逻辑与神经

Iterative algorithms solve problems by taking steps until a solution is reached. Models in the form of Deep Thinking (DT) networks have been demonstrated to learn iterative algorithms in a way that can scale to different sized problems at inference time using recurrent computation and convolutions. However, they are often unstable during training, and have no guarantees of convergence/termination at the solution. This paper addresses the problem of instability by analyzing the growth in intermediate representations, allowing us to build models (referred to as Deep Thinking with Lipschitz Constraints (DT-L)) with many fewer parameters and providing more reliable solutions. Additionally our DT-L formulation provides guarantees of convergence of the learned iterative procedure to a unique solution at inference time. We demonstrate DT-L is capable of robustly learning algorithms which extrapolate to harder problems than in the training set. We benchmark on the traveling salesperson problem to evaluate the capabilities of the modified system in an NP-hard problem where DT fails to learn.

Large language models (LLMs) like transformers demonstrate impressive in-context learning (ICL) capabilities, allowing them to make predictions for new tasks based on prompt exemplars without parameter updates. While existing ICL theories often assume structured training data resembling ICL tasks (e.g., x-y pairs for linear regression), LLMs are typically trained unsupervised on unstructured text, such as web content, which lacks clear parallels to tasks like word analogy. To address this gap, we examine what enables ICL in models trained on unstructured data, focusing on critical sequence model requirements and training data structure. We find that many ICL capabilities can emerge simply from co-occurrence of semantically related word pairs in unstructured data; word analogy completion, for example, can provably arise purely through co-occurrence modeling, using classical language models like continuous bag of words (CBOW), without needing positional information or attention mechanisms. However, positional information becomes crucial for logic reasoning tasks requiring generalization to unseen tokens. Finally, we identify two cases where ICL fails: one in logic reasoning tasks that require generalizing to new, unseen patterns, and another in analogy completion where relevant word pairs appear only in fixed training positions. These findings suggest that LLMs' ICL abilities depend heavily on the structural elements within their training data.

Disentangling the explanatory factors in complex data is a promising approach for generalizable and data-efficient representation learning. While a variety of quantitative metrics for learning and evaluating disentangled representations have been proposed, it remains unclear what properties these metrics truly quantify. In this work, we establish algebraic relationships between logical definitions and quantitative metrics to derive theoretically grounded disentanglement metrics. Concretely, we introduce a compositional approach for converting a higher-order predicate into a real-valued quantity by replacing (i) equality with a strict premetric, (ii) the Heyting algebra of binary truth values with a quantale of continuous values, and (iii) quantifiers with aggregators. The metrics induced by logical definitions have strong theoretical guarantees, and some of them are easily differentiable and can be used as learning objectives directly. Finally, we empirically demonstrate the effectiveness of the proposed metrics by isolating different aspects of disentangled representations.

Circuits based on sum-product structure have become a ubiquitous representation to compactly encode knowledge, from Boolean functions to probability distributions. By imposing constraints on the structure of such circuits, certain inference queries become tractable, such as model counting and most probable configuration. Recent works have explored analyzing probabilistic and causal inference queries as compositions of basic operators to derive tractability conditions. In this paper, we take an algebraic perspective for compositional inference, and show that a large class of queries—including marginal MAP, probabilistic answer set programming inference, and causal backdoor adjustment—correspond to a combination of basic operators over semirings: aggregation, product, and elementwise mapping. Using this framework, we uncover simple and general sufficient conditions for tractable composition of these operators, in terms of circuit properties (e.g., marginal determinism, compatibility) and conditions on the elementwise mappings. Applying our analysis, we derive novel tractability conditions for many such compositional queries. Our results unify tractability conditions for existing problems on circuits, while providing a blueprint for analysing novel compositional inference queries.

Linear temporal logic (LTL) and, more generally, $\omega$-regular objectives are alternatives to the traditional discount sum and average reward objectives in reinforcement learning (RL), offering the advantage of greater comprehensibility and hence explainability. In this work, we study the relationship between these objectives. Our main result is that each RL problem for $\omega$-regular objectives can be reduced to a limit-average reward problem in an optimality-preserving fashion, via (finite-memory) reward machines. Furthermore, we demonstrate the efficacy of this approach by showing that optimal policies for limit-average problems can be found asymptotically by solving a sequence of discount-sum problems approximately. Consequently, we resolve an open problem: optimal policies for LTL and $\omega$-regular objectives can be learned asymptotically.

