Understanding minimal assumptions that enable learning and generalization is perhaps the central question of learning theory. Several celebrated results in statistical learning theory, such as the VC theorem and Littlestone’s characterization of online learnability, establish conditions on the hypothesis class that allow for learning under independent data and adversarial data, respectively. Building upon recent work bridging these extremes, we study sequential decision making under \emph{distributional adversaries} that can adaptively choose data-generating distributions from a fixed family $\mathcal{U}$ and ask when such problems are learnable with sample complexity that behaves like the favorable independent case. We provide a near complete characterization of families $\mathcal{U}$ that admit learnability in terms of a notion known as \emph{generalized smoothness}, i.e., a distribution family admits VC-dimension-dependent regret bounds for every finite-VC hypothesis class if and only if it is generalized smooth. Further, we give universal algorithms that achieve low regret under any generalized smooth adversary without explicit knowledge of $\mathcal{U}$. Finally, when $\mathcal{U}$ is known, we provide refined bounds in terms of a combinatorial parameter, the fragmentation number, that captures how many disjoint regions can carry nontrivial mass under $\mathcal{U}$. These results provide a nearly complete understanding of learnability under distributional adversaries. In addition, building upon the surprising connection between online learning and differential privacy, we show that the generalized smoothness also characterizes private learnability under distributional constraints.

We show that high-accuracy guarantees for log-concave sampling—that is, iteration and query complexities which scale as $\mathrm{poly}\log(1/\delta)$, where $\delta$ is the desired target accuracy—are achievable using stochastic gradients with sub-exponential tails. Notably, this exhibits a separation with the problem of convex optimization, where stochasticity (even additive Gaussian noise) in the gradient oracle incurs $\mathrm{poly}(1/\delta)$ queries. We also give an information-theoretic argument that light-tailed stochastic gradients are necessary for high accuracy: for example, in the bounded variance case, we show that the minimax-optimal query complexity scales as $\Theta(1/\delta)$. Our framework also provides similar high-accuracy guarantees under stochastic zeroth-order (value) queries, and an improved complexity result for sampling from finite-sum potentials.

Self-normalized martingale inequalities lie at the heart of confidence ellipsoids for online least squares and, more broadly, many bandit and reinforcement-learning results. Yet existing vector and scalar results typically rely on bounded covariates and an explicit regularization matrix, producing bounds that are \emph{not scale-invariant}: although the self-normalized quantity is scale-invariant by definition, its standard upper bounds are not. We characterize when scale-invariant upper bounds on self-normalized martingales are possible. Without further assumptions, we prove that nontrivial scale-invariant bounds exist only in dimension $d=1$; moreover, in $d=1$ we obtain $O(\log T)$ scale-invariant self-normalized bounds without any assumptions on the covariates. In contrast, for $d>1$ we show that no nontrivial scale-invariant bound can hold in full generality. We then connect this dichotomy to \emph{doubly-uniform} regret in online linear regression (i.e., regret bounds that are simultaneously independent of the covariate scale and the comparator norm) and use it to resolve the open question of Gaillard, Gerchinovitz, Huard, and Stoltz, \emph{“Uniform regret bounds over $\mathbb{R}^d$ for the sequential linear regression problem with the square loss”} (ALT 2019): in $d=1$ we give an explicit algorithm with $O(\log T)$ doubly-uniform regret, whereas for $d>1$ sublinear doubly-uniform regret is impossible. Finally, under a natural \emph{smoothness} condition (bounded Radon–Nikodym derivatives of the conditional covariate laws with respect to a fixed base measure), we recover sublinear regret for $d>1$ without bounded covariates and derive a self-normalized concentration inequality free of the usual regularization penalties, yielding arguably a first natural scale-invariant bound for adaptive, non-i.i.d. vector martingales.

We study contextual bandits in the stochastic i.i.d. setting, where a learner observes contexts drawn from an unknown distribution, selects actions from a finite set $\mathcal{A}$, and aims to identify an approximately optimal policy from a given class based on bandit feedback. Motivated by the important special case of bandit multiclass classification with zero-one rewards, we focus on the \emph{$s$-sparse} setting in which, for every context, the underlying reward vector has $L_1$-norm at most $s \ll |\mathcal{A}|$. Our main result is the design of algorithms that, with probability at least $1-\delta$, output an $\varepsilon$-optimal policy compared to policy class $\Pi$ using \begin{align*} \widetilde{O} \left( \left( \frac{s}{\varepsilon^2} + \frac{|\mathcal{A}|}{\varepsilon}\right) \log \frac{|\Pi|}{\delta}\right) \end{align*} samples. We further extend this bound to general Natarajan classes and complement it with a matching lower bound (up to logarithmic factors), thereby closing a substantial gap left by prior work (Erez et al., 2024a,b; Erez and Koren, 2025), which incurred an additional $\Theta(|\mathcal{A}|^9)$ dependence. We obtain these results via two complementary approaches. First, we analyze contextual bandits through the lens of contextual decision making with structured observations, designing an exploration-by-optimization algorithm whose sample complexity is governed by the \emph{decision-estimation coefficient} (DEC; Foster et al., 2021, 2022). We show that, with $s$-sparse rewards, the induced model class admits a sharp DEC bound that scales with $s$ and directly yields the optimal rate. Since this approach is largely information-theoretic and involves solving complex min-max optimization problems, we also develop a second, more specialized algorithmic method based on a low-variance exploration technique. This approach leads to concrete, tractable algorithms and naturally extends to contextual combinatorial semi-bandits, leading to improved sample complexity guarantees for bandit multiclass list classification.

Understanding the minimal assumptions necessary for generalization is the fundamental question in learning theory. Unfortunately, most results rely heavily on independence (or some proxy thereof) of the data-generating process, while results for strongly dependent data are far more limited. Towards addressing this gap, we introduce the framework of simulatable processes, where the learner has access to a simulator that approximates the distribution generating the data (which may be an arbitrarily complex and dependent process). Surprisingly, given access to such a simulator, we show that we can recover the same learning guarantees as in the classical setting with independent data, namely, error bounds that depend on the VC dimension. Further, we use this framework to study the power of conditional sampling and show strict statistical and computational advantages in this setting. As a highlight of our framework, we exhibit a single algorithm that simultaneously learns any given VC class under all processes samplable in bounded polynomial time, with regret controlled by the time-bounded Kolmogorov complexity of the process. This provides a significant conceptual broadening of the classical PAC model.

We revisit the sequential variants of linear regression with the squared loss, classification problems with hinge loss, and logistic regression, all characterized by unbounded losses in the setup where no assumptions are made on the magnitude of design vectors and the norm of the optimal vector of parameters. The key distinction from existing results lies in our assumption that the set of design vectors is known in advance (though their order is not), a setup sometimes referred to as transductive online learning. While this assumption might seem similar to fixed design regression or denoising, we demonstrate that the sequential nature of our algorithms allows us to convert our bounds into statistical ones with random design without making any additional assumptions about the distribution of the design vectors-an impossibility for standard denoising results. Our key tools are based on the exponential weights algorithm with carefully chosen transductive (design-dependent) priors, which exploit the full horizon of the design vectors, as well as additional aggregation tools that address the possibly unbounded norm of the vector of the optimal solution. Our classification regret bounds have a feature that is only attributed to bounded losses in the literature: they depend solely on the dimension of the parameter space and on the number of rounds, independent of the design vectors or the norm of the optimal solution. For linear regression with squared loss, we further extend our analysis to the sparse case, providing sparsity regret bounds that depend additionally only on the magnitude of the response variables. We argue that these improved bounds are specific to the transductive setting and unattainable in the worst-case sequential setup. Our algorithms, in several cases, have polynomial-time approximations and reduce to sampling with respect to log-concave measures instead of aggregating over hard-to-construct epsilon-covers of classes.

We study the problem of interactive decision making in which the underlying environment changes over time subject to given constraints. We propose a framework, which we call \textit{hybrid Decision Making with Structured Observations} (hybrid DMSO), that provides an interpolation between the stochastic and adversarial settings of decision making. Within this framework, we can analyze local differentially private decision making, query-based learning (in particular, SQ learning), and robust and smooth decision making under the same umbrella, deriving upper and lower bounds based on variants of the Decision-Estimation Coefficient (DEC). We further establish strong connections between the DEC’s behavior, the SQ dimension, local minimax complexity, learnability, and joint differential privacy.

We study the problem of sequential probability assignment under logarithmic loss, both with and without side information. Our objective is to analyze the \emph{minimax regret}—a notion extensively studied in the literature—in terms of geometric quantities, such as covering numbers and scale-sensitive dimensions. We show that the minimax regret for the case of no side information (equivalently, the Shtarkov sum) can be upper bounded in terms of \emph{sequential square-root entropy}, a notion closely related to Hellinger distance. For the problem of sequential probability assignment with side information, we develop both upper and lower bounds based on the aforementioned entropy. The lower bound matches the upper bound, up to log factors, for classes in the Donsker regime (according to our definition of entropy).

