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Abstract:Inverse propensity-score weighted (IPW) estimators are prevalent in causal inference for estimating average treatment effects in observational studies. Under unconfoundedness, given accurate propensity scores and $n$ samples, the size of confidence intervals of IPW estimators scales down with $n$, and, several of their variants improve the rate of scaling. However, neither IPW estimators nor their variants are robust to inaccuracies: even if a single covariate has an $\varepsilon>0$ additive error in the propensity score, the size of confidence intervals of these estimators can increase arbitrarily. Moreover, even without errors, the rate with which the confidence intervals of these estimators go to zero with $n$ can be arbitrarily slow in the presence of extreme propensity scores (those close to 0 or 1). We introduce a family of Coarse IPW (CIPW) estimators that captures existing IPW estimators and their variants. Each CIPW estimator is an IPW estimator on a coarsened covariate space, where certain covariates are merged. Under mild assumptions, e.g., Lipschitzness in expected outcomes and sparsity of extreme propensity scores, we give an efficient algorithm to find a robust estimator: given $\varepsilon$-inaccurate propensity scores and $n$ samples, its confidence interval size scales with $\varepsilon+1/\sqrt{n}$. In contrast, under the same assumptions, existing estimators' confidence interval sizes are $\Omega(1)$ irrespective of $\varepsilon$ and $n$. Crucially, our estimator is data-dependent and we show that no data-independent CIPW estimator can be robust to inaccuracies.

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Authors:Alkis Kalavasis, Amin Karbasi, Argyris Oikonomou, Katerina Sotiraki, Grigoris Velegkas, Manolis Zampetakis

Abstract:As ML models become increasingly complex and integral to high-stakes domains such as finance and healthcare, they also become more susceptible to sophisticated adversarial attacks. We investigate the threat posed by undetectable backdoors in models developed by insidious external expert firms. When such backdoors exist, they allow the designer of the model to sell information to the users on how to carefully perturb the least significant bits of their input to change the classification outcome to a favorable one. We develop a general strategy to plant a backdoor to neural networks while ensuring that even if the model's weights and architecture are accessible, the existence of the backdoor is still undetectable. To achieve this, we utilize techniques from cryptography such as cryptographic signatures and indistinguishability obfuscation. We further introduce the notion of undetectable backdoors to language models and extend our neural network backdoor attacks to such models based on the existence of steganographic functions.

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Abstract:We study computational aspects of algorithmic replicability, a notion of stability introduced by Impagliazzo, Lei, Pitassi, and Sorrell [2022]. Motivated by a recent line of work that established strong statistical connections between replicability and other notions of learnability such as online learning, private learning, and SQ learning, we aim to understand better the computational connections between replicability and these learning paradigms. Our first result shows that there is a concept class that is efficiently replicably PAC learnable, but, under standard cryptographic assumptions, no efficient online learner exists for this class. Subsequently, we design an efficient replicable learner for PAC learning parities when the marginal distribution is far from uniform, making progress on a question posed by Impagliazzo et al. [2022]. To obtain this result, we design a replicable lifting framework inspired by Blanc, Lange, Malik, and Tan [2023] that transforms in a black-box manner efficient replicable PAC learners under the uniform marginal distribution over the Boolean hypercube to replicable PAC learners under any marginal distribution, with sample and time complexity that depends on a certain measure of the complexity of the distribution. Finally, we show that any pure DP learner can be transformed to a replicable one in time polynomial in the accuracy, confidence parameters and exponential in the representation dimension of the underlying hypothesis class.

