Decision Tree (DT) Learning is a fundamental problem in Interpretable Machine Learning, yet it poses a formidable optimisation challenge. Practical algorithms have recently emerged, primarily leveraging Dynamic Programming and Branch & Bound. However, most of these approaches rely on a Depth-First-Search strategy, which is inefficient when searching for DTs at high depths and requires the definition of a maximum depth hyperparameter. Best-First-Search was also employed by other methods to circumvent these issues. The downside of this strategy is its higher memory consumption, as such, it has to be designed in a fully efficient manner that takes full advantage of the problem's structure. We formulate the problem within an AND/OR graph search framework and we solve it with a novel AO*-type algorithm called Branches. We prove both optimality and complexity guarantees for Branches and we show that it is more efficient than the state of the art theoretically and on a variety of experiments. Furthermore, Branches supports non-binary features unlike the other methods, we show that this property can further induce larger gains in computational efficiency.
We study the trajectory of iterations and the convergence rates of the Expectation-Maximization (EM) algorithm for two-component Mixed Linear Regression (2MLR). The fundamental goal of MLR is to learn the regression models from unlabeled observations. The EM algorithm finds extensive applications in solving the mixture of linear regressions. Recent results have established the super-linear convergence of EM for 2MLR in the noiseless and high SNR settings under some assumptions and its global convergence rate with random initialization has been affirmed. However, the exponent of convergence has not been theoretically estimated and the geometric properties of the trajectory of EM iterations are not well-understood. In this paper, first, using Bessel functions we provide explicit closed-form expressions for the EM updates under all SNR regimes. Then, in the noiseless setting, we completely characterize the behavior of EM iterations by deriving a recurrence relation at the population level and notably show that all the iterations lie on a certain cycloid. Based on this new trajectory-based analysis, we exhibit the theoretical estimate for the exponent of super-linear convergence and further improve the statistical error bound at the finite-sample level. Our analysis provides a new framework for studying the behavior of EM for Mixed Linear Regression.
Recent research in mechanistic interpretability has attempted to reverse-engineer Transformer models by carefully inspecting network weights and activations. However, these approaches require considerable manual effort and still fall short of providing complete, faithful descriptions of the underlying algorithms. In this work, we introduce a procedure for training Transformers that are mechanistically interpretable by design. We build on RASP [Weiss et al., 2021], a programming language that can be compiled into Transformer weights. Instead of compiling human-written programs into Transformers, we design a modified Transformer that can be trained using gradient-based optimization and then automatically converted into a discrete, human-readable program. We refer to these models as Transformer Programs. To validate our approach, we learn Transformer Programs for a variety of problems, including an in-context learning task, a suite of algorithmic problems (e.g. sorting, recognizing Dyck languages), and NLP tasks including named entity recognition and text classification. The Transformer Programs can automatically find reasonable solutions, performing on par with standard Transformers of comparable size; and, more importantly, they are easy to interpret. To demonstrate these advantages, we convert Transformers into Python programs and use off-the-shelf code analysis tools to debug model errors and identify the "circuits" used to solve different sub-problems. We hope that Transformer Programs open a new path toward the goal of intrinsically interpretable machine learning.
Machine learning inference pipelines commonly encountered in data science and industries often require real-time responsiveness due to their user-facing nature. However, meeting this requirement becomes particularly challenging when certain input features require aggregating a large volume of data online. Recent literature on interpretable machine learning reveals that most machine learning models exhibit a notable degree of resilience to variations in input. This suggests that machine learning models can effectively accommodate approximate input features with minimal discernible impact on accuracy. In this paper, we introduce Biathlon, a novel ML serving system that leverages the inherent resilience of models and determines the optimal degree of approximation for each aggregation feature. This approach enables maximum speedup while ensuring a guaranteed bound on accuracy loss. We evaluate Biathlon on real pipelines from both industry applications and data science competitions, demonstrating its ability to meet real-time latency requirements by achieving 5.3x to 16.6x speedup with almost no accuracy loss.