This paper investigates a basic question in reinforcement learning from human feedback (RLHF) from a theoretical perspective: how to efficiently explore in an online manner under preference feedback and general function approximation. We take the initial step towards a theoretical understanding of this problem by proposing a novel algorithm, *Exploratory Preference Optimization* (XPO). This algorithm is elegantly simple---requiring only a one-line modification to (online) Direct Preference Optimization (DPO; Rafailov et al., 2023)---yet provides the strongest known provable guarantees. XPO augments the DPO objective with a novel and principled *exploration bonus*, enabling the algorithm to strategically explore beyond the support of the initial model and preference feedback data. We prove that XPO is provably sample-efficient and converges to a near-optimal policy under natural exploration conditions, regardless of the initial model's coverage. Our analysis builds on the observation that DPO implicitly performs a form of *Bellman error minimization*. It synthesizes previously disparate techniques from language modeling and theoretical reinforcement learning in a serendipitous fashion through the lens of *KL-regularized Markov decision processes*.

Watermarking, the process by which Large Language Model (LLM) servers imbed an imperceptible signal at inference time in order to detect text generated by their own models, has grown in importance due to the significant improvements in natural language processing tasks by modern LLMs. Current approaches are often impractical due to generation latency, detection time, degradation in text quality, or robustness; such problems often arise due to the focus on token level watermarking, which ignores the inherent structure of text. In this work, we introduce a new scheme, GaussMark, that is simple and efficient to implement, has formal statistical guarantees, comes at no cost in generation latency, and embeds the watermark into the weights of the model itself, providing a structural watermark. Our approach is based on Gaussian independence testing and is motivated by recent empirical observations that minor additive corruptions to LLM weights can result in models of identical (or even improved) quality. We provide formal statistical bounds on the validity and power of our procedure and, through an extensive suite of experiments, demonstrate that GaussMark is reliable, efficient, relatively robust to corruption, and can be instantiated with essentially no loss in model quality.

Process and outcome supervision represent two fundamental approaches to reinforcement learning, especially for complex reasoning tasks in large language models. While process supervision offers intuitive advantages for long-term credit assignment, the precise relationship between these paradigms has remained an open question. Conventional wisdom suggests that outcome supervision is fundamentally more challenging due to the trajectory-level coverage problem, leading to significant investment in collecting fine-grained process supervision data. In this paper, we provide a possible theoretical resolution to this debate. Perhaps surprisingly, our main theorem shows that: under standard data coverage assumptions, reinforcement learning through outcome supervision is no more statistically difficult than through process supervision. At the core of this result lies the novel Change of Trajectory Measure Lemma—a powerful technical tool that bridges return-based trajectory measure and step-level distribution shift. Furthermore, for settings with access to a verifier or a rollout capability, we prove that any policy’s advantage function can serve as an optimal process reward model, providing a simple yet powerful connection between outcome and process supervision. These findings suggest that the empirically observed performance gap between outcome and process supervision likely stems from algorithmic limitations rather than inherent statistical difficulties, potentially transforming how we approach data and algorithm design for reinforcement learning.

Reinforcement learning with outcome-based feedback faces a fundamental challenge: when rewards are only observed at trajectory endpoints, how do we assign credit to the right actions? This paper provides the first comprehensive analysis of this problem in online RL with general function approximation. We develop a provably sample-efficient algorithm achieving $\widetilde{O}({C_{\rm cov} H^3}/{\varepsilon^2})$ sample complexity, where $C_{\rm cov}$ is the coverability coefficient of the underlying MDP. By leveraging general function approximation, our approach works effectively in large or infinite state spaces where tabular methods fail, requiring only that value functions and reward functions can be represented by appropriate function classes. Our results also characterize when outcome-based feedback is statistically separated from per-step rewards, revealing an unavoidable exponential separation for certain MDPs. For deterministic MDPs, we show how to eliminate the completeness assumption, dramatically simplifying the algorithm. We further extend our approach to preference-based feedback settings, proving that equivalent statistical efficiency can be achieved even under more limited information. Together, these results constitute a theoretical foundation for understanding the statistical properties of outcome-based reinforcement learning.

Policy-based methods currently dominate reinforcement learning (RL) pipelines for large language model (LLM) reasoning, leaving value-based approaches largely unexplored. We revisit the classical paradigm of Bellman Residual Minimization and introduce Trajectory Bellman Residual Minimization (TBRM), an algorithm that naturally adapts this idea to LLMs, yielding a simple yet effective off-policy algorithm that optimizes a single trajectory-level Bellman objective using the model's own logits as $Q$-values. TBRM removes the need for critics, importance-sampling ratios, or clipping, and can operate with only one rollout per prompt. We prove convergence to the near-optimal KL-regularized policy from arbitrary off-policy data via an improved change-of-trajectory-measure analysis. Experiments on standard mathematical-reasoning benchmarks show that TBRM matches or surpasses policy-based baselines, like PPO and GRPO, with comparable or lower computational and memory overhead. Our results indicate that value-based RL might be a principled and efficient alternative for enhancing reasoning capabilities in LLMs. The codebase for TBRM is publicly available at [https://github.com/rlx-lab/TBRM](https://github.com/rlx-lab/TBRM).

Pre-trained masked language models, such as BERT, perform strongly on a wide variety of NLP tasks and have become ubiquitous in recent years. The typical way to use such models is to fine-tune them on downstream data. In this work, we aim to study how the difference in domains between the pre-trained model and the task effects its final performance. We first devise a simple mechanism to quantify the domain difference (using a cloze task) and use it to partition our dataset. Using these partitions of varying domain discrepancy, we focus on answering key questions around the impact of discrepancy on final performance, robustness to out-of-domain test-time examples and effect of domain-adaptive pre-training. We base our experiments on a large-scale openly available e-commerce dataset, and our findings suggest that in spite of pre-training the performance of BERT degrades on datasets with high domain discrepancy, especially in low resource cases. This effect is somewhat mitigated by continued pre-training for domain adaptation. Furthermore, the domain-gap also makes BERT sensitive to out-of-domain examples during inference, even in high resource tasks, and it is prudent to use as diverse a dataset as possible during fine-tuning to make it robust to domain shift.

In order to circumvent statistical and computational hardness results in sequential decision-making, recent work has considered smoothed online learning, where the distribution of data at each time is assumed to have bounded likeliehood ratio with respect to a base measure when conditioned on the history. While previous works have demonstrated the benefits of smoothness, they have either assumed that the base measure is known to the learner or have presented computationally inefficient algorithms applying only in special cases. This work investigates the more general setting where the base measure is \emph{unknown} to the learner, focusing in particular on the performance of Empirical Risk Minimization (ERM) with square loss when the data are well-specified and smooth. We show that in this setting, ERM is able to achieve sublinear error whenever a class is learnable with iid data; in particular, ERM achieves error scaling as $\tilde O( \sqrt{\mathrm{comp}(\mathcal F) \cdot T} )$, where $\mathrm{comp}(\mathcal{F})$ is the statistical complexity of learning $\mathcal F$ with iid data. In so doing, we prove a novel norm comparison bound for smoothed data that comprises the first sharp norm comparison for dependent data applying to arbitrary, nonlinear function classes. We complement these results with a lower bound indicating that our analysis of ERM is essentially tight, establishing a separation in the performance of ERM between smoothed and iid data.

We study computational and statistical aspects of learning Latent Markov Decision Processes (LMDPs). In this model, the learner interacts with an MDP drawn at the beginning of each epoch from an unknown mixture of MDPs. To sidestep known impossibility results, we consider several notions of $\delta$-separation of the constituent MDPs. The main thrust of this paper is in establishing a nearly-sharp \textit{statistical threshold} for the horizon length necessary for efficient learning. On the computational side, we show that under a weaker assumption of separability under the optimal policy, there is a quasi-polynomial algorithm with time complexity scaling in terms of the statistical threshold. We further show a near-matching time complexity lower bound under the exponential time hypothesis.

We revisit the problem of offline reinforcement learning with value function realizability but without Bellman completeness. Previous work by Xie and Jiang (2021) and Foster et al. (2022) left open the question of whether bounded (all-policy) concentrability coefficient along with trajectory-based offline data admits a polynomial sample complexity. In this work, we provide a negative answer to this question for the task of offline policy evaluation. In addition to addressing this question, we provide a rather complete picture for offline policy evaluation with only value function realizability. Our primary findings are threefold: 1) The sample complexity of offline policy evaluation is governed by the concentrability coefficient in an aggregated Markov Transition Model jointly determined by the function class and the offline data distribution, rather than that in the original MDP. This unifies and generalizes the ideas of Xie and Jiang (2021) and Foster et al. (2022), 2) The concentrability coefficient in the aggregated Markov Transition Model may grow exponentially with the horizon length, even when the concentrability coefficient in the original MDP is small and the offline data is \emph{admissible} (i.e., the data distribution equals the occupancy measure of some policy), 3) Under value function realizability, there is a generic reduction that can convert any hard instance with admissible data to a hard instance with trajectory data, implying that trajectory data offers no extra benefits over admissible data. These three pieces jointly resolve the open problem, though each of them could be of independent interest.

