Probably Approximately Correct Research Articles

Eliciting Kemeny Rankings

We formulate the problem of eliciting agents' preferences with the goal of finding a Kemeny ranking as a Dueling Bandits problem. Here the bandits' arms correspond to alternatives that need to be ranked and the feedback corresponds to a pairwise comparison between alternatives by a randomly sampled agent. We consider both sampling with and without replacement, i.e., the possibility to ask the same agent about some comparison multiple times or not. We find approximation bounds for Kemeny rankings dependant on confidence intervals over estimated winning probabilities of arms. Based on these we state algorithms to find Probably Approximately Correct (PAC) solutions and elaborate on their sample complexity for sampling with or without replacement. Furthermore, if all agents' preferences are strict rankings over the alternatives, we provide means to prune confidence intervals and thereby guide a more efficient elicitation. We formulate several adaptive sampling methods that use look-aheads to estimate how much confidence intervals (and thus approximation guarantees) might be tightened. All described methods are compared on synthetic data.

A PAC Learning Algorithm for LTL and Omega-Regular Objectives in MDPs

Linear temporal logic (LTL) and omega-regular objectives---a superset of LTL---have seen recent use as a way to express non-Markovian objectives in reinforcement learning. We introduce a model-based probably approximately correct (PAC) learning algorithm for omega-regular objectives in Markov decision processes (MDPs). As part of the development of our algorithm, we introduce the epsilon-recurrence time: a measure of the speed at which a policy converges to the satisfaction of the omega-regular objective in the limit. We prove that our algorithm only requires a polynomial number of samples in the relevant parameters, and perform experiments which confirm our theory.

Sim-to-Lab-to-Real: Safe Reinforcement Learning with Shielding and Generalization Guarantees (Abstract Reprint)

Safety is a critical component of autonomous systems and remains a challenge for learning-based policies to be utilized in the real world. In particular, policies learned using reinforcement learning often fail to generalize to novel environments due to unsafe behavior. In this paper, we propose Sim-to-Lab-to-Real to bridge the reality gap with a probabilistically guaranteed safety-aware policy distribution. To improve safety, we apply a dual policy setup where a performance policy is trained using the cumulative task reward and a backup (safety) policy is trained by solving the Safety Bellman Equation based on Hamilton-Jacobi (HJ) reachability analysis. In Sim-to-Lab transfer, we apply a supervisory control scheme to shield unsafe actions during exploration; in Lab-to-Real transfer, we leverage the Probably Approximately Correct (PAC)-Bayes framework to provide lower bounds on the expected performance and safety of policies in unseen environments. Additionally, inheriting from the HJ reachability analysis, the bound accounts for the expectation over the worst-case safety in each environment. We empirically study the proposed framework for ego-vision navigation in two types of indoor environments with varying degrees of photorealism. We also demonstrate strong generalization performance through hardware experiments in real indoor spaces with a quadrupedal robot. See https://sites.google.com/princeton.edu/sim-to-lab-to-real for supplementary material.

SHAP@k: Efficient and Probably Approximately Correct (PAC) Identification of Top-K Features

The SHAP framework provides a principled method to explain the predictions of a model by computing feature importance. Motivated by applications in finance, we introduce the Top-k Identification Problem (TkIP) (and its ordered variant TkIP- O), where the objective is to identify the subset (or ordered subset for TkIP-O) of k features corresponding to the highest SHAP values with PAC guarantees. While any sampling-based method that estimates SHAP values (such as KernelSHAP and SamplingSHAP) can be trivially adapted to solve TkIP, doing so is highly sample inefficient. Instead, we leverage the connection between SHAP values and multi-armed bandits (MAB) to show that both TkIP and TkIP-O can be reduced to variants of problems in MAB literature. This reduction allows us to use insights from the MAB literature to develop sample-efficient variants of KernelSHAP and SamplingSHAP. We propose KernelSHAP@k and SamplingSHAP@k for solving TkIP; along with KernelSHAP-O and SamplingSHAP-O to solve the ordering problem in TkIP-O. We perform extensive experiments using several credit-related datasets to show that our methods offer significant improvements of up to 40× in sample efficiency and 39× in runtime.

