Autoformalization involves automatically translating informal math into formal theorems and proofs that are machine-verifiable. Euclidean geometry provides an interesting and controllable domain for studying autoformalization. In this paper, we introduce a neuro-symbolic framework for autoformalizing Euclidean geometry, which combines domain knowledge, SMT solvers, and large language models (LLMs). One challenge in Euclidean geometry is that informal proofs rely on diagrams, leaving gaps in texts that are hard to formalize. To address this issue, we use theorem provers to fill in such diagrammatic information automatically, so that the LLM only needs to autoformalize the explicit textual steps, making it easier for the model. We also provide automatic semantic evaluation for autoformalized theorem statements. We construct LeanEuclid, an autoformalization benchmark consisting of problems from Euclid's Elements and the UniGeo dataset formalized in the Lean proof assistant. Experiments with GPT-4 and GPT-4V show the capability and limitations of state-of-the-art LLMs on autoformalizing geometry problems. The data and code are available at //github.com/loganrjmurphy/LeanEuclid.
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Modern cyber-physical systems are becoming increasingly complex to model, thus motivating data-driven techniques such as reinforcement learning (RL) to find appropriate control agents. However, most systems are subject to hard constraints such as safety or operational bounds. Typically, to learn to satisfy these constraints, the agent must violate them systematically, which is computationally prohibitive in most systems. Recent efforts aim to utilize feasibility models that assess whether a proposed action is feasible to avoid applying the agent's infeasible action proposals to the system. However, these efforts focus on guaranteeing constraint satisfaction rather than the agent's learning efficiency. To improve the learning process, we introduce action mapping, a novel approach that divides the learning process into two steps: first learn feasibility and subsequently, the objective by mapping actions into the sets of feasible actions. This paper focuses on the feasibility part by learning to generate all feasible actions through self-supervised querying of the feasibility model. We train the agent by formulating the problem as a distribution matching problem and deriving gradient estimators for different divergences. Through an illustrative example, a robotic path planning scenario, and a robotic grasping simulation, we demonstrate the agent's proficiency in generating actions across disconnected feasible action sets. By addressing the feasibility step, this paper makes it possible to focus future work on the objective part of action mapping, paving the way for an RL framework that is both safe and efficient.
Social identities play an important role in the dynamics of human societies, and it can be argued that some sense of identification with a larger cause or idea plays a critical role in making humans act responsibly. Often social activists strive to get populations to identify with some cause or notion -- like green energy, diversity, etc. in order to bring about desired social changes. We explore the problem of designing computational models for social identities in the context of autonomous AI agents. For this, we propose an agent model that enables agents to identify with certain notions and show how this affects collective outcomes. We also contrast between associations of identity with rational preferences. The proposed model is simulated in an application context of urban mobility, where we show how changes in social identity affect mobility patterns and collective outcomes.
A variety of knowledge graph embedding approaches have been developed. Most of them obtain embeddings by learning the structure of the knowledge graph within a link prediction setting. As a result, the embeddings reflect only the structure of a single knowledge graph, and embeddings for different knowledge graphs are not aligned, e.g., they cannot be used to find similar entities across knowledge graphs via nearest neighbor search. However, knowledge graph embedding applications such as entity disambiguation require a more global representation, i.e., a representation that is valid across multiple sources. We propose to learn universal knowledge graph embeddings from large-scale interlinked knowledge sources. To this end, we fuse large knowledge graphs based on the owl:sameAs relation such that every entity is represented by a unique identity. We instantiate our idea by computing universal embeddings based on DBpedia and Wikidata yielding embeddings for about 180 million entities, 15 thousand relations, and 1.2 billion triples. We believe our computed embeddings will support the emerging field of graph foundation models. Moreover, we develop a convenient API to provide embeddings as a service. Experiments on link prediction suggest that universal knowledge graph embeddings encode better semantics compared to embeddings computed on a single knowledge graph. For reproducibility purposes, we provide our source code and datasets open access.
The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $\rho^{(l)} \to \varrho^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}=\sum_{l=1}^{M} \omega^{(l)} F\left(\varrho^{(l)},\sum_s B_s \rho^{(l)} B^{\dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,\sigma)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$ (either exactly or as an approximation) an iterative algorithm is developed to find the global maximum. The result comprises in $N_s$ operators $B_s$ that collectively form an $IN$ to $OUT$ quantum channel $A^{OUT}=\sum_s B_s A^{IN} B_s^{\dagger}$. The work introduces two important generalizations of unitary learning: 1. $IN$/$OUT$ states are represented as density matrices. 2. The mapping itself is formulated as a general quantum channel. This marks a crucial advancement from the commonly studied unitary mapping of pure states $\phi_l=\mathcal{U} \psi_l$ to a general quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping $\varrho^{(l)}=\mathcal{U} \rho^{(l)} \mathcal{U}^{\dagger}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{\rho^{(l)}} \to \sqrt{\varrho^{(l)}}$ mapping, and on a general quantum channel of Kraus rank $N_s$, where quadratic on $B_s$ fidelity is an approximation -- a quantum channel is then built as a hierarchy of unitary mappings. The approach can be applied to study decoherence effects, spontaneous coherence, synchronizing, etc.
