arXiv subjects

Robert E. Schapire

Publications and source records attributed to Robert E. Schapire.

At least 19 recordsLinked to original sources

arXiv2025

Astral Space: Convex Analysis at Infinity

Not all convex functions on $\mathbb{R}^n$ have finite minimizers; some can only be minimized by a sequence as it heads to infinity. In this work, we aim to develop a theory for understanding such minimizers at infinity. We study astral space, a compact extension of $\mathbb{R}^n$ to which such points at infinity have been added. Astral space is constructed to be as small as possible while still ensuring that all linear functions can be continuously extended to the new space. Although astral space includes all of $\mathbb{R}^n$, it is not a vector space, nor even a metric space. However, it is sufficiently well-structured to allow useful and meaningful extensions of concepts of convexity, conjugacy, and subdifferentials. We develop these concepts and analyze various properties of convex functions on astral space, including the detailed structure of their minimizers, exact characterizations of continuity, and convergence of descent algorithms.

math.OC↗

arXiv2024

Lexicographic Optimization: Algorithms and Stability

A lexicographic maximum of a set $X \subseteq \mathbb{R}^n$ is a vector in $X$ whose smallest component is as large as possible, and subject to that requirement, whose second smallest component is as large as possible, and so on for the third smallest component, etc. Lexicographic maximization has numerous practical and theoretical applications, including fair resource allocation, analyzing the implicit regularization of learning algorithms, and characterizing refinements of game-theoretic equilibria. We prove that a minimizer in $X$ of the exponential loss function $L_c(\mathbf{x}) = \sum_i \exp(-c x_i)$ converges to a lexicographic maximum of $X$ as $c \rightarrow \infty$, provided that $X$ is stable in the sense that a well-known iterative method for finding a lexicographic maximum of $X$ cannot be made to fail simply by reducing the required quality of each iterate by an arbitrarily tiny degree. Our result holds for both near and exact minimizers of the exponential loss, while earlier convergence results made much stronger assumptions about the set $X$ and only held for the exact minimizer. We are aware of no previous results showing a connection between the iterative method for computing a lexicographic maximum and exponential loss minimization. We show that every convex polytope is stable, but that there exist compact, convex sets that are not stable. We also provide the first analysis of the convergence rate of an exponential loss minimizer (near or exact) and discover a curious dichotomy: While the two smallest components of the vector converge to the lexicographically maximum values very quickly (at roughly the rate $\frac{\log n}{c}$), all other components can converge arbitrarily slowly.

math.OC↗

arXiv2024

Provable Interactive Learning with Hindsight Instruction Feedback

We study interactive learning in a setting where the agent has to generate a response (e.g., an action or trajectory) given a context and an instruction. In contrast, to typical approaches that train the system using reward or expert supervision on response, we study learning with hindsight instruction where a teacher provides an instruction that is most suitable for the agent's generated response. This hindsight labeling of instruction is often easier to provide than providing expert supervision of the optimal response which may require expert knowledge or can be impractical to elicit. We initiate the theoretical analysis of interactive learning with hindsight labeling. We first provide a lower bound showing that in general, the regret of any algorithm must scale with the size of the agent's response space. We then study a specialized setting where the underlying instruction-response distribution can be decomposed as a low-rank matrix. We introduce an algorithm called LORIL for this setting and show that its regret scales as $\sqrt{T}$ where $T$ is the number of rounds and depends on the intrinsic rank but does not depend on the size of the agent's response space. We provide experiments in two domains showing that LORIL outperforms baselines even when the low-rank assumption is violated.

cs.LG↗

arXiv2022

Provably Sample-Efficient RL with Side Information about Latent Dynamics

We study reinforcement learning (RL) in settings where observations are high-dimensional, but where an RL agent has access to abstract knowledge about the structure of the state space, as is the case, for example, when a robot is tasked to go to a specific room in a building using observations from its own camera, while having access to the floor plan. We formalize this setting as transfer reinforcement learning from an abstract simulator, which we assume is deterministic (such as a simple model of moving around the floor plan), but which is only required to capture the target domain's latent-state dynamics approximately up to unknown (bounded) perturbations (to account for environment stochasticity). Crucially, we assume no prior knowledge about the structure of observations in the target domain except that they can be used to identify the latent states (but the decoding map is unknown). Under these assumptions, we present an algorithm, called TASID, that learns a robust policy in the target domain, with sample complexity that is polynomial in the horizon, and independent of the number of states, which is not possible without access to some prior knowledge. In synthetic experiments, we verify various properties of our algorithm and show that it empirically outperforms transfer RL algorithms that require access to "full simulators" (i.e., those that also simulate observations).

