Machine Learning (cs.LG)

Abstract: Predicting the dynamics of chaotic systems is one of the most challenging tasks for neural networks, and machine learning in general. Here we aim to predict the spatiotemporal chaotic dynamics of a high-dimensional non-linear system. In our attempt we use the TensorFlow library, representing the state of the art for deep neural networks training and prediction. While our results are encouraging, and show that the dynamics of the considered system can be predicted for short time, we also indirectly discovered an unexpected and undesirable behavior of the TensorFlow library. More specifically, the longer term prediction of the system's chaotic behavior quickly deteriorates and blows up due to the nondeterministic behavior of the TensorFlow library. Here we provide numerical evidence of the short time prediction ability, and of the longer term predictability blow up.

Abstract: A rising trend in theoretical deep learning is to understand why deep learning works through Neural Tangent Kernel (NTK) [jgh18], a kernel method that is equivalent to using gradient descent to train a multi-layer infinitely-wide neural network. NTK is a major step forward in the theoretical deep learning because it allows researchers to use traditional mathematical tools to analyze properties of deep neural networks and to explain various neural network techniques from a theoretical view. A natural extension of NTK on graph learning is \textit{Graph Neural Tangent Kernel (GNTK)}, and researchers have already provide GNTK formulation for graph-level regression and show empirically that this kernel method can achieve similar accuracy as GNNs on various bioinformatics datasets [dhs+19]. The remaining question now is whether solving GNTK regression is equivalent to training an infinite-wide multi-layer GNN using gradient descent. In this paper, we provide three new theoretical results. First, we formally prove this equivalence for graph-level regression. Second, we present the first GNTK formulation for node-level regression. Finally, we prove the equivalence for node-level regression.

Abstract: In this paper, we exploit the unique properties of a deterministic projected belief network (D-PBN) to take full advantage of trainable compound activation functions (TCAs). A D-PBN is a type of auto-encoder that operates by "backing up" through a feed-forward neural network. TCAs are activation functions with complex monotonic-increasing shapes that change the distribution of the data so that the linear transformation that follows is more effective. Because a D-PBN operates by "backing up", the TCAs are inverted in the reconstruction process, restoring the original distribution of the data, thus taking advantage of a given TCA in both analysis and reconstruction. In this paper, we show that a D-PBN auto-encoder with TCAs can significantly out-perform standard auto-encoders including variational auto-encoders.

Abstract: By identifying similarities between successive inputs, Self-Supervised Learning (SSL) methods for time series analysis have demonstrated their effectiveness in encoding the inherent static characteristics of temporal data. However, an exclusive emphasis on similarities might result in representations that overlook the dynamic attributes critical for modeling cardiovascular diseases within a confined subject cohort. Introducing Distilled Encoding Beyond Similarities (DEBS), this paper pioneers an SSL approach that transcends mere similarities by integrating dissimilarities among positive pairs. The framework is applied to electrocardiogram (ECG) signals, leading to a notable enhancement of +10\% in the detection accuracy of Atrial Fibrillation (AFib) across diverse subjects. DEBS underscores the potential of attaining a more refined representation by encoding the dynamic characteristics of time series data, tapping into dissimilarities during the optimization process. Broadly, the strategy delineated in this study holds the promise of unearthing novel avenues for advancing SSL methodologies tailored to temporal data.

Abstract: We study the classical problem of approximating a non-decreasing function $f: \mathcal{X} \to \mathcal{Y}$ in $L^p(\mu)$ norm by sequentially querying its values, for known compact real intervals $\mathcal{X}$, $\mathcal{Y}$ and a known probability measure $\mu$ on $\cX$. For any function~$f$ we characterize the minimum number of evaluations of $f$ that algorithms need to guarantee an approximation $\hat{f}$ with an $L^p(\mu)$ error below $\epsilon$ after stopping. Unlike worst-case results that hold uniformly over all $f$, our complexity measure is dependent on each specific function $f$. To address this problem, we introduce GreedyBox, a generalization of an algorithm originally proposed by Novak (1992) for numerical integration. We prove that GreedyBox achieves an optimal sample complexity for any function $f$, up to logarithmic factors. Additionally, we uncover results regarding piecewise-smooth functions. Perhaps as expected, the $L^p(\mu)$ error of GreedyBox decreases much faster for piecewise-$C^2$ functions than predicted by the algorithm (without any knowledge on the smoothness of $f$). A simple modification even achieves optimal minimax approximation rates for such functions, which we compute explicitly. In particular, our findings highlight multiple performance gaps between adaptive and non-adaptive algorithms, smooth and piecewise-smooth functions, as well as monotone or non-monotone functions. Finally, we provide numerical experiments to support our theoretical results.

