Given a Boolean relational specification between inputs and outputs, the problem of functional synthesis is to construct a function that maps each assignment of the input to an assignment of the output such that each tuple of input and output assignments meets the specification. The past decade has witnessed significant improvement in the scalability of functional synthesis tools, allowing them to handle problems with tens of thousands of variables. A common ingredient in these approaches is their reliance on SAT solvers, thereby exploiting the breakthrough advances in SAT solving over the past three decades. While the recent techniques have been shown to perform well in practice, there is little theoretical understanding of the limitations and power of these approaches. The primary contribution of this work is to initiate a systematic theoretical investigation into the power of functional synthesis approaches that rely on NP oracles. We first show that even when small Skolem functions exist, naive bit-by-bit learning approaches fail due to the relational nature of specifications. We establish fundamental limitations of interpolation-based approaches proving that even when small Skolem functions exist, resolution-based interpolation must produce exponential-size circuits. We prove that access to an NP oracle is inherently necessary for efficient synthesis. Our main technical result shows that it is possible to use NP oracles to synthesize small Skolem functions in time polynomial in the size of the specification and the size of the smallest sufficient set of witnesses, establishing positive results for a broad class of relational specifications.

Hand-crafting models of interactive domains is challenging, especially when the dynamics of the domain are stochastic. We thus wish to automatically learn such models instead. In this work, we propose an algorithm to learn stochastic planning models where the distribution over the sets of effects for each action has a small support, but the sets may set values to an arbitrary number of attributes. This class captures many benchmark domains, in contrast to prior work that assumed independence of the effects on individual attributes. Our algorithm has polynomial time and sample complexity when the support size is bounded by a constant. Importantly, our method is safe in that we learn offline from example trajectories and we guarantee that actions are only permitted in states where our model of the dynamics is accurate. Moreover, we guarantee approximate completeness of the model, in the sense that if the examples are achieving goals from some distribution, then with high probability there will exist plans in our learned model that achieve goals from the same distribution

Background: Multi-Agent Planning (MAP) involves coordinating the actions of multiple autonomous agents to achieve shared objectives. A prevalent formalism for MAP is the Multi-Agent Planning Domain Definition Language (MA-PDDL). While effective, existing MA-PDDL solvers typically require complete access to agents’ action models—specifically their preconditions and effects. However, manually creating these models is often intractable, requiring exhaustive domain expertise. Objectives: This work explores an alternative approach: automatically learning agents’ action models from observed transitions. Since learned models may be inaccurate, planning with them can yield invalid or non-executable sequences. To mitigate this, we formalize a requirement for safety, ensuring that plans generated via the learned model remain sound with respect to the real unknown action model. Methods: Previous research introduced the Safe Action Model Learning (SAM) algorithm for single-agent domains. However, SAM is not suitable for MA-PDDL environments where observations include concurrently executed actions, since it cannot naturally disambiguate the individual contributions of the agents to the observed effects. To address this, we introduce Multi-Agent Safe Action Model Learning (MA-SAM), a safe action model learning algorithm designed to handle concurrent multi-agent observations. For scenarios where individual action effects remain ambiguous, we further propose MA-SAM+ , which learns the preconditions and effects of macro-actions representing concurrent execution of subsets of actions. We evaluate both algorithms on domains from the Competition of Distributed and Multi-Agent Planners (CoDMAP) benchmarks and a novel MAP domain inspired by the game Overcooked. Results: We establish a theoretical lower bound on the sample complexity for learning safe action models in multi-agent settings. We prove that MA-SAM does not achieve this lower bound in all cases, identifying specific conditions under which its sample complexity may become unbounded. Empirically, both MA-SAM and MA-SAM+ significantly outperform SAM-based baselines in coverage and applicability rates. While their performance is comparable in many settings, MA-SAM+ highly outperforms MA-SAM in some of the evaluated domains. Conclusions: We present the first algorithms capable of learning safe MA-PDDL action models from concurrently executed actions, providing both theoretical foundations and empirical validation across diverse planning benchmarks

We study "selective" or "conditional" classification problems under an agnostic setting. Classification tasks commonly focus on modeling the relationship between features and categories that captures the vast majority of data. In contrast to common machine learning frameworks, conditional classification intends to model such relationships only on a subset of the data defined by some selection rule. Most work on conditional classification either solves the problem in a realizable setting or does not guarantee the error is bounded compared to an optimal solution. In this work, we consider selective/conditional classification by sparse linear classifiers for subsets defined by halfspaces, and give both positive as well as negative results for Gaussian feature distributions. On the positive side, we present the first PAC-learning algorithm for homogeneous halfspace selectors with error guarantee $\tilde{O}(\sqrt{\mathrm{opt}})$, where $\mathrm{opt}$ is the smallest conditional classification error over the given class of classifiers and homogeneous halfspaces. On the negative side, we find that, under cryptographic assumptions, approximating the conditional classification loss within a small additive error is computationally hard even under Gaussian distribution. We prove that approximating conditional classification is at least as hard as approximating agnostic classification in both additive and multiplicative form.