We propose a generalization of the classical G-optimal design concept to non-linear function classes. The criterion, termed F -design, coincides with G-design in the linear case. We compute the value of the optimal design, termed the F-condition number, for several non-linear function classes. We further provide algorithms to construct designs with a bounded F -condition number. Finally, we employ the F-design in a variety of interactive machine learning tasks, where the design is naturally useful for data collection or exploration. We show that in four diverse settings of confidence band construction, contextual bandits, model-free reinforcement learning, and active learning, F-design can be combined with existing approaches in a black-box manner to yield state-of-the-art results in known problem settings as well as to generalize to novel ones.

The ability to efficiently explore high-dimensional state spaces is essential for the practical success of deep Reinforcement Learning (RL). This paper introduces a new exploration technique called Random Latent Exploration (RLE), that combines the strengths of exploration bonuses and randomized value functions (two popular approaches for effective exploration in deep RL). RLE leverages the idea of perturbing rewards by adding structured random rewards to the original task rewards in certain (random) states of the environment, to encourage the agent to explore the environment during training. RLE is straightforward to implement and performs well in practice. To demonstrate the practical effectiveness of RLE, we evaluate it on the challenging Atari and IsaacGym benchmarks and show that RLE exhibits higher overall scores across all the tasks than other approaches, including action-noise and randomized value function exploration.

We develop a unifying framework for information-theoretic lower bound in statistical estimation and interactive decision making. Classical lower bound techniques---such as Fano's method, Le Cam's method, and Assouad's lemma---are central to the study of minimax risk in statistical estimation, yet are insufficient to provide tight lower bounds for \emph{interactive decision making} algorithms that collect data interactively (e.g., algorithms for bandits and reinforcement learning). Recent work of Foster et al. provides minimax lower bounds for interactive decision making using seemingly different analysis techniques from the classical methods. These results---which are proven using a complexity measure known as the \emph{Decision-Estimation Coefficient} (DEC)---capture difficulties unique to interactive learning, yet do not recover the tightest known lower bounds for passive estimation. We propose a unified view of these distinct methodologies through a new lower bound approach called \emph{interactive Fano method}. As an application, we introduce a novel complexity measure, the \emph{Fractional Covering Number}, which facilitates the new lower bounds for interactive decision making that extend the DEC methodology by incorporating the complexity of estimation. Using the fractional covering number, we (i) provide a unified characterization of learnability for \emph{any} stochastic bandit problem, (ii) close the remaining gap between the upper and lower bounds in Foster et al. (up to polynomial factors) for any interactive decision making problem in which the underlying model class is convex.

The classical theory of statistical estimation aims to estimate a parameter of interest under data generated from a fixed design (''offline estimation''), while the contemporary theory of online learning provides algorithms for estimation under adaptively chosen covariates (''online estimation''). Motivated by connections between estimation and interactive decision making, we ask: is it possible to convert offline estimation algorithms into online estimation algorithms in a black-box fashion? We investigate this question from an information-theoretic perspective by introducing a new framework, Oracle-Efficient Online Estimation (OEOE), where the learner can only interact with the data stream indirectly through a sequence of offline estimators produced by a black-box algorithm operating on the stream. Our main results settle the statistical and computational complexity of online estimation in this framework. $\bullet$ Statistical complexity. We show that information-theoretically, there exist algorithms that achieve near-optimal online estimation error via black-box offline estimation oracles, and give a nearly-tight characterization for minimax rates in the OEOE framework. $\bullet$ Computational complexity. We show that the guarantees above cannot be achieved in a computationally efficient fashion in general, but give a refined characterization for the special case of conditional density estimation: computationally efficient online estimation via black-box offline estimation is possible whenever it is possible via unrestricted algorithms. Finally, we apply our results to give offline oracle-efficient algorithms for interactive decision making.

We consider realizable contextual bandits with general function approximation, investigating how small reward variance can lead to better-than-minimax regret bounds. Unlike in minimax regret bounds, we show that the eluder dimension $d_{\text{elu}}$$-$a measure of the complexity of the function class$-$plays a crucial role in variance-dependent bounds. We consider two types of adversary: (1) Weak adversary: The adversary sets the reward variance before observing the learner's action. In this setting, we prove that a regret of $\Omega( \sqrt{ \min (A, d_{\text{elu}}) \Lambda } + d_{\text{elu}} )$ is unavoidable when $d_{\text{elu}} \leq \sqrt{A T}$, where $A$ is the number of actions, $T$ is the total number of rounds, and $\Lambda$ is the total variance over $T$ rounds. For the $A\leq d_{\text{elu}}$ regime, we derive a nearly matching upper bound $\tilde{O}( \sqrt{ A\Lambda } + d_{\text{elu} } )$ for the special case where the variance is revealed at the beginning of each round. (2) Strong adversary: The adversary sets the reward variance after observing the learner's action. We show that a regret of $\Omega( \sqrt{ d_{\text{elu}} \Lambda } + d_{\text{elu}} )$ is unavoidable when $\sqrt{ d_{\text{elu}} \Lambda } + d_{\text{elu}} \leq \sqrt{A T}$. In this setting, we provide an upper bound of order $\tilde{O}( d_{\text{elu}}\sqrt{ \Lambda } + d_{\text{elu}} )$. Furthermore, we examine the setting where the function class additionally provides distributional information of the reward, as studied by Wang et al. (2024). We demonstrate that the regret bound $\tilde{O}(\sqrt{d_{\text{elu}} \Lambda} + d_{\text{elu}})$ established in their work is unimprovable when $\sqrt{d_{\text{elu}} \Lambda} + d_{\text{elu}}\leq \sqrt{AT}$. However, with a slightly different definition of the total variance and with the assumption that the reward follows a Gaussian distribution, one can achieve a regret of $\tilde{O}(\sqrt{A\Lambda} + d_{\text{elu}})$.

Simulators are a pervasive tool in reinforcement learning, but most existing algorithms cannot efficiently exploit simulator access -- particularly in high-dimensional domains that require general function approximation. We explore the power of simulators through online reinforcement learning with local simulator access (or, local planning), an RL protocol where the agent is allowed to reset to previously observed states and follow their dynamics during training. We use local simulator access to unlock new statistical guarantees that were previously out of reach: - We show that MDPs with low coverability (Xie et al. 2023) -- a general structural condition that subsumes Block MDPs and Low-Rank MDPs -- can be learned in a sample-efficient fashion with only Q⋆-realizability (realizability of the optimal state-value function); existing online RL algorithms require significantly stronger representation conditions. - As a consequence, we show that the notorious Exogenous Block MDP problem (Efroni et al. 2022) is tractable under local simulator access. The results above are achieved through a computationally inefficient algorithm. We complement them with a more computationally efficient algorithm, RVFS (Recursive Value Function Search), which achieves provable sample complexity guarantees under a strengthened statistical assumption known as pushforward coverability. RVFS can be viewed as a principled, provable counterpart to a successful empirical paradigm that combines recursive search (e.g., MCTS) with value function approximation.

Smoothed online learning has emerged as a popular framework to mitigate the substantial loss in statistical and computational complexity that arises when one moves from classical to adversarial learning. Unfortunately, for some spaces, it has been shown that efficient algorithms suffer an exponentially worse regret than that which is minimax optimal, even when the learner has access to an optimization oracle over the space. To mitigate that exponential dependence, this work introduces a new notion of complexity, the generalized bracketing numbers, which marries constraints on the adversary to the size of the space, and shows that an instantiation of Follow-the-Perturbed-Leader can attain low regret with the number of calls to the optimization oracle scaling optimally with respect to average regret. We then instantiate our bounds in several problems of interest, including online prediction and planning of piecewise continuous functions, which has many applications in fields as diverse as econometrics and robotics.

A central problem in the theory of multi-agent reinforcement learning (MARL) is to understand what structural conditions and algorithmic principles lead to sample-efficient learning guarantees, and how these considerations change as we move from few to many agents. We study this question in a general framework for interactive decision making with multiple agents, encompassing Markov games with function approximation and normal-form games with bandit feedback. We focus on equilibrium computation, in which a centralized learning algorithm aims to compute an equilibrium by controlling multiple agents that interact with an (unknown) environment. Our main contributions are:• We provide upper and lower bounds on the optimal sample complexity for multi-agent decision making based on a multi-agent generalization of the Decision-Estimation Coefficient, a complexity measure introduced by Foster et al. (2021) in the single-agent counterpart to our setting. Compared to the best results for the single-agent setting, our upper and lower bounds have additional gaps. We show that no “reasonable” complexity measure can close these gaps, highlighting a striking separation between single and multiple agents.• We show that characterizing the statistical complexity for multi-agent decision making is equivalent to characterizing the statistical complexity of single-agent decision making, but with hidden (unobserved) rewards, a framework that subsumes variants of the partial monitoring problem. As a consequence of this connection, we characterize the statistical complexity for hidden-reward interactive decision making to the best extent possible.Building on this development, we provide several new structural results, including 1) conditions under which the statistical complexity of multi-agent decision making can be reduced to that of single-agent, and 2) conditions under which the so-called curse of multiple agents can be avoided.

