Manolis Zampetakis

Michael I. Jordan, Tianyi Lin, Manolis Zampetakis

We consider the problem of computing an equilibrium in a class of \textit{nonlinear generalized Nash equilibrium problems (NGNEPs)} in which the strategy sets for each player are defined by equality and inequality constraints that may depend on the choices of rival players. While the asymptotic global convergence and local convergence rates of algorithms to solve this problem have been extensively investigated, the analysis of nonasymptotic iteration complexity is still in its infancy. This paper presents two first-order algorithms -- based on the quadratic penalty method (QPM) and augmented Lagrangian method (ALM), respectively -- with an accelerated mirror-prox algorithm as the solver in each inner loop. We establish a global convergence guarantee for solving monotone and strongly monotone NGNEPs and provide nonasymptotic complexity bounds expressed in terms of the number of gradient evaluations. Experimental results demonstrate the efficiency of our algorithms in practice.

Michael I. Jordan, Guy Kornowski, Tianyi Lin et al.

We study the complexity of optimizing nonsmooth nonconvex Lipschitz functions by producing $(δ,ε)$-stationary points. Several recent works have presented randomized algorithms that produce such points using $\tilde O(δ^{-1}ε^{-3})$ first-order oracle calls, independent of the dimension $d$. It has been an open problem as to whether a similar result can be obtained via a deterministic algorithm. We resolve this open problem, showing that randomization is necessary to obtain a dimension-free rate. In particular, we prove a lower bound of $Ω(d)$ for any deterministic algorithm. Moreover, we show that unlike smooth or convex optimization, access to function values is required for any deterministic algorithm to halt within any finite time. On the other hand, we prove that if the function is even slightly smooth, then the dimension-free rate of $\tilde O(δ^{-1}ε^{-3})$ can be obtained by a deterministic algorithm with merely a logarithmic dependence on the smoothness parameter. Motivated by these findings, we turn to study the complexity of deterministically smoothing Lipschitz functions. Though there are efficient black-box randomized smoothings, we start by showing that no such deterministic procedure can smooth functions in a meaningful manner, resolving an open question. We then bypass this impossibility result for the structured case of ReLU neural networks. To that end, in a practical white-box setting in which the optimizer is granted access to the network's architecture, we propose a simple, dimension-free, deterministic smoothing that provably preserves $(δ,ε)$-stationary points. Our method applies to a variety of architectures of arbitrary depth, including ResNets and ConvNets. Combined with our algorithm, this yields the first deterministic dimension-free algorithm for optimizing ReLU networks, circumventing our lower bound.

Constantinos Daskalakis, Patroklos Stefanou, Rui Yao et al.

Truncated linear regression is a classical challenge in Statistics, wherein a label, $y = w^T x + \varepsilon$, and its corresponding feature vector, $x \in \mathbb{R}^k$, are only observed if the label falls in some subset $S \subseteq \mathbb{R}$; otherwise the existence of the pair $(x, y)$ is hidden from observation. Linear regression with truncated observations has remained a challenge, in its general form, since the early works of~\citet{tobin1958estimation,amemiya1973regression}. When the distribution of the error is normal with known variance, recent work of~\citet{daskalakis2019truncatedregression} provides computationally and statistically efficient estimators of the linear model, $w$. In this paper, we provide the first computationally and statistically efficient estimators for truncated linear regression when the noise variance is unknown, estimating both the linear model and the variance of the noise. Our estimator is based on an efficient implementation of Projected Stochastic Gradient Descent on the negative log-likelihood of the truncated sample. Importantly, we show that the error of our estimates is asymptotically normal, and we use this to provide explicit confidence regions for our estimates.

Yeshwanth Cherapanamjeri, Constantinos Daskalakis, Andrew Ilyas et al.

In the classical setting of self-selection, the goal is to learn $k$ models, simultaneously from observations $(x^{(i)}, y^{(i)})$ where $y^{(i)}$ is the output of one of $k$ underlying models on input $x^{(i)}$. In contrast to mixture models, where we observe the output of a randomly selected model, here the observed model depends on the outputs themselves, and is determined by some known selection criterion. For example, we might observe the highest output, the smallest output, or the median output of the $k$ models. In known-index self-selection, the identity of the observed model output is observable; in unknown-index self-selection, it is not. Self-selection has a long history in Econometrics and applications in various theoretical and applied fields, including treatment effect estimation, imitation learning, learning from strategically reported data, and learning from markets at disequilibrium. In this work, we present the first computationally and statistically efficient estimation algorithms for the most standard setting of this problem where the models are linear. In the known-index case, we require poly$(1/\varepsilon, k, d)$ sample and time complexity to estimate all model parameters to accuracy $\varepsilon$ in $d$ dimensions, and can accommodate quite general selection criteria. In the more challenging unknown-index case, even the identifiability of the linear models (from infinitely many samples) was not known. We show three results in this case for the commonly studied $\max$ self-selection criterion: (1) we show that the linear models are indeed identifiable, (2) for general $k$ we provide an algorithm with poly$(d) \exp(\text{poly}(k))$ sample and time complexity to estimate the regression parameters up to error $1/\text{poly}(k)$, and (3) for $k = 2$ we provide an algorithm for any error $\varepsilon$ and poly$(d, 1/\varepsilon)$ sample and time complexity.

Constantinos Daskalakis, Noah Golowich, Stratis Skoulakis et al.