Abstract: The increasing popularity of large language models (LLMs) has paved the way for their application in diverse domains. This paper proposes a benchmarking framework tailored specifically for evaluating LLM performance in the context of Verilog code generation for hardware design and verification. We present a comprehensive evaluation dataset consisting of 156 problems from the Verilog instructional website HDLBits. The evaluation set consists of a diverse set of Verilog code generation tasks, ranging from simple combinational circuits to complex finite state machines. The Verilog code completions can be automatically tested for functional correctness by comparing the transient simulation outputs of the generated design with a golden solution. We also demonstrate that the Verilog code generation capability of pretrained language models could be improved with supervised fine-tuning by bootstrapping with LLM generated synthetic problem-code pairs.

Abstract: We present a novel approach to address the challenge of generalization in offline reinforcement learning (RL), where the agent learns from a fixed dataset without any additional interaction with the environment. Specifically, we aim to improve the agent's ability to generalize to out-of-distribution goals. To achieve this, we propose to learn a dynamics model and check if it is equivariant with respect to a fixed type of transformation, namely translations in the state space. We then use an entropy regularizer to increase the equivariant set and augment the dataset with the resulting transformed samples. Finally, we learn a new policy offline based on the augmented dataset, with an off-the-shelf offline RL algorithm. Our experimental results demonstrate that our approach can greatly improve the test performance of the policy on the considered environments.

Abstract: In recent years, Transformer-based auto-attention mechanisms have been successfully applied to the analysis of a variety of context-reliant data types, from texts to images and beyond, including data from non-Euclidean geometries. In this paper, we present such a mechanism, designed to classify sequences of Symmetric Positive Definite matrices while preserving their Riemannian geometry throughout the analysis. We apply our method to automatic sleep staging on timeseries of EEG-derived covariance matrices from a standard dataset, obtaining high levels of stage-wise performance.

Abstract: Variable importance assessment has become a crucial step in machine-learning applications when using complex learners, such as deep neural networks, on large-scale data. Removal-based importance assessment is currently the reference approach, particularly when statistical guarantees are sought to justify variable inclusion. It is often implemented with variable permutation schemes. On the flip side, these approaches risk misidentifying unimportant variables as important in the presence of correlations among covariates. Here we develop a systematic approach for studying Conditional Permutation Importance (CPI) that is model agnostic and computationally lean, as well as reusable benchmarks of state-of-the-art variable importance estimators. We show theoretically and empirically that $\textit{CPI}$ overcomes the limitations of standard permutation importance by providing accurate type-I error control. When used with a deep neural network, $\textit{CPI}$ consistently showed top accuracy across benchmarks. An empirical benchmark on real-world data analysis in a large-scale medical dataset showed that $\textit{CPI}$ provides a more parsimonious selection of statistically significant variables. Our results suggest that $\textit{CPI}$ can be readily used as drop-in replacement for permutation-based methods.

Abstract: Community question answering (CQA) forums are Internet-based platforms where users ask questions about a topic and other expert users try to provide solutions. Many CQA forums such as Quora, Stackoverflow, Yahoo!Answer, StackExchange exist with a lot of user-generated data. These data are leveraged in automated CQA ranking systems where similar questions (and answers) are presented in response to the query of the user. In this work, we empirically investigate a few aspects of this domain. Firstly, in addition to traditional features like TF-IDF, BM25 etc., we introduce a BERT-based feature that captures the semantic similarity between the question and answer. Secondly, most of the existing research works have focused on features extracted only from the question part; features extracted from answers have not been explored extensively. We combine both types of features in a linear fashion. Thirdly, using our proposed concepts, we conduct an empirical investigation with different rank-learning algorithms, some of which have not been used so far in CQA domain. On three standard CQA datasets, our proposed framework achieves state-of-the-art performance. We also analyze importance of the features we use in our investigation. This work is expected to guide the practitioners to select a better set of features for the CQA retrieval task.

