The no-free-lunch theorems promote a skeptical conclusion that all possible machine learning algorithms equally lack justification. But how could this leave room for a learning theory, that shows that some algorithms are better than others? Drawing parallels to the philosophy of induction, we point out that the no-free-lunch results presuppose a conception of learning algorithms as purely data-driven. On this conception, every algorithm must have an inherent inductive bias, that wants justification. We argue that many standard learning algorithms should rather be understood as model-dependent: in each application they also require for input a model, representing a bias. Generic algorithms themselves, they can be given a model-relative justification.
The important recent book by G. Schurz appreciates that the no-free-lunch theorems (NFL) have major implications for the problem of (meta) induction. Here I review the NFL theorems, emphasizing that they do not only concern the case where there is a uniform prior -- they prove that there are "as many priors" (loosely speaking) for which any induction algorithm $A$ out-generalizes some induction algorithm $B$ as vice-versa. Importantly though, in addition to the NFL theorems, there are many {free lunch} theorems. In particular, the NFL theorems can only be used to compare the {marginal} expecte…
Challenging optimization problems, which elude acceptable solution via conventional calculus methods, arise commonly in different areas of industrial design and practice. Hard optimization problems are those who manifest the following behavior: a) high number of independent input variables; b) very complex or irregular multi-modal fitness; c) computational expensive fitness evaluation. This paper will focus on some theoretical issues that have strong implications for practice. I will stress how an interpretation of the No Free Lunch theorem leads naturally to a general Bayesian optimization fr…
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Louisiana State University（路易斯安那州立大学）
The no-free-lunch (NFL) theorem is a celebrated result in learning theory that limits one's ability to learn a function with a training data set. With the recent rise of quantum machine learning, it is natural to ask whether there is a quantum analog of the NFL theorem, which would restrict a quantum computer's ability to learn a unitary process (the quantum analog of a function) with quantum training data. However, in the quantum setting, the training data can possess entanglement, a strong correlation with no classical analog. In this work, we show that entangled data sets lead to an apparen…
Australian National University（澳大利亚国立大学）；ETH Zürich（苏黎世联邦理工学院）；NICTA（澳大利亚国家信息与通信技术研究院）
The No Free Lunch theorems are often used to argue that domain specific knowledge is required to design successful algorithms. We use algorithmic information theory to argue the case for a universal bias allowing an algorithm to succeed in all interesting problem domains. Additionally, we give a new algorithm for off-line classification, inspired by Solomonoff induction, with good performance on all structured problems under reasonable assumptions. This includes a proof of the efficacy of the well-known heuristic of randomly selecting training data in the hope of reducing misclassification rat…
Hong Kong University of Science and Technology（香港科技大学）；Webank（微众银行）
In a federated learning scenario where multiple parties jointly learn a model from their respective data, there exist two conflicting goals for the choice of appropriate algorithms. On one hand, private and sensitive training data must be kept secure as much as possible in the presence of \textit{semi-honest} partners, while on the other hand, a certain amount of information has to be exchanged among different parties for the sake of learning utility. Such a challenge calls for the privacy-preserving federated learning solution, which maximizes the utility of the learned model and maintains a …
This manuscript presents some new impossibility results on adversarial robustness in machine learning, a very important yet largely open problem. We show that if conditioned on a class label the data distribution satisfies the $W_2$ Talagrand transportation-cost inequality (for example, this condition is satisfied if the conditional distribution has density which is log-concave; is the uniform measure on a compact Riemannian manifold with positive Ricci curvature, any classifier can be adversarially fooled with high probability once the perturbations are slightly greater than the natural noise…
The No Free Lunch theorems prove that under a uniform distribution over induction problems (search problems or learning problems), all induction algorithms perform equally. As I discuss in this chapter, the importance of the theorems arises by using them to analyze scenarios involving {non-uniform} distributions, and to compare different algorithms, without any assumption about the distribution over problems at all. In particular, the theorems prove that {anti}-cross-validation (choosing among a set of candidate algorithms based on which has {worst} out-of-sample behavior) performs as well as …
InfoNCE is a Free Lunch for Semantically guided Graph Contrastive Learning
State Key Laboratory of AI Safety, Institute of Computing Technology, Chinese Academy of Sciences University of Chinese Academy of Sciences（人工智能安全国家重点实验室，计算技术研究所，中国科学院，中国科学院大学）
本文将图对比学习重新表述为 PU 学习问题，提出 IFL-GCL，利用 InfoNCE 提取语义正样本信息并重构损失，在 IID/OOD 场景最高提升 9.05%。
