AI Hilbert: From Data and Background Knowledge to Automated Scientific Discovery

The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. In recent years, data-driven scientific discovery has emerged as a viable competitor in settings with large amounts of experimental data. Unfortunately, data-driven methods often fail to discover valid laws when data is noisy or scarce. Accordingly, recent works combine regression and reasoning to eliminate formulae inconsistent with background theory. However, the problem of searching over the space of formulae consistent with background theory to find one that fits the data best is not well solved. We propose a solution to this problem when all axioms and scientific laws are expressible via polynomial equalities and inequalities and argue that our approach is widely applicable. We further model notions of minimal complexity using binary variables and logical constraints, solve polynomial optimization problems via mixed-integer linear or semidefinite optimization, and automatically prove the validity of our scientific discoveries via Positivestellensatz certificates. Remarkably, the optimization techniques leveraged in this paper allow our approach to run in polynomial time with fully correct background theory, or non-deterministic polynomial (NP) time with partially correct background theory. We experimentally demonstrate that some famous scientific laws, including Kepler's Third Law of Planetary Motion, the Hagen-Poiseuille Equation, and the Radiated Gravitational Wave Power equation, can be automatically derived from sets of partially correct background axioms.

Value-based Fast and Slow AI Nudging

Nudging is a behavioral strategy aimed at influencing people's thoughts and actions. Nudging techniques can be found in many situations in our daily lives, and these nudging techniques can targeted at human fast and unconscious thinking, e.g., by using images to generate fear or the more careful and effortful slow thinking, e.g., by releasing information that makes us reflect on our choices. In this paper, we propose and discuss a value-based AI-human collaborative framework where AI systems nudge humans by proposing decision recommendations. Three different nudging modalities, based on when recommendations are presented to the human, are intended to stimulate human fast thinking, slow thinking, or meta-cognition. Values that are relevant to a specific decision scenario are used to decide when and how to use each of these nudging modalities. Examples of values are decision quality, speed, human upskilling and learning, human agency, and privacy. Several values can be present at the same time, and their priorities can vary over time. The framework treats values as parameters to be instantiated in a specific decision environment.

Understanding the Capabilities of Large Language Models for Automated Planning

Automated planning is concerned with developing efficient algorithms to generate plans or sequences of actions to achieve a specific goal in a given environment. Emerging Large Language Models (LLMs) can answer questions, write high-quality programming code, and predict protein folding, showcasing their versatility in solving various tasks beyond language-based problems. In this paper, we aim to explore how LLMs can also be used for automated planning. To do so, we seek to answer four key questions. Firstly, we want to understand the extent to which LLMs can be used for plan generation. Secondly, we aim to identify which pre-training data is most effective in facilitating plan generation. Thirdly, we investigate whether fine-tuning or prompting is a more effective approach for plan generation. Finally, we explore whether LLMs are capable of plan generalization. By answering these questions, the study seeks to shed light on the capabilities of LLMs in solving complex planning problems and provide insights into the most effective approaches for using LLMs in this context.

Fast and Slow Planning

The concept of Artificial Intelligence has gained a lot of attention over the last decade. In particular, AI-based tools have been employed in several scenarios and are, by now, pervading our everyday life. Nonetheless, most of these systems lack many capabilities that we would naturally consider to be included in a notion of "intelligence". In this work, we present an architecture that, inspired by the cognitive theory known as Thinking Fast and Slow by D. Kahneman, is tasked with solving planning problems in different settings, specifically: classical and multi-agent epistemic. The system proposed is an instance of a more general AI paradigm, referred to as SOFAI (for Slow and Fast AI). SOFAI exploits multiple solving approaches, with different capabilities that characterize them as either fast or slow, and a metacognitive module to regulate them. This combination of components, which roughly reflects the human reasoning process according to D. Kahneman, allowed us to enhance the reasoning process that, in this case, is concerned with planning in two different settings. The behavior of this system is then compared to state-of-the-art solvers, showing that the newly introduced system presents better results in terms of generality, solving a wider set of problems with an acceptable trade-off between solving times and solution accuracy.

Plansformer: Generating Symbolic Plans using Transformers

Large Language Models (LLMs) have been the subject of active research, significantly advancing the field of Natural Language Processing (NLP). From BERT to BLOOM, LLMs have surpassed state-of-the-art results in various natural language tasks such as question answering, summarization, and text generation. Many ongoing efforts focus on understanding LLMs' capabilities, including their knowledge of the world, syntax, and semantics. However, extending the textual prowess of LLMs to symbolic reasoning has been slow and predominantly focused on tackling problems related to the mathematical field. In this paper, we explore the use of LLMs for automated planning - a branch of AI concerned with the realization of action sequences (plans) to achieve a goal, typically executed by intelligent agents, autonomous robots, and unmanned vehicles. We introduce Plansformer; an LLM fine-tuned on planning problems and capable of generating plans with favorable behavior in terms of correctness and length with reduced knowledge-engineering efforts. We also demonstrate the adaptability of Plansformer in solving different planning domains with varying complexities, owing to the transfer learning abilities of LLMs. For one configuration of Plansformer, we achieve ~97% valid plans, out of which ~95% are optimal for Towers of Hanoi - a puzzle-solving domain.