Abstract: In this paper, we consider the intersection of two problems in machine learning: Multi-Source Domain Adaptation (MSDA) and Dataset Distillation (DD). On the one hand, the first considers adapting multiple heterogeneous labeled source domains to an unlabeled target domain. On the other hand, the second attacks the problem of synthesizing a small summary containing all the information about the datasets. We thus consider a new problem called MSDA-DD. To solve it, we adapt previous works in the MSDA literature, such as Wasserstein Barycenter Transport and Dataset Dictionary Learning, as well as DD method Distribution Matching. We thoroughly experiment with this novel problem on four benchmarks (Caltech-Office 10, Tennessee-Eastman Process, Continuous Stirred Tank Reactor, and Case Western Reserve University), where we show that, even with as little as 1 sample per class, one achieves state-of-the-art adaptation performance.

Abstract: In this article, we propose an approach for federated domain adaptation, a setting where distributional shift exists among clients and some have unlabeled data. The proposed framework, FedDaDiL, tackles the resulting challenge through dictionary learning of empirical distributions. In our setting, clients' distributions represent particular domains, and FedDaDiL collectively trains a federated dictionary of empirical distributions. In particular, we build upon the Dataset Dictionary Learning framework by designing collaborative communication protocols and aggregation operations. The chosen protocols keep clients' data private, thus enhancing overall privacy compared to its centralized counterpart. We empirically demonstrate that our approach successfully generates labeled data on the target domain with extensive experiments on (i) Caltech-Office, (ii) TEP, and (iii) CWRU benchmarks. Furthermore, we compare our method to its centralized counterpart and other benchmarks in federated domain adaptation.

Abstract: Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge, especially when encountering noisy observations and no prior knowledge available. This study proposes R-DISCOVER, a framework designed to robustly uncover open-form partial differential equations (PDEs) from limited and noisy data. The framework operates through two alternating update processes: discovering and embedding. The discovering phase employs symbolic representation and a reinforcement learning (RL)-guided hybrid PDE generator to efficiently produce diverse open-form PDEs with tree structures. A neural network-based predictive model fits the system response and serves as the reward evaluator for the generated PDEs. PDEs with superior fits are utilized to iteratively optimize the generator via the RL method and the best-performing PDE is selected by a parameter-free stability metric. The embedding phase integrates the initially identified PDE from the discovering process as a physical constraint into the predictive model for robust training. The traversal of PDE trees automates the construction of the computational graph and the embedding process without human intervention. Numerical experiments demonstrate our framework's capability to uncover governing equations from nonlinear dynamic systems with limited and highly noisy data and outperform other physics-informed neural network-based discovery methods. This work opens new potential for exploring real-world systems with limited understanding.

Abstract: Open-ended learning benefits immensely from the use of symbolic methods for goal representation as they offer ways to structure knowledge for efficient and transferable learning. However, the existing Hierarchical Reinforcement Learning (HRL) approaches relying on symbolic reasoning are often limited as they require a manual goal representation. The challenge in autonomously discovering a symbolic goal representation is that it must preserve critical information, such as the environment dynamics. In this paper, we propose a developmental mechanism for goal discovery via an emergent representation that abstracts (i.e., groups together) sets of environment states that have similar roles in the task. We introduce a Feudal HRL algorithm that concurrently learns both the goal representation and a hierarchical policy. The algorithm uses symbolic reachability analysis for neural networks to approximate the transition relation among sets of states and to refine the goal representation. We evaluate our approach on complex navigation tasks, showing the learned representation is interpretable, transferrable and results in data efficient learning.

Abstract: The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the proposed design. In each equation, a deep neural network is used to approximate the unknown function. Three forms of fractional differential equations have been examined to highlight the method's versatility: a fractional ordinary differential equation, a fractional order integrodifferential equation, and a fractional order partial differential equation. The results show that the proposed architecture solves different forms of fractional differential equations with excellent precision.