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Columbia University（哥伦比亚大学）
Multitask learning and related areas such as multi-source domain adaptation address modern settings where datasets from $N$ related distributions $\{P_t\}$ are to be combined towards improving performance on any single such distribution ${\cal D}$. A perplexing fact remains in the evolving theory on the subject: while we would hope for performance bounds that account for the contribution from multiple tasks, the vast majority of analyses result in bounds that improve at best in the number $n$ of samples per task, but most often do not improve in $N$. As such, it might seem at first that the di…
The fields of artificial intelligence and neuroscience have a long history of fertile bi-directional interactions. On the one hand, important inspiration for the development of artificial intelligence systems has come from the study of natural systems of intelligence, the mammalian neocortex in particular. On the other, important inspiration for models and theories of the brain have emerged from artificial intelligence research. A central question at the intersection of these two areas is concerned with the processes by which neocortex learns, and the extent to which they are analogous to the …
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Universidad del País Vasco UPV/EHU（巴斯克大学）；Universität Siegen（锡根大学）；ICFO-The Institute of Photonic Sciences（光子科学研究所）；University of Borås（布罗斯大学）
In supervised learning, an inductive learning algorithm extracts general rules from observed training instances, then the rules are applied to test instances. We show that this splitting of training and application arises naturally, in the classical setting, from a simple independence requirement with a physical interpretation of being non-signalling. Thus, two seemingly different definitions of inductive learning happen to coincide. This follows from the properties of classical information that break down in the quantum setup. We prove a quantum de Finetti theorem for quantum channels, which …
According to the no-free-lunch theorem, there is no single meta-heuristic algorithm that can optimally solve all optimization problems. This motivates many researchers to continuously develop new optimization algorithms. In this paper, a novel nature-inspired meta-heuristic optimization algorithm called virus spread optimization (VSO) is proposed. VSO loosely mimics the spread of viruses among hosts, and can be effectively applied to solving many challenging and continuous optimization problems. We devise a new representation scheme and viral operations that are radically different from previo…
This paper presents a topological learning-theoretic perspective on causal inference by introducing a series of topologies defined on general spaces of structural causal models (SCMs). As an illustration of the framework we prove a topological causal hierarchy theorem, showing that substantive assumption-free causal inference is possible only in a meager set of SCMs. Thanks to a known correspondence between open sets in the weak topology and statistically verifiable hypotheses, our results show that inductive assumptions sufficient to license valid causal inferences are statistically unverifia…
Academy for Advanced Interdisciplinary Studies（前沿交叉学科研究院）；Peking University（北京大学）；MGH/BWH Center for Advanced Medical Computing and Analysis（麻省总医院/布莱根妇女医院高级医学计算与分析中心）；Massachusetts General Hospital and Harvard Medical School（麻省总医院和哈佛医学院）；Beijing International Center for Mathematical Research（北京国际数学研究中心）
We proved that a trained model in supervised deep learning minimizes the conditional risk for each input (Theorem 2.1). This property provided insights into the behavior of trained models and established a connection between supervised and unsupervised learning in some cases. In addition, when the labels are intractable but can be written as a conditional risk minimizer, we proved an equivalent form of the original supervised learning problem with accessible labels (Theorem 2.2). We demonstrated that many existing works, such as Noise2Score, Noise2Noise and score function estimation can be exp…
Supervised learning, characterized by both discriminative and generative learning, seeks to predict the values of single (or sometimes multiple) predefined target attributes based on a predefined set of predictor attributes. For applications where the information available and predictions to be made may vary from instance to instance, we propose the task of target-agnostic learning where arbitrary disjoint sets of attributes can be used for each of predictors and targets for each to-be-predicted instance. For this task, we survey a wide range of techniques available for handling missing values…
Xidian University（西安电子科技大学）；Tencent AI Healthcare（腾讯AI医疗）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
We propose a self-supervised Gaussian ATtention network for image Clustering (GATCluster). Rather than extracting intermediate features first and then performing the traditional clustering algorithm, GATCluster directly outputs semantic cluster labels without further post-processing. Theoretically, we give a Label Feature Theorem to guarantee the learned features are one-hot encoded vectors, and the trivial solutions are avoided. To train the GATCluster in a completely unsupervised manner, we design four self-learning tasks with the constraints of transformation invariance, separability maximi…