Reasoning under uncertainty is a fundamental challenge in Artificial Intelligence. As with most of these challenges, there is a harsh dilemma between the expressive power of the language used, and the tractability of the computational problem posed by reasoning. Inspired by human reasoning, we introduce a method of first-order relational probabilistic inference that satisfies both criteria, and can handle hybrid (discrete and continuous) variables. Specifically, we extend sum-of-squares logic of expectation to relational settings, demonstrating that lifted reasoning in the bounded-degree fragment for knowledge bases of bounded quantifier rank can be performed in polynomial time, even with an a priori unknown and/or countably infinite set of objects. Crucially, our notion of tractability is framed in proof-theoretic terms, which extends beyond the syntactic properties of the language or queries. We are able to derive the tightest bounds provable by proofs of a given degree and size and establish completeness in our sum-of-squares refutations for fixed degrees.

A common approach for solving planning problems is to model them in a formal language such as the Planning Domain Definition Language (PDDL), and then use an appropriate PDDL planner. Several algorithms for learning PDDL models from observations have been proposed but plans created with these learned models may not be sound. We propose two algorithms for learning PDDL models that are guaranteed to be safe to use even when given observations that include partially observable states. We analyze these algorithms theoretically, characterizing the sample complexity each algorithm requires to guarantee probabilistic completeness. We also show experimentally that our algorithms are often better than FAMA, a state-of-the-art PDDL learning algorithm.

One approach to probabilistic inference involves counting the number of models of a given Boolean formula. Here, we are interested in inferences involving higher-order objects, i.e., functions. We study the following task: Given a Boolean specification between a set of inputs and outputs, count the number of functions of inputs such that the specification is met. Such functions are called Skolem functions. We are motivated by the recent development of scalable approaches to Boolean function synthesis. This stands in relation to our problem analogously to the relationship between Boolean satisfiability and the model counting problem. Yet, counting Skolem functions poses considerable new challenges. From the complexity-theoretic standpoint, counting Skolem functions is not only #P-hard; it is quite unlikely to have an FPRAS (Fully Polynomial Randomized Approximation Scheme) as the problem of synthesizing a Skolem function remains challenging, even given access to an NP oracle. The primary contribution of this work is the first algorithm, SkolemFC, that computes the number of Skolem functions. SkolemFC relies on technical connections between counting functions and propositional model counting: our algorithm makes a linear number of calls to an approximate model counter and computes an estimate of the number of Skolem functions with theoretical guarantees. Our prototype displays impressive scalability, handling benchmarks comparably to state-of-the-art Skolem function synthesis engines, even though counting all such functions ostensibly poses a greater challenge than synthesizing a single function.

Powerful domain-independent planners have been developed to solve various types of planning problems. These planners often require a model of the acting agent's actions, given in some planning domain description language. Manually designing such an action model is a notoriously challenging task. An alternative is to automatically learn action models from observation. Such an action model is called safe if every plan created with it is consistent with the real, unknown action model. Algorithms for learning such safe action models exist, yet they cannot handle domains with conditional or universal effects, which are common constructs in many planning problems. We prove that learning non-trivial safe action models with conditional effects may require an exponential number of samples. Then, we identify reasonable assumptions under which such learning is tractable and propose Conditional-SAM, the first algorithm capable of doing so. We analyze Conditional-SAM theoretically and evaluate it experimentally. Our results show that the action models learned by Conditional-SAM can be used to solve perfectly most of the test set problems in most of the experimented domains.

Group-fair learning methods typically seek to ensure that some measure of prediction efficacy for (often historically) disadvantaged minority groups is comparable to that for the majority of the population. When a principal seeks to adopt a group-fair approach to replace another, the principal may face opposition from those who feel they may be harmed by the switch, and this, in turn, may deter adoption. We propose that a potential mitigation to this concern is to ensure that a group-fair model is also popular, in the sense that, for a majority of the target population, it yields a preferred distribution over outcomes compared with the conventional model. In this paper, we show that state of the art fair learning approaches are often unpopular in this sense. We propose several efficient algorithms for postprocessing an existing group-fair learning scheme to improve its popularity while retaining fairness. Through extensive experiments, we demonstrate that the proposed postprocessing approaches are highly effective in practice.

Powerful domain-independent planners have been developed to solve various types of planning problems. These planners often require a model of the acting agent's actions, given in some planning domain description language. Yet obtaining such an action model is a notoriously hard task. This task is even more challenging in mission-critical domains, where a trial-and-error approach to learning how to act is not an option. In such domains, the action model used to generate plans must be safe, in the sense that plans generated with it must be applicable and achieve their goals. Learning safe action models for planning has been recently explored for domains in which states are sufficiently described with Boolean variables. In this work, we go beyond this limitation and propose the NSAM algorithm. NSAM runs in time that is polynomial in the number of observations and, under certain conditions, is guaranteed to return safe action models. We analyze its worst-case sample complexity, which may be intractable for some domains. Empirically, however, NSAM can quickly learn a safe action model that can solve most problems in the domain.

Abstract The tension between deduction and induction is perhaps the most fundamental issue in areas such as philosophy, cognition, and artificial intelligence. In an influential paper,Valiantrecognized that the challenge of learning should be integrated with deduction. In particular, he proposed a semantics to capture the quality possessed by the output ofprobably 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-calledRepresentation Theoremexplored for single-agent and multi-agent epistemic logics with theonly knowingoperator to reduce modal reasoning to propositional reasoning.