Abstract: Artificial intelligence models and methods commonly lack causal interpretability. Despite the advancements in interpretable machine learning (IML) methods, they frequently assign importance to features which lack causal influence on the outcome variable. Selecting causally relevant features among those identified as relevant by these methods, or even before model training, would offer a solution. Feature selection methods utilizing information theoretical quantities have been successful in identifying statistically relevant features. However, the information theoretical quantities they are based on do not incorporate causality, rendering them unsuitable for such scenarios. To address this challenge, this article proposes information theoretical quantities that incorporate the causal structure of the system, which can be used to evaluate causal importance of features for some given outcome variable. Specifically, we introduce causal versions of entropy and mutual information, termed causal entropy and causal information gain, which are designed to assess how much control a feature provides over the outcome variable. These newly defined quantities capture changes in the entropy of a variable resulting from interventions on other variables. Fundamental results connecting these quantities to the existence of causal effects are derived. The use of causal information gain in feature selection is demonstrated, highlighting its superiority over standard mutual information in revealing which features provide control over a chosen outcome variable. Our investigation paves the way for the development of methods with improved interpretability in domains involving causation.

Abstract: Financial simulators play an important role in enhancing forecasting accuracy, managing risks, and fostering strategic financial decision-making. Despite the development of financial market simulation methodologies, existing frameworks often struggle with adapting to specialized simulation context. We pinpoint the challenges as i) current financial datasets do not contain context labels; ii) current techniques are not designed to generate financial data with context as control, which demands greater precision compared to other modalities; iii) the inherent difficulties in generating context-aligned, high-fidelity data given the non-stationary, noisy nature of financial data. To address these challenges, our contributions are: i) we proposed the Contextual Market Dataset with market dynamics, stock ticker, and history state as context, leveraging a market dynamics modeling method that combines linear regression and Dynamic Time Warping clustering to extract market dynamics; ii) we present Market-GAN, a novel architecture incorporating a Generative Adversarial Networks (GAN) for the controllable generation with context, an autoencoder for learning low-dimension features, and supervisors for knowledge transfer; iii) we introduce a two-stage training scheme to ensure that Market-GAN captures the intrinsic market distribution with multiple objectives. In the pertaining stage, with the use of the autoencoder and supervisors, we prepare the generator with a better initialization for the adversarial training stage. We propose a set of holistic evaluation metrics that consider alignment, fidelity, data usability on downstream tasks, and market facts. We evaluate Market-GAN with the Dow Jones Industrial Average data from 2000 to 2023 and showcase superior performance in comparison to 4 state-of-the-art time-series generative models.

Abstract: Despite the many successful applications of deep learning models for multidimensional signal and image processing, most traditional neural networks process data represented by (multidimensional) arrays of real numbers. The intercorrelation between feature channels is usually expected to be learned from the training data, requiring numerous parameters and careful training. In contrast, vector-valued neural networks are conceived to process arrays of vectors and naturally consider the intercorrelation between feature channels. Consequently, they usually have fewer parameters and often undergo more robust training than traditional neural networks. This paper aims to present a broad framework for vector-valued neural networks, referred to as V-nets. In this context, hypercomplex-valued neural networks are regarded as vector-valued models with additional algebraic properties. Furthermore, this paper explains the relationship between vector-valued and traditional neural networks. Precisely, a vector-valued neural network can be obtained by placing restrictions on a real-valued model to consider the intercorrelation between feature channels. Finally, we show how V-nets, including hypercomplex-valued neural networks, can be implemented in current deep-learning libraries as real-valued networks.