Most curriculum learning methods require an approach to sort the data samples by difficulty, which is often cumbersome to perform. In this work, we propose a novel curriculum learning approach termed Learning Rate Curriculum (LeRaC), which leverages the use of a different learning rate for each layer of a neural network to create a data-agnostic curriculum during the initial training epochs. More specifically, LeRaC assigns higher learning rates to neural layers closer to the input, gradually decreasing the learning rates as the layers are placed farther away from the input. The learning rates increase at various paces during the first training iterations, until they all reach the same value. From this point on, the neural model is trained as usual. This creates a model-level curriculum learning strategy that does not require sorting the examples by difficulty and is compatible with any neural network, generating higher performance levels regardless of the architecture. We conduct comprehensive experiments on 12 data sets from the computer vision (CIFAR-10, CIFAR-100, Tiny ImageNet, ImageNet-200, Food-101, UTKFace, PASCAL VOC), language (BoolQ, QNLI, RTE) and audio (ESC-50, CREMA-D) domains, considering various convolutional (ResNet-18, Wide-ResNet-50, DenseNet-121, YOLOv5), recurrent (LSTM) and transformer (CvT, BERT, SepTr) architectures. We compare our approach with the conventional training regime, as well as with Curriculum by Smoothing (CBS), a state-of-the-art data-agnostic curriculum learning approach. Unlike CBS, our performance improvements over the standard training regime are consistent across all data sets and models. Furthermore, we significantly surpass CBS in terms of training time (there is no additional cost over the standard training regime for LeRaC). Our code is freely available at: //github.com/CroitoruAlin/LeRaC.
A wide range of symbolic analysis and optimization problems can be formalized using polyhedra. Sub-classes of polyhedra, also known as sub-polyhedral domains, are sought for their lower space and time complexity. We introduce the Strided Difference Bound Matrix (SDBM) domain, which represents a sweet spot in the context of optimizing compilers. Its expressiveness and efficient algorithms are particularly well suited to the construction of machine learning compilers. We present decision algorithms, abstract domain operators and computational complexity proofs for SDBM. We also conduct an empirical study with the MLIR compiler framework to validate the domain's practical applicability. We characterize a sub-class of SDBMs that frequently occurs in practice, and demonstrate even faster algorithms on this sub-class.
Being the most classical generative model for serial data, state-space models (SSM) are fundamental in AI and statistical machine learning. In SSM, any form of parameter learning or latent state inference typically involves the computation of complex latent-state posteriors. In this work, we build upon the variational sequential Monte Carlo (VSMC) method, which provides computationally efficient and accurate model parameter estimation and Bayesian latent-state inference by combining particle methods and variational inference. While standard VSMC operates in the offline mode, by re-processing repeatedly a given batch of data, we distribute the approximation of the gradient of the VSMC surrogate ELBO in time using stochastic approximation, allowing for online learning in the presence of streams of data. This results in an algorithm, online VSMC, that is capable of performing efficiently, entirely on-the-fly, both parameter estimation and particle proposal adaptation. In addition, we provide rigorous theoretical results describing the algorithm's convergence properties as the number of data tends to infinity as well as numerical illustrations of its excellent convergence properties and usefulness also in batch-processing settings.
Disentangled Representation Learning (DRL) aims to learn a model capable of identifying and disentangling the underlying factors hidden in the observable data in representation form. The process of separating underlying factors of variation into variables with semantic meaning benefits in learning explainable representations of data, which imitates the meaningful understanding process of humans when observing an object or relation. As a general learning strategy, DRL has demonstrated its power in improving the model explainability, controlability, robustness, as well as generalization capacity in a wide range of scenarios such as computer vision, natural language processing, data mining etc. In this article, we comprehensively review DRL from various aspects including motivations, definitions, methodologies, evaluations, applications and model designs. We discuss works on DRL based on two well-recognized definitions, i.e., Intuitive Definition and Group Theory Definition. We further categorize the methodologies for DRL into four groups, i.e., Traditional Statistical Approaches, Variational Auto-encoder Based Approaches, Generative Adversarial Networks Based Approaches, Hierarchical Approaches and Other Approaches. We also analyze principles to design different DRL models that may benefit different tasks in practical applications. Finally, we point out challenges in DRL as well as potential research directions deserving future investigations. We believe this work may provide insights for promoting the DRL research in the community.
The information bottleneck (IB) method is a technique for extracting information that is relevant for predicting the target random variable from the source random variable, which is typically implemented by optimizing the IB Lagrangian that balances the compression and prediction terms. However, the IB Lagrangian is hard to optimize, and multiple trials for tuning values of Lagrangian multiplier are required. Moreover, we show that the prediction performance strictly decreases as the compression gets stronger during optimizing the IB Lagrangian. In this paper, we implement the IB method from the perspective of supervised disentangling. Specifically, we introduce Disentangled Information Bottleneck (DisenIB) that is consistent on compressing source maximally without target prediction performance loss (maximum compression). Theoretical and experimental results demonstrate that our method is consistent on maximum compression, and performs well in terms of generalization, robustness to adversarial attack, out-of-distribution detection, and supervised disentangling.
Embedding models for deterministic Knowledge Graphs (KG) have been extensively studied, with the purpose of capturing latent semantic relations between entities and incorporating the structured knowledge into machine learning. However, there are many KGs that model uncertain knowledge, which typically model the inherent uncertainty of relations facts with a confidence score, and embedding such uncertain knowledge represents an unresolved challenge. The capturing of uncertain knowledge will benefit many knowledge-driven applications such as question answering and semantic search by providing more natural characterization of the knowledge. In this paper, we propose a novel uncertain KG embedding model UKGE, which aims to preserve both structural and uncertainty information of relation facts in the embedding space. Unlike previous models that characterize relation facts with binary classification techniques, UKGE learns embeddings according to the confidence scores of uncertain relation facts. To further enhance the precision of UKGE, we also introduce probabilistic soft logic to infer confidence scores for unseen relation facts during training. We propose and evaluate two variants of UKGE based on different learning objectives. Experiments are conducted on three real-world uncertain KGs via three tasks, i.e. confidence prediction, relation fact ranking, and relation fact classification. UKGE shows effectiveness in capturing uncertain knowledge by achieving promising results on these tasks, and consistently outperforms baselines on these tasks.
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