Neural ranking models have become increasingly popular for real-world search and recommendation systems in recent years. Unlike their tree-based counterparts, neural models are much less interpretable. That is, it is very difficult to understand their inner workings and answer questions like how do they make their ranking decisions? or what document features do they find important? This is particularly disadvantageous since interpretability is highly important for real-world systems. In this work, we explore feature selection for neural learning-to-rank (LTR). In particular, we investigate six widely-used methods from the field of interpretable machine learning (ML) and introduce our own modification, to select the input features that are most important to the ranking behavior. To understand whether these methods are useful for practitioners, we further study whether they contribute to efficiency enhancement. Our experimental results reveal a large feature redundancy in several LTR benchmarks: the local selection method TabNet can achieve optimal ranking performance with less than 10 features; the global methods, particularly our G-L2X, require slightly more selected features, but exhibit higher potential in improving efficiency. We hope that our analysis of these feature selection methods will bring the fields of interpretable ML and LTR closer together.
Mathematical equations have been unreasonably effective in describing complex natural phenomena across various scientific disciplines. However, discovering such insightful equations from data presents significant challenges due to the necessity of navigating extremely large combinatorial hypothesis spaces. Current methods of equation discovery, commonly known as symbolic regression techniques, largely focus on extracting equations from data alone, often neglecting the domain-specific prior knowledge that scientists typically depend on. They also employ limited representations such as expression trees, constraining the search space and expressiveness of equations. To bridge this gap, we introduce LLM-SR, a novel approach that leverages the extensive scientific knowledge and robust code generation capabilities of Large Language Models (LLMs) to discover scientific equations from data. Specifically, LLM-SR treats equations as programs with mathematical operators and combines LLMs' scientific priors with evolutionary search over equation programs. The LLM iteratively proposes new equation skeleton hypotheses, drawing from its domain knowledge, which are then optimized against data to estimate parameters. We evaluate LLM-SR on four benchmark problems across diverse scientific domains (e.g., physics, biology), which we carefully designed to simulate the discovery process and prevent LLM recitation. Our results demonstrate that LLM-SR discovers physically accurate equations that significantly outperform state-of-the-art symbolic regression baselines, particularly in out-of-domain test settings. We also show that LLM-SR's incorporation of scientific priors enables more efficient equation space exploration than the baselines. Code and data are available: https://github.com/deep-symbolic-mathematics/LLM-SR
Decision Trees are prominent prediction models for interpretable Machine Learning. They have been thoroughly researched, mostly in the batch setting with a fixed labelled dataset, leading to popular algorithms such as C4.5, ID3 and CART. Unfortunately, these methods are of heuristic nature, they rely on greedy splits offering no guarantees of global optimality and often leading to unnecessarily complex and hard-to-interpret Decision Trees. Recent breakthroughs addressed this suboptimality issue in the batch setting, but no such work has considered the online setting with data arriving in a stream. To this end, we devise a new Monte Carlo Tree Search algorithm, Thompson Sampling Decision Trees (TSDT), able to produce optimal Decision Trees in an online setting. We analyse our algorithm and prove its almost sure convergence to the optimal tree. Furthermore, we conduct extensive experiments to validate our findings empirically. The proposed TSDT outperforms existing algorithms on several benchmarks, all while presenting the practical advantage of being tailored to the online setting.
We propose a novel architecture and method of explainable classification with Concept Bottleneck Models (CBMs). While SOTA approaches to Image Classification task work as a black box, there is a growing demand for models that would provide interpreted results. Such a models often learn to predict the distribution over class labels using additional description of this target instances, called concepts. However, existing Bottleneck methods have a number of limitations: their accuracy is lower than that of a standard model and CBMs require an additional set of concepts to leverage. We provide a framework for creating Concept Bottleneck Model from pre-trained multi-modal encoder and new CLIP-like architectures. By introducing a new type of layers known as Concept Bottleneck Layers, we outline three methods for training them: with $\ell_1$-loss, contrastive loss and loss function based on Gumbel-Softmax distribution (Sparse-CBM), while final FC layer is still trained with Cross-Entropy. We show a significant increase in accuracy using sparse hidden layers in CLIP-based bottleneck models. Which means that sparse representation of concepts activation vector is meaningful in Concept Bottleneck Models. Moreover, with our Concept Matrix Search algorithm we can improve CLIP predictions on complex datasets without any additional training or fine-tuning. The code is available at: https://github.com/Andron00e/SparseCBM.
