Improving OOD Generalization of Pre-trained Encoders via Aligned Embedding-Space Ensembles

The quality of self-supervised pre-trained embeddings on out-of-distribution (OOD) data is poor without fine-tuning. A straightforward and simple approach to improving the generalization of pre-trained representation to OOD data is the use of deep ensembles. However, obtaining an effective ensemble in the embedding space with only unlabeled data remains an unsolved problem. We first perform a theoretical analysis that reveals the relationship between individual hyperspherical embedding spaces in an ensemble. We then design a principled method to align these embedding spaces in an unsupervised manner. Experimental results on the MNIST dataset show that our embedding-space ensemble method improves pre-trained embedding quality on in-distribution and OOD data compared to single encoders.

Fast Convergence of Softmax Policy Mirror Ascent

Natural policy gradient (NPG) is a common policy optimization algorithm and can be viewed as mirror ascent in the space of probabilities. Recently, Vaswani et al. [2021] introduced a policy gradient method that corresponds to mirror ascent in the dual space of logits. We refine this algorithm, removing its need for a normalization across actions and analyze the resulting method (referred to as SPMA). For tabular MDPs, we prove that SPMA with a constant step-size matches the linear convergence of NPG and achieves a faster convergence than constant step-size (accelerated) softmax policy gradient. To handle large state-action spaces, we extend SPMA to use a log-linear policy parameterization. Unlike that for NPG, generalizing SPMA to the linear function approximation (FA) setting does not require compatible function approximation. Unlike MDPO, a practical generalization of NPG, SPMA with linear FA only requires solving convex softmax classification problems. We prove that SPMA achieves linear convergence to the neighbourhood of the optimal value function. We extend SPMA to handle non-linear FA and evaluate its empirical performance on the MuJoCo and Atari benchmarks. Our results demonstrate that SPMA consistently achieves similar or better performance compared to MDPO, PPO and TRPO.

Towards Principled, Practical Policy Gradient for Bandits and Tabular MDPs

We consider (stochastic) softmax policy gradient (PG) methods for bandits and tabular Markov decision processes (MDPs). While the PG objective is non-concave, recent research has used the objective's smoothness and gradient domination properties to achieve convergence to an optimal policy. However, these theoretical results require setting the algorithm parameters according to unknown problem-dependent quantities (e.g. the optimal action or the true reward vector in a bandit problem). To address this issue, we borrow ideas from the optimization literature to design practical, principled PG methods in both the exact and stochastic settings. In the exact setting, we employ an Armijo line-search to set the step-size for softmax PG and empirically demonstrate a linear convergence rate. In the stochastic setting, we utilize exponentially decreasing step-sizes, and characterize the convergence rate of the resulting algorithm. We show that the proposed algorithm offers similar theoretical guarantees as the state-of-the art results, but does not require the knowledge of oracle-like quantities. For the multi-armed bandit setting, our techniques result in a theoretically-principled PG algorithm that does not require explicit exploration, the knowledge of the reward gap, the reward distributions, or the noise. Finally, we empirically compare the proposed methods to PG approaches that require oracle knowledge, and demonstrate competitive performance.

From Inverse Optimization to Feasibility to ERM

Inverse optimization involves inferring unknown parameters of an optimization problem from known solutions, and is widely used in fields such as transportation, power systems and healthcare. We study the contextual inverse optimization setting that utilizes additional contextual information to better predict the unknown problem parameters. We focus on contextual inverse linear programming (CILP), addressing the challenges posed by the non-differentiable nature of LPs. For a linear prediction model, we reduce CILP to a convex feasibility problem allowing the use of standard algorithms such as alternating projections. The resulting algorithm for CILP is equipped with a linear convergence guarantee without additional assumptions such as degeneracy or interpolation. Next, we reduce CILP to empirical risk minimization (ERM) on a smooth, convex loss that satisfies the Polyak-Lojasiewicz condition. This reduction enables the use of scalable first-order optimization methods to solve large non-convex problems, while maintaining theoretical guarantees in the convex setting. Finally, we experimentally validate our approach on both synthetic and real-world problems, and demonstrate improved performance compared to existing methods.

