arXiv subjects

Nathan Kallus

Publications and source records attributed to Nathan Kallus.

At least 127 records · Page 7Linked to original sources

arXiv2022

Provably Efficient Reinforcement Learning in Partially Observable Dynamical Systems

We study Reinforcement Learning for partially observable dynamical systems using function approximation. We propose a new \textit{Partially Observable Bilinear Actor-Critic framework}, that is general enough to include models such as observable tabular Partially Observable Markov Decision Processes (POMDPs), observable Linear-Quadratic-Gaussian (LQG), Predictive State Representations (PSRs), as well as a newly introduced model Hilbert Space Embeddings of POMDPs and observable POMDPs with latent low-rank transition. Under this framework, we propose an actor-critic style algorithm that is capable of performing agnostic policy learning. Given a policy class that consists of memory based policies (that look at a fixed-length window of recent observations), and a value function class that consists of functions taking both memory and future observations as inputs, our algorithm learns to compete against the best memory-based policy in the given policy class. For certain examples such as undercomplete observable tabular POMDPs, observable LQGs and observable POMDPs with latent low-rank transition, by implicitly leveraging their special properties, our algorithm is even capable of competing against the globally optimal policy without paying an exponential dependence on the horizon in its sample complexity.

cs.LG↗

arXiv2022

Computationally Efficient PAC RL in POMDPs with Latent Determinism and Conditional Embeddings

We study reinforcement learning with function approximation for large-scale Partially Observable Markov Decision Processes (POMDPs) where the state space and observation space are large or even continuous. Particularly, we consider Hilbert space embeddings of POMDP where the feature of latent states and the feature of observations admit a conditional Hilbert space embedding of the observation emission process, and the latent state transition is deterministic. Under the function approximation setup where the optimal latent state-action $Q$-function is linear in the state feature, and the optimal $Q$-function has a gap in actions, we provide a \emph{computationally and statistically efficient} algorithm for finding the \emph{exact optimal} policy. We show our algorithm's computational and statistical complexities scale polynomially with respect to the horizon and the intrinsic dimension of the feature on the observation space. Furthermore, we show both the deterministic latent transitions and gap assumptions are necessary to avoid statistical complexity exponential in horizon or dimension. Since our guarantee does not have an explicit dependence on the size of the state and observation spaces, our algorithm provably scales to large-scale POMDPs.

cs.LG↗

arXiv2022

Estimating Structural Disparities for Face Models

In machine learning, disparity metrics are often defined by measuring the difference in the performance or outcome of a model, across different sub-populations (groups) of datapoints. Thus, the inputs to disparity quantification consist of a model's predictions $\hat{y}$, the ground-truth labels for the predictions $y$, and group labels $g$ for the data points. Performance of the model for each group is calculated by comparing $\hat{y}$ and $y$ for the datapoints within a specific group, and as a result, disparity of performance across the different groups can be calculated. In many real world scenarios however, group labels ($g$) may not be available at scale during training and validation time, or collecting them might not be feasible or desirable as they could often be sensitive information. As a result, evaluating disparity metrics across categorical groups would not be feasible. On the other hand, in many scenarios noisy groupings may be obtainable using some form of a proxy, which would allow measuring disparity metrics across sub-populations. Here we explore performing such analysis on computer vision models trained on human faces, and on tasks such as face attribute prediction and affect estimation. Our experiments indicate that embeddings resulting from an off-the-shelf face recognition model, could meaningfully serve as a proxy for such estimation.

cs.CV↗

arXiv2022

Stochastic Optimization Forests

We study contextual stochastic optimization problems, where we leverage rich auxiliary observations (e.g., product characteristics) to improve decision making with uncertain variables (e.g., demand). We show how to train forest decision policies for this problem by growing trees that choose splits to directly optimize the downstream decision quality, rather than splitting to improve prediction accuracy as in the standard random forest algorithm. We realize this seemingly computationally intractable problem by developing approximate splitting criteria that utilize optimization perturbation analysis to eschew burdensome re-optimization for every candidate split, so that our method scales to large-scale problems. We prove that our splitting criteria consistently approximate the true risk and that our method achieves asymptotic optimality. We extensively validate our method empirically, demonstrating the value of optimization-aware construction of forests and the success of our efficient approximations. We show that our approximate splitting criteria can reduce running time hundredfold, while achieving performance close to forest algorithms that exactly re-optimize for every candidate split.