We study the design of sample-efficient algorithms for reinforcement learning in the presence of rich, high-dimensional observations, formalized via the Block MDP problem. Existing algorithms suffer from either 1) computational intractability, 2) strong statistical assumptions that are not necessarily satisfied in practice, or 3) suboptimal sample complexity. We address these issues by providing the first computationally efficient algorithm that attains rate-optimal sample complexity with respect to the desired accuracy level, with minimal statistical assumptions. Our algorithm, MusIK, combines exploration with representation learning based on multi-step inverse kinematics, a learning objective in which the aim is to predict the current action from the current observation and observations in the (potentially distant) future. MusIK is simple and flexible, and can efficiently take advantage of general-purpose function approximation. Our analysis of MusIK leverages several new techniques tailored to non-optimistic algorithms for reward-free exploration, which we anticipate will find broader use.

We consider the problem of interactive decision making, encompassing structured bandits and reinforcement learning with general function approximation. Recently, Foster et al. (2021) introduced the Decision-Estimation Coefficient, a measure of statistical complexity that lower bounds the optimal regret for interactive decision making, as well as a meta-algorithm, Estimation-to-Decisions, which achieves upper bounds in terms of the same quantity. Estimation-to-Decisions is a reduction, which lifts algorithms for (supervised) online estimation into algorithms for decision making. In this paper, we show that by combining Estimation-to-Decisions with a specialized form of "optimistic" estimation introduced by Zhang (2022), it is possible to obtain guarantees that improve upon those of Foster et al. (2021) by accommodating more lenient notions of estimation error. We use this approach to derive regret bounds for model-free reinforcement learning with value function approximation, and give structural results showing when it can and cannot help more generally.

We study the problem of agnostic PAC reinforcement learning (RL): given a policy class $\Pi$, how many rounds of interaction with an unknown MDP (with a potentially large state and action space) are required to learn an $\epsilon$-suboptimal policy with respect to \(\Pi\)? Towards that end, we introduce a new complexity measure, called the \emph{spanning capacity}, that depends solely on the set \(\Pi\) and is independent of the MDP dynamics. With a generative model, we show that the spanning capacity characterizes PAC learnability for every policy class $\Pi$. However, for online RL, the situation is more subtle. We show there exists a policy class $\Pi$ with a bounded spanning capacity that requires a superpolynomial number of samples to learn. This reveals a surprising separation for agnostic learnability between generative access and online access models (as well as between deterministic/stochastic MDPs under online access). On the positive side, we identify an additional \emph{sunflower} structure which in conjunction with bounded spanning capacity enables statistically efficient online RL via a new algorithm called POPLER, which takes inspiration from classical importance sampling methods as well as recent developments for reachable-state identification and policy evaluation in reward-free exploration.

It is well known that Empirical Risk Minimization (ERM) may attain minimax suboptimal rates in terms of the mean squared error (Birgé and Massart, 1993). In this paper, we prove that, under relatively mild assumptions, the suboptimality of ERM must be due to its bias. Namely, the variance error term of ERM (in terms of the bias and variance decomposition) enjoys the minimax rate. In the fixed design setting, we provide an elementary proof of this result using the probabilistic method. Then, we extend our proof to the random design setting for various models. In addition, we provide a simple proof of Chatterjee’s admissibility theorem (Chatterjee, 2014, Theorem 1.4), which states that in the fixed design setting, ERM cannot be ruled out as an optimal method, and then we extend this result to the random design setting. We also show that our estimates imply stability of ERM, complementing the main result of Caponnetto and Rakhlin (2006) for non-Donsker classes. Finally, we highlight the somewhat irregular nature of the loss landscape of ERM in the non-Donsker regime, by showing that functions can be close to ERM, in terms of $L_2$ distance, while still being far from almost-minimizers of the empirical loss.

Classical analysis of convex and non-convex optimization methods often requires the Lipschitz continuity of the gradient, which limits the analysis to functions bounded by quadratics. Recent work relaxed this requirement to a non-uniform smoothness condition with the Hessian norm bounded by an affine function of the gradient norm, and proved convergence in the non-convex setting via gradient clipping, assuming bounded noise. In this paper, we further generalize this non-uniform smoothness condition and develop a simple, yet powerful analysis technique that bounds the gradients along the trajectory, thereby leading to stronger results for both convex and non-convex optimization problems. In particular, we obtain the classical convergence rates for (stochastic) gradient descent and Nesterov's accelerated gradient method in the convex and/or non-convex setting under this general smoothness condition. The new analysis approach does not require gradient clipping and allows heavy-tailed noise with bounded variance in the stochastic setting.

In this paper, we provide a rigorous proof of convergence of the Adaptive Moment Estimate (Adam) algorithm for a wide class of optimization objectives. Despite the popularity and efficiency of the Adam algorithm in training deep neural networks, its theoretical properties are not yet fully understood, and existing convergence proofs require unrealistically strong assumptions, such as globally bounded gradients, to show the convergence to stationary points. In this paper, we show that Adam provably converges to $\epsilon$-stationary points with $\mathcal{O}(\epsilon^{-4})$ gradient complexity under far more realistic conditions. The key to our analysis is a new proof of boundedness of gradients along the optimization trajectory of Adam, under a generalized smoothness assumption according to which the local smoothness (i.e., Hessian norm when it exists) is bounded by a sub-quadratic function of the gradient norm. Moreover, we propose a variance-reduced version of Adam with an accelerated gradient complexity of $\mathcal{O}(\epsilon^{-3})$.

A major challenge in reinforcement learning is to develop practical, sample-efficient algorithms for exploration in high-dimensional domains where generalization and function approximation is required. Low-Rank Markov Decision Processes---where transition probabilities admit a low-rank factorization based on an unknown feature embedding---offer a simple, yet expressive framework for RL with function approximation, yet existing algorithms either (1) are computationally intractable, or (2) require restrictive statistical assumptions such as latent variable structure or access to model-based function approximation. In this work, we propose the first provably sample-efficient algorithm for exploration in Low-Rank MDPs that is both computationally efficient and model-free, allowing for general function approximation while requiring no structural assumptions beyond a reachability condition that we show is substantially weaker than that assumed in prior work. Our algorithm, SpanRL, uses the notion of a barycentric spanner for the feature embedding as an efficiently computable basis for exploration, performing efficient spanner computation by interleaving representation learning and policy optimization subroutines. Our analysis---which is appealingly simple and modular---carefully combines several techniques, including a new approach to error-tolerant barycentric spanner computation, and a new analysis of a certain minimax representation learning objective found in prior work.

Much of modern learning theory has been split between two regimes: the classical offline setting, where data arrive independently, and the online setting, where data arrive adversarially. While the former model is often both computationally and statistically tractable, the latter requires no distributional assumptions. In an attempt to achieve the best of both worlds, previous work proposed the smooth online setting where each sample is drawn from an adversarially chosen distribution, which is smooth, i.e., it has a bounded density with respect to a fixed dominating measure. Existing results for the smooth setting were known only for binary-valued function classes and were computation- ally expensive in general; in this paper, we fill these lacunae. In particular, we provide tight bounds on the minimax regret of learning a nonparametric function class, with nearly optimal dependence on both the horizon and smoothness parameters. Furthermore, we provide the first oracle-efficient, no-regret algorithms in this setting. In particular, we propose an oracle-efficient improper algorithm whose regret achieves optimal dependence on the horizon and a proper algorithm requiring only a single oracle call per round whose regret has the optimal horizon dependence in the classification setting and is sublinear in general. Both algorithms have exponentially worse dependence on the smoothness parameter of the adversary than the minimax rate. We then prove a lower bound on the oracle complexity of any proper learning algorithm, which matches the oracle-efficient upper bounds up to a polynomial factor, thus demonstrating the existence of a statistical-computational gap in smooth online learning. Finally, we apply our results to the contextual bandit setting to show that if a function class is learnable in the classical setting, then there is an oracle-efficient, no-regret algorithm for contextual bandits in the case that contexts arrive in a smooth manner.

We revisit the classic online portfolio selection problem, where at each round a learner selects a distribution over a set of portfolios to allocate its wealth. It is known that for this problem a logarithmic regret with respect to Cover’s loss is achievable using the Universal Portfolio Selection algorithm, for example. However, all existing algorithms that achieve a logarithmic regret for this problem have per-round time and space complexities that scale polynomially with the total number of rounds, making them impractical. In this paper, we build on the recent work by Luo et al. 2018 and present the first practical online portfolio selection algorithm with a logarithmic regret and whose per-round time and space complexities depend only logarithmically on the horizon. Behind our approach are two key technical novelties. We first show that the Damped Online Newton steps can approximate mirror descent iterates well, even when dealing with time-varying regularizers. Second, we present a new meta-algorithm that achieves an adaptive, logarithmic regret (i.e. a logarithmic regret on any sub-interval) for mixable losses.

