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We study the problem of learning under arbitrary distribution shift, where the learner is trained on a labeled set from one distribution but evaluated on a different, potentially adversarially generated test distribution. We focus on two frameworks: PQ learning [GKKM'20], allowing abstention on adversarially generated parts of the test distribution, and TDS learning [KSV'23], permitting abstention on the entire test distribution if distribution shift is detected. All prior known algorithms either rely on learning primitives that are computationally hard even for simple function classes, or end up abstaining entirely even in the presence of a tiny amount of distribution shift. We address both these challenges for natural function classes, including intersections of halfspaces and decision trees, and standard training distributions, including Gaussians. For PQ learning, we give efficient learning algorithms, while for TDS learning, our algorithms can tolerate moderate amounts of distribution shift. At the core of our approach is an improved analysis of spectral outlier-removal techniques from learning with nasty noise. Our analysis can (1) handle arbitrarily large fraction of outliers, which is crucial for handling arbitrary distribution shifts, and (2) obtain stronger bounds on polynomial moments of the distribution after outlier removal, yielding new insights into polynomial regression under distribution shifts. Lastly, our techniques lead to novel results for tolerant testable learning [RV'23], and learning with nasty noise. 

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by \(\tfrac{1}{\sigma }\)times that of the uniform distribution; nature then samples an input from this distribution. Here, σ is a parameter that interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold foradaptiveadversaries that can base their choice of input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm’s current state; this appears to rule out the standard proof approaches in smoothed analysis. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems:(1)Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of σ-smooth distributions and the hypothesis class has VC dimensiond. We bound the regret by\(\tilde{O}(\sqrt {T d\ln (1/\sigma)} + d\ln (T/\sigma))\)and provide a near-matching lower bound. Our result shows that under smoothed analysis, learnability against adaptive adversaries is characterized by the finiteness of the VC dimension. This is as opposed to the worst-case analysis, where online learnability is characterized by Littlestone dimension (which is infinite even in the extremely restricted case of one-dimensional threshold functions). Our results fully answer an open question of Rakhlin et al. [64].(2)Online discrepancy minimization: We consider the setting of the online Komlós problem, where the input is generated from an adaptive sequence of σ-smooth and isotropic distributions on the ℓ₂unit ball. We bound the ℓ_∞norm of the discrepancy vector by\(\tilde{O}(\ln ^2(\frac{nT}{\sigma }))\). This is as opposed to the worst-case analysis, where the tight discrepancy bound is\(\Theta (\sqrt {T/n})\). We show such\(\mathrm{polylog}(nT/\sigma)\)discrepancy guarantees are not achievable for non-isotropic σ-smooth distributions.(3)Dispersion in online optimization: We consider online optimization with piecewise Lipschitz functions where functions with ℓ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is\(({\sigma }/{\sqrt {T\ell }}, \tilde{O}(\sqrt {T\ell }))\)-dispersed. That is, every ball of radius\({\sigma }/{\sqrt {T\ell }}\)is split by\(\tilde{O}(\sqrt {T\ell })\)of the partitions made by these functions. This result matches the dispersion parameters of Balcan et al. [13] for oblivious smooth adversaries, up to logarithmic factors. On the other hand, worst-case sequences are trivially (0,T)-dispersed.¹ 

This paper addresses the challenge of learning under arbitrary distribution shift, where models are trained on data from one distribution but evaluated on a potentially adversarially generated test distribution. The authors study two frameworks: PQ learning [Goldwasser, A. Kalai, Y. Kalai, Montasser NeurIPS 2020], which allows abstention on adversarial portions of the test distribution, and TDS learning [Klivans, Stavropoulos, Vasilyan COLT 2024], which permits abstention on the entire test set if a distribution shift is detected. Previous algorithms either relied on computationally hard primitives (even for simple classes) or abstained entirely with even small shifts. This work develops efficient algorithms for natural function classes such as intersections of halfspaces and decision trees, under standard training distributions like Gaussians. The key innovation is an improved analysis of spectral outlier-removal techniques from learning with nasty noise, enabling: (1) handling arbitrarily large fractions of outliers, crucial for arbitrary shifts, and (2) stronger bounds on polynomial moments post outlier-removal, yielding new insights into polynomial regression under distribution shift. The techniques also produce novel results for tolerant testable learning [Rubinfeld and Vasilyan STOC 2023] and learning with nasty noise. 

