Recent studies have discovered that Chain-of-Thought prompting (CoT) can dramatically improve the performance of Large Language Models (LLMs), particularly when dealing with complex tasks involving mathematics or reasoning. Despite the enormous empirical success, the underlying mechanisms behind CoT and how it unlocks the potential of LLMs remain elusive. In this paper, we take a first step towards theoretically answering these questions. Specifically, we examine the capacity of LLMs with CoT in solving fundamental mathematical and decision-making problems. We start by giving an impossibility result showing that any bounded-depth Transformer cannot directly output correct answers for basic arithmetic/equation tasks unless the model size grows super-polynomially with respect to the input length. In contrast, we then prove by construction that autoregressive Transformers of a constant size suffice to solve both tasks by generating CoT derivations using a commonly-used math language format. Moreover, we show LLMs with CoT are capable of solving a general class of decision-making problems known as Dynamic Programming, thus justifying its power in tackling complex real-world tasks. Finally, extensive experiments on four tasks show that, while Transformers always fail to predict the answers directly, they can consistently learn to generate correct solutions step-by-step given sufficient CoT demonstrations.
相關內容
Providing a model that achieves a strong predictive performance and at the same time is interpretable by humans is one of the most difficult challenges in machine learning research due to the conflicting nature of these two objectives. To address this challenge, we propose a modification of the Radial Basis Function Neural Network model by equipping its Gaussian kernel with a learnable precision matrix. We show that precious information is contained in the spectrum of the precision matrix that can be extracted once the training of the model is completed. In particular, the eigenvectors explain the directions of maximum sensitivity of the model revealing the active subspace and suggesting potential applications for supervised dimensionality reduction. At the same time, the eigenvectors highlight the relationship in terms of absolute variation between the input and the latent variables, thereby allowing us to extract a ranking of the input variables based on their importance to the prediction task enhancing the model interpretability. We conducted numerical experiments for regression, classification, and feature selection tasks, comparing our model against popular machine learning models and the state-of-the-art deep learning-based embedding feature selection techniques. Our results demonstrate that the proposed model does not only yield an attractive prediction performance with respect to the competitors but also provides meaningful and interpretable results that potentially could assist the decision-making process in real-world applications. A PyTorch implementation of the model is available on GitHub at the following link. //github.com/dannyzx/GRBF-NNs
Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications
We study the problem of estimating the convex hull of the image $f(X)\subset\mathbb{R}^n$ of a compact set $X\subset\mathbb{R}^m$ with smooth boundary through a smooth function $f:\mathbb{R}^m\to\mathbb{R}^n$. Assuming that $f$ is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of $f(X)$ and the convex hull of the images $f(x_i)$ of $M$ sampled inputs $x_i$ on the boundary of $X$. When applied to the problem of geometric inference from a random sample, our results give tighter and more general error bounds than the state of the art. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.
In this article we present new results about the expressivity of Graph Neural Networks (GNNs). We prove that for any GNN with piecewise polynomial activations, whose architecture size does not grow with the graph input sizes, there exists a pair of non-isomorphic rooted trees of depth two such that the GNN cannot distinguish their root vertex up to an arbitrary number of iterations. The proof relies on tools from the algebra of symmetric polynomials. In contrast, it was already known that unbounded GNNs (those whose size is allowed to change with the graph sizes) with piecewise polynomial activations can distinguish these vertices in only two iterations. Our results imply a strict separation between bounded and unbounded size GNNs, answering an open question formulated by [Grohe, 2021]. We next prove that if one allows activations that are not piecewise polynomial, then in two iterations a single neuron perceptron can distinguish the root vertices of any pair of nonisomorphic trees of depth two (our results hold for activations like the sigmoid, hyperbolic tan and others). This shows how the power of graph neural networks can change drastically if one changes the activation function of the neural networks. The proof of this result utilizes the Lindemann-Weierstrauss theorem from transcendental number theory.
The two-trials rule for drug approval requires "at least two adequate and well-controlled studies, each convincing on its own, to establish effectiveness". This is usually employed by requiring two significant pivotal trials and is the standard regulatory requirement to provide evidence for a new drug's efficacy. However, there is need to develop suitable alternatives to this rule for a number of reasons, including the possible availability of data from more than two trials. I consider the case of up to 3 studies and stress the importance to control the partial Type-I error rate, where only some studies have a true null effect, while maintaining the overall Type-I error rate of the two-trials rule, where all studies have a null effect. Some less-known $p$-value combination methods are useful to achieve this: Pearson's method, Edgington's method and the recently proposed harmonic mean $\chi^2$-test. I study their properties and discuss how they can be extended to a sequential assessment of success while still ensuring overall Type-I error control. I compare the different methods in terms of partial Type-I error rate, project power and the expected number of studies required. Edgington's method is eventually recommended as it is easy to implement and communicate, has only moderate partial Type-I error rate inflation but substantially increased project power.
Deep neural networks (DNNs) are increasingly being deployed to perform safety-critical tasks. The opacity of DNNs, which prevents humans from reasoning about them, presents new safety and security challenges. To address these challenges, the verification community has begun developing techniques for rigorously analyzing DNNs, with numerous verification algorithms proposed in recent years. While a significant amount of work has gone into developing these verification algorithms, little work has been devoted to rigorously studying the computability and complexity of the underlying theoretical problems. Here, we seek to contribute to the bridging of this gap. We focus on two kinds of DNNs: those that employ piecewise-linear activation functions (e.g., ReLU), and those that employ piecewise-smooth activation functions (e.g., Sigmoids). We prove the two following theorems: 1) The decidability of verifying DNNs with a particular set of piecewise-smooth activation functions is equivalent to a well-known, open problem formulated by Tarski; and 2) The DNN verification problem for any quantifier-free linear arithmetic specification can be reduced to the DNN reachability problem, whose approximation is NP-complete. These results answer two fundamental questions about the computability and complexity of DNN verification, and the ways it is affected by the network's activation functions and error tolerance; and could help guide future efforts in developing DNN verification tools.
