The impossibility of simultaneously cloning non-orthogonal states lies at the foundations of quantum theory. Even when allowing for approximation errors, cloning an arbitrary unknown pure state requires as many initial copies as needed to fully learn the state. Rather than arbitrary unknown states, modern quantum learning theory often considers structured classes of states and exploits such structure to develop learning algorithms that outperform general-state tomography. This raises the question: How do the sample complexities of learning and cloning relate for such structured classes? We answer this question for an important class of states. Namely, for $n$-qubit stabilizer states, we show that the optimal sample complexity of cloning is $\Theta(n)$. Thus, also for this structured class of states, cloning is as hard as learning. To prove this result, we use representation-theoretic tools in the recently proposed Abelian State Hidden Subgroup framework and a new structured version of the recently introduced random purification channel to relate stabilizer state cloning to a variant of the sample amplification problem for probability distributions that was recently introduced in classical learning theory. This allows us to obtain our cloning lower bounds by proving new sample amplification lower bounds for classes of distributions with an underlying linear structure. Our results provide a more fine-grained perspective on No-Cloning theorems, opening up connections from foundations to quantum learning theory and quantum cryptography.

The Fundamental Theorem of PAC Learning asserts that learnability of a concept class $H$ is equivalent to the *uniform convergence* of empirical error in $H$ to its mean, or equivalently, to the problem of *density estimation*, learnability of the underlying marginal distribution with respect to events in $H$. This seminal equivalence relies strongly on PAC learning’s ‘distribution-free’ assumption, that the adversary may choose any marginal distribution over data. Unfortunately, the distribution-free model is known to be overly adversarial in practice, failing to predict the success of modern machine learning algorithms, but without the Fundamental Theorem our theoretical understanding of learning under distributional constraints remains highly limited. In this work, we revisit the connection between PAC learning, uniform convergence, and density estimation beyond the distribution-free setting when the adversary is restricted to choosing a marginal distribution from a known family $\mathscr{P}$. We prove that while the traditional Fundamental Theorem fails, a finer-grained connection between the three fundamental notions continues to hold: 1. PAC-Learning is strictly sandwiched between two relaxed models of density estimation, differing only in whether the learner knows the set of well-estimated events in $H$. 2. Under reasonable assumptions on $H$ and $\mathscr{P}$, density estimation is equivalent to *uniform estimation*, a weakening of uniform convergence allowing non-empirical estimators. Together, our results give a clearer picture of how the Fundamental Theorem extends beyond the distribution-free setting and shed new light on the classically challenging problem of learning under arbitrary distributional assumptions.

A fundamental question in reinforcement learning theory is: suppose the optimal value functions are linear in given features, can we learn them efficiently? This problem’s counterpart in supervised learning, linear regression, can be solved both statistically and computationally efficiently. Therefore, it was quite surprising when a recent work \cite{kane2022computational} showed a computational-statistical gap for linear reinforcement learning: even though there are polynomial sample-complexity algorithms, unless NP = RP, there are no polynomial time algorithms for this setting.In this work, we build on their result to show a computational lower bound, which is exponential in feature dimension and horizon, for linear reinforcement learning under the Randomized Exponential Time Hypothesis. To prove this we build a round-based game where in each round the learner is searching for an unknown vector in a unit hypercube. The rewards in this game are chosen such that if the learner achieves large reward, then the learner’s actions can be used to simulate solving a variant of 3-SAT, where (a) each variable shows up in a bounded number of clauses (b) if an instance has no solutions then it also has no solutions that satisfy more than (1-$\epsilon$)-fraction of clauses. We use standard reductions to show this 3-SAT variant is approximately as hard as 3-SAT. Finally, we also show a lower bound optimized for horizon dependence that almost matches the best known upper bound of $\exp(\sqrt{H})$.

This paper is concerned with the computational and statistical complexity of learning the Hidden Markov model (HMM). Although HMMs are some of the most widely used tools in sequential and time series modeling, they are cryptographically hard to learn in the standard setting where one has access to i.i.d. samples of observation sequences. In this paper, we depart from this setup and consider an \emph{interactive access model}, in which the algorithm can query for samples from the \emph{conditional distributions} of the HMMs. We show that interactive access to the HMM enables computationally efficient learning algorithms, thereby bypassing cryptographic hardness.Specifically, we obtain efficient algorithms for learning HMMs in two settings: \begin{enumerate} \item An easier setting where we have query access to the exact conditional probabilities. Here our algorithm runs in polynomial time and makes polynomially many queries to approximate any HMM in total variation distance. \item A harder setting where we can only obtain samples from the conditional distributions. Here the performance of the algorithm depends on a new parameter, called the fidelity of the HMM. We show that this captures cryptographically hard instances and all previously known positive results. \end{enumerate} We also show that these results extend to a broader class of distributions with latent low rank structure. Our algorithms can be viewed as generalizations and robustifications of Angluin’s $L^*$ algorithm for learning deterministic finite automata from membership queries.