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Abstract:We study the fundamental problem of transfer learning where a learning algorithm collects data from some source distribution $P$ but needs to perform well with respect to a different target distribution $Q$. A standard change of measure argument implies that transfer learning happens when the density ratio $dQ/dP$ is bounded. Yet, prior thought-provoking works by Kpotufe and Martinet (COLT, 2018) and Hanneke and Kpotufe (NeurIPS, 2019) demonstrate cases where the ratio $dQ/dP$ is unbounded, but transfer learning is possible. In this work, we focus on transfer learning over the class of low-degree polynomial estimators. Our main result is a general transfer inequality over the domain $\mathbb{R}^n$, proving that non-trivial transfer learning for low-degree polynomials is possible under very mild assumptions, going well beyond the classical assumption that $dQ/dP$ is bounded. For instance, it always applies if $Q$ is a log-concave measure and the inverse ratio $dP/dQ$ is bounded. To demonstrate the applicability of our inequality, we obtain new results in the settings of: (1) the classical truncated regression setting, where $dQ/dP$ equals infinity, and (2) the more recent out-of-distribution generalization setting for in-context learning linear functions with transformers. We also provide a discrete analogue of our transfer inequality on the Boolean Hypercube $\{-1,1\}^n$, and study its connections with the recent problem of Generalization on the Unseen of Abbe, Bengio, Lotfi and Rizk (ICML, 2023). Our main conceptual contribution is that the maximum influence of the error of the estimator $\widehat{f}-f^*$ under $Q$, $\mathrm{I}_{\max}(\widehat{f}-f^*)$, acts as a sufficient condition for transferability; when $\mathrm{I}_{\max}(\widehat{f}-f^*)$ is appropriately bounded, transfer is possible over the Boolean domain.

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Abstract:We provide efficient replicable algorithms for the problem of learning large-margin halfspaces. Our results improve upon the algorithms provided by Impagliazzo, Lei, Pitassi, and Sorrell [STOC, 2022]. We design the first dimension-independent replicable algorithms for this task which runs in polynomial time, is proper, and has strictly improved sample complexity compared to the one achieved by Impagliazzo et al. [2022] with respect to all the relevant parameters. Moreover, our first algorithm has sample complexity that is optimal with respect to the accuracy parameter $\epsilon$. We also design an SGD-based replicable algorithm that, in some parameters' regimes, achieves better sample and time complexity than our first algorithm. Departing from the requirement of polynomial time algorithms, using the DP-to-Replicability reduction of Bun, Gaboardi, Hopkins, Impagliazzo, Lei, Pitassi, Sorrell, and Sivakumar [STOC, 2023], we show how to obtain a replicable algorithm for large-margin halfspaces with improved sample complexity with respect to the margin parameter $\tau$, but running time doubly exponential in $1/\tau^2$ and worse sample complexity dependence on $\epsilon$ than one of our previous algorithms. We then design an improved algorithm with better sample complexity than all three of our previous algorithms and running time exponential in $1/\tau^{2}$.

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Abstract:We consider the problem of estimating the parameters of a Markov Random Field with hard-constraints using a single sample. As our main running examples, we use the $k$-SAT and the proper coloring models, as well as general $H$-coloring models; for all of these we obtain both positive and negative results. In contrast to the soft-constrained case, we show in particular that single-sample estimation is not always possible, and that the existence of an estimator is related to the existence of non-satisfiable instances. Our algorithms are based on the pseudo-likelihood estimator. We show variance bounds for this estimator using coupling techniques inspired, in the case of $k$-SAT, by Moitra's sampling algorithm (JACM, 2019); our positive results for colorings build on this new coupling approach. For $q$-colorings on graphs with maximum degree $d$, we give a linear-time estimator when $q>d+1$, whereas the problem is non-identifiable when $q\leq d+1$. For general $H$-colorings, we show that standard conditions that guarantee sampling, such as Dobrushin's condition, are insufficient for one-sample learning; on the positive side, we provide a general condition that is sufficient to guarantee linear-time learning and obtain applications for proper colorings and permissive models. For the $k$-SAT model on formulas with maximum degree $d$, we provide a linear-time estimator when $k\gtrsim 6.45\log d$, whereas the problem becomes non-identifiable when $k\lesssim \log d$.