Realizable Learning is All You Need

The equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory. With variants ranging from classical settings like PAC learning and regression to recent trends such as adversarially robust learning, it's surprising that we still lack a unified theory; traditional proofs of the equivalence tend to be disparate, and rely on strong model-specific assumptions like uniform convergence and sample compression. In this work, we give the first model-independent framework explaining the equivalence of realizable and agnostic learnability: a three-line blackbox reduction that simplifies, unifies, and extends our understanding across a wide variety of settings. This includes models with no known characterization of learnability such as learning with arbitrary distributional assumptions and more general loss functions, as well as a host of other popular settings such as robust learning, partial learning, fair learning, and the statistical query model. More generally, we argue that the equivalence of realizable and agnostic learning is actually a special case of a broader phenomenon we call property generalization: any desirable property of a learning algorithm (e.g. noise tolerance, privacy, stability) that can be satisfied over finite hypothesis classes extends (possibly in some variation) to any learnable hypothesis class.

Robust Gaussian Process Regression With Input Uncertainty: A PAC-Bayes Perspective.

The Gaussian process (GP) algorithm is considered as a powerful nonparametric-learning approach, which can provide uncertainty measurements on the predictions. The standard GP requires clearly observed data, unexpected perturbations in the input may lead to learned regression model mismatching. Besides, GP also suffers from the lack of good generalization performance guarantees. To deal with data uncertainty and provide a numerical generalization performance guarantee on the unknown data distribution, this article proposes a novel robust noisy input GP (NIGP) algorithm based on the probably approximately correct (PAC) Bayes theory. Furthermore, to reduce the computational complexity, we develop a sparse NIGP algorithm, and then develop a sparse PAC-Bayes NIGP approach. Compared with NIGP algorithms, instead of maximizing the marginal log likelihood, one can optimize the PAC-Bayes bound to pursue a tighter generalization error upper bound. Experiments verify that the NIGP algorithms can attain greater accuracy. Besides, the PAC-NIGP algorithms proposed herein can achieve both robust performance and improved generalization error upper bound in the face of both uncertain input and output data.

PARLE-G: Provable Automated Representation and Analysis Framework for Learnability Evaluation of Generic PUF Compositions

Besides enormous research efforts in the design of Physically Unclonable Functions (PUFs), its vulnerabilities are still being exploited using machine learning (ML) based model-building attacks. Due to inherent complicacy in exploring and manually converging to a strong PUF composition, the challenge of building ML-attack resistant PUFs continues. Hence, it becomes imperative to develop an automated framework that can formally assess the learnability of different PUF constructions and compositions to guide the designer to explore resilient PUFs. In this work, we present an automated analysis framework (PARLE-G), to formally represent and evaluate the Probably Approximately Correct (PAC) learnability of PUF constructions and their compositions. A high-level specification language PUF-G has been developed to structurally represent any PUF composition comprising a specified set of primitive components and composition operations. The tool takes a PUF design represented in PUF-G language as input and returns its PAC learnability result, identifying a suitable PAC learning algorithm and the PAC model parameters based on the input PUF design. PUF designs proven to be learnable by PARLE-G are segregated into different classes based on the asymptotic complexity of their learnability bounds. Such automated analysis helps a designer to make informed design choices, thereby strengthening a PUF construction from the architectural level.

Robust PAC m : Training Ensemble Models Under Misspecification and Outliers.