A central problem in online learning and decision making---from bandits to reinforcement learning---is to understand what modeling assumptions lead to sample-efficient learning guarantees. We consider a general adversarial decision making framework that encompasses (structured) bandit problems with adversarial rewards and reinforcement learning problems with adversarial dynamics. Our main result is to show---via new upper and lower bounds---that the Decision-Estimation Coefficient, a complexity measure introduced by Foster et al. in the stochastic counterpart to our setting, is necessary and sufficient to obtain low regret for adversarial decision making. However, compared to the stochastic setting, one must apply the Decision-Estimation Coefficient to the convex hull of the class of models (or, hypotheses) under consideration. This establishes that the price of accommodating adversarial rewards or dynamics is governed by the behavior of the model class under convexification, and recovers a number of existing results --both positive and negative. En route to obtaining these guarantees, we provide new structural results that connect the Decision-Estimation Coefficient to variants of other well-known complexity measures, including the Information Ratio of Russo and Van Roy and the Exploration-by-Optimization objective of Lattimore and György.

It has long been thought that high-dimensional data encountered in many practical machine learning tasks have low-dimensional structure, i.e., the manifold hypothesis holds. A natural question, thus, is to estimate the intrinsic dimension of a given population distribution from a finite sample. We introduce a new estimator of the intrinsic dimension and provide finite sample, non-asymptotic guarantees. We then apply our techniques to get new sample complexity bounds for Generative Adversarial Networks (GANs) depending only on the intrinsic dimension of the data.

We introduce the technique of generic chaining and majorizing measures for controlling sequential Rademacher complexity. We relate majorizing measures to the notion of fractional covering numbers, which we show to be dominated in terms of sequential scale-sensitive dimensions in a horizon-independent way, and, under additional complexity assumptions establish a tight control on worst-case sequential Rademacher complexity in terms of the integral of sequential scale-sensitive dimension. Finally, we establish a tight contraction inequality for worst-case sequential Rademacher complexity. The above constitutes the resolution of a number of outstanding open problems in extending the classical theory of empirical processes to the sequential case, and, in turn, establishes sharp results for online learning.

In the classical multi-armed bandit problem, instance-dependent algorithms attain improved performance on "easy" problems with a gap between the best and second-best arm. Are similar guarantees possible for contextual bandits? While positive results are known for certain special cases, there is no general theory characterizing when and how instance-dependent regret bounds for contextual bandits can be achieved for rich, general classes of policies. We introduce a family of complexity measures that are both sufficient and necessary to obtain instance-dependent regret bounds. We then introduce new oracle-efficient algorithms which adapt to the gap whenever possible, while also attaining the minimax rate in the worst case. Finally, we provide structural results that tie together a number of complexity measures previously proposed throughout contextual bandits, reinforcement learning, and active learning and elucidate their role in determining the optimal instance-dependent regret. In a large-scale empirical evaluation, we find that our approach often gives superior results for challenging exploration problems. Turning our focus to reinforcement learning with function approximation, we develop new oracle-efficient algorithms for reinforcement learning with rich observations that obtain optimal gap-dependent sample complexity.

We study the minimal squared error of the Empirical Risk Minimization (ERM) procedure in the task of regression, both in random and fixed design settings. Our sharp lower bounds shed light on the possibility (or impossibility) of adapting to simplicity of the model generating the data. In the fixed design setting, we show that the error is governed by the global complexity of the entire class. In contrast, in random design, ERM may only adapt to simpler models if the local neighborhoods around the regression function are nearly as complex as the class itself, a somewhat counter-intuitive conclusion. We provide sharp lower bounds for performance of ERM in Donsker and non-Donsker classes. We also discuss our results through the lens of recent studies on interpolation in overparameterized models.

Motivated by modern applications, such as online advertisement and recommender systems, we study the top-$k$ extreme contextual bandits problem, where the total number of arms can be enormous, and the learner is allowed to select $k$ arms and observe all or some of the rewards for the chosen arms. We first propose an algorithm for the non-extreme realizable setting, utilizing the Inverse Gap Weighting strategy for selecting multiple arms. We show that our algorithm has a regret guarantee of $O(k\sqrt{(A-k+1)T \log (|F|T)})$, where $A$ is the total number of arms and $F$ is the class containing the regression function, while only requiring $\tilde{O}(A)$ computation per time step. In the extreme setting, where the total number of arms can be in the millions, we propose a practically-motivated arm hierarchy model that induces a certain structure in mean rewards to ensure statistical and computational efficiency. The hierarchical structure allows for an exponential reduction in the number of relevant arms for each context, thus resulting in a regret guarantee of $O(k\sqrt{(\log A-k+1)T \log (|F|T)})$. Finally, we implement our algorithm using a hierarchical linear function class and show superior performance with respect to well-known benchmarks on simulated bandit feedback experiments using extreme multi-label classification datasets. On a dataset with three million arms, our reduction scheme has an average inference time of only 7.9 milliseconds, which is a 100x improvement.

We address the problem of learning the parameters of a stable linear time invariant (LTI) system with unknown latent space dimension, or order, from a single time--series of noisy input-output data. We focus on learning the best lower order approximation allowed by finite data. Motivated by subspace algorithms in systems theory, where the doubly infinite system Hankel matrix captures both order and good lower order approximations, we construct a Hankel-like matrix from noisy finite data using ordinary least squares. This circumvents the non-convexities that arise in system identification, and allows accurate estimation of the underlying LTI system. Our results rely on careful analysis of self-normalized martingale difference terms that helps bound identification error up to logarithmic factors of the lower bound. We provide a data-dependent scheme for order selection and find an accurate realization of system parameters, corresponding to that order, by an approach that is closely related to the Ho-Kalman subspace algorithm. We demonstrate that the proposed model order selection procedure is not overly conservative, i.e., for the given data length it is not possible to estimate higher order models or find higher order approximations with reasonable accuracy.

We develop a technique for establishing lower bounds on the sample complexity of Least Squares (or, Empirical Risk Minimization) for large classes of functions. As an application, we settle an open problem regarding optimality of Least Squares in estimating a convex set from noisy support function measurements in dimension $d\geq 6$. Specifically, we establish that Least Squares is mimimax sub-optimal, and achieves a rate of $\tilde{\Theta}_d(n^{-2/(d-1)})$ whereas the minimax rate is $\Theta_d(n^{-4/(d+3)})$.

We study the risk of minimum-norm interpolants of data in Reproducing Kernel Hilbert Spaces. Our upper bounds on the risk are of a multiple-descent shape for the various scalings of $d = n^{\alpha}$, $\alpha\in(0,1)$, for the input dimension $d$ and sample size $n$. Empirical evidence supports our finding that minimum-norm interpolants in RKHS can exhibit this unusual non-monotonicity in sample size; furthermore, locations of the peaks in our experiments match our theoretical predictions. Since gradient flow on appropriately initialized wide neural networks converges to a minimum-norm interpolant with respect to a certain kernel, our analysis also yields novel estimation and generalization guarantees for these over-parametrized models. At the heart of our analysis is a study of spectral properties of the random kernel matrix restricted to a filtration of eigen-spaces of the population covariance operator, and may be of independent interest.

A fundamental challenge in contextual bandits is to develop flexible, general-purpose algorithms with computational requirements no worse than classical supervised learning tasks such as classification and regression. Algorithms based on regression have shown promising empirical success, but theoretical guarantees have remained elusive except in special cases. We provide the first universal and optimal reduction from contextual bandits to online regression. We show how to transform any oracle for online regression with a given value function class into an algorithm for contextual bandits with the induced policy class, with no overhead in runtime or memory requirements. We characterize the minimax rates for contextual bandits with general, potentially nonparametric function classes, and show that our algorithm is minimax optimal whenever the oracle obtains the optimal rate for regression. Compared to previous results, our algorithm requires no distributional assumptions beyond realizability, and works even when contexts are chosen adversarially.

We introduce a new problem setting for continuous control called the LQR with Rich Observations, or RichLQR. In our setting, the environment is summarized by a low-dimensional continuous latent state with linear dynamics and quadratic costs, but the agent operates on high-dimensional, nonlinear observations such as images from a camera. To enable sample-efficient learning, we assume that the learner has access to a class of decoder functions (e.g., neural networks) that is flexible enough to capture the mapping from observations to latent states. We introduce a new algorithm, RichID, which learns a near-optimal policy for the RichLQR with sample complexity scaling only with the dimension of the latent state space and the capacity of the decoder function class. RichID is oracle-efficient and accesses the decoder class only through calls to a least-squares regression oracle. To our knowledge, our results constitute the first provable sample complexity guarantee for continuous control with an unknown nonlinearity in the system model.

MOTIVATION: Imaging mass spectrometry (imaging MS) is a prominent technique for capturing distributions of molecules in tissue sections. Various computational methods for imaging MS rely on quantifying spatial correlations between ion images, referred to as co-localization. However, no comprehensive evaluation of co-localization measures has ever been performed; this leads to arbitrary choices and hinders method development. RESULTS: We present ColocML, a machine learning approach addressing this gap. With the help of 42 imaging MS experts from nine laboratories, we created a gold standard of 2210 pairs of ion images ranked by their co-localization. We evaluated existing co-localization measures and developed novel measures using term frequency-inverse document frequency and deep neural networks. The semi-supervised deep learning Pi model and the cosine score applied after median thresholding performed the best (Spearman 0.797 and 0.794 with expert rankings, respectively). We illustrate these measures by inferring co-localization properties of 10 273 molecules from 3685 public METASPACE datasets. AVAILABILITY AND IMPLEMENTATION: https://github.com/metaspace2020/coloc. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

We study the misclassification error for community detection in general heterogeneous stochastic block models (SBM) with noisy or partial label information. We establish a connection between the misclassification rate and the notion of minimum energy on the local neighborhood of the SBM. We develop an optimally weighted message passing algorithm to reconstruct labels for SBM based on the minimum energy flow and the eigenvectors of a certain Markov transition matrix. The general SBM considered in this paper allows for unequal-size communities, degree heterogeneity, and different connection probabilities among blocks. We focus on how to optimally weigh the message passing to improve misclassification.