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Abstract:Deep Neural Networks and Reinforcement Learning methods have empirically shown great promise in tackling challenging combinatorial problems. In those methods a deep neural network is used as a solution generator which is then trained by gradient-based methods (e.g., policy gradient) to successively obtain better solution distributions. In this work we introduce a novel theoretical framework for analyzing the effectiveness of such methods. We ask whether there exist generative models that (i) are expressive enough to generate approximately optimal solutions; (ii) have a tractable, i.e, polynomial in the size of the input, number of parameters; (iii) their optimization landscape is benign in the sense that it does not contain sub-optimal stationary points. Our main contribution is a positive answer to this question. Our result holds for a broad class of combinatorial problems including Max- and Min-Cut, Max-$k$-CSP, Maximum-Weight-Bipartite-Matching, and the Traveling Salesman Problem. As a byproduct of our analysis we introduce a novel regularization process over vanilla gradient descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.

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Abstract:In this work, we aim to characterize the statistical complexity of realizable regression both in the PAC learning setting and the online learning setting. Previous work had established the sufficiency of finiteness of the fat shattering dimension for PAC learnability and the necessity of finiteness of the scaled Natarajan dimension, but little progress had been made towards a more complete characterization since the work of Simon 1997 (SICOMP '97). To this end, we first introduce a minimax instance optimal learner for realizable regression and propose a novel dimension that both qualitatively and quantitatively characterizes which classes of real-valued predictors are learnable. We then identify a combinatorial dimension related to the Graph dimension that characterizes ERM learnability in the realizable setting. Finally, we establish a necessary condition for learnability based on a combinatorial dimension related to the DS dimension, and conjecture that it may also be sufficient in this context. Additionally, in the context of online learning we provide a dimension that characterizes the minimax instance optimal cumulative loss up to a constant factor and design an optimal online learner for realizable regression, thus resolving an open question raised by Daskalakis and Golowich in STOC '22.

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Abstract:When two different parties use the same learning rule on their own data, how can we test whether the distributions of the two outcomes are similar? In this paper, we study the similarity of outcomes of learning rules through the lens of the Total Variation (TV) distance of distributions. We say that a learning rule is TV indistinguishable if the expected TV distance between the posterior distributions of its outputs, executed on two training data sets drawn independently from the same distribution, is small. We first investigate the learnability of hypothesis classes using TV indistinguishable learners. Our main results are information-theoretic equivalences between TV indistinguishability and existing algorithmic stability notions such as replicability and approximate differential privacy. Then, we provide statistical amplification and boosting algorithms for TV indistinguishable learners.

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Abstract:In this work, we study how to efficiently obtain perfect samples from a discrete distribution $\mathcal{D}$ given access only to pairwise comparisons of elements of its support. Specifically, we assume access to samples $(x, S)$, where $S$ is drawn from a distribution over sets $\mathcal{Q}$ (indicating the elements being compared), and $x$ is drawn from the conditional distribution $\mathcal{D}_S$ (indicating the winner of the comparison) and aim to output a clean sample $y$ distributed according to $\mathcal{D}$. We mainly focus on the case of pairwise comparisons where all sets $S$ have size 2. We design a Markov chain whose stationary distribution coincides with $\mathcal{D}$ and give an algorithm to obtain exact samples using the technique of Coupling from the Past. However, the sample complexity of this algorithm depends on the structure of the distribution $\mathcal{D}$ and can be even exponential in the support of $\mathcal{D}$ in many natural scenarios. Our main contribution is to provide an efficient exact sampling algorithm whose complexity does not depend on the structure of $\mathcal{D}$. To this end, we give a parametric Markov chain that mixes significantly faster given a good approximation to the stationary distribution. We can obtain such an approximation using an efficient learning from pairwise comparisons algorithm (Shah et al., JMLR 17, 2016). Our technique for speeding up sampling from a Markov chain whose stationary distribution is approximately known is simple, general and possibly of independent interest.

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