We consider the problem of learning action models for planning in unknown stochastic environments that can be defined using the Probabilistic Planning Domain Description Language (PPDDL). As input, we are given a set of previously executed trajectories, and the main challenge is to learn an action model that has a similar goal achievement probability to the policies used to create these trajectories. To this end, we introduce a variant of PPDDL in which there is uncertainty about the transition probabilities, specified by an interval for each factor that contains the respective true transition probabilities. Then, we present SAM+, an algorithm that learns such an imprecise-PPDDL environment model. SAM+ has a polynomial time and sample complexity, and guarantees that with high probability, the true environment is indeed captured by the defined intervals. We prove that the action model SAM+ outputs has a goal achievement probability that is almost as good or better than that of the policies used to produced the training trajectories. Then, we show how to produce a PPDDL model based on this imprecise-PPDDL model that has similar properties.

Many reinforcement learning (RL) environments in practice feature enormous state spaces that may be described compactly by a "factored" structure, that may be modeled by Factored Markov Decision Processes (FMDPs). We present the first polynomial time algorithm for RL in Factored State MDPs (generalizing FMDPs) that neither relies on an oracle planner nor requires a linear transition model; it only requires a linear value function with a suitable local basis with respect to the factorization, permitting efficient variable elimination. With this assumption, we can solve this family of Factored State MDPs in polynomial time by constructing an efficient separation oracle for convex optimization. Importantly, and in contrast to prior work on FMDPs, we do not assume that the transitions on various factors are conditionally independent.

Often machine learning and statistical models will attempt to describe the majority of the data. However, there may be situations where only a fraction of the data can be fit well by a linear regression model. Here, we are interested in a case where such inliers can be identified by a Disjunctive Normal Form (DNF) formula. We give a polynomial time algorithm for the conditional linear regression task, which identifies a DNF condition together with the linear predictor on the corresponding portion of the data. In this work, we improve on previous algorithms by removing a requirement that the covariances of the data satisfying each of the terms of the condition have to all be very similar in spectral norm to the covariance of the overall condition.

Abstract Quantified information flow (QIF) has emerged as a rigorous approach to quantitatively measure confidentiality; the information-theoretic underpinning of QIF allows the end-users to link the computed quantities with the computational effort required on the part of the adversary to gain access to desired confidential information. In this work, we focus on the estimation of Shannon entropy for a given program $$\varPi $$ Π . As a first step, we focus on the case wherein a Boolean formula $$\varphi (X,Y)$$ φ ( X , Y ) captures the relationship between inputs X and output Y of $$\varPi $$ Π . Such formulas $$\varphi (X,Y)$$ φ ( X , Y ) have the property that for every valuation to X, there exists exactly one valuation to Y such that $$\varphi $$ φ is satisfied. The existing techniques require $$\mathcal {O}(2^m)$$ O ( 2 m ) model counting queries, where $$m = |Y|$$ m = | Y | . We propose the first efficient algorithmic technique, called $$\mathsf {EntropyEstimation}$$ EntropyEstimation to estimate the Shannon entropy of $$\varphi $$ φ with PAC-style guarantees, i.e., the computed estimate is guaranteed to lie within a $$(1\pm \varepsilon )$$ ( 1 ± ε ) -factor of the ground truth with confidence at least $$1-\delta $$ 1 - δ . Furthermore, $$\mathsf {EntropyEstimation}$$ EntropyEstimation makes only $$\mathcal {O}(\frac{min(m,n)}{\varepsilon ^2})$$ O ( m i n ( m , n ) ε 2 ) counting and sampling queries, where $$m = |Y|$$ m = | Y | , and $$n = |X|$$ n = | X | , thereby achieving a significant reduction in the number of model counting queries. We demonstrate the practical efficiency of our algorithmic framework via a detailed experimental evaluation. Our evaluation demonstrates that the proposed framework scales to the formulas beyond the reach of the previously known approaches.

Determinantal Point Processes (DPPs) are a widely used probabilistic model for negatively correlated sets. DPPs are used in Machine Learning applications to select a diverse, yet representative subset of data. In these applications, the parameters of the DPP need to be fit to match the data; typically, we seek a set of parameters that maximize the likelihood of the data. The algorithms used for this task either optimize over a limited family of DPPs, or else use local improvement heuristics that do not provide theoretical guarantees of optimality. It is natural to ask if there exist efficient algorithms for finding a maximum likelihood DPP model for a given data set. In seminal work on DPPs in Machine Learning, Kulesza conjectured in his PhD Thesis (2012) that the problem is NP-complete. In this work we prove Kulesza’s conjecture: we prove moreover, that even computing a $1-\frac{1}{\mathrm{poly} \log N}$-approximation to the maximum log-likelihood of a DPP on a set of $N$ items is NP-complete. At the same time, we also obtain the first polynomial-time algorithm obtaining a nontrivial worst-case approximation to the optimal likelihood: we present a polynomial-time $1/\log m$-approximation algorithm (for data sets of size $m$), which moreover obtains a $1-\frac{1}{\log N}$-approximation if all $N$ elements appear in a $O(1/N)$-fraction of the subsets. In terms of techniques, the hardness result reduces to solving a gap instance of a “vector coloring" problem on a hypergraph obtained from an adaptation of the constructions of Bogdanov, Obata and Trevisan (FOCS 2002), using the strong expanders of Alon and Capalbo (FOCS 2007).