Standard Bayesian learning is known to have suboptimal generalization capabilities under misspecification and in the presence of outliers. Probably approximately correct (PAC)-Bayes theory demonstrates that the free energy criterion minimized by Bayesian learning is a bound on the generalization error for Gibbs predictors (i.e., for single models drawn at random from the posterior) under the assumption of sampling distributions uncontaminated by outliers. This viewpoint provides a justification for the limitations of Bayesian learning when the model is misspecified, requiring ensembling, and when data are affected by outliers. In recent work, PAC-Bayes bounds-referred to as PAC m -were derived to introduce free energy metrics that account for the performance of ensemble predictors, obtaining enhanced performance under misspecification. This work presents a novel robust free energy criterion that combines the generalized logarithm score function with PAC m ensemble bounds. The proposed free energy training criterion produces predictive distributions that are able to concurrently counteract the detrimental effects of misspecification-with respect to both likelihood and prior distribution-and outliers.

A Survey on Scenario Theory, Complexity, and Compression-Based Learning and Generalization.

This work investigates formal generalization error bounds that apply to support vector machines (SVMs) in realizable and agnostic learning problems. We focus on recently observed parallels between probably approximately correct (PAC)-learning bounds, such as compression and complexity-based bounds, and novel error guarantees derived within scenario theory. Scenario theory provides nonasymptotic and distributional-free error bounds for models trained by solving data-driven decision-making problems. Relevant theorems and assumptions are reviewed and discussed. We propose a numerical comparison of the tightness and effectiveness of theoretical error bounds for support vector classifiers trained on several randomized experiments from 13 real-life problems. This analysis allows for a fair comparison of different approaches from both conceptual and experimental standpoints. Based on the numerical results, we argue that the error guarantees derived from scenario theory are often tighter for realizable problems and always yield informative results, i.e., probability bounds tighter than a vacuous [0,1] interval. This work promotes scenario theory as an alternative tool for model selection, structural-risk minimization, and generalization error analysis of SVMs. In this way, we hope to bring the communities of scenario and statistical learning theory closer, so that they can benefit from each other's insights.

PAC learning halfspaces in non-interactive local differential privacy model with public unlabeled data

In this paper, we study the problem of PAC learning halfspaces in the non-interactive local differential privacy model (NLDP). To breach the barrier of exponential sample complexity, previous results studied a relaxed setting where the server has access to some additional public but unlabeled data. We continue in this direction. Specifically, we consider the problem under the standard setting instead of the large margin setting studied before. Under different mild assumptions on the underlying data distribution, we propose two approaches that are based on the Massart noise model and self-supervised learning and show that it is possible to achieve sample complexities that are only linear in the dimension and polynomial in other terms for both private and public data, which significantly improve the previous results. Our methods could also be used for other private PAC learning problems.

Knowledge representation and acquisition in the era of large language models: Reflections on learning to reason via PAC-Semantics

Human beings are known for their remarkable ability to comprehend, analyse, and interpret common sense knowledge. This ability is critical for exhibiting intelligent behaviour, often defined as a mapping from beliefs to actions, which has led to attempts to formalize and capture explicit representations in the form of databases, knowledge bases, and ontologies in AI agents.But in the era of large language models (LLMs), this emphasis might seem unnecessary. After all, these models already capture the extent of human knowledge and can infer appropriate things from it (presumably) as per some innate logical rules. The question then is whether they can also be trained to perform mathematical computations.Although the consensus on the reliability of such models is still being studied, early results do seem to suggest they do not offer logically and mathematically consistent results. In this short summary article, we articulate the motivations for still caring about logical/symbolic artefacts and representations, and report on recent progress in learning to reason via the so-called probably approximately correct (PAC)-semantics.

Data-Driven Reachability and Support Estimation With Christoffel Functions

We present algorithms for estimating the forward reachable set of a dynamical system using only a finite collection of independent and identically distributed samples. The produced estimate is the sublevel set of a function called an empirical inverse Christoffel function: empirical inverse Christoffel functions are known to provide good approximations to the support of probability distributions. In addition to reachability analysis, the same approach can be applied to general problems of estimating the support of a random variable, which has applications in data science towards detection of novelties and outliers in data sets. In applications where safety is a concern, having a guarantee of accuracy that holds on finite data sets is critical. In this paper, we prove such bounds for our algorithms under the Probably Approximately Correct (PAC) framework. In addition to applying classical Vapnik- Chervonenkis (VC) dimension bound arguments, we apply the PAC-Bayes theorem by leveraging a formal connection between kernelized empirical inverse Christoffel functions and Gaussian process regression models.