cs.LG↗

arXiv2021

Bayesian decision-making under misspecified priors with applications to meta-learning

Thompson sampling and other Bayesian sequential decision-making algorithms are among the most popular approaches to tackle explore/exploit trade-offs in (contextual) bandits. The choice of prior in these algorithms offers flexibility to encode domain knowledge but can also lead to poor performance when misspecified. In this paper, we demonstrate that performance degrades gracefully with misspecification. We prove that the expected reward accrued by Thompson sampling (TS) with a misspecified prior differs by at most $\tilde{\mathcal{O}}(H^2 ε)$ from TS with a well specified prior, where $ε$ is the total-variation distance between priors and $H$ is the learning horizon. Our bound does not require the prior to have any parametric form. For priors with bounded support, our bound is independent of the cardinality or structure of the action space, and we show that it is tight up to universal constants in the worst case. Building on our sensitivity analysis, we establish generic PAC guarantees for algorithms in the recently studied Bayesian meta-learning setting and derive corollaries for various families of priors. Our results generalize along two axes: (1) they apply to a broader family of Bayesian decision-making algorithms, including a Monte-Carlo implementation of the knowledge gradient algorithm (KG), and (2) they apply to Bayesian POMDPs, the most general Bayesian decision-making setting, encompassing contextual bandits as a special case. Through numerical simulations, we illustrate how prior misspecification and the deployment of one-step look-ahead (as in KG) can impact the convergence of meta-learning in multi-armed and contextual bandits with structured and correlated priors.

cs.LG↗

arXiv2020

Gradient descent follows the regularization path for general losses

Recent work across many machine learning disciplines has highlighted that standard descent methods, even without explicit regularization, do not merely minimize the training error, but also exhibit an implicit bias. This bias is typically towards a certain regularized solution, and relies upon the details of the learning process, for instance the use of the cross-entropy loss. In this work, we show that for empirical risk minimization over linear predictors with arbitrary convex, strictly decreasing losses, if the risk does not attain its infimum, then the gradient-descent path and the algorithm-independent regularization path converge to the same direction (whenever either converges to a direction). Using this result, we provide a justification for the widely-used exponentially-tailed losses (such as the exponential loss or the logistic loss): while this convergence to a direction for exponentially-tailed losses is necessarily to the maximum-margin direction, other losses such as polynomially-tailed losses may induce convergence to a direction with a poor margin.

cs.LG↗

arXiv2019

Oracle-Efficient Online Learning and Auction Design

We consider the design of computationally efficient online learning algorithms in an adversarial setting in which the learner has access to an offline optimization oracle. We present an algorithm called Generalized Follow-the-Perturbed-Leader and provide conditions under which it is oracle-efficient while achieving vanishing regret. Our results make significant progress on an open problem raised by Hazan and Koren, who showed that oracle-efficient algorithms do not exist in general and asked whether one can identify properties under which oracle-efficient online learning may be possible. Our auction-design framework considers an auctioneer learning an optimal auction for a sequence of adversarially selected valuations with the goal of achieving revenue that is almost as good as the optimal auction in hindsight, among a class of auctions. We give oracle-efficient learning results for: (1) VCG auctions with bidder-specific reserves in single-parameter settings, (2) envy-free item pricing in multi-item auctions, and (3) s-level auctions of Morgenstern and Roughgarden for single-item settings. The last result leads to an approximation of the overall optimal Myerson auction when bidders' valuations are drawn according to a fast-mixing Markov process, extending prior work that only gave such guarantees for the i.i.d. setting. Finally, we derive various extensions, including: (1) oracle-efficient algorithms for the contextual learning setting in which the learner has access to side information (such as bidder demographics), (2) learning with approximate oracles such as those based on Maximal-in-Range algorithms, and (3) no-regret bidding in simultaneous auctions, resolving an open problem of Daskalakis and Syrgkanis.

cs.LG↗

arXiv2019

On Oracle-Efficient PAC RL with Rich Observations

We study the computational tractability of PAC reinforcement learning with rich observations. We present new provably sample-efficient algorithms for environments with deterministic hidden state dynamics and stochastic rich observations. These methods operate in an oracle model of computation -- accessing policy and value function classes exclusively through standard optimization primitives -- and therefore represent computationally efficient alternatives to prior algorithms that require enumeration. With stochastic hidden state dynamics, we prove that the only known sample-efficient algorithm, OLIVE, cannot be implemented in the oracle model. We also present several examples that illustrate fundamental challenges of tractable PAC reinforcement learning in such general settings.