We introduce and study the model of list learning with attribute noise. Learning with attribute noise was introduced by Shackelford and Volper (COLT, 1988) as a variant of PAC learning, in which the algorithm has access to noisy examples and uncorrupted labels, and the goal is to recover an accurate hypothesis. Sloan (COLT, 1988) and Goldman and Sloan (Algorithmica, 1995) discovered information-theoretic limits to learning in this model, which have impeded further progress. In this article we extend the model to that of list learning, drawing inspiration from the list-decoding model in coding theory, and its recent variant studied in the context of learning. On the positive side, we show that sparse conjunctions can be efficiently list learned under some assumptions on the underlying ground-truth distribution. On the negative side, our results show that even in the list-learning model, efficient learning of parities and majorities is not possible regardless of the representation used.

Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs are strictly more expressive efficient than many existing tractable probabilistic models, including determinantal point processes (DPPs), probabilistic circuits (PCs) such as sum-product networks, and tractable graphical models. We contend that PGCs are not just a theoretical framework that unifies vastly different existing models, but also show great potential in modeling realistic data. We exhibit a simple class of PGCs that are not trivially subsumed by simple combinations of PCs and DPPs, and obtain competitive performance on a suite of density estimation benchmarks. We also highlight PGCs’ connection to the theory of strongly Rayleigh distributions.

Robust learning in expressive languages with real-world data continues to be a challenging task. Numerous conventional methods appeal to heuristics without any assurances of robustness. While probably approximately correct (PAC) Semantics offers strong guarantees, learning explicit representations is not tractable, even in propositional logic. However, recent work on so-called “implicit" learning has shown tremendous promise in terms of obtaining polynomial-time results for fragments of first-order logic. In this work, we extend implicit learning in PAC-Semantics to handle noisy data in the form of intervals and threshold uncertainty in the language of linear arithmetic. We prove that our extended framework keeps the existing polynomial-time complexity guarantees. Furthermore, we provide the first empirical investigation of this hitherto purely theoretical framework. Using benchmark problems, we show that our implicit approach to learning optimal linear programming objective constraints significantly outperforms an explicit approach in practice.

Creating a domain model, even for classical, domain-independent planning, is a notoriously hard knowledge-engineering task. A natural approach to solve this problem is to learn a domain model from observations. However, model learning approaches frequently do not provide safety guarantees: the learned model may assume actions are applicable when they are not, and may incorrectly capture actions' effects. This may result in generating plans that will fail when executed. In some domains such failures are not acceptable, due to the cost of failure or inability to replan online after failure. In such settings, all learning must be done offline, based on some observations collected, e.g., by some other agents or a human. Through this learning, the task is to generate a plan that is guaranteed to be successful. This is called the model-free planning problem. Prior work proposed an algorithm for solving the model-free planning problem in classical planning. However, they were limited to learning grounded domains, and thus they could not scale. We generalize this prior work and propose the first safe model-free planning algorithm for lifted domains. We prove the correctness of our approach, and provide a statistical analysis showing that the number of trajectories needed to solve future problems with high probability is linear in the potential size of the domain model. We also present experiments on twelve IPC domains showing that our approach is able to learn the real action model in all cases with at most two trajectories.

In many real-world scenarios, the time it takes for a mobile agent, e.g., a robot, to move from one location to another may vary due to exogenous events and be difficult to predict accurately. Planning in such scenarios is challenging, especially in the context of Multi-Agent Pathfinding (MAPF), where the goal is to find paths to multiple agents and temporal coordination is necessary to avoid collisions. In this work, we consider a MAPF problem with this form of time uncertainty, where we are only given upper and lower bounds on the time it takes each agent to move. The objective is to find a safe solution, which is a solution that can be executed by all agents and is guaranteed to avoid collisions. We propose two complete and optimal algorithms for finding safe solutions based on well-known MAPF algorithms, namely, A* with Operator Decomposition (A* + OD) and Conflict-Based Search (CBS). Experimentally, we observe that on several standard MAPF grids the CBS-based algorithm performs better. We also explore the option of online replanning in this context, i.e., modifying the agents' plans during execution, to reduce the overall execution cost. We consider two online settings: (a) when an agent can sense the current time and its current location, and (b) when the agents can also communicate seamlessly during execution. For each setting, we propose a replanning algorithm and analyze its behavior theoretically and empirically. Our experimental evaluation confirms that indeed online replanning in both settings can significantly reduce solution cost.