On the non-efficient PAC learnability of conjunctive queries

This note serves three purposes: (i) we provide a self-contained exposition of the fact that conjunctive queries are not efficiently learnable in the Probably-Approximately-Correct (PAC) model, paying clear attention to the complicating fact that this concept class lacks the polynomial-size fitting property, a property that is tacitly assumed in much of the computational learning theory literature; (ii) we establish a strong negative PAC learnability result that applies to many restricted classes of conjunctive queries (CQs), including acyclic CQs for a wide range of notions of acyclicity; (iii) we show that CQs (and UCQs) are efficiently PAC learnable with membership queries.

Learnability with PAC Semantics for Multi-agent Beliefs

AbstractThe tension between deduction and induction is perhaps the most fundamental issue in areas such as philosophy, cognition, and artificial intelligence. In an influential paper, Valiant recognized that the challenge of learning should be integrated with deduction. In particular, he proposed a semantics to capture the quality possessed by the output of probably approximately correct (PAC) learning algorithms when formulated in a logic. Although weaker than classical entailment, it allows for a powerful model-theoretic framework for answering queries. In this paper, we provide a new technical foundation to demonstrate PAC learning with multi-agent epistemic logics. To circumvent the negative results in the literature on the difficulty of robust learning with the PAC semantics, we consider so-called implicit learning where we are able to incorporate observations to the background theory in service of deciding the entailment of an epistemic query. We prove correctness of the learning procedure and discuss results on the sample complexity, that is how many observations we will need to provably assert that the query is entailed given a user-specified error bound. Finally, we investigate under what circumstances this algorithm can be made efficient. On the last point, given that reasoning in epistemic logics especially in multi-agent epistemic logics is PSPACE-complete, it might seem like there is no hope for this problem. We leverage some recent results on the so-called Representation Theorem explored for single-agent and multi-agent epistemic logics with the only knowing operator to reduce modal reasoning to propositional reasoning.

Tree Learning: Optimal Sample Complexity and Algorithms

We study the problem of learning a hierarchical tree representation of data from labeled samples, taken from an arbitrary (and possibly adversarial) distribution. Consider a collection of data tuples labeled according to their hierarchical structure. The smallest number of such tuples required in order to be able to accurately label subsequent tuples is of interest for data collection in machine learning. We present optimal sample complexity bounds for this problem in several learning settings, including (agnostic) PAC learning and online learning. Our results are based on tight bounds of the Natarajan and Littlestone dimensions of the associated problem. The corresponding tree classifiers can be constructed efficiently in near-linear time.

PAC Learning and Stabilizing Hedonic Games: Towards a Unifying Approach.

We study PAC learnability and PAC stabilizability of Hedonic Games (HGs), i.e., efficiently inferring preferences or core-stable partitions from samples. We first expand the known learnability/stabilizability landscape for some of the most prominent HGs classes, providing results for Friends and Enemies Games, Bottom Responsive, and Anonymous HGs. Then, having a broader view in mind, we attempt to shed light on the structural properties leading to learnability/stabilizability, or lack thereof, for specific HGs classes. Along this path, we focus on the fully expressive Hedonic Coalition Nets representation of HGs. We identify two sets of conditions that lead to efficient learnability, and which encompass all of the known positive learnability results. On the side of stability, we reveal that, while the freedom of choosing an ad hoc adversarial distribution is the most obvious hurdle to achieving PAC stability, it is not the only one. First, we show a distribution independent necessary condition for PAC stability. Then, we focus on W-games, where players have individual preferences over other players and evaluate coalitions based on the least preferred member. We prove that these games are PAC stabilizable under the class of bounded distributions, which assign positive probability mass to all coalitions. Finally, we discuss why such a result is not easily extendable to other HGs classes even in this promising scenario. Namely, we establish a purely computational property necessary for achieving PAC stability.