We prove asymptotic convergence for a general class of k-means algorithms performed over streaming data from a distribution–the centers asymptotically converge to the set of stationary points of the k-means objective function. To do so, we show that online k-means over a distribution can be interpreted as stochastic gradient descent with a stochastic learning rate schedule. Then, we prove convergence by extending techniques used in optimization literature to handle settings where center-specific learning rates may depend on the past trajectory of the centers.

We consider a lifelong learning scenario in which a learner faces a neverending and arbitrary stream of facts and has to decide which ones to retain in its limited memory. We introduce a mathematical model based on the online learning framework, in which the learner measures itself against a collection of experts that are also memory-constrained and that reflect different policies for what to remember. Interspersed with the stream of facts are occasional questions, and on each of these the learner incurs a loss if it has not remembered the corresponding fact. Its goal is to do almost as well as the best expert in hindsight, while using roughly the same amount of memory. We identify difficulties with using the multiplicative weights update algorithm in this memory-constrained scenario, and design an alternative scheme whose regret guarantees are close to the best possible.

The equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory. With variants ranging from classical settings like PAC learning and regression to recent trends such as adversarially robust and private learning, it’s surprising we still lack a unified theory; traditional proofs of the equivalence tend to be disparate, and rely on strong model-specific assumptions like uniform convergence and sample compression. In this work, we give the first model-independent framework explaining the equivalence of realizable and agnostic learnability: a three-line blackbox reduction that simplifies, unifies, and extends our understanding across a wide variety of settings. This includes models with no known characterization of learnability such as learning with arbitrary distributional assumptions or general loss, as well as a host of other popular settings such as robust learning, partial learning, fair learning, and the statistical query model. More generally, we argue that the equivalence of realizable and agnostic learning is actually a special case of a broader phenomenon we call property generalization: any desirable property of a learning algorithm (e.g. noise tolerance, privacy, stability) that can be satisfied over finite hypothesis classes extends (possibly in some variation) to any learnable hypothesis class.

Reinforcement learning with function approximation has recently achieved tremendous results in applications with large state spaces. This empirical success has motivated a growing body of theoretical work proposing necessary and sufficient conditions under which efficient reinforcement learning is possible. From this line of work, a remarkably simple minimal sufficient condition has emerged for sample efficient reinforcement learning: MDPs with optimal value function V* and Q* linear in some known low-dimensional features. In this setting, recent works have designed sample efficient algorithms which require a number of samples polynomial in the feature dimension and independent of the size of state space. They however leave finding computationally efficient algorithms as future work and this is considered a major open problem in the community. In this work, we make progress on this open problem by presenting the first computational lower bound for RL with linear function approximation: unless NP=RP, no randomized polynomial time algorithm exists for deterministic transition MDPs with a constant number of actions and linear optimal value functions. To prove this, we show a reduction from Unique-Sat, where we convert a CNF formula into an MDP with deterministic transitions, constant number of actions and low dimensional linear optimal value functions. This result also exhibits the first computational-statistical gap in reinforcement learning with linear function approximation, as the underlying statistical problem is information-theoretically solvable with a polynomial number of queries, but no computationally efficient algorithm exists unless NP=RP. Finally, we also prove a quasi-polynomial time lower bound under the Randomized Exponential Time Hypothesis.

This work introduces Bilinear Classes, a new structural framework, which permit generalization in reinforcement learning in a wide variety of settings through the use of function approximation. The framework incorporates nearly all existing models in which a polynomial sample complexity is achievable, and, notably, also includes new models, such as the Linear Q*/V* model in which both the optimal Q-function and the optimal V-function are linear in some known feature space. Our main result provides an RL algorithm which has polynomial sample complexity for Bilinear Classes; notably, this sample complexity is stated in terms of a reduction to the generalization error of an underlying supervised learning sub-problem. These bounds nearly match the best known sample complexity bounds for existing models. Furthermore, this framework also extends to the infinite dimensional (RKHS) setting: for the the Linear Q*/V* model, linear MDPs, and linear mixture MDPs, we provide sample complexities that have no explicit dependence on the explicit feature dimension (which could be infinite), but instead depends only on information theoretic quantities.

Policy gradient methods are among the most effective methods in challenging reinforcement learning problems with large state and/or action spaces. However, little is known about even their most basic theoretical convergence properties, including: if and how fast they converge to a globally optimal solution or how they cope with approximation error due to using a restricted class of parametric policies. This work provides provable characterizations of the computational, approximation, and sample size properties of policy gradient methods in the context of discounted Markov Decision Processes (MDPs). We focus on both: "tabular" policy parameterizations, where the optimal policy is contained in the class and where we show global convergence to the optimal policy; and parametric policy classes (considering both log-linear and neural policy classes), which may not contain the optimal policy and where we provide agnostic learning results. One central contribution of this work is in providing approximation guarantees that are average case --- which avoid explicit worst-case dependencies on the size of state space --- by making a formal connection to supervised learning under distribution shift. This characterization shows an important interplay between estimation error, approximation error, and exploration (as characterized through a precisely defined condition number).