Min-max optimization problems involving nonconvex-nonconcave objectives have found important applications in adversarial training and other multi-agent learning settings. Yet, no known gradient descent-based method is guaranteed to converge to (even local notions of) min-max equilibrium in the nonconvex-nonconcave setting. For all known methods, there exist relatively simple objectives for which they cycle or exhibit other undesirable behavior different from converging to a point, let alone to some game-theoretically meaningful one~\cite{flokas2019poincare,hsieh2021limits}. The only known convergence guarantees hold under the strong assumption that the initialization is very close to a local min-max equilibrium~\cite{wang2019solving}. Moreover, the afore-described challenges are not just theoretical curiosities. All known methods are unstable in practice, even in simple settings. We propose the first method that is guaranteed to converge to a local min-max equilibrium for smooth nonconvex-nonconcave objectives. Our method is second-order and provably escapes limit cycles as long as it is initialized at an easy-to-find initial point. Both the definition of our method and its convergence analysis are motivated by the topological nature of the problem. In particular, our method is not designed to decrease some potential function, such as the distance of its iterate from the set of local min-max equilibria or the projected gradient of the objective, but is designed to satisfy a topological property that guarantees the avoidance of cycles and implies its convergence.

Jane H. Lee, Anay Mehrotra, Manolis Zampetakis

We study the estimation of distributional parameters when samples are shown only if they fall in some unknown set $S \subseteq \mathbb{R}^d$. Kontonis, Tzamos, and Zampetakis (FOCS'19) gave a $d^{\mathrm{poly}(1/\varepsilon)}$ time algorithm for finding $\varepsilon$-accurate parameters for the special case of Gaussian distributions with diagonal covariance matrix. Recently, Diakonikolas, Kane, Pittas, and Zarifis (COLT'24) showed that this exponential dependence on $1/\varepsilon$ is necessary even when $S$ belongs to some well-behaved classes. These works leave the following open problems which we address in this work: Can we estimate the parameters of any Gaussian or even extend beyond Gaussians? Can we design $\mathrm{poly}(d/\varepsilon)$ time algorithms when $S$ is a simple set such as a halfspace? We make progress on both of these questions by providing the following results: 1. Toward the first question, we give a $d^{\mathrm{poly}(\ell/\varepsilon)}$ time algorithm for any exponential family that satisfies some structural assumptions and any unknown set $S$ that is $\varepsilon$-approximable by degree-$\ell$ polynomials. This result has two important applications: 1a) The first algorithm for estimating arbitrary Gaussian distributions from samples truncated to an unknown $S$; and 1b) The first algorithm for linear regression with unknown truncation and Gaussian features. 2. To address the second question, we provide an algorithm with runtime $\mathrm{poly}(d/\varepsilon)$ that works for a set of exponential families (containing all Gaussians) when $S$ is a halfspace or an axis-aligned rectangle. Along the way, we develop tools that may be of independent interest, including, a reduction from PAC learning with positive and unlabeled samples to PAC learning with positive and negative samples that is robust to certain covariate shifts.

Anay Mehrotra, Phuc Tran, Van H. Vu et al.

A central goal of modern causal inference is estimating heterogeneous treatment effects to answer questions like "how does an intervention affect each unit," rather than only on average. We study this problem with panel-data where we observe $n$ units across $m$ times under unknown, non-uniform treatment assignments. The data in this setting is naturally represented as a matrix of all unit--time treatment effects. Estimating heterogeneous treatment effects can then be expressed as obtaining a good estimation of each row's average in this matrix. This allows us to formulate the problem as matrix completion, which can be solved under natural low-rankness assumptions. However, existing matrix-completion guarantees are not powerful enough to get meaningful bounds for the per-row guarantee required for estimating the heterogeneous treatment effect; roughly speaking, they are only useful for estimating average treatment effect bounds, as also illustrated in a recent line of work. We give a simple, computationally efficient estimator that, without knowledge of the propensities and under standard low-rankness and regularity assumptions, achieves a row-wise $\ell_2$ error of $\tilde{O}(\sqrt{\frac{1}{n} + \frac{n}{m^2}})$. Technically, our analysis establishes the first sharp row-wise $\ell_2$-perturbation bound for low-rank approximation, complementing existing spectral-, Frobenius-, and entrywise perturbation theory.

Alexandros Hollender, Manolis Zampetakis

Finding approximate stationary points, i.e., points where the gradient is approximately zero, of non-convex but smooth objective functions $f$ over unrestricted $d$-dimensional domains is one of the most fundamental problems in classical non-convex optimization. Nevertheless, the computational and query complexity of this problem are still not well understood when the dimension $d$ of the problem is independent of the approximation error. In this paper, we show the following computational and query complexity results: 1. The problem of finding approximate stationary points over unrestricted domains is PLS-complete. 2. For $d = 2$, we provide a zero-order algorithm for finding $\varepsilon$-approximate stationary points that requires at most $O(1/\varepsilon)$ value queries to the objective function. 3. We show that any algorithm needs at least $Ω(1/\varepsilon)$ queries to the objective function and/or its gradient to find $\varepsilon$-approximate stationary points when $d=2$. Combined with the above, this characterizes the query complexity of this problem to be $Θ(1/\varepsilon)$. 4. For $d = 2$, we provide a zero-order algorithm for finding $\varepsilon$-KKT points in constrained optimization problems that requires at most $O(1/\sqrt{\varepsilon})$ value queries to the objective function. This closes the gap between the works of Bubeck and Mikulincer [2020] and Vavasis [1993] and characterizes the query complexity of this problem to be $Θ(1/\sqrt{\varepsilon})$. 5. Combining our results with the recent result of Fearnley et al. [2022], we show that finding approximate KKT points in constrained optimization is reducible to finding approximate stationary points in unconstrained optimization but the converse is impossible.