cs.LG↗

arXiv2018

Practical Contextual Bandits with Regression Oracles

A major challenge in contextual bandits is to design general-purpose algorithms that are both practically useful and theoretically well-founded. We present a new technique that has the empirical and computational advantages of realizability-based approaches combined with the flexibility of agnostic methods. Our algorithms leverage the availability of a regression oracle for the value-function class, a more realistic and reasonable oracle than the classification oracles over policies typically assumed by agnostic methods. Our approach generalizes both UCB and LinUCB to far more expressive possible model classes and achieves low regret under certain distributional assumptions. In an extensive empirical evaluation, compared to both realizability-based and agnostic baselines, we find that our approach typically gives comparable or superior results.

cs.LG↗

arXiv2017

Corralling a Band of Bandit Algorithms

We study the problem of combining multiple bandit algorithms (that is, online learning algorithms with partial feedback) with the goal of creating a master algorithm that performs almost as well as the best base algorithm if it were to be run on its own. The main challenge is that when run with a master, base algorithms unavoidably receive much less feedback and it is thus critical that the master not starve a base algorithm that might perform uncompetitively initially but would eventually outperform others if given enough feedback. We address this difficulty by devising a version of Online Mirror Descent with a special mirror map together with a sophisticated learning rate scheme. We show that this approach manages to achieve a more delicate balance between exploiting and exploring base algorithms than previous works yielding superior regret bounds. Our results are applicable to many settings, such as multi-armed bandits, contextual bandits, and convex bandits. As examples, we present two main applications. The first is to create an algorithm that enjoys worst-case robustness while at the same time performing much better when the environment is relatively easy. The second is to create an algorithm that works simultaneously under different assumptions of the environment, such as different priors or different loss structures.

cs.LG↗

arXiv2017

Unsupervised Domain Adaptation Using Approximate Label Matching

Domain adaptation addresses the problem created when training data is generated by a so-called source distribution, but test data is generated by a significantly different target distribution. In this work, we present approximate label matching (ALM), a new unsupervised domain adaptation technique that creates and leverages a rough labeling on the test samples, then uses these noisy labels to learn a transformation that aligns the source and target samples. We show that the transformation estimated by ALM has favorable properties compared to transformations estimated by other methods, which do not use any kind of target labeling. Our model is regularized by requiring that a classifier trained to discriminate source from transformed target samples cannot distinguish between the two. We experiment with ALM on simulated and real data, and show that it outperforms techniques commonly used in the field.

cs.LG↗

arXiv2016

Contextual Decision Processes with Low Bellman Rank are PAC-Learnable

This paper studies systematic exploration for reinforcement learning with rich observations and function approximation. We introduce a new model called contextual decision processes, that unifies and generalizes most prior settings. Our first contribution is a complexity measure, the Bellman rank, that we show enables tractable learning of near-optimal behavior in these processes and is naturally small for many well-studied reinforcement learning settings. Our second contribution is a new reinforcement learning algorithm that engages in systematic exploration to learn contextual decision processes with low Bellman rank. Our algorithm provably learns near-optimal behavior with a number of samples that is polynomial in all relevant parameters but independent of the number of unique observations. The approach uses Bellman error minimization with optimistic exploration and provides new insights into efficient exploration for reinforcement learning with function approximation.

cs.LG↗

arXiv2016

Improved Regret Bounds for Oracle-Based Adversarial Contextual Bandits

We give an oracle-based algorithm for the adversarial contextual bandit problem, where either contexts are drawn i.i.d. or the sequence of contexts is known a priori, but where the losses are picked adversarially. Our algorithm is computationally efficient, assuming access to an offline optimization oracle, and enjoys a regret of order $O((KT)^{\frac{2}{3}}(\log N)^{\frac{1}{3}})$, where $K$ is the number of actions, $T$ is the number of iterations and $N$ is the number of baseline policies. Our result is the first to break the $O(T^{\frac{3}{4}})$ barrier that is achieved by recently introduced algorithms. Breaking this barrier was left as a major open problem. Our analysis is based on the recent relaxation based approach of (Rakhlin and Sridharan, 2016).