Assuming that the term 'metaverse' could be understood as a computer-based implementation of multiverse applications, we started to look in the present work for a logic that would be powerful enough to handle the situations arising both in the real and in the fictional underlying application domains. Realizing that first-order logic fails to account for the unstable behavior of even the most simpleminded information system domains, we resorted to non-conventional extensions, in an attempt to sketch a minimal composite logic strategy. The discussion was kept at a rather informal level, always trying to convey the intuition behind the theoretical notions in natural language terms, and appealing to an AI agent, namely ChatGPT, in the hope that algorithmic and common-sense approaches can be usefully combined.
Recent works have empirically analyzed in-context learning and shown that transformers trained on synthetic linear regression tasks can learn to implement ridge regression, which is the Bayes-optimal predictor, given sufficient capacity [Aky\"urek et al., 2023], while one-layer transformers with linear self-attention and no MLP layer will learn to implement one step of gradient descent (GD) on a least-squares linear regression objective [von Oswald et al., 2022]. However, the theory behind these observations remains poorly understood. We theoretically study transformers with a single layer of linear self-attention, trained on synthetic noisy linear regression data. First, we mathematically show that when the covariates are drawn from a standard Gaussian distribution, the one-layer transformer which minimizes the pre-training loss will implement a single step of GD on the least-squares linear regression objective. Then, we find that changing the distribution of the covariates and weight vector to a non-isotropic Gaussian distribution has a strong impact on the learned algorithm: the global minimizer of the pre-training loss now implements a single step of $\textit{pre-conditioned}$ GD. However, if only the distribution of the responses is changed, then this does not have a large effect on the learned algorithm: even when the response comes from a more general family of $\textit{nonlinear}$ functions, the global minimizer of the pre-training loss still implements a single step of GD on a least-squares linear regression objective.
There is abundant observational data in the software engineering domain, whereas running large-scale controlled experiments is often practically impossible. Thus, most empirical studies can only report statistical correlations -- instead of potentially more insightful and robust causal relations. To support analyzing purely observational data for causal relations, and to assess any differences between purely predictive and causal models of the same data, this paper discusses some novel techniques based on structural causal models (such as directed acyclic graphs of causal Bayesian networks). Using these techniques, one can rigorously express, and partially validate, causal hypotheses; and then use the causal information to guide the construction of a statistical model that captures genuine causal relations -- such that correlation does imply causation. We apply these ideas to analyzing public data about programmer performance in Code Jam, a large world-wide coding contest organized by Google every year. Specifically, we look at the impact of different programming languages on a participant's performance in the contest. While the overall effect associated with programming languages is weak compared to other variables -- regardless of whether we consider correlational or causal links -- we found considerable differences between a purely associational and a causal analysis of the very same data. The takeaway message is that even an imperfect causal analysis of observational data can help answer the salient research questions more precisely and more robustly than with just purely predictive techniques -- where genuine causal effects may be confounded.
The generalization mystery in deep learning is the following: Why do over-parameterized neural networks trained with gradient descent (GD) generalize well on real datasets even though they are capable of fitting random datasets of comparable size? Furthermore, from among all solutions that fit the training data, how does GD find one that generalizes well (when such a well-generalizing solution exists)? We argue that the answer to both questions lies in the interaction of the gradients of different examples during training. Intuitively, if the per-example gradients are well-aligned, that is, if they are coherent, then one may expect GD to be (algorithmically) stable, and hence generalize well. We formalize this argument with an easy to compute and interpretable metric for coherence, and show that the metric takes on very different values on real and random datasets for several common vision networks. The theory also explains a number of other phenomena in deep learning, such as why some examples are reliably learned earlier than others, why early stopping works, and why it is possible to learn from noisy labels. Moreover, since the theory provides a causal explanation of how GD finds a well-generalizing solution when one exists, it motivates a class of simple modifications to GD that attenuate memorization and improve generalization. Generalization in deep learning is an extremely broad phenomenon, and therefore, it requires an equally general explanation. We conclude with a survey of alternative lines of attack on this problem, and argue that the proposed approach is the most viable one on this basis.
Deep Learning algorithms have achieved the state-of-the-art performance for Image Classification and have been used even in security-critical applications, such as biometric recognition systems and self-driving cars. However, recent works have shown those algorithms, which can even surpass the human capabilities, are vulnerable to adversarial examples. In Computer Vision, adversarial examples are images containing subtle perturbations generated by malicious optimization algorithms in order to fool classifiers. As an attempt to mitigate these vulnerabilities, numerous countermeasures have been constantly proposed in literature. Nevertheless, devising an efficient defense mechanism has proven to be a difficult task, since many approaches have already shown to be ineffective to adaptive attackers. Thus, this self-containing paper aims to provide all readerships with a review of the latest research progress on Adversarial Machine Learning in Image Classification, however with a defender's perspective. Here, novel taxonomies for categorizing adversarial attacks and defenses are introduced and discussions about the existence of adversarial examples are provided. Further, in contrast to exisiting surveys, it is also given relevant guidance that should be taken into consideration by researchers when devising and evaluating defenses. Finally, based on the reviewed literature, it is discussed some promising paths for future research.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191