a precedent for exploring adaptive settings in other complex algorithmic domains.

Practical Implications

The paper's results are significant for several practical applications where inputs are neither worst-case nor purely stochastic but exhibit characteristics of both. The smoothed analysis approach provides a realistic assessment of algorithmic performance, which is invaluable in domains such as machine learning, operations management, and network design. By ensuring that guarantees hold under the nuanced conditions imposed by adaptive adversaries, this work offers practitioners a way to anticipate and mitigate risks associated with adaptive behavior in real-world applications.

Future Directions

The research suggests multiple future avenues, most notably:

• Extending the coupling method to address other online and offline problems influenced by adaptive adversaries.
• Exploring the bounds of complexity classes when adversaries have varied degrees of adaptivity.
• Applying the insights from this work to develop algorithmic frameworks that can predict and counteract adaptive adversarial strategies in real-time.

The paper represents a significant step forward in the ongoing effort to reconcile theoretical robustness with practical applicability in algorithmic performance analyses.