We consider the challenge of black-box optimization within hybrid discrete-continuous and variable-length spaces, a problem that arises in various applications, such as decision tree learning and symbolic regression. We propose DisCo-DSO (Discrete-Continuous Deep Symbolic Optimization), a novel approach that uses a generative model to learn a joint distribution over discrete and continuous design variables to sample new hybrid designs. In contrast to standard decoupled approaches, in which the discrete and continuous variables are optimized separately, our joint optimization approach uses fewer objective function evaluations, is robust against non-differentiable objectives, and learns from prior samples to guide the search, leading to significant improvement in performance and sample efficiency. Our experiments on a diverse set of optimization tasks demonstrate that the advantages of DisCo-DSO become increasingly evident as problem complexity grows. In particular, we illustrate DisCo-DSO's superiority over the state-of-the-art methods for interpretable reinforcement learning with decision trees.

Symbolic regression searches for analytic expressions that accurately describe studied phenomena. The main promise of this approach is that it may return an interpretable model that can be insightful to users, while maintaining high accuracy. The current standard for benchmarking these algorithms is SRBench, which evaluates methods on hundreds of datasets that are a mix of real-world and simulated processes spanning multiple domains. At present, the ability of SRBench to evaluate interpretability is limited to measuring the size of expressions on real-world data, and the exactness of model forms on synthetic data. In practice, model size is only one of many factors used by subject experts to determine how interpretable a model truly is. Furthermore, SRBench does not characterize algorithm performance on specific, challenging sub-tasks of regression such as feature selection and evasion of local minima. In this work, we propose and evaluate an approach to benchmarking SR algorithms that addresses these limitations of SRBench by 1) incorporating expert evaluations of interpretability on a domain-specific task, and 2) evaluating algorithms over distinct properties of data science tasks. We evaluate 12 modern symbolic regression algorithms on these benchmarks and present an in-depth analysis of the results, discuss current challenges of symbolic regression algorithms and highlight possible improvements for the benchmark itself.

Finite element simulations of physical systems governed by partial differential equations (PDE) crucially depend on adaptive mesh refinement (AMR) to allocate computational budget to regions where higher resolution is required. Existing scalable AMR methods make heuristic refinement decisions based on instantaneous error estimation and thus do not aim for long-term optimality over an entire simulation. We propose a novel formulation of AMR as a Markov decision process and apply deep reinforcement learning (RL) to train refinement policies directly from simulation. AMR poses a challenge for RL as both the state dimension and available action set changes at every step, which we solve by proposing new policy architectures with differing generality and inductive bias. The model sizes of these policy architectures are independent of the mesh size and hence can be deployed on larger simulations than those used at training time. We demonstrate in comprehensive experiments on static function estimation and time-dependent equations that RL policies can be trained on problems without using ground truth solutions, are competitive with a widely-used error estimator, and generalize to larger and unseen test problems.

The last few years have witnessed a surge in methods for symbolic regression, from advances in traditional evolutionary approaches to novel deep learning-based systems. Individual works typically focus on advancing the state-of-the-art for one particular class of solution strategies, and there have been few attempts to investigate the benefits of hybridizing or integrating multiple strategies. In this work, we identify five classes of symbolic regression solution strategies---recursive problem simplification, neural-guided search, large-scale pre-training, genetic programming, and linear models---and propose a strategy to hybridize them into a single modular, unified symbolic regression framework. Based on empirical evaluation using SRBench, a new community tool for benchmarking symbolic regression methods, our unified framework achieves state-of-the-art performance in its ability to (1) symbolically recover analytical expressions, (2) fit datasets with high accuracy, and (3) balance accuracy-complexity trade-offs, across 252 ground-truth and black-box benchmark problems, in both noiseless settings and across various noise levels. Finally, we provide practical use case-based guidance for constructing hybrid symbolic regression algorithms, supported by extensive, combinatorial ablation studies.