We show that classical learning methods interpolating the training data can achieve optimal rates for the problems of nonparametric regression and prediction with square loss.

We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. We introduce a new notion of capacity — the Fisher-Rao norm — that possesses desirable invariance properties and is motivated by Information Geometry. We discover an analytical characterization of the new capacity measure, through which we establish norm-comparison inequalities and further show that the new measure serves as an umbrella for several existing norm-based complexity measures. We discuss upper bounds on the generalization error induced by the proposed measure. Extensive numerical experiments on CIFAR-10 support our theoretical findings. Our theoretical analysis rests on a key structural lemma about partial derivatives of multi-layer rectifier networks.

We show that minimum-norm interpolation in the Reproducing Kernel Hilbert Space corresponding to the Laplace kernel is not consistent if input dimension is constant. The lower bound holds for any choice of kernel bandwidth, even if selected based on data. The result supports the empirical observation that minimum-norm interpolation (that is, exact fit to training data) in RKHS generalizes well for some high-dimensional datasets, but not for low-dimensional ones.

We derive finite time error bounds for estimating general linear time-invariant (LTI) systems from a single observed trajectory using the method of least squares. We provide the first analysis of the general case when eigenvalues of the LTI system are arbitrarily distributed in three regimes: stable, marginally stable, and explosive. Our analysis yields sharp upper bounds for each of these cases separately. We observe that although the underlying process behaves quite differently in each of these three regimes, the systematic analysis of a self–normalized martingale difference term helps bound identification error up to logarithmic factors of the lower bound. On the other hand, we demonstrate that the least squares solution may be statistically inconsistent under certain conditions even when the signal-to-noise ratio is high.

We uncover a fairly general principle in online learning: If a regret inequality can be (approximately) expressed as a function of certain "sufficient statistics" for the data sequence, then there exists a special Burkholder function that 1) can be used algorithmically to achieve the regret bound and 2) only depends on these sufficient statistics, not the entire data sequence, so that the online strategy is only required to keep the sufficient statistics in memory. This characterization is achieved by bringing the full power of the Burkholder Method—originally developed for certifying probabilistic martingale inequalities—to bear on the online learning setting. To demonstrate the scope and effectiveness of the Burkholder method, we develop a novel online strategy for matrix prediction that attains a regret bound corresponding to the variance term in matrix concentration inequalities. We also present a linear-time/space prediction strategy for parameter-free supervised learning with linear classes and general smooth norms.

We study the sample complexity of learning neural networks, by providing new bounds on their Rademacher complexity assuming norm constraints on the parameter matrix of each layer. Compared to previous work, these complexity bounds have improved dependence on the network depth, and under some additional assumptions, are fully independent of the network size (both depth and width). These results are derived using some novel techniques, which may be of independent interest.

Accurate detection and localization for angiodysplasia lesions is an important problem in early stage diagnostics of gastrointestinal bleeding and anemia. Gold standard for angiodysplasia detection and localization is performed using wireless capsule endoscopy. This pill-like device is able to produce thousand of high enough resolution images during one passage through gastrointestinal tract. In this paper we present our solution for MICCAI 2017 Endoscopic Vision SubChallenge: Angiodysplasia Detection and Localization its further improvements over the state-of-the-art results using several deep neural network architectures. It addresses the binary segmentation problem, where every pixel in an image is labeled as an angiodysplasia lesions or background. Then, we analyze connected components of each predicted mask. Based on the analysis we developed a classifier that predict angiodysplasia lesions (binary variable) and a detector for their localization (center of a component). In this setting, our approach demonstrates one of the top results in every task subcategory for angiodysplasia detection and localization thereby providing state-of-the-art performance for these problems. The source code for our solution is made publicly available at https://github.com/ternaus/angiodysplasia-segmentation.

Semantic segmentation of robotic instruments is an important problem for the robot-assisted surgery. One of the main challenges is to correctly detect an instrument's position for the tracking and pose estimation in the vicinity of surgical scenes. Accurate pixel-wise instrument segmentation is needed to address this challenge. In this paper we describe our deep learning-based approach for robotic instrument segmentation. Our approach demonstrates an improvement over the state-of-the-art results using several deep neural network architectures. It addressed the binary segmentation problem, where every pixel in an image is labeled as an instrument or background from the surgery video feed. In addition, we address a multi-class segmentation problem, in which we distinguish between different instruments or different parts of an instrument from the background. In this setting, our approach outperforms other methods for automatic instrument segmentation thereby providing state-of-the-art results for these problems. The source code for our solution is made publicly available at https://github.com/ternaus/robot-surgery-segmentation.

We develop computationally efficient algorithms for online multi-class prediction. Our construction is based on carefully-chosen data-dependent surrogate loss functions, and the new methods enjoy strong mistake bound guarantees. To illustrate the technique, we study the combinatorial problem of node classification and develop a prediction strategy that is linear-time per round. In contrast, the offline benchmark is NP-hard to compute in general. We demonstrate the empirical performance of the method on several datasets.

We develop a new family of algorithms for the online learning setting with regret against any data sequence bounded by the empirical Rademacher complexity of that sequence. To develop a general theory of when this type of adaptive regret bound is achievable we establish a connection to the theory of decoupling inequalities for martingales in Banach spaces. When the hypothesis class is a set of linear functions bounded in some norm, such a regret bound is achievable if and only if the norm satisfies certain decoupling inequalities for martingales. Donald Burkholder’s celebrated geometric characterization of decoupling inequalities (1984) states that such an inequality holds if and only if there exists a special function called a Burkholder function satisfying certain restricted concavity properties. Our online learning algorithms are efficient in terms of queries to this function. We realize our general theory by giving new efficient and adaptive algorithms for classes including $\ell_p$ norms, group norms, and reproducing kernel Hilbert spaces. The empirical Rademacher complexity regret bound implies — when used in the i.i.d. setting — a data-dependent complexity bound for excess risk after online-to-batch conversion. To showcase the power of the empirical Rademacher complexity regret bound, we derive improved rates for a supervised learning generalization of the online learning with low rank experts task and for the online matrix prediction task. In addition to obtaining tight data-dependent regret bounds, our algorithms enjoy improved efficiency over previous techniques based on Rademacher complexity, automatically work in the infinite horizon setting, and adapt to scale. To obtain such adaptive methods, we introduce novel machinery, and the resulting algorithms are not based on the standard tools of online convex optimization. We conclude with a number of open problems and new directions, both algorithmic and information-theoretic.

Stochastic Gradient Langevin Dynamics (SGLD) is a popular variant of Stochastic Gradient Descent, where properly scaled isotropic Gaussian noise is added to an unbiased estimate of the gradient at each iteration. This modest change allows SGLD to escape local minima and suffices to guarantee asymptotic convergence to global minimizers for sufficiently regular non-convex objectives. The present work provides a nonasymptotic analysis in the context of non-convex learning problems, giving finite-time guarantees for SGLD to find approximate minimizers of both empirical and population risks. As in the asymptotic setting, our analysis relates the discrete-time SGLD Markov chain to a continuous-time diffusion process. A new tool that drives the results is the use of weighted transportation cost inequalities to quantify the rate of convergence of SGLD to a stationary distribution in the Euclidean $2$-Wasserstein distance.

We study an equivalence of (i) deterministic pathwise statements appearing in the online learning literature (termed \emphregret bounds), (ii) high-probability tail bounds for the supremum of a collection of martingales (of a specific form arising from uniform laws of large numbers), and (iii) in-expectation bounds for the supremum. By virtue of the equivalence, we prove exponential tail bounds for norms of Banach space valued martingales via deterministic regret bounds for the online mirror descent algorithm with an adaptive step size. We show that the phenomenon extends beyond the setting of online linear optimization and present the equivalence for the supervised online learning setting.

This paper addresses the multi-armed bandit problem in a multi-player framework. Players explore a finite set of arms with stochastic rewards, and the reward distribution of each arm is player-dependent. The goal is to find the best global arm, i.e., the one with the largest expected reward when averaged out among players. To achieve this goal, we develop a distributed variant of the well-known UCB1 algorithm. Confined to a network structure, players exchange information locally to estimate the global rewards, while using a confidence bound relying on the network characteristics. Then, at each round, each player votes for an arm, and the majority vote is played as the network action. The whole network gains the reward of the network action, hoping to maximize the global welfare. The performance of the algorithm is measured via the notion of network regret. We prove that the regret scales logarithmically with respect to time horizon and inversely in the spectral gap of the network. Our algorithm is optimal in the sense that in a complete network it scales down the regret of its single-player counterpart by the network size. We demonstrate numerical experiments to verify our theoretical results.

