Transfer Learning Beyond Bounded Density Ratios

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.

Tree of Attacks: Jailbreaking Black-Box LLMs Automatically

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 LLM to iteratively refine candidate (attack) prompts using tree-of-thoughts reasoning until one of the generated prompts jailbreaks the target. Crucially, before sending prompts to the target, TAP assesses them and prunes the ones unlikely to result in jailbreaks. Using tree-of-thought reasoning allows TAP to navigate a large search space of prompts and pruning reduces the total number of queries sent to the target. In empirical evaluations, we observe that TAP generates prompts that jailbreak state-of-the-art LLMs (including GPT4 and GPT4-Turbo) for more than 80% of the prompts using only a small number of queries. This significantly improves upon the previous state-of-the-art black-box method for generating jailbreaks.

The Computational Complexity of Finding Stationary Points in Non-Convex Optimization

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 $\Omega(1/\varepsilon)$ 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 $\Theta(1/\varepsilon)$. 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 $\Theta(1/\sqrt{\varepsilon})$. 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.

Deterministic Nonsmooth Nonconvex Optimization

We study the complexity of optimizing nonsmooth nonconvex Lipschitz functions by producing $(\delta,\epsilon)$-stationary points. Several recent works have presented randomized algorithms that produce such points using $\tilde O(\delta^{-1}\epsilon^{-3})$ 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 $\Omega(d)$ 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 $\tilde O(\delta^{-1}\epsilon^{-3})$ 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 $(\delta,\epsilon)$-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.

Learning and Covering Sums of Independent Random Variables with Unbounded Support

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 $\text{poly}(1/\epsilon)$ 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 $\epsilon > 0$, and every $k$-parameter family $\mathcal{E}$ that satisfies some structural assumptions, there exists an algorithm with $\tilde{O}(k) \cdot \text{poly}(1/\epsilon)$ samples that learns a sum of $n$ arbitrary members of $\mathcal{E}$ within $\epsilon$ 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.

STay-ON-the-Ridge: Guaranteed Convergence to Local Minimax Equilibrium in Nonconvex-Nonconcave Games

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.

Efficient Truncated Linear Regression with Unknown Noise Variance

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.

What Makes A Good Fisherman? Linear Regression under Self-Selection Bias

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.

Estimation of Standard Auction Models

We provide efficient estimation methods for first- and second-price auctions under independent (asymmetric) private values and partial observability. Given a finite set of observations, each comprising the identity of the winner and the price they paid in a sequence of identical auctions, we provide algorithms for non-parametrically estimating the bid distribution of each bidder, as well as their value distributions under equilibrium assumptions. We provide finite-sample estimation bounds which are uniform in that their error rates do not depend on the bid/value distributions being estimated. Our estimation guarantees advance a body of work in Econometrics wherein only identification results have been obtained, unless the setting is symmetric, parametric, or all bids are observable. Our guarantees also provide computationally and statistically effective alternatives to classical techniques from reliability theory. Finally, our results are immediately applicable to Dutch and English auctions.

First-Order Algorithms for Nonlinear Generalized Nash Equilibrium Problems

We consider the problem of computing an equilibrium in a class of 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 rate of solution procedures have been studied in this setting, the analysis of iteration complexity is still in its infancy. Our contribution is to provide two simple first-order algorithmic frameworks based on the quadratic penalty method and the augmented Lagrangian method, respectively, with an accelerated mirror-prox algorithm as the inner loop. We provide nonasymptotic theoretical guarantees for these algorithms. More specifically, we establish the global convergence rate of our algorithms for solving (strongly) monotone NGNEPs and we provide iteration complexity bounds expressed in terms of the number of gradient evaluations. Experimental results demonstrate the efficiency of our algorithms.