Alkis Kalavasis, Konstantinos Stavropoulos, Manolis Zampetakis

We study the problem of covering and learning sums $X = X_1 + \cdots + X_n$ of independent integer-valued random variables $X_i$ (SIIRVs) with unbounded, or even infinite, support. De et al. at FOCS 2018, showed that the maximum value of the collective support of $X_i$'s necessarily appears in the sample complexity of learning $X$. In this work, we address two questions: (i) Are there general families of SIIRVs with unbounded support that can be learned with sample complexity independent of both $n$ and the maximal element of the support? (ii) Are there general families of SIIRVs with unbounded support that admit proper sparse covers in total variation distance? As for question (i), we provide a set of simple conditions that allow the unbounded SIIRV to be learned with complexity $\text{poly}(1/ε)$ bypassing the aforementioned lower bound. We further address question (ii) in the general setting where each variable $X_i$ has unimodal probability mass function and is a different member of some, possibly multi-parameter, exponential family $\mathcal{E}$ that satisfies some structural properties. These properties allow $\mathcal{E}$ to contain heavy tailed and non log-concave distributions. Moreover, we show that for every $ε> 0$, and every $k$-parameter family $\mathcal{E}$ that satisfies some structural assumptions, there exists an algorithm with $\tilde{O}(k) \cdot \text{poly}(1/ε)$ samples that learns a sum of $n$ arbitrary members of $\mathcal{E}$ within $ε$ in TV distance. The output of the learning algorithm is also a sum of random variables whose distribution lies in the family $\mathcal{E}$. En route, we prove that any discrete unimodal exponential family with bounded constant-degree central moments can be approximated by the family corresponding to a bounded subset of the initial (unbounded) parameter space.

Alkis Kalavasis, Pravesh K. Kothari, Shuchen Li et al.

In this work, we give a ${\rm poly}(d,k)$ time and sample algorithm for efficiently learning the parameters of a mixture of $k$ spherical distributions in $d$ dimensions. Unlike all previous methods, our techniques apply to heavy-tailed distributions and include examples that do not even have finite covariances. Our method succeeds whenever the cluster distributions have a characteristic function with sufficiently heavy tails. Such distributions include the Laplace distribution but crucially exclude Gaussians. All previous methods for learning mixture models relied implicitly or explicitly on the low-degree moments. Even for the case of Laplace distributions, we prove that any such algorithm must use super-polynomially many samples. Our method thus adds to the short list of techniques that bypass the limitations of the method of moments. Somewhat surprisingly, our algorithm does not require any minimum separation between the cluster means. This is in stark contrast to spherical Gaussian mixtures where a minimum $\ell_2$-separation is provably necessary even information-theoretically [Regev and Vijayaraghavan '17]. Our methods compose well with existing techniques and allow obtaining ''best of both worlds" guarantees for mixtures where every component either has a heavy-tailed characteristic function or has a sub-Gaussian tail with a light-tailed characteristic function. Our algorithm is based on a new approach to learning mixture models via efficient high-dimensional sparse Fourier transforms. We believe that this method will find more applications to statistical estimation. As an example, we give an algorithm for consistent robust mean estimation against noise-oblivious adversaries, a model practically motivated by the literature on multiple hypothesis testing. It was formally proposed in a recent Master's thesis by one of the authors, and has already inspired follow-up works.

Alkis Kalavasis, Anay Mehrotra, Manolis Zampetakis et al.

Coarse data arise when learners observe only partial information about samples; namely, a set containing the sample rather than its exact value. This occurs naturally through measurement rounding, sensor limitations, and lag in economic systems. We study Gaussian mean estimation from coarse data, where each true sample $x$ is drawn from a $d$-dimensional Gaussian distribution with identity covariance, but is revealed only through the set of a partition containing $x$. When the coarse samples, roughly speaking, have ``low'' information, the mean cannot be uniquely recovered from observed samples (i.e., the problem is not identifiable). Recent work by Fotakis, Kalavasis, Kontonis, and Tzamos [FKKT21] established that sample-efficient mean estimation is possible when the unknown mean is identifiable and the partition consists of only convex sets. Moreover, they showed that without convexity, mean estimation becomes NP-hard. However, two fundamental questions remained open: (1) When is the mean identifiable under convex partitions? (2) Is computationally efficient estimation possible under identifiability and convex partitions? This work resolves both questions. [...]

Anay Mehrotra, Manolis Zampetakis, Paul Kassianik et al.

While Large Language Models (LLMs) display versatile functionality, they continue to generate harmful, biased, and toxic content, as demonstrated by the prevalence of human-designed jailbreaks. In this work, we present Tree of Attacks with Pruning (TAP), an automated method for generating jailbreaks that only requires black-box access to the target LLM. TAP utilizes an attacker LLM to iteratively refine candidate (attack) prompts until one of the refined prompts jailbreaks the target. In addition, before sending prompts to the target, TAP assesses them and prunes the ones unlikely to result in jailbreaks, reducing the number of queries sent to the target LLM. In empirical evaluations, we observe that TAP generates prompts that jailbreak state-of-the-art LLMs (including GPT4-Turbo and GPT4o) for more than 80% of the prompts. This significantly improves upon the previous state-of-the-art black-box methods for generating jailbreaks while using a smaller number of queries than them. Furthermore, TAP is also capable of jailbreaking LLMs protected by state-of-the-art guardrails, e.g., LlamaGuard.

Steve Hanneke, Anay Mehrotra, Grigoris Velegkas et al.