Deep reinforcement learning (DRL) has proven successful for many difficult control problems by learning policies represented by neural networks. However, the complexity of neural network-based policies{—}involving thousands of composed non-linear operators{—}can render them problematic to understand, trust, and deploy. In contrast, simple policies comprising short symbolic expressions can facilitate human understanding, while also being transparent and exhibiting predictable behavior. To this end, we propose deep symbolic policy, a novel approach to directly search the space of symbolic policies. We use an autoregressive recurrent neural network to generate control policies represented by tractable mathematical expressions, employing a risk-seeking policy gradient to maximize performance of the generated policies. To scale to environments with multi-dimensional action spaces, we propose an "anchoring" algorithm that distills pre-trained neural network-based policies into fully symbolic policies, one action dimension at a time. We also introduce two novel methods to improve exploration in DRL-based combinatorial optimization, building on ideas of entropy regularization and distribution initialization. Despite their dramatically reduced complexity, we demonstrate that discovered symbolic policies outperform seven state-of-the-art DRL algorithms in terms of average rank and average normalized episodic reward across eight benchmark environments.

Symbolic regression is the process of identifying mathematical expressions that fit observed output from a black-box process. It is a discrete optimization problem generally believed to be NP-hard. Prior approaches to solving the problem include neural-guided search (e.g. using reinforcement learning) and genetic programming. In this work, we introduce a hybrid neural-guided/genetic programming approach to symbolic regression and other combinatorial optimization problems. We propose a neural-guided component used to seed the starting population of a random restart genetic programming component, gradually learning better starting populations. On a number of common benchmark tasks to recover underlying expressions from a dataset, our method recovers 65% more expressions than a recently published top-performing model using the same experimental setup. We demonstrate that running many genetic programming generations without interdependence on the neural-guided component performs better for symbolic regression than alternative formulations where the two are more strongly coupled. Finally, we introduce a new set of 22 symbolic regression benchmark problems with increased difficulty over existing benchmarks. Source code is provided at www.github.com/brendenpetersen/deep-symbolic-optimization.

Discovering tractable mathematical expressions that best explain a dataset is a long-standing challenge in artificial intelligence. This problem, known as symbolic regression, is relevant when one seeks to generate new physical knowledge and insights. Since practitioners are primarily interested in knowledge generation, the ability to interact with a symbolic regression algorithm would be highly valuable. Thus, we present an interactive symbolic regression framework that allows users not only to configure runs, but also to control the system during training. The interface provides real-time visualization and diagnostics to help guide the user as they control the algorithm on the fly.

Acetaminophen-induced liver injury in mice is a model for drug-induced liver injury in humans. A precondition for improved strategies to disrupt and/or reverse the damage is a credible explanatory mechanism for how toxicity phenomena emerge and converge to cause hepatic necrosis. The Target Phenomenon in mice is that necrosis begins adjacent to the lobule's central vein (CV) and progresses outward. An explanatory mechanism remains elusive. Evidence supports that location dependent differences in NAPQI (the reactive metabolite) formation within hepatic lobules (NAPQI zonation) are necessary and sufficient prerequisites to account for that phenomenon. We call that the NZ-mechanism hypothesis. Challenging that hypothesis in mice is infeasible because 1) influential variables cannot be controlled, and 2) it would require sequential intracellular measurements at different lobular locations within the same mouse. Virtual hepatocytes use independently configured periportal-to-CV gradients to exhibit lobule-location dependent behaviors. Employing NZ-mechanism achieved quantitative validation targets for acetaminophen clearance and metabolism but failed to achieve the Target Phenomenon. We posited that, in order to do so, at least one additional feature must exhibit zonation by decreasing in the CV direction. We instantiated and explored two alternatives: 1) a glutathione depletion threshold diminishes in the CV direction; and 2) ability to repair mitochondrial damage diminishes in the CV direction. Inclusion of one or the other feature into NZ-mechanism failed to achieve the Target Phenomenon. However, inclusion of both features enabled successfully achieving the Target Phenomenon. The merged mechanism provides a multilevel, multiscale causal explanation of key temporal features of acetaminophen hepatotoxicity in mice. We discovered that variants of the merged mechanism provide plausible quantitative explanations for the considerable variation in 24-hour necrosis scores among 37 genetically diverse mouse strains following a single toxic acetaminophen dose.