We initiate the study of smoothed analysis for the sequential probability assignment problem with contexts. We study information-theoretically optimal minmax rates as well as a framework for algorithmic reduction involving the maximum likelihood estimator oracle. Our approach establishes a general-purpose reduction from minimax rates for sequential probability assignment for smoothed adversaries to minimax rates for transductive learning. This leads to optimal (logarithmic) fast rates for parametric classes and classes with finite VC dimension. On the algorithmic front, we develop an algorithm that efficiently taps into the MLE oracle, for general classes of functions. We show that under general conditions this algorithmic approach yields sublinear regret. 

We initiate the study of smoothed analysis for the sequential probability assignment problem with contexts. We study information-theoretically optimal minmax rates as well as a framework for algorithmic reduction involving the maximum likelihood estimator oracle. Our approach establishes a general-purpose reduction from minimax rates for sequential probability assignment for smoothed adversaries to minimax rates for transductive learning. This leads to optimal (logarithmic) fast rates for parametric classes and classes with finite VC dimension. On the algorithmic front, we develop an algorithm that efficiently taps into the MLE oracle, for general classes of functions. We show that under general conditions this algorithmic approach yields sublinear regret. 

A fundamental shortcoming of the concept of Nash equilibrium is its computational intractability: approximating Nash equilibria in normal-form games is PPAD-hard. In this paper, inspired by the ideas of smoothed analysis, we introduce a relaxed variant of Nash equilibrium called σ-smooth Nash equilibrium, for a {smoothness parameter} σ. In a σ-smooth Nash equilibrium, players only need to achieve utility at least as high as their best deviation to a σ-smooth strategy, which is a distribution that does not put too much mass (as parametrized by σ) on any fixed action. We distinguish two variants of σ-smooth Nash equilibria: strong σ-smooth Nash equilibria, in which players are required to play σ-smooth strategies under equilibrium play, and weak σ-smooth Nash equilibria, where there is no such requirement. We show that both weak and strong σ-smooth Nash equilibria have superior computational properties to Nash equilibria: when σ as well as an approximation parameter ϵ and the number of players are all constants, there is a {constant-time} randomized algorithm to find a weak ϵ-approximate σ-smooth Nash equilibrium in normal-form games. In the same parameter regime, there is a polynomial-time deterministic algorithm to find a strong ϵ-approximate σ-smooth Nash equilibrium in a normal-form game. These results stand in contrast to the optimal algorithm for computing ϵ-approximate Nash equilibria, which cannot run in faster than quasipolynomial-time, subject to complexity-theoretic assumptions. We complement our upper bounds by showing that when either σ or ϵ is an inverse polynomial, finding a weak ϵ-approximate σ-smooth Nash equilibria becomes computationally intractable. Our results are the first to propose a variant of Nash equilibrium which is computationally tractable, allows players to act independently, and which, as we discuss, is justified by an extensive line of work on individual choice behavior in the economics literature. 

Systems driven far from equilibrium often retain structural memories of their processing history. This memory has, in some cases, been shown to dramatically alter the material response. For example, work hardening in crystalline metals can alter the hardness, yield strength, and tensile strength to prevent catastrophic failure. Whether memory of processing history can be similarly exploited in flowing systems, where significantly larger changes in structure should be possible, remains poorly understood. Here, we demonstrate a promising route to embedding such useful memories. We build on work showing that exposing a sheared dense suspension to acoustic perturbations of different power allows for dramatically tuning the sheared suspension viscosity and underlying structure. We find that, for sufficiently dense suspensions, upon removing the acoustic perturbations, the suspension shear jams with shear stress contributions from the maximum compressive and maximum extensive axes that reflect or “remember” the acoustic training. Because the contributions from these two orthogonal axes to the total shear stress are antagonistic, it is possible to tune the resulting suspension response in surprising ways. For example, we show that differently trained sheared suspensions exhibit (1) different susceptibility to the same acoustic perturbation, (2) orders of magnitude changes in their instantaneous viscosities upon shear reversal, and (3) even a shear stress that increases in magnitude upon shear cessation. We work through these examples to explain the underlying mechanisms governing each behavior. Then, to illustrate the power of this approach for controlling suspension properties, we demonstrate that flowing states well below the shear jamming threshold can be shear jammed via acoustic training. Collectively, our work paves the way for using acoustically induced memory in dense suspensions to generate rapidly and widely tunable materials. Published by the American Physical Society2024