Valiant's 1984 paper is widely credited with introducing the PAC learning model, but it, in fact, introduced a different model: unlike PAC learning, the learner receives only positives, may issue membership queries, and must output a hypothesis with no false positives. Prior work characterized variants, including the case without queries. We revisit Valiant's original model and ask: *Which classes are learnable in it?* For every finite domain, including Valiant's Boolean-hypercube setting, we show that a class is learnable if and only if every realizable positive sample can be certified by a poly-size adaptive query-compression scheme. This is a new variant of sample compression where the learner certifies samples via a short interaction with the membership oracle. Our characterization shows that learnability in Valiant's model is strictly sandwiched between learnability in the PAC model and the variant of Valiant's model without membership queries. This is one of the rare cases where introducing membership queries changes the set of learnable classes, and not just the sample or computational complexity. Next, we study the natural extension of the model to arbitrary domains. While we do not obtain an exact characterization, our techniques readily generalize and show that the same strict sandwiching persists. Finally, we show that $d$-dimensional halfspaces, which are not learnable without queries, are learnable with queries: we give a $\mathrm{poly}(d) \tilde{O}(1/ε)$ sample and $\mathrm{poly}(d) \mathrm{polylog}(1/ε)$ query algorithm, and prove that at least $Ω(d)$ samples or queries are necessary. To our knowledge, this is the first algorithm for halfspaces in Valiant's model. Together, these results uncover a surprisingly rich theory behind Valiant's original notion of learnability and introduce ideas that may be of independent interest in learning theory.

Jane H. Lee, Anay Mehrotra, Manolis Zampetakis

$L_1$-Approximating polynomials, i.e., polynomials that approximate indicator functions in $L_1$-norm under certain distributions, are widely used in computational learning theory. We study the existence of \textit{non-negative} $L_1$-approximating polynomials with respect to Gaussian distributions. This is a stronger requirement than $L_1$-approximation but weaker than sandwiching polynomials (which themselves have many applications). These non-negative approximating polynomials have recently found uses in smoothed learning from positive-only examples. In this short note, we prove that every class of sets with Gaussian surface area (GSA) at most $Γ$ under the standard Gaussian admits degree-$k$ non-negative polynomials that $\eps$-approximate its indicator functions in $L_1$-norm, for $k=\tilde{O}(Γ^2/\varepsilon^2)$. Equivalently, finite GSA implies $L_1$-approximation with the stronger pointwise guarantee that the approximating polynomial has range contained in $[0,\infty)$. Up to a constant-factor, this matches the degree of the best currently known Gaussian $L_1$-approximation degree bound without the non-negativity constraint.

Alkis Kalavasis, Ilias Zadik, Manolis Zampetakis

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.

Idan Attias, Yuval Dagan, Constantinos Daskalakis et al.

We propose a new algorithm that finds an $\varepsilon$-approximate fixed point of a smooth function from the $n$-dimensional $\ell_2$ unit ball to itself. We use the general framework of finding approximate solutions to a variational inequality, a problem that subsumes fixed point computation and the computation of a Nash Equilibrium. The algorithm's runtime is bounded by $e^{O(n)}/\varepsilon$, under the smoothed-analysis framework. This is the first known algorithm in such a generality whose runtime is faster than $(1/\varepsilon)^{O(n)}$, which is a time that suffices for an exhaustive search. We complement this result with a lower bound of $e^{Ω(n)}$ on the query complexity for finding an $O(1)$-approximate fixed point on the unit ball, which holds even in the smoothed-analysis model, yet without the assumption that the function is smooth. Existing lower bounds are only known for the hypercube, and adapting them to the ball does not give non-trivial results even for finding $O(1/\sqrt{n})$-approximate fixed points.

Jane H. Lee, Anay Mehrotra, Manolis Zampetakis

We study the problem of learning binary classifiers from positive and unlabeled data when the unlabeled data distribution is shifted, which we call Positive and Imperfect Unlabeled (PIU) Learning. In the absence of covariate shifts, i.e., with perfect unlabeled data, Denis (1998) reduced this problem to learning under Massart noise; however, that reduction fails under even slight shifts. Our main results on PIU learning are the characterizations of the sample complexity of PIU learning and a computationally and sample-efficient algorithm achieving a misclassification error $\varepsilon$. We further show that our results lead to new algorithms for several related problems. 1. Learning from smooth distributions: We give algorithms that learn interesting concept classes from only positive samples under smooth feature distributions, bypassing known existing impossibility results and contributing to recent advances in smoothened learning (Haghtalab et al, J.ACM'24) (Chandrasekaran et al., COLT'24). 2. Learning with a list of unlabeled distributions: We design new algorithms that apply to a broad class of concept classes under the assumption that we are given a list of unlabeled distributions, one of which--unknown to the learner--is $O(1)$-close to the true feature distribution. 3. Estimation in the presence of unknown truncation: We give the first polynomial sample and time algorithm for estimating the parameters of an exponential family distribution from samples truncated to an unknown set approximable by polynomials in $L_1$-norm. This improves the algorithm by Lee et al. (FOCS'24) that requires approximation in $L_2$-norm. 4. Detecting truncation: We present new algorithms for detecting whether given samples have been truncated (or not) for a broad class of non-product distributions, including non-product distributions, improving the algorithm by De et al. (STOC'24).

Vikram Kher, Argyris Oikonomou, Manolis Zampetakis

The remarkable success of machine learning (ML) in predictive tasks has led scientists to incorporate ML predictions as a core component of the scientific discovery pipeline. This was exemplified by the landmark achievement of AlphaFold (Jumper et al. (2021)). In this paper, we study how ML predictions can be safely used in statistical analysis of data towards scientific discovery. In particular, we follow the framework introduced by Angelopoulos et al. (2023). In this framework, we assume access to a small set of $n$ gold-standard labeled samples, a much larger set of $N$ unlabeled samples, and a ML model that can be used to impute the labels of the unlabeled data points. We introduce two new learning-augmented estimators: (1) Prediction-Augmented Residual Tree (PART), and (2) Prediction-Augmented Quadrature (PAQ). Both estimators have significant advantages over existing estimators like PPI and PPI++ introduced by Angelopoulos et al. (2023) and Angelopoulos et al. (2024), respectively. PART is a decision-tree based estimator built using a greedy criterion. We first characterize PART's asymptotic distribution and demonstrate how to construct valid confidence intervals. Then we show that PART outperforms existing methods in real-world datasets from ecology, astronomy, and census reports, among other domains. This leads to estimators with higher confidence, which is the result of using both the gold-standard samples and the machine learning predictions. Finally, we provide a formal proof of the advantage of PART by exploring PAQ, an estimation that arises when considering the limit of PART when the depth its tree grows to infinity. Under appropriate assumptions in the input data we show that the variance of PAQ shrinks at rate of $O(N^{-1} + n^{-4})$, improving significantly on the $O(N^{-1}+n^{-1})$ rate of existing methods.

