Episodic Bandits with Stochastic Experts

We study a version of the contextual bandit problem where an agent is given soft control of a node in a graph-structured environment through a set of stochastic expert policies. The agent interacts with the environment over episodes, with each episode having different context distributions; this results in the `best expert' changing across episodes. Our goal is to develop an agent that tracks the best expert over episodes. We introduce the Empirical Divergence-based UCB (ED-UCB) algorithm in this setting where the agent does not have any knowledge of the expert policies or changes in context distributions. With mild assumptions, we show that bootstrapping from $\tilde{O}(N\log(NT^2\sqrt{E}))$ samples results in a regret of $\tilde{O}(E(N+1) + \frac{N\sqrt{E}}{T^2})$. If the expert policies are known to the agent a priori, then we can improve the regret to $\tilde{O}(EN)$ without requiring any bootstrapping. Our analysis also tightens pre-existing logarithmic regret bounds to a problem-dependent constant in the non-episodic setting when expert policies are known. We finally empirically validate our findings through simulations.

Finite-Sample Analysis of Off-Policy TD-Learning via Generalized Bellman Operators

In temporal difference (TD) learning, off-policy sampling is known to be more practical than on-policy sampling, and by decoupling learning from data collection, it enables data reuse. It is known that policy evaluation (including multi-step off-policy importance sampling) has the interpretation of solving a generalized Bellman equation. In this paper, we derive finite-sample bounds for any general off-policy TD-like stochastic approximation algorithm that solves for the fixed-point of this generalized Bellman operator. Our key step is to show that the generalized Bellman operator is simultaneously a contraction mapping with respect to a weighted $\ell_p$-norm for each $p$ in $[1,\infty)$, with a common contraction factor. Off-policy TD-learning is known to suffer from high variance due to the product of importance sampling ratios. A number of algorithms (e.g. $Q^\pi(\lambda)$, Tree-Backup$(\lambda)$, Retrace$(\lambda)$, and $Q$-trace) have been proposed in the literature to address this issue. Our results immediately imply finite-sample bounds of these algorithms. In particular, we provide first-known finite-sample guarantees for $Q^\pi(\lambda)$, Tree-Backup$(\lambda)$, and Retrace$(\lambda)$, and improve the best known bounds of $Q$-trace in [19]. Moreover, we show the bias-variance trade-offs in each of these algorithms.

Finding Valid Adjustments under Non-ignorability with Minimal DAG Knowledge

Treatment effect estimation from observational data is a fundamental problem in causal inference. There are two very different schools of thought that have tackled this problem. On the one hand, the Pearlian framework commonly assumes structural knowledge (provided by an expert) in the form of Directed Acyclic Graphs (DAGs) and provides graphical criteria such as the back-door criterion to identify the valid adjustment sets. On the other hand, the potential outcomes (PO) framework commonly assumes that all the observed features satisfy ignorability (i.e., no hidden confounding), which in general is untestable. In this work, we take steps to bridge these two frameworks. We show that even if we know only one parent of the treatment variable (provided by an expert), then quite remarkably it suffices to test a broad class of (but not all) back-door criteria. Importantly, we also cover the non-trivial case where the entire set of observed features is not ignorable (generalizing the PO framework) without requiring all the parents of the treatment variable to be observed. Our key technical idea involves a more general result -- Given a synthetic sub-sampling (or environment) variable that is a function of the parent variable, we show that an invariance test involving this sub-sampling variable is equivalent to testing a broad class of back-door criteria. We demonstrate our approach on synthetic data as well as real causal effect estimation benchmarks.

Treatment Effect Estimation using Invariant Risk Minimization

Inferring causal individual treatment effect (ITE) from observational data is a challenging problem whose difficulty is exacerbated by the presence of treatment assignment bias. In this work, we propose a new way to estimate the ITE using the domain generalization framework of invariant risk minimization (IRM). IRM uses data from multiple domains, learns predictors that do not exploit spurious domain-dependent factors, and generalizes better to unseen domains. We propose an IRM-based ITE estimator aimed at tackling treatment assignment bias when there is little support overlap between the control group and the treatment group. We accomplish this by creating diversity: given a single dataset, we split the data into multiple domains artificially. These diverse domains are then exploited by IRM to more effectively generalize regression-based models to data regions that lack support overlap. We show gains over classical regression approaches to ITE estimation in settings when support mismatch is more pronounced.

Efficient Encrypted Inference on Ensembles of Decision Trees

Data privacy concerns often prevent the use of cloud-based machine learning services for sensitive personal data. While homomorphic encryption (HE) offers a potential solution by enabling computations on encrypted data, the challenge is to obtain accurate machine learning models that work within the multiplicative depth constraints of a leveled HE scheme. Existing approaches for encrypted inference either make ad-hoc simplifications to a pre-trained model (e.g., replace hard comparisons in a decision tree with soft comparators) at the cost of accuracy or directly train a new depth-constrained model using the original training set. In this work, we propose a framework to transfer knowledge extracted by complex decision tree ensembles to shallow neural networks (referred to as DTNets) that are highly conducive to encrypted inference. Our approach minimizes the accuracy loss by searching for the best DTNet architecture that operates within the given depth constraints and training this DTNet using only synthetic data sampled from the training data distribution. Extensive experiments on real-world datasets demonstrate that these characteristics are critical in ensuring that DTNet accuracy approaches that of the original tree ensemble. Our system is highly scalable and can perform efficient inference on batched encrypted (134 bits of security) data with amortized time in milliseconds. This is approximately three orders of magnitude faster than the standard approach of applying soft comparison at the internal nodes of the ensemble trees.