We propose a computationally efficient random walk on a convex body which rapidly mixes with respect to a fixed log-concave distribution and closely tracks a time-varying log-concave distribution. We develop general theoretical guarantees on the required number of steps; this number can be calculated on the fly according to the distance from and the shape of the next distribution. We then illustrate the technique on several examples. Within the context of exponential families, the proposed method produces samples from a posterior distribution which is updated as data arrive in a streaming fashion. The sampling technique can be used to track time-varying truncated distributions, as well as to obtain samples from a changing mixture model, fitted in a streaming fashion to data. In the setting of linear optimization, the proposed method has oracle complexity with best known dependence on the dimension for certain geometries. In the context of online learning and repeated games, the algorithm is an efficient method for implementing no-regret mixture forecasting strategies. Remarkably, in some of these examples, only one step of the random walk is needed to track the next distribution.

We present efficient algorithms for the problem of contextual bandits with i.i.d. covariates, an arbitrary sequence of rewards, and an arbitrary class of policies. Our algorithm BISTRO requires d calls to the empirical risk minimization (ERM) oracle per round, where d is the number of actions. The method uses unlabeled data to make the problem computationally simple. When the ERM problem itself is computationally hard, we extend the approach by employing multiplicative approximation algorithms for the ERM. The integrality gap of the relaxation only enters in the regret bound rather than the benchmark. Finally, we show that the adversarial version of the contextual bandit problem is learnable (and efficient) whenever the full-information supervised online learning problem has a non-trivial regret bound (and efficient).

Machine learning algorithms can be viewed as stochastic transformations that map training data to hypotheses. Following Bousquet and Elisseeff, we say that such an algorithm is stable if its output does not depend too much on any individual training example. Since stability is closely connected to generalization capabilities of learning algorithms, it is of theoretical and practical interest to obtain sharp quantitative estimates on the generalization bias of machine learning algorithms in terms of their stability properties. We propose several information-theoretic measures of algorithmic stability and use them to upper-bound the generalization bias of learning algorithms. Our framework is complementary to the information-theoretic methodology developed recently by Russo and Zou.

Recent literature on online learning has focused on developing adaptive algorithms that take advantage of a regularity of the sequence of observations, yet retain worst-case performance guarantees. A complementary direction is to develop prediction methods that perform well against complex benchmarks. In this paper, we address these two directions together. We present a fully adaptive method that competes with dynamic benchmarks in which regret guarantee scales with regularity of the sequence of cost functions and comparators. Notably, the regret bound adapts to the smaller complexity measure in the problem environment. Finally, we apply our results to drifting zero-sum, two-player games where both players achieve no regret guarantees against best sequences of actions in hindsight.

We consider the problem of optimizing an approximately convex function over a bounded convex set in \mathbbR^n using only function evaluations. The problem is reduced to sampling from an \emphapproximately log-concave distribution using the Hit-and-Run method, which is shown to have the same \mathcalO^* complexity as sampling from log-concave distributions. In addition to extend the analysis for log-concave distributions to approximate log-concave distributions, the implementation of the 1-dimensional sampler of the Hit-and-Run walk requires new methods and analysis. The algorithm then is based on simulated annealing which does not relies on first order conditions which makes it essentially immune to local minima. We then apply the method to different motivating problems. In the context of zeroth order stochastic convex optimization, the proposed method produces an ε-minimizer after \mathcalO^*(n^7.5ε^-2) noisy function evaluations by inducing a \mathcalO(ε/n)-approximately log concave distribution. We also consider in detail the case when the “amount of non-convexity” decays towards the optimum of the function. Other applications of the method discussed in this work include private computation of empirical risk minimizers, two-stage stochastic programming, and approximate dynamic programming for online learning.

We consider regression with square loss and general classes of functions without the boundedness assumption. We introduce a notion of offset Rademacher complexity that provides a transparent way to study localization both in expectation and in high probability. For any (possibly non-convex) class, the excess loss of a two-step estimator is shown to be upper bounded by this offset complexity through a novel geometric inequality. In the convex case, the estimator reduces to an empirical risk minimizer. The method recovers the results of \citepRakSriTsy15 for the bounded case while also providing guarantees without the boundedness assumption.

We study online prediction where regret of the algorithm is measured against a benchmark defined via evolving constraints. This framework captures online prediction on graphs, as well as other prediction problems with combinatorial structure. A key aspect here is that finding the optimal benchmark predictor (even in hindsight, given all the data) might be computationally hard due to the combinatorial nature of the constraints. Despite this, we provide polynomial-time prediction algorithms that achieve low regret against combinatorial benchmark sets. We do so by building improper learning algorithms based on two ideas that work together. The first is to alleviate part of the computational burden through random playout, and the second is to employ Lasserre semidefinite hierarchies to approximate the resulting integer program. Interestingly, for our prediction algorithms, we only need to compute the values of the semidefinite programs and not the rounded solutions. However, the integrality gap for Lasserre hierarchy does enter the generic regret bound in terms of Rademacher complexity of the benchmark set. This establishes a trade-off between the computation time and the regret bound of the algorithm.

We propose a general framework for studying adaptive regret bounds in the online learning setting, subsuming model selection and data-dependent bounds. Given a data- or model-dependent bound we ask, “Does there exist some algorithm achieving this bound?” We show that modifications to recently introduced sequential complexity measures can be used to answer this question by providing sufficient conditions under which adaptive rates can be achieved. In particular each adaptive rate induces a set of so-called offset complexity measures, and obtaining small upper bounds on these quantities is sufficient to demonstrate achievability. A cornerstone of our analysis technique is the use of one-sided tail inequalities to bound suprema of offset random processes.Our framework recovers and improves a wide variety of adaptive bounds including quantile bounds, second order data-dependent bounds, and small loss bounds. In addition we derive a new type of adaptive bound for online linear optimization based on the spectral norm, as well as a new online PAC-Bayes theorem.

We establish optimal rates for online regression for arbitrary classes of regression functions in terms of the sequential entropy introduced in (Rakhlin et al., 2010). The optimal rates are shown to exhibit a phase transition analogous to the i.i.d./statistical learning case, studied in (Rakhlin et al., 2014b). In the frequently encountered situation when sequential entropy and i.i.d. empirical entropy match, our results point to the interesting phenomenon that the rates for statistical learning with squared loss and online nonparametric regression are the same. In addition to a non-algorithmic study of minimax regret, we exhibit a generic forecaster that enjoys the established optimal rates. We also provide a recipe for designing online regression algorithms that can be computationally efficient. We illustrate the techniques by deriving existing and new forecasters for the case of finite experts and for online linear regression.

We introduce a formalism of localization for online learning problems, which, similarly to statistical learning theory, can be used to obtain fast rates. In particular, we introduce local sequential Rademacher complexities and other local measures. Based on the idea of relaxations for deriving algorithms, we provide a template method that takes advantage of localization. Furthermore, we build a general adaptive method that can take advantage of the suboptimality of the observed sequence. We illustrate the utility of the introduced concepts on several problems. Among them is a novel upper bound on regret in terms of classical Rademacher complexity when the data are i.i.d.

We study the problem of online learning with a notion of regret defined with respect to a set of strategies. We develop tools for analyzing the minimax rates and for deriving regret-minimization algorithms in this scenario. While the standard methods for minimizing the usual notion of regret fail, through our analysis we demonstrate existence of regret-minimization methods that compete with such sets of strategies as: autoregressive algorithms, strategies based on statistical models, regularized least squares, and follow the regularized leader strategies. In several cases we also derive efficient learning algorithms.

We present methods for online linear optimization that take advantage of benign (as opposed to worst-case) sequences. Specifically if the sequence encountered by the learner is described well by a known “predictable process”, the algorithms presented enjoy tighter bounds as compared to the typical worst case bounds. Additionally, the methods achieve the usual worst-case regret bounds if the sequence is not benign. Our approach can be seen as a way of adding \emphprior knowledge about the sequence within the paradigm of online learning. The setting is shown to encompass partial and side information. Variance and path-length bounds can be seen as particular examples of online learning with simple predictable sequences.We further extend our methods to include competing with a set of possible predictable processes (models), that is “learning” the predictable process itself concurrently with using it to obtain better regret guarantees. We show that such model selection is possible under various assumptions on the available feedback.

We discuss two scenarios of universal prediction, as well as some recent advances in the study of minimax regret and algorithmic development. We then propose an intermediate scenario, the Semi-Probabilistic Setting, and make progress towards understanding the associated minimax regret.

We provide several applications of Optimistic Mirror Descent, an online learning algorithm based on the idea of predictable sequences. First, we recover the Mirror-Prox algorithm, prove an extension to Holder-smooth functions, and apply the results to saddle-point type problems. Second, we prove that a version of Optimistic Mirror Descent (which has a close relation to the Exponential Weights algorithm) can be used by two strongly-uncoupled players in a finite zero-sum matrix game to converge to the minimax equilibrium at the rate of O(log T / T). This addresses a question of Daskalakis et al, 2011. Further, we consider a partial information version of the problem. We then apply the results to approximate convex programming and show a simple algorithm for the approximate Max-Flow problem.