The recently developed Prior-Data Fitted Networks (PFNs) have shown very promising results for applications in low-data regimes. The TabPFN model, a special case of PFNs for tabular data, is able to achieve state-of-the-art performance on a variety of classification tasks while producing posterior predictive distributions in mere seconds by in-context learning without the need for learning parameters or hyperparameter tuning. This makes TabPFN a very attractive option for a wide range of domain applications. However, a major drawback of the method is its lack of interpretability. Therefore, we propose several adaptations of popular interpretability methods that we specifically design for TabPFN. By taking advantage of the unique properties of the model, our adaptations allow for more efficient computations than existing implementations. In particular, we show how in-context learning facilitates the estimation of Shapley values by avoiding approximate retraining and enables the use of Leave-One-Covariate-Out (LOCO) even when working with large-scale Transformers. In addition, we demonstrate how data valuation methods can be used to address scalability challenges of TabPFN. Our proposed methods are implemented in a package tabpfn_iml and made available at https://github.com/david-rundel/tabpfn_iml.
With the spread and rapid advancement of black box machine learning models, the field of interpretable machine learning (IML) or explainable artificial intelligence (XAI) has become increasingly important over the last decade. This is particularly relevant for survival analysis, where the adoption of IML techniques promotes transparency, accountability and fairness in sensitive areas, such as clinical decision making processes, the development of targeted therapies, interventions or in other medical or healthcare related contexts. More specifically, explainability can uncover a survival model's potential biases and limitations and provide more mathematically sound ways to understand how and which features are influential for prediction or constitute risk factors. However, the lack of readily available IML methods may have deterred medical practitioners and policy makers in public health from leveraging the full potential of machine learning for predicting time-to-event data. We present a comprehensive review of the limited existing amount of work on IML methods for survival analysis within the context of the general IML taxonomy. In addition, we formally detail how commonly used IML methods, such as such as individual conditional expectation (ICE), partial dependence plots (PDP), accumulated local effects (ALE), different feature importance measures or Friedman's H-interaction statistics can be adapted to survival outcomes. An application of several IML methods to real data on data on under-5 year mortality of Ghanaian children from the Demographic and Health Surveys (DHS) Program serves as a tutorial or guide for researchers, on how to utilize the techniques in practice to facilitate understanding of model decisions or predictions.
Prediction of the Solar Energetic Particle (SEP) events garner increasing interest as space missions extend beyond Earth's protective magnetosphere. These events, which are, in most cases, products of magnetic reconnection-driven processes during solar flares or fast coronal-mass-ejection-driven shock waves, pose significant radiation hazards to aviation, space-based electronics, and particularly, space exploration. In this work, we utilize the recently developed dataset that combines the Solar Dynamics Observatory/Helioseismic and Magnetic Imager's (SDO/HMI) Space weather HMI Active Region Patches (SHARP) and the Solar and Heliospheric Observatory/Michelson Doppler Imager's (SoHO/MDI) Space Weather MDI Active Region Patches (SMARP). We employ a suite of machine learning strategies, including Support Vector Machines (SVM) and regression models, to evaluate the predictive potential of this new data product for a forecast of post-solar flare SEP events. Our study indicates that despite the augmented volume of data, the prediction accuracy reaches 0.7 +- 0.1, which aligns with but does not exceed these published benchmarks. A linear SVM model with training and testing configurations that mimic an operational setting (positive-negative imbalance) reveals a slight increase (+ 0.04 +- 0.05) in the accuracy of a 14-hour SEP forecast compared to previous studies. This outcome emphasizes the imperative for more sophisticated, physics-informed models to better understand the underlying processes leading to SEP events.
Interpretable machine learning has exploded as an area of interest over the last decade, sparked by the rise of increasingly large datasets and deep neural networks. Simultaneously, large language models (LLMs) have demonstrated remarkable capabilities across a wide array of tasks, offering a chance to rethink opportunities in interpretable machine learning. Notably, the capability to explain in natural language allows LLMs to expand the scale and complexity of patterns that can be given to a human. However, these new capabilities raise new challenges, such as hallucinated explanations and immense computational costs. In this position paper, we start by reviewing existing methods to evaluate the emerging field of LLM interpretation (both interpreting LLMs and using LLMs for explanation). We contend that, despite their limitations, LLMs hold the opportunity to redefine interpretability with a more ambitious scope across many applications, including in auditing LLMs themselves. We highlight two emerging research priorities for LLM interpretation: using LLMs to directly analyze new datasets and to generate interactive explanations.