Work in machine learning and statistics commonly focuses on building models that capture the vast majority of data, possibly ignoring a segment of the population as outliers. However, there may not exist a good, simple model for the distribution, so we seek to find a small subset where there exists such a model. We give a computationally efficient algorithm with theoretical analysis for the conditional linear regression task, which is the joint task of identifying a significant portion of the data distribution, described by a k-DNF, along with a linear predictor on that portion with a small loss. In contrast to work in robust statistics on small subsets, our loss bounds do not feature a dependence on the density of the portion we fit, and compared to previous work on conditional linear regression, our algorithm’s running time scales polynomially with the sparsity of the linear predictor. We also demonstrate empirically that our algorithm can leverage this advantage to obtain a k-DNF with a better linear predictor in practice.

To deploy knowledge-based systems in the real world, the challenge of knowledge acquisition must be addressed. Knowledge engineering by hand is a daunting task, so machine learning has been widely proposed as an alternative. However, machine learning has difficulty acquiring rules that feature the kind of exceptions that are prevalent in real-world knowledge. Moreover, it is conjectured to be impossible to reliably learn representations featuring a desirable level of expressiveness. Works by Khardon and Roth and by Juba proposed solutions to such problems by learning to reason directly, bypassing the intractable step of producing an explicit representation of the learned knowledge. These works focused on Boolean, propositional logics. In this work, we consider such implicit learning to reason for arithmetic theories, including logics considered with satisfiability modulo theory (SMT) solvers. We show that for standard fragments of linear arithmetic, we can learn to reason efficiently. These results are consequences of a more general finding: we show that there is an efficient reduction from the learning to reason problem for a logic to any sound and complete solver for that logic.

We introduce a tensor factorization approach to unsupervised feature learning of hyper-spectral imagery, and demonstrate its effectiveness on land type classification of publicly available datasets. The results show that this approach can produce state of the art accuracy, compared to other methods for feature learning in the classification task.

Practitioners of data mining and machine learning have long observed that the imbalance of classes in a data set negatively impacts the quality of classifiers trained on that data. Numerous techniques for coping with such imbalances have been proposed, but nearly all lack any theoretical grounding. By contrast, the standard theoretical analysis of machine learning admits no dependence on the imbalance of classes at all. The basic theorems of statistical learning establish the number of examples needed to estimate the accuracy of a classifier as a function of its complexity (VC-dimension) and the confidence desired; the class imbalance does not enter these formulas anywhere. In this work, we consider the measures of classifier performance in terms of precision and recall, a measure that is widely suggested as more appropriate to the classification of imbalanced data. We observe that whenever the precision is moderately large, the worse of the precision and recall is within a small constant factor of the accuracy weighted by the class imbalance. A corollary of this observation is that a larger number of examples is necessary and sufficient to address class imbalance, a finding we also illustrate empirically.

Model-based diagnosis (MBD) is difficult to use in practice because it requires a model of the diagnosed system, which is often very hard to obtain. We explore theoretically how observing the system when it is in a normal state can provide information about the system that is sufficient to learn a partial system model that allows automated diagnosis. We analyze the number of observations needed to learn a model capable of finding faulty components in most cases. Then, we explore how knowing the system topology can help us to learn a useful model from the normal observations for settings in which many of the internal system variables cannot be observed. Unlike other data-driven methods, our learned model is safe, in the sense that subsystems identified as faulty are guaranteed to truly be faulty.

We consider the following conditional linear regression problem: the task is to identify both (i) a $k$-DNF condition $c$ and (ii) a linear rule $f$ such that the probability of $c$ is (approximately) at least some given bound $\mu$, and minimizing the $l_p$ loss of $f$ at predicting the target $z$ in the distribution conditioned on $c$. Thus, the task is to identify a portion of the distribution on which a linear rule can provide a good fit. Algorithms for this task are useful in cases where portions of the distribution are not modeled well by simple, learnable rules, but on other portions such rules perform well. The prior state-of-the-art for such algorithms could only guarantee finding a condition of probability $O(\mu/n^k )$ when a condition of probability $\mu$ exists, and achieved a $O(n^k)$-approximation to the target loss. Here, we give efficient algorithms for solving this task with a condition $c$ that nearly matches the probability of the ideal condition, while also improving the approximation to the target loss to a $O (n^{k/2})$ factor. We also give an algorithm for finding a k-DNF reference class for prediction at a given query point, that obtains a sparse regression fit that has loss within $O(n^k)$ of optimal among all sparse regression parameters and sufficiently large $k$-DNF reference classes containing the query point.

We consider several closely related variants of PAC-learning in which false-positive and false-negative errors are treated differently. In these models we seek to guarantee a given, low rate of false-positive errors and asfew false-negative errors as possible given that we meet the false-positived constraint. Bshouty and Burroughs first observed that learning conjunctions in such models would enable PAC-learning of DNF in the usual distribution-free model; in turn, results of Daniely and Shalev-Shwartz establish that learning of DNF would imply algorithms for refuting random k-SAT using far fewer constraints than believed possible. Such algorithms would violate a slight strengthening of Feige’s R3SAT assumption, and would violate the RCSP hypothesis of Barak et al. We show here that actually, an algorithm for learning conjunctions in this model would have much more far-reaching consequences: it gives refutation algorithms for all predicates that are falsified by one of the uniform constant strings. To our knowledge, this is the first hardness result of improper learning for such a large class of natural average-case problems with natural distributions.