math.OC↗

arXiv2021

Control Variates for Slate Off-Policy Evaluation

We study the problem of off-policy evaluation from batched contextual bandit data with multidimensional actions, often termed slates. The problem is common to recommender systems and user-interface optimization, and it is particularly challenging because of the combinatorially-sized action space. Swaminathan et al. (2017) have proposed the pseudoinverse (PI) estimator under the assumption that the conditional mean rewards are additive in actions. Using control variates, we consider a large class of unbiased estimators that includes as specific cases the PI estimator and (asymptotically) its self-normalized variant. By optimizing over this class, we obtain new estimators with risk improvement guarantees over both the PI and the self-normalized PI estimators. Experiments with real-world recommender data as well as synthetic data validate these improvements in practice.

cs.LG↗

arXiv2021

Stateful Offline Contextual Policy Evaluation and Learning

We study off-policy evaluation and learning from sequential data in a structured class of Markov decision processes that arise from repeated interactions with an exogenous sequence of arrivals with contexts, which generate unknown individual-level responses to agent actions. This model can be thought of as an offline generalization of contextual bandits with resource constraints. We formalize the relevant causal structure of problems such as dynamic personalized pricing and other operations management problems in the presence of potentially high-dimensional user types. The key insight is that an individual-level response is often not causally affected by the state variable and can therefore easily be generalized across timesteps and states. When this is true, we study implications for (doubly robust) off-policy evaluation and learning by instead leveraging single time-step evaluation, estimating the expectation over a single arrival via data from a population, for fitted-value iteration in a marginal MDP. We study sample complexity and analyze error amplification that leads to the persistence, rather than attenuation, of confounding error over time. In simulations of dynamic and capacitated pricing, we show improved out-of-sample policy performance in this class of relevant problems.

cs.LG↗

arXiv2021

Residual Overfit Method of Exploration

Exploration is a crucial aspect of bandit and reinforcement learning algorithms. The uncertainty quantification necessary for exploration often comes from either closed-form expressions based on simple models or resampling and posterior approximations that are computationally intensive. We propose instead an approximate exploration methodology based on fitting only two point estimates, one tuned and one overfit. The approach, which we term the residual overfit method of exploration (ROME), drives exploration towards actions where the overfit model exhibits the most overfitting compared to the tuned model. The intuition is that overfitting occurs the most at actions and contexts with insufficient data to form accurate predictions of the reward. We justify this intuition formally from both a frequentist and a Bayesian information theoretic perspective. The result is a method that generalizes to a wide variety of models and avoids the computational overhead of resampling or posterior approximations. We compare ROME against a set of established contextual bandit methods on three datasets and find it to be one of the best performing.

cs.LG↗

arXiv2021

Fast Rates for Contextual Linear Optimization

Incorporating side observations in decision making can reduce uncertainty and boost performance, but it also requires we tackle a potentially complex predictive relationship. While one may use off-the-shelf machine learning methods to separately learn a predictive model and plug it in, a variety of recent methods instead integrate estimation and optimization by fitting the model to directly optimize downstream decision performance. Surprisingly, in the case of contextual linear optimization, we show that the naive plug-in approach actually achieves regret convergence rates that are significantly faster than methods that directly optimize downstream decision performance. We show this by leveraging the fact that specific problem instances do not have arbitrarily bad near-dual-degeneracy. While there are other pros and cons to consider as we discuss and illustrate numerically, our results highlight a nuanced landscape for the enterprise to integrate estimation and optimization. Our results are overall positive for practice: predictive models are easy and fast to train using existing tools, simple to interpret, and, as we show, lead to decisions that perform very well.

stat.ML↗

arXiv2021

Controlling for Unmeasured Confounding in Panel Data Using Minimal Bridge Functions: From Two-Way Fixed Effects to Factor Models

We develop a new approach for identifying and estimating average causal effects in panel data under a linear factor model with unmeasured confounders. Compared to other methods tackling factor models such as synthetic controls and matrix completion, our method does not require the number of time periods to grow infinitely. Instead, we draw inspiration from the two-way fixed effect model as a special case of the linear factor model, where a simple difference-in-differences transformation identifies the effect. We show that analogous, albeit more complex, transformations exist in the more general linear factor model, providing a new means to identify the effect in that model. In fact many such transformations exist, called bridge functions, all identifying the same causal effect estimand. This poses a unique challenge for estimation and inference, which we solve by targeting the minimal bridge function using a regularized estimation approach. We prove that our resulting average causal effect estimator is root-N consistent and asymptotically normal, and we provide asymptotically valid confidence intervals. Finally, we provide extensions for the case of a linear factor model with time-varying unmeasured confounders.