Yang Cai, Alkis Kalavasis, Katerina Mamali et al.

Most of the widely used estimators of the average treatment effect (ATE) in causal inference rely on the assumptions of unconfoundedness and overlap. Unconfoundedness requires that the observed covariates account for all correlations between the outcome and treatment. Overlap requires the existence of randomness in treatment decisions for all individuals. Nevertheless, many types of studies frequently violate unconfoundedness or overlap, for instance, observational studies with deterministic treatment decisions - popularly known as Regression Discontinuity designs - violate overlap. In this paper, we initiate the study of general conditions that enable the identification of the average treatment effect, extending beyond unconfoundedness and overlap. In particular, following the paradigm of statistical learning theory, we provide an interpretable condition that is sufficient and necessary for the identification of ATE. Moreover, this condition also characterizes the identification of the average treatment effect on the treated (ATT) and can be used to characterize other treatment effects as well. To illustrate the utility of our condition, we present several well-studied scenarios where our condition is satisfied and, hence, we prove that ATE can be identified in regimes that prior works could not capture. For example, under mild assumptions on the data distributions, this holds for the models proposed by Tan (2006) and Rosenbaum (2002), and the Regression Discontinuity design model introduced by Thistlethwaite and Campbell (1960). For each of these scenarios, we also show that, under natural additional assumptions, ATE can be estimated from finite samples. We believe these findings open new avenues for bridging learning-theoretic insights and causal inference methodologies, particularly in observational studies with complex treatment mechanisms.

Manolis Zampetakis, Felix Zhou

We introduce a novel framework for differentially private (DP) statistical estimation via data truncation, addressing a key challenge in DP estimation when the data support is unbounded. Traditional approaches rely on problem-specific sensitivity analysis, limiting their applicability. By leveraging techniques from truncated statistics, we develop computationally efficient DP estimators for exponential family distributions, including Gaussian mean and covariance estimation, achieving near-optimal sample complexity. Previous works on exponential families only consider bounded or one-dimensional families. Our approach mitigates sensitivity through truncation while carefully correcting for the introduced bias using maximum likelihood estimation and DP stochastic gradient descent. Along the way, we establish improved uniform convergence guarantees for the log-likelihood function of exponential families, which may be of independent interest. Our results provide a general blueprint for DP algorithm design via truncated statistics.

Emanuel Tewolde, Brian Hu Zhang, Caspar Oesterheld et al.

We investigate optimal decision making under imperfect recall, that is, when an agent forgets information it once held before. An example is the absentminded driver game, as well as team games in which the members have limited communication capabilities. In the framework of extensive-form games with imperfect recall, we analyze the computational complexities of finding equilibria in multiplayer settings across three different solution concepts: Nash, multiselves based on evidential decision theory (EDT), and multiselves based on causal decision theory (CDT). We are interested in both exact and approximate solution computation. As special cases, we consider (1) single-player games, (2) two-player zero-sum games and relationships to maximin values, and (3) games without exogenous stochasticity (chance nodes). We relate these problems to the complexity classes P, PPAD, PLS, $Σ_2^P$ , $\exists$R, and $\exists \forall$R.

Alkis Kalavasis, Amin Karbasi, Argyris Oikonomou et al.

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, as defined in Goldwasser et al. (FOCS '22), in models developed by insidious external expert firms. When such backdoors exist, they allow the designer of the model to sell information on how to slightly perturb their input to change the outcome of the model. We develop a general strategy to plant backdoors to obfuscated neural networks, that satisfy the security properties of the celebrated notion of indistinguishability obfuscation. Applying obfuscation before releasing neural networks is a strategy that is well motivated to protect sensitive information of the external expert firm. Our method to plant backdoors ensures that even if the weights and architecture of the obfuscated model are accessible, the existence of the backdoor is still undetectable. Finally, we 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.

Tatjana Chavdarova, Michael I. Jordan, Manolis Zampetakis

Several widely-used first-order saddle-point optimization methods yield an identical continuous-time ordinary differential equation (ODE) that is identical to that of the Gradient Descent Ascent (GDA) method when derived naively. However, the convergence properties of these methods are qualitatively different, even on simple bilinear games. Thus the ODE perspective, which has proved powerful in analyzing single-objective optimization methods, has not played a similar role in saddle-point optimization. We adopt a framework studied in fluid dynamics -- known as High-Resolution Differential Equations (HRDEs) -- to design differential equation models for several saddle-point optimization methods. Critically, these HRDEs are distinct for various saddle-point optimization methods. Moreover, in bilinear games, the convergence properties of the HRDEs match the qualitative features of the corresponding discrete methods. Additionally, we show that the HRDE of Optimistic Gradient Descent Ascent (OGDA) exhibits \emph{last-iterate convergence} for general monotone variational inequalities. Finally, we provide rates of convergence for the \emph{best-iterate convergence} of the OGDA method, relying solely on the first-order smoothness of the monotone operator.