cs.LG↗

arXiv2016

Exploratory Gradient Boosting for Reinforcement Learning in Complex Domains

High-dimensional observations and complex real-world dynamics present major challenges in reinforcement learning for both function approximation and exploration. We address both of these challenges with two complementary techniques: First, we develop a gradient-boosting style, non-parametric function approximator for learning on $Q$-function residuals. And second, we propose an exploration strategy inspired by the principles of state abstraction and information acquisition under uncertainty. We demonstrate the empirical effectiveness of these techniques, first, as a preliminary check, on two standard tasks (Blackjack and $n$-Chain), and then on two much larger and more realistic tasks with high-dimensional observation spaces. Specifically, we introduce two benchmarks built within the game Minecraft where the observations are pixel arrays of the agent's visual field. A combination of our two algorithmic techniques performs competitively on the standard reinforcement-learning tasks while consistently and substantially outperforming baselines on the two tasks with high-dimensional observation spaces. The new function approximator, exploration strategy, and evaluation benchmarks are each of independent interest in the pursuit of reinforcement-learning methods that scale to real-world domains.

cs.AI↗

arXiv2016

Efficient Algorithms for Adversarial Contextual Learning

We provide the first oracle efficient sublinear regret algorithms for adversarial versions of the contextual bandit problem. In this problem, the learner repeatedly makes an action on the basis of a context and receives reward for the chosen action, with the goal of achieving reward competitive with a large class of policies. We analyze two settings: i) in the transductive setting the learner knows the set of contexts a priori, ii) in the small separator setting, there exists a small set of contexts such that any two policies behave differently in one of the contexts in the set. Our algorithms fall into the follow the perturbed leader family \cite{Kalai2005} and achieve regret $O(T^{3/4}\sqrt{K\log(N)})$ in the transductive setting and $O(T^{2/3} d^{3/4} K\sqrt{\log(N)})$ in the separator setting, where $K$ is the number of actions, $N$ is the number of baseline policies, and $d$ is the size of the separator. We actually solve the more general adversarial contextual semi-bandit linear optimization problem, whilst in the full information setting we address the even more general contextual combinatorial optimization. We provide several extensions and implications of our algorithms, such as switching regret and efficient learning with predictable sequences.

cs.LG↗

arXiv2016

Efficient and Parsimonious Agnostic Active Learning

We develop a new active learning algorithm for the streaming setting satisfying three important properties: 1) It provably works for any classifier representation and classification problem including those with severe noise. 2) It is efficiently implementable with an ERM oracle. 3) It is more aggressive than all previous approaches satisfying 1 and 2. To do this we create an algorithm based on a newly defined optimization problem and analyze it. We also conduct the first experimental analysis of all efficient agnostic active learning algorithms, evaluating their strengths and weaknesses in different settings.

cs.LG↗

arXiv2015

Fast Convergence of Regularized Learning in Games

We show that natural classes of regularized learning algorithms with a form of recency bias achieve faster convergence rates to approximate efficiency and to coarse correlated equilibria in multiplayer normal form games. When each player in a game uses an algorithm from our class, their individual regret decays at $O(T^{-3/4})$, while the sum of utilities converges to an approximate optimum at $O(T^{-1})$--an improvement upon the worst case $O(T^{-1/2})$ rates. We show a black-box reduction for any algorithm in the class to achieve $\tilde{O}(T^{-1/2})$ rates against an adversary, while maintaining the faster rates against algorithms in the class. Our results extend those of [Rakhlin and Shridharan 2013] and [Daskalakis et al. 2014], who only analyzed two-player zero-sum games for specific algorithms.

cs.GT↗

arXiv2015

Contextual Dueling Bandits

We consider the problem of learning to choose actions using contextual information when provided with limited feedback in the form of relative pairwise comparisons. We study this problem in the dueling-bandits framework of Yue et al. (2009), which we extend to incorporate context. Roughly, the learner's goal is to find the best policy, or way of behaving, in some space of policies, although "best" is not always so clearly defined. Here, we propose a new and natural solution concept, rooted in game theory, called a von Neumann winner, a randomized policy that beats or ties every other policy. We show that this notion overcomes important limitations of existing solutions, particularly the Condorcet winner which has typically been used in the past, but which requires strong and often unrealistic assumptions. We then present three efficient algorithms for online learning in our setting, and for approximating a von Neumann winner from batch-like data. The first of these algorithms achieves particularly low regret, even when data is adversarial, although its time and space requirements are linear in the size of the policy space. The other two algorithms require time and space only logarithmic in the size of the policy space when provided access to an oracle for solving classification problems on the space.

cs.LG↗