stat.ME↗

arXiv2021

Risk Minimization from Adaptively Collected Data: Guarantees for Supervised and Policy Learning

Empirical risk minimization (ERM) is the workhorse of machine learning, whether for classification and regression or for off-policy policy learning, but its model-agnostic guarantees can fail when we use adaptively collected data, such as the result of running a contextual bandit algorithm. We study a generic importance sampling weighted ERM algorithm for using adaptively collected data to minimize the average of a loss function over a hypothesis class and provide first-of-their-kind generalization guarantees and fast convergence rates. Our results are based on a new maximal inequality that carefully leverages the importance sampling structure to obtain rates with the right dependence on the exploration rate in the data. For regression, we provide fast rates that leverage the strong convexity of squared-error loss. For policy learning, we provide rate-optimal regret guarantees that close an open gap in the existing literature whenever exploration decays to zero, as is the case for bandit-collected data. An empirical investigation validates our theory.

stat.ML↗

arXiv2021

Post-Contextual-Bandit Inference

Contextual bandit algorithms are increasingly replacing non-adaptive A/B tests in e-commerce, healthcare, and policymaking because they can both improve outcomes for study participants and increase the chance of identifying good or even best policies. To support credible inference on novel interventions at the end of the study, nonetheless, we still want to construct valid confidence intervals on average treatment effects, subgroup effects, or value of new policies. The adaptive nature of the data collected by contextual bandit algorithms, however, makes this difficult: standard estimators are no longer asymptotically normally distributed and classic confidence intervals fail to provide correct coverage. While this has been addressed in non-contextual settings by using stabilized estimators, the contextual setting poses unique challenges that we tackle for the first time in this paper. We propose the Contextual Adaptive Doubly Robust (CADR) estimator, the first estimator for policy value that is asymptotically normal under contextual adaptive data collection. The main technical challenge in constructing CADR is designing adaptive and consistent conditional standard deviation estimators for stabilization. Extensive numerical experiments using 57 OpenML datasets demonstrate that confidence intervals based on CADR uniquely provide correct coverage.

stat.ML↗

arXiv2020

Fairness, Welfare, and Equity in Personalized Pricing

We study the interplay of fairness, welfare, and equity considerations in personalized pricing based on customer features. Sellers are increasingly able to conduct price personalization based on predictive modeling of demand conditional on covariates: setting customized interest rates, targeted discounts of consumer goods, and personalized subsidies of scarce resources with positive externalities like vaccines and bed nets. These different application areas may lead to different concerns around fairness, welfare, and equity on different objectives: price burdens on consumers, price envy, firm revenue, access to a good, equal access, and distributional consequences when the good in question further impacts downstream outcomes of interest. We conduct a comprehensive literature review in order to disentangle these different normative considerations and propose a taxonomy of different objectives with mathematical definitions. We focus on observational metrics that do not assume access to an underlying valuation distribution which is either unobserved due to binary feedback or ill-defined due to overriding behavioral concerns regarding interpreting revealed preferences. In the setting of personalized pricing for the provision of goods with positive benefits, we discuss how price optimization may provide unambiguous benefit by achieving a "triple bottom line": personalized pricing enables expanding access, which in turn may lead to gains in welfare due to heterogeneous utility, and improve revenue or budget utilization. We empirically demonstrate the potential benefits of personalized pricing in two settings: pricing subsidies for an elective vaccine, and the effects of personalized interest rates on downstream outcomes in microcredit.

cs.LG↗

arXiv2020

Rejoinder: New Objectives for Policy Learning

I provide a rejoinder for discussion of "More Efficient Policy Learning via Optimal Retargeting" to appear in the Journal of the American Statistical Association with discussion by Oliver Dukes and Stijn Vansteelandt; Sijia Li, Xiudi Li, and Alex Luedtkeand; and Muxuan Liang and Yingqi Zhao.

stat.ML↗

arXiv2020

More Efficient Policy Learning via Optimal Retargeting

Policy learning can be used to extract individualized treatment regimes from observational data in healthcare, civics, e-commerce, and beyond. One big hurdle to policy learning is a commonplace lack of overlap in the data for different actions, which can lead to unwieldy policy evaluation and poorly performing learned policies. We study a solution to this problem based on retargeting, that is, changing the population on which policies are optimized. We first argue that at the population level, retargeting may induce little to no bias. We then characterize the optimal reference policy and retargeting weights in both binary-action and multi-action settings. We do this in terms of the asymptotic efficient estimation variance of the new learning objective. Extensive empirical results in a simulation study and a case study of personalized job counseling demonstrate that retargeting is a fairly easy way to significantly improve any policy learning procedure applied to observational data.