Bayesian Experimental Design for Symbolic Discovery

This study concerns the formulation and application of Bayesian optimal experimental design to symbolic discovery, which is the inference from observational data of predictive models taking general functional forms. We apply constrained first-order methods to optimize an appropriate selection criterion, using Hamiltonian Monte Carlo to sample from the prior. A step for computing the predictive distribution, involving convolution, is computed via either numerical integration, or via fast transform methods.

Towards Quantum Advantage on Noisy Quantum Computers

Topological data analysis (TDA) is a powerful technique for extracting complex and valuable shape-related summaries of high-dimensional data. However, the computational demands of classical TDA algorithms are exorbitant, and quickly become impractical for high-order characteristics. Quantum computing promises exponential speedup for certain problems. Yet, many existing quantum algorithms with notable asymptotic speedups require a degree of fault tolerance that is currently unavailable. In this paper, we present NISQ-TDA, the first fully implemented end-to-end quantum machine learning algorithm needing only a linear circuit-depth, that is applicable to non-handcrafted high-dimensional classical data, with potential speedup under stringent conditions. The algorithm neither suffers from the data-loading problem nor does it need to store the input data on the quantum computer explicitly. Our approach includes three key innovations: (a) an efficient realization of the full boundary operator as a sum of Pauli operators; (b) a quantum rejection sampling and projection approach to restrict a uniform superposition to the simplices of the desired order in the complex; and (c) a stochastic rank estimation method to estimate the topological features in the form of approximate Betti numbers. We present theoretical results that establish additive error guarantees for NISQ-TDA, and the circuit and computational time and depth complexities for exponentially scaled output estimates, up to the error tolerance. The algorithm was successfully executed on quantum computing devices, as well as on noisy quantum simulators, applied to small datasets. Preliminary empirical results suggest that the algorithm is robust to noise.

Exponential advantage on noisy quantum computers

Quantum computing offers the potential of exponential speedup over classical computation for certain problems. However, many of the existing algorithms with provable speedups require currently unavailable fault-tolerant quantum computers. We present NISQ-TDA, the first fully implemented quantum machine learning algorithm with provable exponential speedup on arbitrary classical (non-handcrafted) data and needing only a linear circuit depth. We report the successful execution of our NISQ-TDA algorithm, applied to small datasets run on quantum computing devices, as well as on noisy quantum simulators. We empirically confirm that the algorithm is robust to noise, and provide target depths and noise levels to realize near-term, non-fault-tolerant quantum advantage on real-world problems. Our unique data-loading projection method is the main source of noise robustness, introducing a new self-correcting data-loading approach.

Distributed Adversarial Training to Robustify Deep Neural Networks at Scale

Current deep neural networks (DNNs) are vulnerable to adversarial attacks, where adversarial perturbations to the inputs can change or manipulate classification. To defend against such attacks, an effective and popular approach, known as adversarial training (AT), has been shown to mitigate the negative impact of adversarial attacks by virtue of a min-max robust training method. While effective, it remains unclear whether it can successfully be adapted to the distributed learning context. The power of distributed optimization over multiple machines enables us to scale up robust training over large models and datasets. Spurred by that, we propose distributed adversarial training (DAT), a large-batch adversarial training framework implemented over multiple machines. We show that DAT is general, which supports training over labeled and unlabeled data, multiple types of attack generation methods, and gradient compression operations favored for distributed optimization. Theoretically, we provide, under standard conditions in the optimization theory, the convergence rate of DAT to the first-order stationary points in general non-convex settings. Empirically, we demonstrate that DAT either matches or outperforms state-of-the-art robust accuracies and achieves a graceful training speedup (e.g., on ResNet-50 under ImageNet). Codes are available at https://github.com/dat-2022/dat.

PCENet: High Dimensional Surrogate Modeling for Learning Uncertainty

Learning data representations under uncertainty is an important task that emerges in numerous machine learning applications. However, uncertainty quantification (UQ) techniques are computationally intensive and become prohibitively expensive for high-dimensional data. In this paper, we present a novel surrogate model for representation learning and uncertainty quantification, which aims to deal with data of moderate to high dimensions. The proposed model combines a neural network approach for dimensionality reduction of the (potentially high-dimensional) data, with a surrogate model method for learning the data distribution. We first employ a variational autoencoder (VAE) to learn a low-dimensional representation of the data distribution. We then propose to harness polynomial chaos expansion (PCE) formulation to map this distribution to the output target. The coefficients of PCE are learned from the distribution representation of the training data using a maximum mean discrepancy (MMD) approach. Our model enables us to (a) learn a representation of the data, (b) estimate uncertainty in the high-dimensional data system, and (c) match high order moments of the output distribution; without any prior statistical assumptions on the data. Numerical experimental results are presented to illustrate the performance of the proposed method.