Temporal-difference (TD) learning is widely regarded as one of the most popular algorithms in reinforcement learning (RL). Despite its widespread use, it has only been recently that researchers have begun to actively study its finite time behavior, including the finite time bound on mean squared error and sample complexity. On the empirical side, experience replay has been a key ingredient in the success of deep RL algorithms, but its theoretical effects on RL have yet to be fully understood. In this paper, we present a simple decomposition of the Markovian noise terms and provide finite-time error bounds for TD-learning with experience replay. Specifically, under the Markovian observation model, we demonstrate that for both the averaged iterate and final iterate cases, the error term induced by a constant step-size can be effectively controlled by the size of the replay buffer and the mini-batch sampled from the experience replay buffer.
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The derivation of mathematical results in specialised fields, using Large Language Models (LLMs), is an emerging research direction that can help identify models' limitations, and potentially support mathematical discovery. In this paper, we leverage a symbolic engine to generate derivations of equations at scale, and investigate the capabilities of LLMs when deriving goal equations from premises. Specifically, we employ in-context learning for GPT and fine-tune a range of T5 models to compare the robustness and generalisation of pre-training strategies to specialised models. Empirical results show that fine-tuned FLAN-T5-large (MathT5) outperforms GPT models on all static and out-of-distribution test sets in conventional scores. However, an in-depth analysis reveals that the fine-tuned models are more sensitive to perturbations involving unseen symbols and (to a lesser extent) changes to equation structure. In addition, we analyse 1.7K equations, and over 200 derivations, to highlight common reasoning errors such as the inclusion of incorrect, irrelevant, and redundant equations. Finally, we explore the suitability of existing metrics for evaluating mathematical derivations and find evidence that, while they can capture general properties such as sensitivity to perturbations, they fail to highlight fine-grained reasoning errors and essential differences between models. Overall, this work demonstrates that training models on synthetic data may improve their math capabilities beyond much larger LLMs, but current metrics are not appropriately assessing the quality of generated mathematical text.
A modern challenge of Artificial Intelligence is learning multiple patterns at once (i.e.parallel learning). While this can not be accomplished by standard Hebbian associative neural networks, in this paper we show how the Multitasking Hebbian Network (a variation on theme of the Hopfield model working on sparse data-sets) is naturally able to perform this complex task. We focus on systems processing in parallel a finite (up to logarithmic growth in the size of the network) amount of patterns, mirroring the low-storage level of standard associative neural networks at work with pattern recognition. For mild dilution in the patterns, the network handles them hierarchically, distributing the amplitudes of their signals as power-laws w.r.t. their information content (hierarchical regime), while, for strong dilution, all the signals pertaining to all the patterns are raised with the same strength (parallel regime). Further, confined to the low-storage setting (i.e., far from the spin glass limit), the presence of a teacher neither alters the multitasking performances nor changes the thresholds for learning: the latter are the same whatever the training protocol is supervised or unsupervised. Results obtained through statistical mechanics, signal-to-noise technique and Monte Carlo simulations are overall in perfect agreement and carry interesting insights on multiple learning at once: for instance, whenever the cost-function of the model is minimized in parallel on several patterns (in its description via Statistical Mechanics), the same happens to the standard sum-squared error Loss function (typically used in Machine Learning).
Federated learning (FL) is a privacy-preserving collaborative learning framework, and differential privacy can be applied to further enhance its privacy protection. Existing FL systems typically adopt Federated Average (FedAvg) as the training algorithm and implement differential privacy with a Gaussian mechanism. However, the inherent privacy-utility trade-off in these systems severely degrades the training performance if a tight privacy budget is enforced. Besides, the Gaussian mechanism requires model weights to be of high-precision. To improve communication efficiency and achieve a better privacy-utility trade-off, we propose a communication-efficient FL training algorithm with differential privacy guarantee. Specifically, we propose to adopt binary neural networks (BNNs) and introduce discrete noise in the FL setting. Binary model parameters are uploaded for higher communication efficiency and discrete noise is added to achieve the client-level differential privacy protection. The achieved performance guarantee is rigorously proved, and it is shown to depend on the level of discrete noise. Experimental results based on MNIST and Fashion-MNIST datasets will demonstrate that the proposed training algorithm achieves client-level privacy protection with performance gain while enjoying the benefits of low communication overhead from binary model updates.
Covert communication is focused on hiding the mere existence of communication from unwanted listeners via the physical layer. In this work, we consider the problem of perfect covert communication in wireless networks. Specifically, harnessing an Intelligent Reflecting Surface (IRS), we turn our attention to schemes that allow the transmitter to completely hide the communication, with zero energy at the unwanted listener (Willie) and hence zero probability of detection. Applications of such schemes go beyond simple covertness, as we prevent detectability or decoding even when the codebook, timings, and channel characteristics are known to Willie. That is, perfect covertness also ensures Willie is unable to decode, even assuming communication took place and knowing the codebook. We define perfect covertness, give a necessary and sufficient condition for it in IRS-assisted communication, and define the optimization problem. For $N=2$ IRS elements, we analyze the probability of finding a solution and derive its closed form. We then investigate the problem of $N>2$ IRS elements by analyzing the probability of such a zero-detection solution. We prove that this probability converges to $1$ as the number of IRS tends to infinity. We provide an iterative algorithm to find a perfectly covert solution and prove its convergence. The results are also supported by simulations, showing that a small amount of IRS elements allows for a positive rate at the legitimate user yet with zero probability of detection at an unwanted listener.