A Parameterized Theory of PAC Learning

Probably Approximately Correct (i.e., PAC) learning is a core concept of sample complexity theory, and efficient PAC learnability is often seen as a natural counterpart to the class P in classical computational complexity. But while the nascent theory of parameterized complexity has allowed us to push beyond the P-NP "dichotomy" in classical computational complexity and identify the exact boundaries of tractability for numerous problems, there is no analogue in the domain of sample complexity that could push beyond efficient PAC learnability. As our core contribution, we fill this gap by developing a theory of parameterized PAC learning which allows us to shed new light on several recent PAC learning results that incorporated elements of parameterized complexity. Within the theory, we identify not one but two notions of fixed-parameter learnability that both form distinct counterparts to the class FPT - the core concept at the center of the parameterized complexity paradigm - and develop the machinery required to exclude fixed-parameter learnability. We then showcase the applications of this theory to identify refined boundaries of tractability for CNF and DNF learning as well as for a range of learning problems on graphs.

On the Complexity of PAC Learning in Hilbert Spaces

We study the problem of binary classification from the point of view of learning convex polyhedra in Hilbert spaces, to which one can reduce any binary classification problem. The problem of learning convex polyhedra in finite-dimensional spaces is sufficiently well studied in the literature. We generalize this problem to that in a Hilbert space and propose an algorithm for learning a polyhedron which correctly classifies at least 1 − ε of the distribution, with a probability of at least 1 − δ, where ε and δ are given parameters. Also, as a corollary, we improve some previous bounds for polyhedral classification in finite-dimensional spaces.

Clifford Circuits can be Properly PAC Learned if and only if RP=NP

Given a dataset of input states, measurements, and probabilities, is it possible to efficiently predict the measurement probabilities associated with a quantum circuit? Recent work of Caro and Datta \cite{2020Caro} studied the problem of PAC learning quantum circuits in an information theoretic sense, leaving open questions of computational efficiency. In particular, one candidate class of circuits for which an efficient learner might have been possible was that of Clifford circuits, since the corresponding set of states generated by such circuits, called stabilizer states, are known to be efficiently PAC learnable \cite{rocchetto2018stabiliser}. Here we provide a negative result, showing that proper learning of CNOT circuits with 1/ poly(n) error is hard for classical learners unless RP = NP, ruling out the possibility of strong learners under standard complexity theoretic assumptions. As the classical analogue and subset of Clifford circuits, this naturally leads to a hardness result for Clifford circuits as well. Additionally, we show that if RP = NP then there would exist efficient proper learning algorithms for CNOT and Clifford circuits. By similar arguments, we also find that an efficient proper quantum learner for such circuits exists if and only if NP ⊆ RQP. We leave open the problem of hardness for improper learning or O(1) error to future work.

Hypothesizing an algorithm from one example: the role of specificity.

Statistical machine learning usually achieves high-accuracy models by employing tens of thousands of examples. By contrast, both children and adult humans typically learn new concepts from either one or a small number of instances. The high data efficiency of human learning is not easily explained in terms of standard formal frameworks for machine learning, including Gold's learning-in-the-limit framework and Valiant's probably approximately correct (PAC) model. This paper explores ways in which this apparent disparity between human and machine learning can be reconciled by considering algorithms involving a preference for specificity combined with program minimality. It is shown how this can be efficiently enacted using hierarchical search based on identification of certificates and push-down automata to support hypothesizing compactly expressed maximal efficiency algorithms. Early results of a new system called DeepLog indicate that such approaches can support efficient top-down construction of relatively complex logic programs from a single example. This article is part of a discussion meeting issue 'Cognitive artificial intelligence'.