Wenshuo Guo, Michael I. Jordan, Manolis Zampetakis

We study the problem of learning revenue-optimal multi-bidder auctions from samples when the samples of bidders' valuations can be adversarially corrupted or drawn from distributions that are adversarially perturbed. First, we prove tight upper bounds on the revenue we can obtain with a corrupted distribution under a population model, for both regular valuation distributions and distributions with monotone hazard rate (MHR). We then propose new algorithms that, given only an ``approximate distribution'' for the bidder's valuation, can learn a mechanism whose revenue is nearly optimal simultaneously for all ``true distributions'' that are $α$-close to the original distribution in Kolmogorov-Smirnov distance. The proposed algorithms operate beyond the setting of bounded distributions that have been studied in prior works, and are guaranteed to obtain a fraction $1-O(α)$ of the optimal revenue under the true distribution when the distributions are MHR. Moreover, they are guaranteed to yield at least a fraction $1-O(\sqrtα)$ of the optimal revenue when the distributions are regular. We prove that these upper bounds cannot be further improved, by providing matching lower bounds. Lastly, we derive sample complexity upper bounds for learning a near-optimal auction for both MHR and regular distributions.

Constantinos Daskalakis, Themis Gouleakis, Christos Tzamos et al.

We provide a computationally and statistically efficient estimator for the classical problem of truncated linear regression, where the dependent variable $y = w^T x + ε$ and its corresponding vector of covariates $x \in R^k$ are only revealed if the dependent variable falls in some subset $S \subseteq R$; otherwise the existence of the pair $(x, y)$ is hidden. This problem has remained a challenge since the early works of [Tobin 1958, Amemiya 1973, Hausman and Wise 1977], its applications are abundant, and its history dates back even further to the work of Galton, Pearson, Lee, and Fisher. While consistent estimators of the regression coefficients have been identified, the error rates are not well-understood, especially in high dimensions. Under a thickness assumption about the covariance matrix of the covariates in the revealed sample, we provide a computationally efficient estimator for the coefficient vector $w$ from $n$ revealed samples that attains $l_2$ error $\tilde{O}(\sqrt{k/n})$. Our estimator uses Projected Stochastic Gradient Descent (PSGD) without replacement on the negative log-likelihood of the truncated sample. For the statistically efficient estimation we only need oracle access to the set $S$.In order to achieve computational efficiency we need to assume that $S$ is a union of a finite number of intervals but still can be complicated. PSGD without replacement must be restricted to an appropriately defined convex cone to guarantee that the negative log-likelihood is strongly convex, which in turn is established using concentration of matrices on variables with sub-exponential tails. We perform experiments on simulated data to illustrate the accuracy of our estimator. As a corollary, we show that SGD learns the parameters of single-layer neural networks with noisy activation functions.

Alessandro Epasto, Mohammad Mahdian, Vahab Mirrokni et al.

A soft-max function has two main efficiency measures: (1) approximation - which corresponds to how well it approximates the maximum function, (2) smoothness - which shows how sensitive it is to changes of its input. Our goal is to identify the optimal approximation-smoothness tradeoffs for different measures of approximation and smoothness. This leads to novel soft-max functions, each of which is optimal for a different application. The most commonly used soft-max function, called exponential mechanism, has optimal tradeoff between approximation measured in terms of expected additive approximation and smoothness measured with respect to Rényi Divergence. We introduce a soft-max function, called "piecewise linear soft-max", with optimal tradeoff between approximation, measured in terms of worst-case additive approximation and smoothness, measured with respect to $\ell_q$-norm. The worst-case approximation guarantee of the piecewise linear mechanism enforces sparsity in the output of our soft-max function, a property that is known to be important in Machine Learning applications [Martins et al. '16, Laha et al. '18] and is not satisfied by the exponential mechanism. Moreover, the $\ell_q$-smoothness is suitable for applications in Mechanism Design and Game Theory where the piecewise linear mechanism outperforms the exponential mechanism. Finally, we investigate another soft-max function, called power mechanism, with optimal tradeoff between expected \textit{multiplicative} approximation and smoothness with respect to the Rényi Divergence, which provides improved theoretical and practical results in differentially private submodular optimization.

Constantinos Daskalakis, Stratis Skoulakis, Manolis Zampetakis

Despite its important applications in Machine Learning, min-max optimization of nonconvex-nonconcave objectives remains elusive. Not only are there no known first-order methods converging even to approximate local min-max points, but the computational complexity of identifying them is also poorly understood. In this paper, we provide a characterization of the computational complexity of the problem, as well as of the limitations of first-order methods in constrained min-max optimization problems with nonconvex-nonconcave objectives and linear constraints. As a warm-up, we show that, even when the objective is a Lipschitz and smooth differentiable function, deciding whether a min-max point exists, in fact even deciding whether an approximate min-max point exists, is NP-hard. More importantly, we show that an approximate local min-max point of large enough approximation is guaranteed to exist, but finding one such point is PPAD-complete. The same is true of computing an approximate fixed point of Gradient Descent/Ascent. An important byproduct of our proof is to establish an unconditional hardness result in the Nemirovsky-Yudin model. We show that, given oracle access to some function $f : P \to [-1, 1]$ and its gradient $\nabla f$, where $P \subseteq [0, 1]^d$ is a known convex polytope, every algorithm that finds a $\varepsilon$-approximate local min-max point needs to make a number of queries that is exponential in at least one of $1/\varepsilon$, $L$, $G$, or $d$, where $L$ and $G$ are respectively the smoothness and Lipschitzness of $f$ and $d$ is the dimension. This comes in sharp contrast to minimization problems, where finding approximate local minima in the same setting can be done with Projected Gradient Descent using $O(L/\varepsilon)$ many queries. Our result is the first to show an exponential separation between these two fundamental optimization problems.