Symbolic regression aims to derive interpretable symbolic expressions from data in order to better understand and interpret data. %which plays an important role in knowledge discovery and interpretable machine learning. In this study, a symbolic network called PruneSymNet is proposed for symbolic regression. This is a novel neural network whose activation function consists of common elementary functions and operators. The whole network is differentiable and can be trained by gradient descent method. Each subnetwork in the network corresponds to an expression, and our goal is to extract such subnetworks to get the desired symbolic expression. Therefore, a greedy pruning algorithm is proposed to prune the network into a subnetwork while ensuring the accuracy of data fitting. The proposed greedy pruning algorithm preserves the edge with the least loss in each pruning, but greedy algorithm often can not get the optimal solution. In order to alleviate this problem, we combine beam search during pruning to obtain multiple candidate expressions each time, and finally select the expression with the smallest loss as the final result. It was tested on the public data set and compared with the current popular algorithms. The results showed that the proposed algorithm had better accuracy.
Recently, interpretable machine learning has re-explored concept bottleneck models (CBM). An advantage of this model class is the user's ability to intervene on predicted concept values, affecting the downstream output. In this work, we introduce a method to perform such concept-based interventions on pretrained neural networks, which are not interpretable by design, only given a small validation set with concept labels. Furthermore, we formalise the notion of intervenability as a measure of the effectiveness of concept-based interventions and leverage this definition to fine-tune black boxes. Empirically, we explore the intervenability of black-box classifiers on synthetic tabular and natural image benchmarks. We focus on backbone architectures of varying complexity, from simple, fully connected neural nets to Stable Diffusion. We demonstrate that the proposed fine-tuning improves intervention effectiveness and often yields better-calibrated predictions. To showcase the practical utility of our techniques, we apply them to deep chest X-ray classifiers and show that fine-tuned black boxes are more intervenable than CBMs. Lastly, we establish that our methods are still effective under vision-language-model-based concept annotations, alleviating the need for a human-annotated validation set.
Explainable AI (XAI) and interpretable machine learning methods help to build trust in model predictions and derived insights, yet also present a perverse incentive for analysts to manipulate XAI metrics to support pre-specified conclusions. This paper introduces the concept of X-hacking, a form of p-hacking applied to XAI metrics such as Shap values. We show how an automated machine learning pipeline can be used to search for 'defensible' models that produce a desired explanation while maintaining superior predictive performance to a common baseline. We formulate the trade-off between explanation and accuracy as a multi-objective optimization problem and illustrate the feasibility and severity of X-hacking empirically on familiar real-world datasets. Finally, we suggest possible methods for detection and prevention, and discuss ethical implications for the credibility and reproducibility of XAI research.
Air pollution stands as the fourth leading cause of death globally. While extensive research has been conducted in this domain, most approaches rely on large datasets when it comes to prediction. This limits their applicability in low-resource settings though more vulnerable. This study addresses this gap by proposing a novel machine learning approach for accurate air quality prediction using two months of air quality data. By leveraging the World Weather Repository, the meteorological, air pollutant, and Air Quality Index features from 197 capital cities were considered to predict air quality for the next day. The evaluation of several machine learning models demonstrates the effectiveness of the Random Forest algorithm in generating reliable predictions, particularly when applied to classification rather than regression, approach which enhances the model's generalizability by 42%, achieving a cross-validation score of 0.38 for regression and 0.89 for classification. To instill confidence in the predictions, interpretable machine learning was considered. Finally, a cost estimation comparing the implementation of this solution in high-resource and low-resource settings is presented including a tentative of technology licensing business model. This research highlights the potential for resource-limited countries to independently predict air quality while awaiting larger datasets to further refine their predictions.