We consider the problem of answering queries about formulas of first-order logic based on background knowledge partially represented explicitly as other formulas, and partially represented as examples independently drawn from a fixed probability distribution. PAC semantics, introduced by Valiant, is one rigorous, general proposal for learning to reason in formal languages: although weaker than classical entailment, it allows for a powerful model theoretic framework for answering queries while requiring minimal assumptions about the form of the distribution in question. To date, however, the most significant limitation of that approach, and more generally most machine learning approaches with robustness guarantees, is that the logical language is ultimately essentially propositional, with finitely many atoms. Indeed, the theoretical findings on the learning of relational theories in such generality have been resoundingly negative. This is despite the fact that first-order logic is widely argued to be most appropriate for representing human knowledge. In this work, we present a new theoretical approach to robustly learning to reason in first-order logic, and consider universally quantified clauses over a countably infinite domain. Our results exploit symmetries exhibited by constants in the language, and generalize the notion of implicit learnability to show how queries can be computed against (implicitly) learned first-order background knowledge.

We consider the problem of one-way communication when the recipient does not know exactly the distribution that the messages are drawn from, but has a “prior” distribution that is known to be close to the source distribution, a problem first considered by Juba et al. We consider the question of how much longer the messages need to be in order to cope with the uncertainty about the receiver's prior and the source distribution, respectively, as compared with the standard source coding problem. We consider two variants of this uncertain priors problem: the original setting of Juba et al. in which the receiver is required to correctly recover the message with probability 1, and a setting introduced by Haramaty and Sudan, in which the receiver is permitted to fail with some probability ∈. In both settings, we obtain lower bounds that are tight up to logarithmically smaller terms. In the latter setting, we furthermore present a variant of the coding scheme of Juba et al. with an overhead of log α + log 1/∈ + 1 bits, thus also establishing the nearly tight upper bound.

Our work extends Juba’s formulation of learning abductive reasoning from examples, in which both the relative plausibility of various explanations, as well as which explanations are valid, are learned directly from data. We extend the formulation to consider partially observed examples, along with declarative background knowledge about the missing data. We show that it is possible to use implicitly learned rules together with the explicitly given declarative knowledge to support hypotheses in the course of abduction. We observe that when a small explanation exists, it is possible to obtain a much-improved guarantee in the challenging exception-tolerant setting.

Anomaly detection is the well-studied task of identifying when data is atypical in some way with respect to its source. In this work, by contrast, we are interested in finding possible descriptions of what may be causing anomalies. We propose a new task, attaching semantics drawn from metadata to a portion of the anomalous examples from some data source. Such a partial description of the anomalous data in terms of the meta-data is useful both because it may help to explain what causes the identified anomalies, and also because it may help to identify the truly unusual examples that defy such simple categorization. This is especially significant when the data set is too large for a human analyst to inspect the anomalies manually. The challenge is that anomalies are, by definition, relatively rare, and so we are seeking to learn a precise characterization of a rare event. We examine algorithms for this task in a webcam domain, generating human-understandable explanations for a pixellevel characterization of anomalies. We find that using a recently proposed algorithm that prioritizes precision over recall, it is possible to attach good descriptions to a moderate fraction of the anomalies in webcam data so long as the data set is fairly large.

Inference from an observed or hypothesized condition to a plausible cause or explanation for this condition is known as abduction. For many tasks, the acquisition of the necessary knowledge by machine learning has been widely found to be highly effective. However, the semantics of learned knowledge are weaker than the usual classical semantics, and this necessitates new formulations of many tasks. We focus on a recently introduced formulation of the abductive inference task that is thus adapted to the semantics of machine learning. A key problem is that we cannot expect that our causes or explanations will be perfect, and they must tolerate some error due to the world being more complicated than our formalization allows. This is a version of the qualification problem, and in machine learning, this is known as agnostic learning. In the work by Juba that introduced the task of learning to make abductive inferences, an algorithm is given for producing k-DNF explanations that tolerates such exceptions: if the best possible k-DNF explanation fails to justify the condition with probability ε, then the algorithm is promised to find a k-DNF explanation that fails to justify the condition with probability at most O(nkε), where n is the number of propositional attributes used to describe the domain. Here, we present an improved algorithm for this task. When the best k- DNF fails with probability ε, our algorithm finds a k-DNF that fails with probability at most O ̃(nk/2ε) (i.e., suppressing logarithmic factors in n and 1/ε). We also examine the empirical advantage of this new algorithm over the previous algorithm in two test domains, one of explaining conditions generated by a “noisy” k-DNF rule, and another of explaining conditions that are actually generated by a linear threshold rule.

In this paper, we introduce a multi-agent multi-armed bandit-based model for ad hoc teamwork with expensive communication. The goal of the team is to maximize the total reward gained from pulling arms of a bandit over a number of epochs. In each epoch, each agent decides whether to pull an arm, or to broadcast the reward it obtained in the previous epoch to the team and forgo pulling an arm. These decisions must be made only on the basis of the agent’s private information and the public information broadcast prior to that epoch. We first benchmark the achievable utility by analyzing an idealized version of this problem where a central authority has complete knowledge of rewards acquired from all arms in all epochs and uses a multiplicative weights update algorithm for allocating arms to agents. We then introduce an algorithm for the decentralized setting that uses a value-of-information based communication strategy and an exploration-exploitation strategy based on the centralized algorithm, and show experimentally that it converges rapidly to the performance of the centralized method.