Policy gradient (PG) methods are among the most effective methods in challenging reinforcement learning problems with large state and/or action spaces. However, little is known about even their most basic theoretical convergence properties, including: if and how fast they converge to a globally optimal solution (say with a sufficiently rich policy class); how they cope with approximation error due to using a restricted class of parametric policies; or their finite sample behavior. Such characterizations are important not only to compare these methods to their approximate value function counterparts (where such issues are relatively well understood, at least in the worst case), but also to help with more principled approaches to algorithm design. This work provides provable characterizations of computational, approximation, and sample size issues with regards to policy gradient methods in the context of discounted Markov Decision Processes (MDPs). We focus on both: 1) “tabular” policy parameterizations, where the optimal policy is contained in the class and where we show global convergence to the optimal policy, and 2) restricted policy classes, which may not contain the optimal policy and where we provide agnostic learning results. In the \emph{tabular setting}, our main results are: 1) convergence rate to global optimum for direct parameterization and projected gradient ascent 2) an asymptotic convergence to global optimum for softmax policy parameterization and PG; and a convergence rate with additional entropy regularization, and 3) dimension-free convergence to global optimum for softmax policy parameterization and Natural Policy Gradient (NPG) method with exact gradients. In \emph{function approximation}, we further analyze NPG with exact as well as inexact gradients under certain smoothness assumptions on the policy parameterization and establish rates of convergence in terms of the quality of the initial state distribution. One insight of this work is in formalizing how a favorable initial state distribution provides a means to circumvent worst-case exploration issues. Overall, these results place PG methods under a solid theoretical footing, analogous to the global convergence guarantees of iterative value function based algorithms.

With the explosion of massive, widely available unlabeled data in the past years, finding label and time efficient, robust learning algorithms has become ever more important in theory and in practice. We study the paradigm of active learning, in which algorithms with access to large pools of data may adaptively choose what samples to label in the hope of exponentially increasing efficiency. By introducing comparisons, an additional type of query comparing two points, we provide the first time and query efficient algorithms for learning non-homogeneous linear separators robust to bounded (Massart) noise. We further provide algorithms for a generalization of the popular Tsybakov low noise condition, and show how comparisons provide a strong reliability guarantee that is often impractical or impossible with only labels - returning a classifier that makes no errors with high probability.

Given a finite set X ⊂ Rdand a binary linear classifier c: Rd→ {0,1}, how many queries of the form c(x) are required to learn the label of every point in X? Known as point location, this problem has inspired over 35 years of research in the pursuit of an optimal algorithm. Building on the prior work of Kane, Lovett, and Moran (ICALP 2018), we provide the first nearly optimal solution, a randomized linear decision tree of depth Õ(dlog(|X|)), improving on the previous best of Õ(d2log(|X|)) from Ezra and Sharir (Discrete and Computational Geometry, 2019). As a corollary, we also provide the first nearly optimal algorithm for actively learning halfspaces in the membership query model. En route to these results, building on the work of Carlen, Lieb, and Loss (J. Geometric Analysis 2004), as well as Dvir, Saraf, and Wigderson (STOC 2014), we prove a novel characterization of Barthe's Theorem (Inventiones Mathematicae, 1998) of independent interest. In particular, we show that X may be transformed into approximate isotropic position if and only if there exists no k-dimensional subspace with more than a k/d-fraction of X, and provide a similar characterization for exact isotropic position. The below is an extended abstract. The full work can be found at https://arxiv.org/abs/2004.11380.

The current paper studies the problem of agnostic $Q$-learning with function approximation in deterministic systems where the optimal $Q$-function is approximable by a function in the class $\mathcal{F}$ with approximation error $\delta \ge 0$. We propose a novel recursion-based algorithm and show that if $\delta = O\left(\rho/\sqrt{\dim_E}\right)$, then one can find the optimal policy using $O(\dim_E)$ trajectories, where $\rho$ is the gap between the optimal $Q$-value of the best actions and that of the second-best actions and $\dim_E$ is the Eluder dimension of $\mathcal{F}$. Our result has two implications: \begin{enumerate} \item In conjunction with the lower bound in [Du et al., 2020], our upper bound suggests that the condition $\delta = \widetilde{\Theta}\left(\rho/\sqrt{\dim_E}\right)$ is necessary and sufficient for algorithms with polynomial sample complexity. \item In conjunction with the obvious lower bound in the tabular case, our upper bound suggests that the sample complexity $\widetilde{\Theta}\left(\dim_E\right)$ is tight in the agnostic setting. \end{enumerate} Therefore, we help address the open problem on agnostic $Q$-learning proposed in [Wen and Van Roy, 2013]. We further extend our algorithm to the stochastic reward setting and obtain similar results.