This paper addresses the problem of online learning in a dynamic setting. We consider a social network in which each individual observes a private signal about the underlying state of the world and communicates with her neighbors at each time period. Unlike many existing approaches, the underlying state is dynamic, and evolves according to a geometric random walk. We view the scenario as an optimization problem where agents aim to learn the true state while suffering the smallest possible loss. Based on the decomposition of the global loss function, we introduce two update mechanisms, each of which generates an estimate of the true state. We establish a tight bound on the rate of change of the underlying state, under which individuals can track the parameter with a bounded variance. Then, we characterize explicit expressions for the steady state mean-square deviation(MSD) of the estimates from the truth, per individual. We observe that only one of the estimators recovers the optimal MSD, which underscores the impact of the objective function decomposition on the learning quality. Finally, we provide an upper bound on the regret of the proposed methods, measured as an average of errors in estimating the parameter in a finite time.

We show a principled way of deriving online learning algorithms from a minimax analysis. Various upper bounds on the minimax value, previously thought to be non-constructive, are shown to yield algorithms. This allows us to seamlessly recover known methods and to derive new ones, also capturing such ''unorthodox'' methods as Follow the Perturbed Leader and the R^2 forecaster. Understanding the inherent complexity of the learning problem thus leads to the development of algorithms. To illustrate our approach, we present several new algorithms, including a family of randomized methods that use the idea of a ''random play out''. New versions of the Follow-the-Perturbed-Leader algorithms are presented, as well as methods based on the Littlestone's dimension, efficient methods for matrix completion with trace norm, and algorithms for the problems of transductive learning and prediction with static experts.

The analysis of quantitative properties, such as timing and power, is central to the design of reliable embedded software and systems. However, the verification of such properties on a program is made difficult by their heavy dependence on the program’s environment, such as the processor it runs on. Modeling the environment by hand can be tedious, error prone, and time consuming. In this article, we present a new game-theoretic approach to analyzing quantitative properties that is based on performing systematic measurements to automatically learn a model of the environment. We model the problem as a game between our algorithm (player) and the environment of the program (adversary) in which the player seeks to accurately predict the property of interest, while the adversary sets environment states and parameters. To solve this problem, we employ a randomized strategy that repeatedly tests the program along a linear-sized set of program paths called basis paths, using the resulting measurements to infer a weighted-graph model of the environment from which quantitative properties can be predicted. Test cases are automatically generated using satisfiability modulo theories (SMT) solving. We prove that our algorithm can, under certain assumptions and with arbitrarily high probability, accurately predict properties such as worst-case execution time or estimate the distribution of execution times. Experimental results for execution time analysis demonstrate that our approach is efficient, accurate, and highly portable.

We study the problem of predicting individual sequences with linear loss with full and partial (or bandit) feed- back. Our main contribution is the first efficient algorithm for the problem of online linear optimization in the bandit setting which achieves the optimal Õ(√(T)) regret. In addition, for the full-information setting, we give a novel regret minimization algorithm. These results are made possible by the introduction of interior-point methods for convex optimization to online learning.

This paper addresses the problem of minimizing a convex, Lipschitz function $f$ over a convex, compact set $X$ under a stochastic bandit feedback model. In this model, the algorithm is allowed to observe noisy realizations of the function value $f(x)$ at any query point $x \in X$. We demonstrate a generalization of the ellipsoid algorithm that incurs $O(\poly(d)\sqrt{T})$ regret. Since any algorithm has regret at least $\Omega(\sqrt{T})$ on this problem, our algorithm is optimal in terms of the scaling with $T$.

We develop unified information-theoretic machinery for deriving lower bounds for passive and active learning schemes. Our bounds involve the so-called Alexander's capacity function. The supremum of this function has been recently rediscovered by Hanneke in the context of active learning under the name of "disagreement coefficient." For passive learning, our lower bounds match the upper bounds of Gine and Koltchinskii up to constants and generalize analogous results of Massart and Nedelec. For active learning, we provide first known lower bounds based on the capacity function rather than the disagreement coefficient.

Learning theory has largely focused on two main learning scenarios: the classical statistical setting where instances are drawn i.i.d. from a fixed distribution, and the adversarial scenario whereby at every time step the worst instance is revealed to the player. It can be argued that in the real world neither of these assumptions is reasonable. We define the minimax value of a game where the adversary is restricted in his moves, capturing stochastic and non-stochastic assumptions on data. Building on the sequential symmetrization approach, we define a notion of distribution-dependent Rademacher complexity for the spectrum of problems ranging from i.i.d. to worst-case. The bounds let us immediately deduce variation-type bounds. We study a smoothed online learning scenario and show that exponentially small amount of noise can make function classes with infinite Littlestone dimension learnable.

We study the intrinsic limitations of sequential convex optimization through the lens of feedback information theory. In the oracle model of optimization, an algorithm queries an oracle for noisy information about the unknown objective function and the goal is to (approximately) minimize every function in a given class using as few queries as possible. We show that, in order for a function to be optimized, the algorithm must be able to accumulate enough information about the objective. This, in turn, puts limits on the speed of optimization under specific assumptions on the oracle and the type of feedback. Our techniques are akin to the ones used in statistical literature to obtain minimax lower bounds on the risks of estimation procedures; the notable difference is that, unlike in the case of i.i.d. data, a sequential optimization algorithm can gather observations in a controlled manner, so that the amount of information at each step is allowed to change in time. In particular, we show that optimization algorithms often obey the law of diminishing returns: the signal-to-noise ratio drops as the optimization algorithm approaches the optimum. To underscore the generality of the tools, we use our approach to derive fundamental lower bounds for a certain active learning problem. Overall, the present work connects the intuitive notions of “information” in optimization, experimental design, estimation, and active learning to the quantitative notion of Shannon information.

We propose a computationally efficient random walk on a convex body which rapidly mixes to a time-varying Gibbs distribution. In the setting of online convex optimization and repeated games, the algorithm yields low regret and presents a novel efficient method for implementing mixture forecasting strategies.

We develop a theory of online learning by defining several complexity measures. Among them are analogues of Rademacher complexity, covering numbers and fat-shattering dimension from statistical learning theory. Relationship among these complexity measures, their connection to online learning, and tools for bounding them are provided. We apply these results to various learning problems. We provide a complete characterization of online learnability in the supervised setting.

Estimating the worst-case execution time (WCET) of tasks is a key step in the design of reliable real-time software and systems. In this paper, we present a new, game-theoretic approach to estimating WCET based on performing directed measurements on the target platform. We model the estimation problem as a game between our algorithm (player) and the environment of the program (adversary), where the player seeks to find the longest path through the program while the adversary sets environment parameters to thwart the player. We present both theoretical and experimental results demonstrating the utility of our approach. On the theoretical side, we prove that our algorithm can converge to find the longest path with high probability. Experimental results indicate that our approach is competitive with an existing technique based on static analysis and integer programming. Moreover, the approach can be easily applied to even complex hardware/software platforms.

We consider the problem of choosing, sequentially, a map which assigns elements of a set A to a few elements of a set B. On each round, the algorithm suffers some cost associated with the chosen assignment, and the goal is to minimize the cumulative loss of these choices relative to the best map on the entire sequence. Even though the offline problem of finding the best map is provably hard, we show that there is an equivalent online approximation algorithm, Randomized Map Prediction (RMP), that is efficient and performs nearly as well. While drawing upon results from the “Online Prediction with Expert Advice ” setting, we show how RMP can be utilized as an online approach to several standard batch problems. We apply RMP to online clustering as well as online feature selection and, surprisingly, RMP often outperforms the standard batch algorithms on these problems. 1.

We phrase K -means clustering as an empirical risk minimization procedure over a class HK and explicitly calculate the covering number for this class. Next, we show that stability of K -means clustering is characterized by the geometry of HK with respect to the underlying distribution. We prove that in the case of a unique global minimizer, the clustering solution is stable with respect to complete changes of the data, while for the case of multiple minimizers, the change of (n1/2 ) samples defines the transition between stability and instability. While for a finite number of minimizers this result follows from multinomial distribution estimates, the case of infinite minimizers requires more refined tools. We conclude by proving that stability of the functions in HK implies stability of the actual centers of the clusters. Since stability is often used for selecting the number of clusters in practice, we hope that our analysis serves as a starting point for finding theoretically grounded recipes for the choice of K .

We study some stability properties of algorithms which minimize (or almost-minimize) empirical error over Donsker classes of functions. We show that, as the number n of samples grows, the L2-diameter of the set of almost-minimizers of empirical error with tolerance ξ(n)=o(n-1/2) converges to zero in probability. Hence, even in the case of multiple minimizers of expected error, as n increases it becomes less and less likely that adding a sample (or a number of samples) to the training set will result in a large jump to a new hypothesis. Moreover, under some assumptions on the entropy of the class, along with an assumption of Komlos-Major-Tusnady type, we derive a power rate of decay for the diameter of almost-minimizers. This rate, through an application of a uniform ratio limit inequality, is shown to govern the closeness of the expected errors of the almost-minimizers. In fact, under the above assumptions, the expected errors of almost-minimizers become closer with a rate strictly faster than n-1/2.