We address a generalization of the bandit with knapsacks problem, where a learner aims to maximize rewards while satisfying an arbitrary set of long-term constraints. Our goal is to design best-of-both-worlds algorithms that perform optimally under both stochastic and adversarial constraints. Previous works address this problem via primal-dual methods, and require some stringent assumptions, namely the Slater's condition, and in adversarial settings, they either assume knowledge of a lower bound on the Slater's parameter, or impose strong requirements on the primal and dual regret minimizers such as requiring weak adaptivity. We propose an alternative and more natural approach based on optimistic estimations of the constraints. Surprisingly, we show that estimating the constraints with an UCB-like approach guarantees optimal performances. Our algorithm consists of two main components: (i) a regret minimizer working on moving strategy sets and (ii) an estimate of the feasible set as an optimistic weighted empirical mean of previous samples. The key challenge in this approach is designing adaptive weights that meet the different requirements for stochastic and adversarial constraints. Our algorithm is significantly simpler than previous approaches, and has a cleaner analysis. Moreover, ours is the first best-of-both-worlds algorithm providing bounds logarithmic in the number of constraints. Additionally, in stochastic settings, it provides $\widetilde O(\sqrt{T})$ regret without Slater's condition.

Exploring the integration of if-then logic rules within neural network architectures presents an intriguing area. This integration seamlessly transforms the rule learning task into neural network training using backpropagation and stochastic gradient descent. From a well-trained sparse and shallow neural network, one can interpret each layer and neuron through the language of logic rules, and a global explanatory rule set can be directly extracted. However, ensuring interpretability may impose constraints on the flexibility, depth, and width of neural networks. In this paper, we propose HyperLogic: a novel framework leveraging hypernetworks to generate weights of the main network. HyperLogic can unveil multiple diverse rule sets, each capable of capturing heterogeneous patterns in data. This provides a simple yet effective method to increase model flexibility and preserve interpretability. We theoretically analyzed the benefits of the HyperLogic by examining the approximation error and generalization capabilities under two types of regularization terms: sparsity and diversity regularizations. Experiments on real data demonstrate that our method can learn more diverse, accurate, and concise rules.

The distribution of pedestrian trajectories is highly complex and influenced by the scene, nearby pedestrians, and subjective intentions. This complexity presents challenges for modeling and generalizing trajectory prediction. Previous methods modeled the feature space of future trajectories based on the high-dimensional feature space of historical trajectories, but this approach is suboptimal because it overlooks the similarity between historical and future trajectories. Our proposed method, TrajCLIP, utilizes contrastive learning and idempotent generative networks to address this issue. By pairing historical and future trajectories and applying contrastive learning on the encoded feature space, we enforce same-space consistency constraints. To manage complex distributions, we use idempotent loss and tightness loss to control over-expansion in the latent space. Additionally, we have developed a trajectory interpolation algorithm and synthetic trajectory data to enhance model capacity and improve generalization. Experimental results on public datasets demonstrate that TrajCLIP achieves state-of-the-art performance and excels in scene-to-scene transfer, few-shot transfer, and online learning tasks.

In pioneering work from 2019, Barceló and coauthors identified logics that precisely match the expressive power of constant iteration-depth graph neural networks (GNNs) relative to properties definable in first-order logic. In this article, we give exact logical characterizations of recurrent GNNs in two scenarios: (1) in the setting with floating-point numbers and (2) with reals. For floats, the formalism matching recurrent GNNs is a rule-based modal logic with counting, while for reals we use a suitable infinitary modal logic, also with counting. These results give exact matches between logics and GNNs in the recurrent setting without relativising to a background logic in either case, but using some natural assumptions about floating-point arithmetic. Applying our characterizations, we also prove that, relative to graph properties definable in monadic second-order logic (MSO), our infinitary and rule-based logics are equally expressive. This implies that recurrent GNNs with reals and floats have the same expressive power over MSO-definable properties and shows that, for such properties, also recurrent GNNs with reals are characterized by a (finitary!) rule-based modal logic. In the general case, in contrast, the expressive power with floats is weaker than with reals. In addition to logic-oriented results, we also characterize recurrent GNNs, with both reals and floats, via distributed automata, drawing links to distributed computing models.

We consider the problem of sampling from a $d$-dimensional log-concave distribution $\pi(\theta) \propto \exp(-f(\theta))$ for $L$-Lipschitz $f$, constrained to a convex body (described by $n$ hyperplanes) equipped with a barrier function, contained in a ball of radius $R$ with a $w$-warm start. We propose a \emph{robust} sampling framework that computes spectral approximations to the Hessian of the barrier functions in each iteration. We prove that for the polytope constraints, sampling with the Lee-Sidford barrier function mixes within $\widetilde O((d^2+dL^2R^2)\log(w/\delta))$ steps with a per step cost of $\widetilde O(nd^{\omega-1})$, where $\omega\approx 2.37$ is the fast matrix multiplication exponent. Compared to the prior work of Mangoubi and Vishnoi, our approach gives faster mixing time as we are able to design a generalized soft-threshold Dikin walk beyond log-barrier. We further extend our result to show how to sample from a $d$-dimensional spectrahedron, the constrained set of a semidefinite program, specified by the set $\{x\in \mathbb{R}^d: \sum_{i=1}^d x_i A_i \succeq C \}$ where $A_1,\ldots,A_d, C$ are $n\times n$ real symmetric matrices. We design a walk that mixes in $\widetilde O((nd+dL^2R^2)\log(w/\delta))$ steps with a per iteration cost of $\widetilde O(n^\omega+n^2d^{3\omega-5})$. We improve the mixing time bound of prior best Dikin walk due to Narayanan and Rakhlin that mixes in $\widetilde O((n^2d^3+n^2dL^2R^2)\log(w/\delta))$ steps.