In this paper we explore the theoretical boundaries of planning in a setting where no model of the agent's actions is given. Instead of an action model, a set of successfully executed plans are given and the task is to generate a plan that is safe, i.e., guaranteed to achieve the goal without failing. To this end, we show how to learn a conservative model of the world in which actions are guaranteed to be applicable. This conservative model is then given to an off-the-shelf classical planner, resulting in a plan that is guaranteed to achieve the goal. However, this reduction from a model-free planning to a model-based planning is not complete: in some cases a plan will not be found even when such exists. We analyze the relation between the number of observed plans and the likelihood that our conservative approach will indeed fail to solve a solvable problem. Our analysis show that the number of trajectories needed scales gracefully.

Machine learning and statistics typically focus on building models that capture the vast majority of the data, possibly ignoring a small subset of data as "noise" or "outliers." By contrast, here we consider the problem of jointly identifying a significant (but perhaps small) segment of a population in which there is a highly sparse linear regression fit, together with the coefficients for the linear fit. We contend that such tasks are of interest both because the models themselves may be able to achieve better predictions in such special cases, but also because they may aid our understanding of the data. We give algorithms for such problems under the sup norm, when this unknown segment of the population is described by a k-DNF condition and the regression fit is s-sparse for constant k and s. For the variants of this problem when the regression fit is not so sparse or using expected error, we also give a preliminary algorithm and highlight the question as a challenge for future work.

We consider a new formulation of abduction in which degrees of "plausibility" of explanations, along with the rules of the domain, are learned from concrete examples (settings of attributes). Our version of abduction thus falls in the "learning to reason" framework of Khardon and Roth. Such approaches enable us to capture a natural notion of "plausibility" in a domain while avoiding the extremely difficult problem of specifying an explicit representation of what is "plausible." We specifically consider the question of which syntactic classes of formulas have efficient algorithms for abduction. We find that the class of k-DNF explanations can be found in polynomial time for any fixed k; but, we also find evidence that even weak versions of our abduction task are intractable for the usual class of conjunctions. This evidence is provided by a connection to the usual, inductive PAC-learning model proposed by Valiant. We also consider an exception-tolerant variant of abduction. We observe that it is possible for polynomial-time algorithms to tolerate a few adversarially chosen exceptions, again for the class of k-DNF explanations. All of the algorithms we study are particularly simple, and indeed are variants of a rule proposed by Mill.

AC^0 o MOD_2 circuits are AC^0 circuits augmented with a layer of parity gates just above the input layer. We study AC^0 o MOD2 circuit lower bounds for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have highlighted this problem as a frontier problem in circuit complexity that arose both as a first step towards solving natural special cases of the matrix rigidity problem and as a candidate for constructing pseudorandom generators of minimal complexity. We give the first superlinear lower bound for the Boolean Inner Product function against AC^0 o MOD2 of depth four or greater. Specifically, we prove a superlinear lower bound for circuits of arbitrary constant depth, and an ~Omega(n^2) lower bound for the special case of depth-4 AC^0 o MOD_2. Our proof of the depth-4 lower bound employs a new "moment-matching" inequality for bounded, nonnegative integer-valued random variables that may be of independent interest: we prove an optimal bound on the maximum difference between two discrete distributions’ values at 0, given that their first d moments match.

We formulate a new variant of the problem of planning in an unknown environment, for which we can provide algorithms with reasonable theoretical guarantees in spite of large state spaces and time horizons, partial observability, and complex dynamics. In this variant, an agent is given a collection of example traces produced by a reference policy, which may, for example, capture the agent's past behavior. The agent is (only) asked to find policies that are supported by regularities in the dynamics that are observable on these example traces. We describe an efficient algorithm that uses such common sense knowledge reflected in the example traces to construct decision tree policies for goal-oriented factored POMDPs. More precisely, our algorithm (provably) succeeds at finding a policy for a given input goal when (1) there is a CNF that is almost always observed satisfied on the traces of the POMDP, capturing a sufficient approximation of its dynamics and (2) for a decision tree policy of bounded complexity, there exist small- space resolution proofs that the goal is achieved on each branch using the aforementioned CNF capturing the common sense rules. Such a CNF always exists for noisy STRIPS domains, for example. Our results thus essentially establish that the possession of a suitable exploration policy for collecting the necessary examples is the fundamental obstacle to learning to act in such environments.