A Lyapunov Theory for Finite-Sample Guarantees of Asynchronous Q-Learning and TD-Learning Variants

This paper develops an unified framework to study finite-sample convergence guarantees of a large class of value-based asynchronous Reinforcement Learning (RL) algorithms. We do this by first reformulating the RL algorithms as Markovian Stochastic Approximation (SA) algorithms to solve fixed-point equations. We then develop a Lyapunov analysis and derive mean-square error bounds on the convergence of the Markovian SA. Based on this central result, we establish finite-sample mean-square convergence bounds for asynchronous RL algorithms such as $Q$-learning, $n$-step TD, TD$(\lambda)$, and off-policy TD algorithms including V-trace. As a by-product, by analyzing the performance bounds of the TD$(\lambda)$ (and $n$-step TD) algorithm for general $\lambda$ (and $n$), we demonstrate a bias-variance trade-off, i.e., efficiency of bootstrapping in RL. This was first posed as an open problem in [37].

Stochastic Linear Bandits with Protected Subspace

We study a variant of the stochastic linear bandit problem wherein we optimize a linear objective function but rewards are accrued only orthogonal to an unknown subspace (which we interpret as a \textit{protected space}) given only zero-order stochastic oracle access to both the objective itself and protected subspace. In particular, at each round, the learner must choose whether to query the objective or the protected subspace alongside choosing an action. Our algorithm, derived from the OFUL principle, uses some of the queries to get an estimate of the protected space, and (in almost all rounds) plays optimistically with respect to a confidence set for this space. We provide a $\tilde{O}(sd\sqrt{T})$ regret upper bound in the case where the action space is the complete unit ball in $\mathbb{R}^d$, $s < d$ is the dimension of the protected subspace, and $T$ is the time horizon. Moreover, we demonstrate that a discrete action space can lead to linear regret with an optimistic algorithm, reinforcing the sub-optimality of optimism in certain settings. We also show that protection constraints imply that for certain settings, no consistent algorithm can have a regret smaller than $\Omega(T^{3/4}).$ We finally empirically validate our results with synthetic and real datasets.

Active Structure Learning of Causal DAGs via Directed Clique Tree

A growing body of work has begun to study intervention design for efficient structure learning of causal directed acyclic graphs (DAGs). A typical setting is a causally sufficient setting, i.e. a system with no latent confounders, selection bias, or feedback, when the essential graph of the observational equivalence class (EC) is given as an input and interventions are assumed to be noiseless. Most existing works focus on worst-case or average-case lower bounds for the number of interventions required to orient a DAG. These worst-case lower bounds only establish that the largest clique in the essential graph could make it difficult to learn the true DAG. In this work, we develop a universal lower bound for single-node interventions that establishes that the largest clique is always a fundamental impediment to structure learning. Specifically, we present a decomposition of a DAG into independently orientable components through directed clique trees and use it to prove that the number of single-node interventions necessary to orient any DAG in an EC is at least the sum of half the size of the largest cliques in each chain component of the essential graph. Moreover, we present a two-phase intervention design algorithm that, under certain conditions on the chordal skeleton, matches the optimal number of interventions up to a multiplicative logarithmic factor in the number of maximal cliques. We show via synthetic experiments that our algorithm can scale to much larger graphs than most of the related work and achieves better worst-case performance than other scalable approaches. A code base to recreate these results can be found at https://github.com/csquires/dct-policy

Empirical or Invariant Risk Minimization? A Sample Complexity Perspective

Recently, invariant risk minimization (IRM) was proposed as a promising solution to address out-of-distribution (OOD) generalization. However, it is unclear when IRM should be preferred over the widely-employed empirical risk minimization (ERM) framework. In this work, we analyze both these frameworks from the perspective of sample complexity, thus taking a firm step towards answering this important question. We find that depending on the type of data generation mechanism, the two approaches might have very different finite sample and asymptotic behavior. For example, in the covariate shift setting we see that the two approaches not only arrive at the same asymptotic solution, but also have similar finite sample behavior with no clear winner. For other distribution shifts such as those involving confounders or anti-causal variables, however, the two approaches arrive at different asymptotic solutions where IRM is guaranteed to be close to the desired OOD solutions in the finite sample regime, while ERM is biased even asymptotically. We further investigate how different factors -- the number of environments, complexity of the model, and IRM penalty weight -- impact the sample complexity of IRM in relation to its distance from the OOD solutions

Linear Regression Games: Convergence Guarantees to Approximate Out-of-Distribution Solutions

Recently, invariant risk minimization (IRM) (Arjovsky et al.) was proposed as a promising solution to address out-of-distribution (OOD) generalization. In Ahuja et al., it was shown that solving for the Nash equilibria of a new class of "ensemble-games" is equivalent to solving IRM. In this work, we extend the framework in Ahuja et al. for linear regressions by projecting the ensemble-game on an $\ell_{\infty}$ ball. We show that such projections help achieve non-trivial OOD guarantees despite not achieving perfect invariance. For linear models with confounders, we prove that Nash equilibria of these games are closer to the ideal OOD solutions than the standard empirical risk minimization (ERM) and we also provide learning algorithms that provably converge to these Nash Equilibria. Empirical comparisons of the proposed approach with the state-of-the-art show consistent gains in achieving OOD solutions in several settings involving anti-causal variables and confounders.