Training neural networks on NP-complete problems typically demands very large amounts of training data and often needs to be coupled with computationally expensive symbolic verifiers to ensure output correctness. In this paper, we present NeuRes, a neuro-symbolic approach to address both challenges for propositional satisfiability, being the quintessential NP-complete problem. By combining certificate-driven training and expert iteration, our model learns better representations than models trained for classification only, with a much higher data efficiency -- requiring orders of magnitude less training data. NeuRes employs propositional resolution as a proof system to generate proofs of unsatisfiability and to accelerate the process of finding satisfying truth assignments, exploring both possibilities in parallel. To realize this, we propose an attention-based architecture that autoregressively selects pairs of clauses from a dynamic formula embedding to derive new clauses. Furthermore, we employ expert iteration whereby model-generated proofs progressively replace longer teacher proofs as the new ground truth. This enables our model to reduce a dataset of proofs generated by an advanced solver by $\sim$$32$% after training on it with no extra guidance. This shows that NeuRes is not limited by the optimality of the teacher algorithm owing to its self-improving workflow. We show that our model achieves far better performance than NeuroSAT in terms of both correctly classified and proven instances.

While neural networks (NNs) have a large potential as autonomous controllers for Cyber-Physical Systems, verifying the safety of neural network based control systems (NNCSs) poses significant challenges for the practical use of NNs— especially when safety is needed for unbounded time horizons. One reason for this is the intractability of analyzing NNs, ODEs and hybrid systems. To this end, we introduce VerSAILLE (Verifiably Safe AI via Logically Linked Envelopes): The first general approach that allows reusing control theory literature for NNCS verification. By joining forces, we can exploit the efficiency of NN verification tools while retaining the rigor of differential dynamic logic (dL). Based on a provably safe control envelope in dL, we derive a specification for the NN which is proven with NN verification tools. We show that a proof of the NN’s adherence to the specification is then mirrored by a dL proof on the infinite-time safety of the NNCS. The NN verification properties resulting from hybrid systems typically contain nonlinear arithmetic over formulas with arbitrary logical structure while efficient NN verification tools merely support linear constraints. To overcome this divide, we present Mosaic: An efficient, sound and complete verification approach for polynomial real arithmetic properties on piece-wise linear NNs. Mosaic partitions complex NN verification queries into simple queries and lifts off-the-shelf linear constraint tools to the nonlinear setting in a completeness-preserving manner by combining approximation with exact reasoning for counterexample regions. In our evaluation we demonstrate the versatility of VerSAILLE and Mosaic: We prove infinite-time safety on the classical Vertical Airborne Collision Avoidance NNCS verification benchmark for some scenarios while (exhaustively) enumerating counterexample regions in unsafe scenarios. We also show that our approach significantly outperforms the State-of-the-Art tools in closed-loop NNV

As learned control policies become increasingly common in autonomous systems, there is increasing need to ensure that they are interpretable and can be checked by human stakeholders. Formal specifications have been proposed as ways to produce human-interpretable policies for autonomous systems that can still be learned from examples. Previous work showed that despite claims of interpretability, humans are unable to use formal specifications presented in a variety of ways to validate even simple robot behaviors. This work uses active learning, a standard pedagogical method, to attempt to improve humans' ability to validate policies in signal temporal logic (STL). Results show that overall validation accuracy is not high, at 65\% $\pm$ 15% (mean $\pm$ standard deviation), and that the three conditions of no active learning, active learning, and active learning with feedback do not significantly differ from each other. Our results suggest that the utility of formal specifications for human interpretability is still unsupported but point to other avenues of development which may enable improvements in system validation.