Explanations in Computer Vision are often desired, but most Deep Neural Networks can only provide saliency maps with questionable faithfulness. Self-Explaining Neural Networks (SENN) extract interpretable concepts with fidelity, diversity, and grounding to combine them linearly for decision-making. While they can explain what was recognized, initial realizations lack accuracy and general applicability. We propose the Quantized-Self-Explaining Neural Network Q-SENN. Q-SENN satisfies or exceeds the desiderata of SENN while being applicable to more complex datasets and maintaining most or all of the accuracy of an uninterpretable baseline model, out-performing previous work in all considered metrics. Q-SENN describes the relationship between every class and feature as either positive, negative or neutral instead of an arbitrary number of possible relations, enforcing more binary human-friendly features. Since every class is assigned just 5 interpretable features on average, Q-SENN shows convincing local and global interpretability. Additionally, we propose a feature alignment method, capable of aligning learned features with human language-based concepts without additional supervision. Thus, what is learned can be more easily verbalized. The code is published: https://github.com/ThomasNorr/Q-SENN
In Federated Learning, clients train models on local data and send updates to a central server, which aggregates them into a global model using a fusion algorithm. This collaborative yet privacy-preserving training comes at a cost. FL developers face significant challenges in attributing global model predictions to specific clients. Localizing responsible clients is a crucial step towards (a) excluding clients primarily responsible for incorrect predictions and (b) encouraging clients who contributed high-quality models to continue participating in the future. Existing ML debugging approaches are inherently inapplicable as they are designed for single-model, centralized training. We introduce TraceFL, a fine-grained neuron provenance capturing mechanism that identifies clients responsible for a global model's prediction by tracking the flow of information from individual clients to the global model. Since inference on different inputs activates a different set of neurons of the global model, TraceFL dynamically quantifies the significance of the global model's neurons in a given prediction, identifying the most crucial neurons in the global model. It then maps them to the corresponding neurons in every participating client to determine each client's contribution, ultimately localizing the responsible client. We evaluate TraceFL on six datasets, including two real-world medical imaging datasets and four neural networks, including advanced models such as GPT. TraceFL achieves 99% accuracy in localizing the responsible client in FL tasks spanning both image and text classification tasks. At a time when state-of-the-artML debugging approaches are mostly domain-specific (e.g., image classification only), TraceFL is the first technique to enable highly accurate automated reasoning across a wide range of FL applications.
In the age of music streaming platforms, the task of automatically tagging music audio has garnered significant attention, driving researchers to devise methods aimed at enhancing performance metrics on standard datasets. Most recent approaches rely on deep neural networks, which, despite their impressive performance, possess opacity, making it challenging to elucidate their output for a given input. While the issue of interpretability has been emphasized in other fields like medicine, it has not received attention in music-related tasks. In this study, we explored the relevance of interpretability in the context of automatic music tagging. We constructed a workflow that incorporates three different information extraction techniques: a) leveraging symbolic knowledge, b) utilizing auxiliary deep neural networks, and c) employing signal processing to extract perceptual features from audio files. These features were subsequently used to train an interpretable machine-learning model for tag prediction. We conducted experiments on two datasets, namely the MTG-Jamendo dataset and the GTZAN dataset. Our method surpassed the performance of baseline models in both tasks and, in certain instances, demonstrated competitiveness with the current state-of-the-art. We conclude that there are use cases where the deterioration in performance is outweighed by the value of interpretability.
Symbolic regression (SR) is an area of interpretable machine learning that aims to identify mathematical expressions, often composed of simple functions, that best fit in a given set of covariates $X$ and response $y$. In recent years, deep symbolic regression (DSR) has emerged as a popular method in the field by leveraging deep reinforcement learning to solve the complicated combinatorial search problem. In this work, we propose an alternative framework (GFN-SR) to approach SR with deep learning. We model the construction of an expression tree as traversing through a directed acyclic graph (DAG) so that GFlowNet can learn a stochastic policy to generate such trees sequentially. Enhanced with an adaptive reward baseline, our method is capable of generating a diverse set of best-fitting expressions. Notably, we observe that GFN-SR outperforms other SR algorithms in noisy data regimes, owing to its ability to learn a distribution of rewards over a space of candidate solutions.