Ridge estimation is an important manifold learning technique. The goal of this paper is to examine the effects of nonlinear transformations on the ridge sets. The main result proves the inclusion relationship between ridges: $\cR(f\circ p)\subseteq \cR(p)$, provided that the transformation $f$ is strictly increasing and concave on the range of the function $p$. Additionally, given an underlying true manifold $\cM$, we show that the Hausdorff distance between $\cR(f\circ p)$ and its projection onto $\cM$ is smaller than the Hausdorff distance between $\cR(p)$ and the corresponding projection. This motivates us to apply an increasing and concave transformation before the ridge estimation. In specific, we show that the power transformations $f^{q}(y)=y^q/q,-\infty<q\leq 1$ are increasing and concave on $\RR_+$, and thus we can use such power transformations when $p$ is strictly positive. Numerical experiments demonstrate the advantages of the proposed methods.
Clinical decisions are often guided by clinical prediction models or diagnostic tests. Decision curve analysis (DCA) combines classical assessment of predictive performance with the consequences of using these strategies for clinical decision-making. In DCA, the best decision strategy is the one that maximizes the so-called net benefit: the net number of true positives (or negatives) provided by a given strategy. In this decision-analytic approach, often only point estimates are published. If uncertainty is reported, a risk-neutral interpretation is recommended: it motivates further research without changing the conclusions based on currently-available data. However, when it comes to new decision strategies, replacing the current Standard of Care must be carefully considered -- prematurely implementing a suboptimal strategy poses potentially irrecoverable costs. In this risk-averse setting, quantifying uncertainty may also inform whether the available data provides enough evidence to change current clinical practice. Here, we employ Bayesian approaches to DCA addressing four fundamental concerns when evaluating clinical decision strategies: (i) which strategies are clinically useful, (ii) what is the best available decision strategy, (iii) pairwise comparisons between strategies, and (iv) the expected net benefit loss associated with the current level of uncertainty. While often consistent with frequentist point estimates, fully Bayesian DCA allows for an intuitive probabilistic interpretation framework and the incorporation of prior evidence. We evaluate the methods using simulation and provide a comprehensive case study. Software implementation is available in the bayesDCA R package. Ultimately, the Bayesian DCA workflow may help clinicians and health policymakers adopt better-informed decisions.
Recently, contrastive learning (CL) has emerged as a successful method for unsupervised graph representation learning. Most graph CL methods first perform stochastic augmentation on the input graph to obtain two graph views and maximize the agreement of representations in the two views. Despite the prosperous development of graph CL methods, the design of graph augmentation schemes -- a crucial component in CL -- remains rarely explored. We argue that the data augmentation schemes should preserve intrinsic structures and attributes of graphs, which will force the model to learn representations that are insensitive to perturbation on unimportant nodes and edges. However, most existing methods adopt uniform data augmentation schemes, like uniformly dropping edges and uniformly shuffling features, leading to suboptimal performance. In this paper, we propose a novel graph contrastive representation learning method with adaptive augmentation that incorporates various priors for topological and semantic aspects of the graph. Specifically, on the topology level, we design augmentation schemes based on node centrality measures to highlight important connective structures. On the node attribute level, we corrupt node features by adding more noise to unimportant node features, to enforce the model to recognize underlying semantic information. We perform extensive experiments of node classification on a variety of real-world datasets. Experimental results demonstrate that our proposed method consistently outperforms existing state-of-the-art baselines and even surpasses some supervised counterparts, which validates the effectiveness of the proposed contrastive framework with adaptive augmentation.
Graph Neural Networks (GNNs) have proven to be useful for many different practical applications. However, many existing GNN models have implicitly assumed homophily among the nodes connected in the graph, and therefore have largely overlooked the important setting of heterophily, where most connected nodes are from different classes. In this work, we propose a novel framework called CPGNN that generalizes GNNs for graphs with either homophily or heterophily. The proposed framework incorporates an interpretable compatibility matrix for modeling the heterophily or homophily level in the graph, which can be learned in an end-to-end fashion, enabling it to go beyond the assumption of strong homophily. Theoretically, we show that replacing the compatibility matrix in our framework with the identity (which represents pure homophily) reduces to GCN. Our extensive experiments demonstrate the effectiveness of our approach in more realistic and challenging experimental settings with significantly less training data compared to previous works: CPGNN variants achieve state-of-the-art results in heterophily settings with or without contextual node features, while maintaining comparable performance in homophily settings.
Graph Neural Networks (GNN) is an emerging field for learning on non-Euclidean data. Recently, there has been increased interest in designing GNN that scales to large graphs. Most existing methods use "graph sampling" or "layer-wise sampling" techniques to reduce training time. However, these methods still suffer from degrading performance and scalability problems when applying to graphs with billions of edges. This paper presents GBP, a scalable GNN that utilizes a localized bidirectional propagation process from both the feature vectors and the training/testing nodes. Theoretical analysis shows that GBP is the first method that achieves sub-linear time complexity for both the precomputation and the training phases. An extensive empirical study demonstrates that GBP achieves state-of-the-art performance with significantly less training/testing time. Most notably, GBP can deliver superior performance on a graph with over 60 million nodes and 1.8 billion edges in less than half an hour on a single machine.
It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.
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