We consider the proof search (--automatizability--) problem for propositional proof systems in the context of knowledge discovery (or data mining and analytics). Discovered knowledge necessarily features a weaker semantics than usually employed in mathematical logic, and in this work we find that these weaker semantics may result in a proof search problem that seems easier than the classical problem, but that is nevertheless nontrivial. Specifically, if we consider a knowledge discovery task corresponding to the unsupervised learning of parities over the uniform distribution from partial information, then we find the following:

We consider a model of teaching in which the learners are consistent and have bounded state, but are otherwise arbitrary. The teacher is non-interactive and "massively open": the teacher broadcasts a sequence of examples of an arbitrary target concept, intended for every possible on-line learning algorithm to learn from. We focus on the problem of designing interesting teachers: efficient sequences of examples that lead all capable and consistent learners to learn concepts, regardless of the underlying algorithm used by the learner. We use two measures of teaching efficiency: the number of mistakes made by the worst-case learner, and the maximum length of the example sequence needed for the worst-case learner. Our results are summarized as follows: Given a uniform random sequence of examples of an n-bit concept function, learners (capable of consistently learning the concept) with s(n) bits of state are guaranteed to make only O(n ⋅ s(n)) mistakes and exactly learn the concept, with high probability. This theorem has interesting corollaries; for instance, every concept c has a sequence of examples can teach c to all capable consistent on-line learners implementable with s(n)-size circuits, such that every learner makes only ~O(s(n)^2) mistakes. That is, all resource-bounded algorithms capable of consistently learning a concept can be simultaneously taught that concept with few mistakes, on a single example sequence. We also show how to efficiently generate such a sequence of examples on-line: using Nisan's pseudorandom generator, each example in the sequence can be generated with polynomial-time overhead per example, with an O(n ⋅ s(n))-bit initial seed. To justify our use of randomness, we prove that any non-trivial derandomization of our sequences would imply new circuit lower bounds. For instance, if there is a deterministic 2n O(1) time algorithm that generates a sequence of examples, such that all consistent and capable polynomial-size circuit learners learn the all-zeroes concept with less than 2n mistakes, then EXP ⊄ P. We present examples illustrating that the key differences in our model -- our focus on mistakes rather than the total number of examples, and our use of a state bound -- must be considered together to obtain our results. We show that for every consistent s(n)-state bounded learner A, and every n-bit concept that A is capable of learning, there is a custom "tutoring" sequence of only O(n ⋅ s(n)) examples that teaches A the concept. That is, in principle, there are no slow learners, only bad teachers: if a state-bounded learner is capable of learning a concept at all, then it can always be taught that concept quickly via some short sequence of examples.

We put forward a general theory of goal-oriented communication , where communication is not an end in itself, but rather a means to achieving some goals of the communicating parties. Focusing on goals provides a framework for addressing the problem of potential “misunderstanding” during communication, where the misunderstanding arises from lack of initial agreement on what protocol and/or language is being used in communication. In this context, “reliable communication” means overcoming any initial misunderstanding between parties towards achieving a given goal. Despite the enormous diversity among the goals of communication, we propose a simple model that captures all goals. In the simplest form of communication we consider, two parties, a user and a server , attempt to communicate with each other in order to achieve some goal of the user. We show that any goal of communication can be modeled mathematically by introducing a third party, which we call the referee , who hypothetically monitors the conversation between the user and the server and determines whether or not the goal has been achieved. Potential misunderstanding between the players is captured by allowing each player (the user/server) to come from a (potentially infinite) class of players such that each player is unaware which instantiation of the other it is talking to. We identify a main concept, which we call sensing , that allows goals to be achieved even under misunderstanding. Informally, sensing captures the user's ability (potentially using help from the server) to simulate the referee's assessment on whether the communication is achieving the goal. We show that when the user can sense progress, the goal of communication can be achieved despite initial misunderstanding. We also show that in certain settings sensing is necessary for overcoming such initial misunderstanding. Our results significantly extend the scope of the investigation started by Juba and Sudan (STOC 2008) who studied the foregoing phenomenon in the case of a single specific goal. Our study shows that their main suggestion, that misunderstanding can be detected and possibly corrected by focusing on the goal, can be proved in full generality.

We put forward a general theory of goal-oriented communication, where communication is not an end in itself, but rather a means to achieving some goals of the communicating parties. Focusing on goals provides a framework for addressing the problem of potential "misunderstanding" during communication, where the misunderstanding arises from lack of initial agreement on what protocol and/or language is being used in communication. Despite the enormous diversity among the goals of communication, we propose a simple model that captures all goals.

We present the first end-user protocol, guaranteeing the delivery of messages, that automatically adapts to any new packet format that is obtained by applying a short, efficient function to packets from an earlier protocol.

Is it possible for two intelligent beings to communicate meaningfully, without any common language or background? This question has interest on its own, but is especially relevant in the context of modern computational infrastructures where an increase in the diversity of computers is making the task of inter-computer interaction increasingly burdensome. Computers spend a substantial amount of time updating their software to increase their knowledge of other computing devices. In turn, for any pair of communicating devices, one has to design software that enables the two to talk to each other. Is it possible instead to let the two computing entities use their intelligence (universality as computers) to learn each others' behavior and attain a common understanding? What is 'common understanding?' We explore this question in this paper.

We show that it is possible to use data compression on independently obtained hypotheses from various tasks to algorithmically provide guarantees that the tasks are sufficiently related to benefit from multitask learning. We give uniform bounds in terms of the empirical average error for the true average error of the n hypotheses provided by deterministic learning algorithms drawing independent samples from a set of n unknown computable task distributions over finite sets.