Abstract: Focus in Explainable AI is shifting from explanations defined in terms of low-level elements, such as input features, to explanations encoded in terms of interpretable concepts learned from data. How to reliably acquire such concepts is, however, still fundamentally unclear. An agreed-upon notion of concept interpretability is missing, with the result that concepts used by both post-hoc explainers and concept-based neural networks are acquired through a variety of mutually incompatible strategies. Critically, most of these neglect the human side of the problem: a representation is understandable only insofar as it can be understood by the human at the receiving end. The key challenge in Human-interpretable Representation Learning (HRL) is how to model and operationalize this human element. In this work, we propose a mathematical framework for acquiring interpretable representations suitable for both post-hoc explainers and concept-based neural networks. Our formalization of HRL builds on recent advances in causal representation learning and explicitly models a human stakeholder as an external observer. This allows us to derive a principled notion of alignment between the machine representation and the vocabulary of concepts understood by the human. In doing so, we link alignment and interpretability through a simple and intuitive name transfer game, and clarify the relationship between alignment and a well-known property of representations, namely disentanglment. We also show that alignment is linked to the issue of undesirable correlations among concepts, also known as concept leakage, and to content-style separation, all through a general information-theoretic reformulation of these properties. Our conceptualization aims to bridge the gap between the human and algorithmic sides of interpretability and establish a stepping stone for new research on human-interpretable representations.

Abstract: The task of preserving privacy while ensuring efficient communication is a fundamental challenge in federated learning. In this work, we tackle this challenge in the trusted aggregator model, and propose a solution that achieves both objectives simultaneously. We show that employing a quantization scheme based on subtractive dithering at the clients can effectively replicate the normal noise addition process at the aggregator. This implies that we can guarantee the same level of differential privacy against other clients while substantially reducing the amount of communication required, as opposed to transmitting full precision gradients and using central noise addition. We also experimentally demonstrate that the accuracy of our proposed approach matches that of the full precision gradient method.

Abstract: Directed graphs are a natural model for many phenomena, in particular scientific knowledge graphs such as molecular interaction or chemical reaction networks that define cellular signaling relationships. In these situations, source nodes typically have distinct biophysical properties from sinks. Due to their ordered and unidirectional relationships, many such networks also have hierarchical and multiscale structure. However, the majority of methods performing node- and edge-level tasks in machine learning do not take these properties into account, and thus have not been leveraged effectively for scientific tasks such as cellular signaling network inference. We propose a new framework called Directed Scattering Autoencoder (DSAE) which uses a directed version of a geometric scattering transform, combined with the non-linear dimensionality reduction properties of an autoencoder and the geometric properties of the hyperbolic space to learn latent hierarchies. We show this method outperforms numerous others on tasks such as embedding directed graphs and learning cellular signaling networks.

Abstract: We introduce beta diffusion, a novel generative modeling method that integrates demasking and denoising to generate data within bounded ranges. Using scaled and shifted beta distributions, beta diffusion utilizes multiplicative transitions over time to create both forward and reverse diffusion processes, maintaining beta distributions in both the forward marginals and the reverse conditionals, given the data at any point in time. Unlike traditional diffusion-based generative models relying on additive Gaussian noise and reweighted evidence lower bounds (ELBOs), beta diffusion is multiplicative and optimized with KL-divergence upper bounds (KLUBs) derived from the convexity of the KL divergence. We demonstrate that the proposed KLUBs are more effective for optimizing beta diffusion compared to negative ELBOs, which can also be derived as the KLUBs of the same KL divergence with its two arguments swapped. The loss function of beta diffusion, expressed in terms of Bregman divergence, further supports the efficacy of KLUBs for optimization. Experimental results on both synthetic data and natural images demonstrate the unique capabilities of beta diffusion in generative modeling of range-bounded data and validate the effectiveness of KLUBs in optimizing diffusion models, thereby making them valuable additions to the family of diffusion-based generative models and the optimization techniques used to train them.

Abstract: In the present paper we introduce new optimization algorithms for the task of density ratio estimation. More precisely, we consider extending the well-known KMM method using the construction of a suitable loss function, in order to encompass more general situations involving the estimation of density ratio with respect to subsets of the training data and test data, respectively. The associated codes can be found at https://github.com/CDAlecsa/Generalized-KMM.

Abstract: Current physics-informed (standard or operator) neural networks still rely on accurately learning the initial conditions of the system they are solving. In contrast, standard numerical methods evolve such initial conditions without needing to learn these. In this study, we propose to improve current physics-informed deep learning strategies such that initial conditions do not need to be learned and are represented exactly in the predicted solution. Moreover, this method guarantees that when a DeepONet is applied multiple times to time step a solution, the resulting function is continuous.