stat.ML↗

arXiv2020

Efficient Evaluation of Natural Stochastic Policies in Offline Reinforcement Learning

We study the efficient off-policy evaluation of natural stochastic policies, which are defined in terms of deviations from the behavior policy. This is a departure from the literature on off-policy evaluation where most work consider the evaluation of explicitly specified policies. Crucially, offline reinforcement learning with natural stochastic policies can help alleviate issues of weak overlap, lead to policies that build upon current practice, and improve policies' implementability in practice. Compared with the classic case of a pre-specified evaluation policy, when evaluating natural stochastic policies, the efficiency bound, which measures the best-achievable estimation error, is inflated since the evaluation policy itself is unknown. In this paper, we derive the efficiency bounds of two major types of natural stochastic policies: tilting policies and modified treatment policies. We then propose efficient nonparametric estimators that attain the efficiency bounds under very lax conditions. These also enjoy a (partial) double robustness property.

cs.LG↗

arXiv2020

Optimal Off-Policy Evaluation from Multiple Logging Policies

We study off-policy evaluation (OPE) from multiple logging policies, each generating a dataset of fixed size, i.e., stratified sampling. Previous work noted that in this setting the ordering of the variances of different importance sampling estimators is instance-dependent, which brings up a dilemma as to which importance sampling weights to use. In this paper, we resolve this dilemma by finding the OPE estimator for multiple loggers with minimum variance for any instance, i.e., the efficient one. In particular, we establish the efficiency bound under stratified sampling and propose an estimator achieving this bound when given consistent $q$-estimates. To guard against misspecification of $q$-functions, we also provide a way to choose the control variate in a hypothesis class to minimize variance. Extensive experiments demonstrate the benefits of our methods' efficiently leveraging of the stratified sampling of off-policy data from multiple loggers.

cs.LG↗

arXiv2020

Smooth Contextual Bandits: Bridging the Parametric and Non-differentiable Regret Regimes

We study a nonparametric contextual bandit problem where the expected reward functions belong to a Hölder class with smoothness parameter $β$. We show how this interpolates between two extremes that were previously studied in isolation: non-differentiable bandits ($β\leq1$), where rate-optimal regret is achieved by running separate non-contextual bandits in different context regions, and parametric-response bandits (satisfying $β=\infty$), where rate-optimal regret can be achieved with minimal or no exploration due to infinite extrapolatability. We develop a novel algorithm that carefully adjusts to all smoothness settings and we prove its regret is rate-optimal by establishing matching upper and lower bounds, recovering the existing results at the two extremes. In this sense, our work bridges the gap between the existing literature on parametric and non-differentiable contextual bandit problems and between bandit algorithms that exclusively use global or local information, shedding light on the crucial interplay of complexity and regret in contextual bandits.

stat.ML↗

arXiv2020

Data-Pooling in Stochastic Optimization

Managing large-scale systems often involves simultaneously solving thousands of unrelated stochastic optimization problems, each with limited data. Intuition suggests one can decouple these unrelated problems and solve them separately without loss of generality. We propose a novel data-pooling algorithm called Shrunken-SAA that disproves this intuition. In particular, we prove that combining data across problems can outperform decoupling, even when there is no a priori structure linking the problems and data are drawn independently. Our approach does not require strong distributional assumptions and applies to constrained, possibly non-convex, non-smooth optimization problems such as vehicle-routing, economic lot-sizing or facility location. We compare and contrast our results to a similar phenomenon in statistics (Stein's Phenomenon), highlighting unique features that arise in the optimization setting that are not present in estimation. We further prove that as the number of problems grows large, Shrunken-SAA learns if pooling can improve upon decoupling and the optimal amount to pool, even if the average amount of data per problem is fixed and bounded. Importantly, we highlight a simple intuition based on stability that highlights when and why data-pooling offers a benefit, elucidating this perhaps surprising phenomenon. This intuition further suggests that data-pooling offers the most benefits when there are many problems, each of which has a small amount of relevant data. Finally, we demonstrate the practical benefits of data-pooling using real data from a chain of retail drug stores in the context of inventory management.

math.OC↗