Vehicle Routing Problems (VRPs) can model many real-world scenarios and often involve complex constraints. While recent neural methods excel in constructing solutions based on feasibility masking, they struggle with handling complex constraints, especially when obtaining the masking itself is NP-hard. In this paper, we propose a novel Proactive Infeasibility Prevention (PIP) framework to advance the capabilities of neural methods towards more complex VRPs. Our PIP integrates the Lagrangian multiplier as a basis to enhance constraint awareness and introduces preventative infeasibility masking to proactively steer the solution construction process. Moreover, we present PIP-D, which employs an auxiliary decoder and two adaptive strategies to learn and predict these tailored masks, potentially enhancing performance while significantly reducing computational costs during training. To verify our PIP designs, we conduct extensive experiments on the highly challenging Traveling Salesman Problem with Time Window (TSPTW), and TSP with Draft Limit (TSPDL) variants under different constraint hardness levels. Notably, our PIP is generic to boost many neural methods, and exhibits both a significant reduction in infeasible rate and a substantial improvement in solution quality.

The design of interpretable deep learning models working in relational domains poses an open challenge: interpretable deep learning methods, such as Concept Bottleneck Models (CBMs), are not designed to solve relational problems, while relational deep learning models, such as Graph Neural Networks (GNNs), are not as interpretable as CBMs. To overcome these limitations, we propose Relational Concept Bottleneck Models (R-CBMs), a family of relational deep learning methods providing interpretable task predictions. As special cases, we show that R-CBMs are capable of both representing standard CBMs and message passing GNNs. To evaluate the effectiveness and versatility of these models, we designed a class of experimental problems, ranging from image classification to link prediction in knowledge graphs. In particular we show that R-CBMs (i) match generalization performance of existing relational black-boxes, (ii) support the generation of quantified concept-based explanations, (iii) effectively respond to test-time interventions, and (iv) withstand demanding settings including out-of-distribution scenarios, limited training data regimes, and scarce concept supervisions.

The expressive power of transformers over inputs of unbounded size can be studied through their ability to recognize classes of formal languages. In this paper, we establish exact characterizations of transformers with hard attention (in which all attention is focused on exactly one position) and attention masking (in which each position only attends to positions on one side). With strict masking (each position cannot attend to itself) and without position embeddings, these transformers are expressively equivalent to linear temporal logic (LTL), which defines exactly the star-free languages. A key technique is the use of Boolean RASP as a convenient intermediate language between transformers and LTL. We then take numerous results known for LTL and apply them to transformers, showing how position embeddings, strict masking, and depth all increase expressive power.

Computational intensiveness of deep learning has motivated low-precision arithmetic designs. However, the current quantized/binarized training approaches are limited by: (1) significant performance loss due to arbitrary approximations of the latent weight gradient through its discretization/binarization function, and (2) training computational intensiveness due to the reliance on full-precision latent weights. This paper proposes a novel mathematical principle by introducing the notion of Boolean variation such that neurons made of Boolean weights and/or activations can be trained ---for the first time--- natively in Boolean domain instead of latent-weight gradient descent and real arithmetic. We explore its convergence, conduct extensively experimental benchmarking, and provide consistent complexity evaluation by considering chip architecture, memory hierarchy, dataflow, and arithmetic precision. Our approach achieves baseline full-precision accuracy in ImageNet classification and surpasses state-of-the-art results in semantic segmentation, with notable performance in image super-resolution, and natural language understanding with transformer-based models. Moreover, it significantly reduces energy consumption during both training and inference.

Reward Machines provide an automaton-inspired structure for specifying instructions, safety constraints, and other temporally extended reward-worthy behaviour. By exposing the underlying structure of a reward function, they enable the decomposition of an RL task, leading to impressive gains in sample efficiency. Although Reward Machines and similar formal specifications have a rich history of application towards sequential decision-making problems, prior frameworks have traditionally ignored ambiguity and uncertainty when interpreting the domain-specific vocabulary forming the building blocks of the reward function. Such uncertainty critically arises in many real-world settings due to factors like partial observability or noisy sensors. In this work, we explore the use of Reward Machines for Deep RL in noisy and uncertain environments. We characterize this problem as a POMDP and propose a suite of RL algorithms that exploit task structure under uncertain interpretation of the domain-specific vocabulary. Through theory and experiments, we expose pitfalls in naive approaches to this problem while simultaneously demonstrating how task structure can be successfully leveraged under noisy interpretations of the vocabulary.

With the increasing inference cost of machine learning models, there is a growing interest in models with fast and efficient inference. Recently, an approach for learning logic gate networks directly via a differentiable relaxation was proposed. Logic gate networks are faster than conventional neural network approaches because their inference only requires logic gate operators such as NAND, OR, and XOR, which are the underlying building blocks of current hardware and can be efficiently executed. We build on this idea, extending it by deep logic gate tree convolutions, logical OR pooling, and residual initializations. This allows scaling logic gate networks up by over one order of magnitude and utilizing the paradigm of convolution. On CIFAR-10, we achieve an accuracy of 86.29% using only 61 million logic gates, which improves over the SOTA while being 29x smaller.