Constantinos Daskalakis, Dhruv Rohatgi, Manolis Zampetakis

As in standard linear regression, in truncated linear regression, we are given access to observations $(A_i, y_i)_i$ whose dependent variable equals $y_i= A_i^{\rm T} \cdot x^* + η_i$, where $x^*$ is some fixed unknown vector of interest and $η_i$ is independent noise; except we are only given an observation if its dependent variable $y_i$ lies in some "truncation set" $S \subset \mathbb{R}$. The goal is to recover $x^*$ under some favorable conditions on the $A_i$'s and the noise distribution. We prove that there exists a computationally and statistically efficient method for recovering $k$-sparse $n$-dimensional vectors $x^*$ from $m$ truncated samples, which attains an optimal $\ell_2$ reconstruction error of $O(\sqrt{(k \log n)/m})$. As a corollary, our guarantees imply a computationally efficient and information-theoretically optimal algorithm for compressed sensing with truncation, which may arise from measurement saturation effects. Our result follows from a statistical and computational analysis of the Stochastic Gradient Descent (SGD) algorithm for solving a natural adaptation of the LASSO optimization problem that accommodates truncation. This generalizes the works of both: (1) [Daskalakis et al. 2018], where no regularization is needed due to the low-dimensionality of the data, and (2) [Wainright 2009], where the objective function is simple due to the absence of truncation. In order to deal with both truncation and high-dimensionality at the same time, we develop new techniques that not only generalize the existing ones but we believe are of independent interest.

Vasilis Syrgkanis, Manolis Zampetakis

We analyze the finite sample mean squared error (MSE) performance of regression trees and forests in the high dimensional regime with binary features, under a sparsity constraint. We prove that if only $r$ of the $d$ features are relevant for the mean outcome function, then shallow trees built greedily via the CART empirical MSE criterion achieve MSE rates that depend only logarithmically on the ambient dimension $d$. We prove upper bounds, whose exact dependence on the number relevant variables $r$ depends on the correlation among the features and on the degree of relevance. For strongly relevant features, we also show that fully grown honest forests achieve fast MSE rates and their predictions are also asymptotically normal, enabling asymptotically valid inference that adapts to the sparsity of the regression function.

Constantinos Daskalakis, Dhruv Rohatgi, Manolis Zampetakis

Generative neural networks have been empirically found very promising in providing effective structural priors for compressed sensing, since they can be trained to span low-dimensional data manifolds in high-dimensional signal spaces. Despite the non-convexity of the resulting optimization problem, it has also been shown theoretically that, for neural networks with random Gaussian weights, a signal in the range of the network can be efficiently, approximately recovered from a few noisy measurements. However, a major bottleneck of these theoretical guarantees is a network expansivity condition: that each layer of the neural network must be larger than the previous by a logarithmic factor. Our main contribution is to break this strong expansivity assumption, showing that constant expansivity suffices to get efficient recovery algorithms, besides it also being information-theoretically necessary. To overcome the theoretical bottleneck in existing approaches we prove a novel uniform concentration theorem for random functions that might not be Lipschitz but satisfy a relaxed notion which we call "pseudo-Lipschitzness." Using this theorem we can show that a matrix concentration inequality known as the Weight Distribution Condition (WDC), which was previously only known to hold for Gaussian matrices with logarithmic aspect ratio, in fact holds for constant aspect ratios too. Since the WDC is a fundamental matrix concentration inequality in the heart of all existing theoretical guarantees on this problem, our tighter bound immediately yields improvements in all known results in the literature on compressed sensing with deep generative priors, including one-bit recovery, phase retrieval, low-rank matrix recovery, and more.

Vasilis Kontonis, Christos Tzamos, Manolis Zampetakis

We study the problem of estimating the parameters of a Gaussian distribution when samples are only shown if they fall in some (unknown) subset $S \subseteq \R^d$. This core problem in truncated statistics has long history going back to Galton, Lee, Pearson and Fisher. Recent work by Daskalakis et al. (FOCS'18), provides the first efficient algorithm that works for arbitrary sets in high dimension when the set is known, but leaves as an open problem the more challenging and relevant case of unknown truncation set. Our main result is a computationally and sample efficient algorithm for estimating the parameters of the Gaussian under arbitrary unknown truncation sets whose performance decays with a natural measure of complexity of the set, namely its Gaussian surface area. Notably, this algorithm works for large families of sets including intersections of halfspaces, polynomial threshold functions and general convex sets. We show that our algorithm closely captures the tradeoff between the complexity of the set and the number of samples needed to learn the parameters by exhibiting a set with small Gaussian surface area for which it is information theoretically impossible to learn the true Gaussian with few samples.

Róbert Busa-Fekete, Dimitris Fotakis, Balázs Szörényi et al.

The Mallows model, introduced in the seminal paper of Mallows 1957, is one of the most fundamental ranking distribution over the symmetric group $S_m$. To analyze more complex ranking data, several studies considered the Generalized Mallows model defined by Fligner and Verducci 1986. Despite the significant research interest of ranking distributions, the exact sample complexity of estimating the parameters of a Mallows and a Generalized Mallows Model is not well-understood. The main result of the paper is a tight sample complexity bound for learning Mallows and Generalized Mallows Model. We approach the learning problem by analyzing a more general model which interpolates between the single parameter Mallows Model and the $m$ parameter Mallows model. We call our model Mallows Block Model -- referring to the Block Models that are a popular model in theoretical statistics. Our sample complexity analysis gives tight bound for learning the Mallows Block Model for any number of blocks. We provide essentially matching lower bounds for our sample complexity results. As a corollary of our analysis, it turns out that, if the central ranking is known, one single sample from the Mallows Block Model is sufficient to estimate the spread parameters with error that goes to zero as the size of the permutations goes to infinity. In addition, we calculate the exact rate of the parameter estimation error.

