In this paper, we consider the problem of replicable realizable PAC learning. We construct a particularly hard learning problem and show a sample complexity lower bound with a close to $(\log|H|)^{3/2}$ dependence on the size of the hypothesis class $H$. Our proof uses several novel techniques and works by defining a particular Cayley graph associated with $H$ and analyzing a suitable random walk on this graph by examining the spectral properties of its adjacency matrix.
Furthermore, we show an almost matching upper bound for the lower bound instance, meaning if a stronger lower bound exists, one would have to consider a different instance of the problem.
Boston University（波士顿大学）；UCSD（加州大学圣地亚哥分校）；Columbia University（哥伦比亚大学）；University of Toronto（多伦多大学）；IAS（高等研究院）；University of Pennsylvania（宾夕法尼亚大学）
The notion of replicable algorithms was introduced in Impagliazzo et al. [STOC '22] to describe randomized algorithms that are stable under the resampling of their inputs. More precisely, a replicable algorithm gives the same output with high probability when its randomness is fixed and it is run on a new i.i.d. sample drawn from the same distribution. Using replicable algorithms for data analysis can facilitate the verification of published results by ensuring that the results of an analysis will be the same with high probability, even when that analysis is performed on a new data set. In thi…
We study a recent model of collaborative PAC learning where $k$ players with $k$ different tasks collaborate to learn a single classifier that works for all tasks. Previous work showed that when there is a classifier that has very small error on all tasks, there is a collaborative algorithm that finds a single classifier for all tasks and has $O((\ln (k))^2)$ times the worst-case sample complexity for learning a single task. In this work, we design new algorithms for both the realizable and the non-realizable setting, having sample complexity only $O(\ln (k))$ times the worst-case sample compl…
This work establishes a new upper bound on the number of samples sufficient for PAC learning in the realizable case. The bound matches known lower bounds up to numerical constant factors. This solves a long-standing open problem on the sample complexity of PAC learning. The technique and analysis build on a recent breakthrough by Hans Simon.
In recent years crowdsourcing has become the method of choice for gathering labeled training data for learning algorithms. Standard approaches to crowdsourcing view the process of acquiring labeled data separately from the process of learning a classifier from the gathered data. This can give rise to computational and statistical challenges. For example, in most cases there are no known computationally efficient learning algorithms that are robust to the high level of noise that exists in crowdsourced data, and efforts to eliminate noise through voting often require a large number of queries p…
Tel Aviv University（特拉维夫大学）；Google AI（谷歌人工智能）；Princeton（普林斯顿大学）
PAC-Bayes is a useful framework for deriving generalization bounds which was introduced by McAllester ('98). This framework has the flexibility of deriving distribution- and algorithm-dependent bounds, which are often tighter than VC-related uniform convergence bounds. In this manuscript we present a limitation for the PAC-Bayes framework. We demonstrate an easy learning task that is not amenable to a PAC-Bayes analysis. Specifically, we consider the task of linear classification in 1D; it is well-known that this task is learnable using just $O(\log(1/δ)/ε)$ examples. On the other hand, we sho…
We give the first sample complexity characterizations for outcome indistinguishability, a theoretical framework of machine learning recently introduced by Dwork, Kim, Reingold, Rothblum, and Yona (STOC 2021). In outcome indistinguishability, the goal of the learner is to output a predictor that cannot be distinguished from the target predictor by a class $D$ of distinguishers examining the outcomes generated according to the predictors' predictions. In the distribution-specific and realizable setting where the learner is given the data distribution together with a predictor class $P$ containin…
In many learning theory problems, a central role is played by a hypothesis class: we might assume that the data is labeled according to a hypothesis in the class (usually referred to as the realizable setting), or we might evaluate the learned model by comparing it with the best hypothesis in the class (the agnostic setting). Taking a step beyond these classic setups that involve only a single hypothesis class, we introduce comparative learning as a combination of the realizable and agnostic settings in PAC learning: given two binary hypothesis classes $S$ and $B$, we assume that the data is l…
The sample complexity of learning a Boolean-valued function class is precisely characterized by its Rademacher complexity. This has little bearing, however, on the sample complexity of \emph{efficient} agnostic learning. We introduce \emph{refutation complexity}, a natural computational analog of Rademacher complexity of a Boolean concept class and show that it exactly characterizes the sample complexity of \emph{efficient} agnostic learning. Informally, refutation complexity of a class $\mathcal{C}$ is the minimum number of example-label pairs required to efficiently distinguish between the c…
Google Research, Brain Team（谷歌研究院大脑团队）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
The classical PAC sample complexity bounds are stated for any Empirical Risk Minimizer (ERM) and contain an extra logarithmic factor $\log(1/ε)$ which is known to be necessary for ERM in general. It has been recently shown by Hanneke (2016) that the optimal sample complexity of PAC learning for any VC class C is achieved by a particular improper learning algorithm, which outputs a specific majority-vote of hypotheses in C. This leaves the question of when this bound can be achieved by proper learning algorithms, which are restricted to always output a hypothesis from C. In this paper we aim to…
University of California, San Diego（加州大学圣迭戈分校）；Microsoft Research, New England（微软新英格兰研究院）；University of Texas, Austin（得克萨斯大学奥斯汀分校）
An active learner is given a class of models, a large set of unlabeled examples, and the ability to interactively query labels of a subset of these examples; the goal of the learner is to learn a model in the class that fits the data well. Previous theoretical work has rigorously characterized label complexity of active learning, but most of this work has focused on the PAC or the agnostic PAC model. In this paper, we shift our attention to a more general setting -- maximum likelihood estimation. Provided certain conditions hold on the model class, we provide a two-stage active learning algori…
Let~$\cH$ be a class of boolean functions and consider a {\it composed class} $\cH'$ that is derived from~$\cH$ using some arbitrary aggregation rule (for example, $\cH'$ may be the class of all 3-wise majority-votes of functions in $\cH$). We upper bound the Littlestone dimension of~$\cH'$ in terms of that of~$\cH$. As a corollary, we derive closure properties for online learning and private PAC learning. The derived bounds on the Littlestone dimension exhibit an undesirable exponential dependence. For private learning, we prove close to optimal bounds that circumvents this suboptimal depende…
In this paper we study the problem of multiclass classification with a bounded number of different labels $k$, in the realizable setting. We extend the traditional PAC model to a) distribution-dependent learning rates, and b) learning rates under data-dependent assumptions. First, we consider the universal learning setting (Bousquet, Hanneke, Moran, van Handel and Yehudayoff, STOC '21), for which we provide a complete characterization of the achievable learning rates that holds for every fixed distribution. In particular, we show the following trichotomy: for any concept class, the optimal lea…
In this paper we study online Reinforcement Learning (RL) in partially observable dynamical systems. We focus on the Predictive State Representations (PSRs) model, which is an expressive model that captures other well-known models such as Partially Observable Markov Decision Processes (POMDP). PSR represents the states using a set of predictions of future observations and is defined entirely using observable quantities. We develop a novel model-based algorithm for PSRs that can learn a near optimal policy in sample complexity scaling polynomially with respect to all the relevant parameters of …
Agnostic Learning under Targeted Poisoning: Optimal Rates and the Role of Randomness
Department of Mathematics, Technion（技术学院数学系）；Departments of Mathematics, Computer Science, and Data and Decision Sciences, Technion and Google Research（技术学院数学系、计算机科学系和数据与决策科学系，以及谷歌研究）