Constantinos Daskalakis, Themis Gouleakis, Christos Tzamos et al.

We provide an efficient algorithm for the classical problem, going back to Galton, Pearson, and Fisher, of estimating, with arbitrary accuracy the parameters of a multivariate normal distribution from truncated samples. Truncated samples from a $d$-variate normal ${\cal N}(\mathbfμ,\mathbfΣ)$ means a samples is only revealed if it falls in some subset $S \subseteq \mathbb{R}^d$; otherwise the samples are hidden and their count in proportion to the revealed samples is also hidden. We show that the mean $\mathbfμ$ and covariance matrix $\mathbfΣ$ can be estimated with arbitrary accuracy in polynomial-time, as long as we have oracle access to $S$, and $S$ has non-trivial measure under the unknown $d$-variate normal distribution. Additionally we show that without oracle access to $S$, any non-trivial estimation is impossible.

Katerina Sotiraki, Manolis Zampetakis, Giorgos Zirdelis

Polynomial Pigeonhole Principle (PPP) is an important subclass of TFNP with profound connections to the complexity of the fundamental cryptographic primitives: collision-resistant hash functions and one-way permutations. In contrast to most of the other subclasses of TFNP, no complete problem is known for PPP. Our work identifies the first PPP-complete problem without any circuit or Turing Machine given explicitly in the input, and thus we answer a longstanding open question from [Papadimitriou1994]. Specifically, we show that constrained-SIS (cSIS), a generalized version of the well-known Short Integer Solution problem (SIS) from lattice-based cryptography, is PPP-complete. In order to give intuition behind our reduction for constrained-SIS, we identify another PPP-complete problem with a circuit in the input but closely related to lattice problems. We call this problem BLICHFELDT and it is the computational problem associated with Blichfeldt's fundamental theorem in the theory of lattices. Building on the inherent connection of PPP with collision-resistant hash functions, we use our completeness result to construct the first natural hash function family that captures the hardness of all collision-resistant hash functions in a worst-case sense, i.e. it is natural and universal in the worst-case. The close resemblance of our hash function family with SIS, leads us to the first candidate collision-resistant hash function that is both natural and universal in an average-case sense. Finally, our results enrich our understanding of the connections between PPP, lattice problems and other concrete cryptographic assumptions, such as the discrete logarithm problem over general groups.

Constantinos Daskalakis, Christos Tzamos, Manolis Zampetakis

Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics. Our first result is a strong converse of Banach's theorem, showing that it is a universal analysis tool for establishing global convergence of iterative methods to unique fixed points, and for bounding their convergence rate. In other words, we show that, whenever an iterative map globally converges to a unique fixed point, there exists a metric under which the iterative map is contracting and which can be used to bound the number of iterations until convergence. We illustrate our approach in the widely used power method, providing a new way of bounding its convergence rate through contraction arguments. We next consider the computational complexity of Banach's fixed point theorem. Making the proof of our converse theorem constructive, we show that computing a fixed point whose existence is guaranteed by Banach's fixed point theorem is CLS-complete. We thus provide the first natural complete problem for the class CLS, which was defined in [Daskalakis, Papadimitriou 2011] to capture the complexity of problems such as P-matrix LCP, computing KKT-points, and finding mixed Nash equilibria in congestion and network coordination games.

Constantinos Daskalakis, Christos Tzamos, Manolis Zampetakis

The Expectation-Maximization (EM) algorithm is a widely used method for maximum likelihood estimation in models with latent variables. For estimating mixtures of Gaussians, its iteration can be viewed as a soft version of the k-means clustering algorithm. Despite its wide use and applications, there are essentially no known convergence guarantees for this method. We provide global convergence guarantees for mixtures of two Gaussians with known covariance matrices. We show that the population version of EM, where the algorithm is given access to infinitely many samples from the mixture, converges geometrically to the correct mean vectors, and provide simple, closed-form expressions for the convergence rate. As a simple illustration, we show that, in one dimension, ten steps of the EM algorithm initialized at infinity result in less than 1\% error estimation of the means. In the finite sample regime, we show that, under a random initialization, $\tilde{O}(d/ε^2)$ samples suffice to compute the unknown vectors to within $ε$ in Mahalanobis distance, where $d$ is the dimension. In particular, the error rate of the EM based estimator is $\tilde{O}\left(\sqrt{d \over n}\right)$ where $n$ is the number of samples, which is optimal up to logarithmic factors.

Themistoklis Gouleakis, Christos Tzamos, Manolis Zampetakis

A conditional sampling oracle for a probability distribution D returns samples from the conditional distribution of D restricted to a specified subset of the domain. A recent line of work (Chakraborty et al. 2013 and Cannone et al. 2014) has shown that having access to such a conditional sampling oracle requires only polylogarithmic or even constant number of samples to solve distribution testing problems like identity and uniformity. This significantly improves over the standard sampling model where polynomially many samples are necessary. Inspired by these results, we introduce a computational model based on conditional sampling to develop sublinear algorithms with exponentially faster runtimes compared to standard sublinear algorithms. We focus on geometric optimization problems over points in high dimensional Euclidean space. Access to these points is provided via a conditional sampling oracle that takes as input a succinct representation of a subset of the domain and outputs a uniformly random point in that subset. We study two well studied problems: k-means clustering and estimating the weight of the minimum spanning tree. In contrast to prior algorithms for the classic model, our algorithms have time, space and sample complexity that is polynomial in the dimension and polylogarithmic in the number of points. Finally, we comment on the applicability of the model and compare with existing ones like streaming, parallel and distributed computational models.