Noise-adaptive (Accelerated) Stochastic Heavy-Ball Momentum

We analyze the convergence of stochastic heavy ball (SHB) momentum in the smooth, strongly-convex setting. Kidambi et al. (2018) show that SHB (with small mini-batches) cannot attain an accelerated rate of convergence even for quadratics, and conjecture that the practical gain of SHB is a by-product of mini-batching. We substantiate this claim by showing that SHB can obtain an accelerated rate when the mini-batch size is larger than some threshold. In particular, for strongly-convex quadratics with condition number $\kappa$, we prove that SHB with the standard step-size and momentum parameters results in an $O\left(\exp(-\frac{T}{\sqrt{\kappa}}) + \sigma \right)$ convergence rate, where $T$ is the number of iterations and $\sigma^2$ is the variance in the stochastic gradients. To ensure convergence to the minimizer, we propose a multi-stage approach that results in a noise-adaptive $O\left(\exp\left(-\frac{T}{\sqrt{\kappa}} \right) + \frac{\sigma}{T}\right)$ rate. For general strongly-convex functions, we use the averaging interpretation of SHB along with exponential step-sizes to prove an $O\left(\exp\left(-\frac{T}{\kappa} \right) + \frac{\sigma^2}{T} \right)$ convergence to the minimizer in a noise-adaptive manner. Finally, we empirically demonstrate the effectiveness of the proposed algorithms.

Decision-Aware Actor-Critic with Function Approximation and Theoretical Guarantees

Actor-critic (AC) methods are widely used in reinforcement learning (RL) and benefit from the flexibility of using any policy gradient method as the actor and value-based method as the critic. The critic is usually trained by minimizing the TD error, an objective that is potentially decorrelated with the true goal of achieving a high reward with the actor. We address this mismatch by designing a joint objective for training the actor and critic in a decision-aware fashion. We use the proposed objective to design a generic, AC algorithm that can easily handle any function approximation. We explicitly characterize the conditions under which the resulting algorithm guarantees monotonic policy improvement, regardless of the choice of the policy and critic parameterization. Instantiating the generic algorithm results in an actor that involves maximizing a sequence of surrogate functions (similar to TRPO, PPO) and a critic that involves minimizing a closely connected objective. Using simple bandit examples, we provably establish the benefit of the proposed critic objective over the standard squared error. Finally, we empirically demonstrate the benefit of our decision-aware actor-critic framework on simple RL problems.

Target-based Surrogates for Stochastic Optimization

We consider minimizing functions for which it is expensive to compute the (possibly stochastic) gradient. Such functions are prevalent in reinforcement learning, imitation learning and adversarial training. Our target optimization framework uses the (expensive) gradient computation to construct surrogate functions in a target space (e.g. the logits output by a linear model for classification) that can be minimized efficiently. This allows for multiple parameter updates to the model, amortizing the cost of gradient computation. In the full-batch setting, we prove that our surrogate is a global upper-bound on the loss, and can be (locally) minimized using a black-box optimization algorithm. We prove that the resulting majorization-minimization algorithm ensures convergence to a stationary point of the loss. Next, we instantiate our framework in the stochastic setting and propose the $SSO$ algorithm, which can be viewed as projected stochastic gradient descent in the target space. This connection enables us to prove theoretical guarantees for $SSO$ when minimizing convex functions. Our framework allows the use of standard stochastic optimization algorithms to construct surrogates which can be minimized by any deterministic optimization method. To evaluate our framework, we consider a suite of supervised learning and imitation learning problems. Our experiments indicate the benefits of target optimization and the effectiveness of $SSO$.

Improved Policy Optimization for Online Imitation Learning

We consider online imitation learning (OIL), where the task is to find a policy that imitates the behavior of an expert via active interaction with the environment. We aim to bridge the gap between the theory and practice of policy optimization algorithms for OIL by analyzing one of the most popular OIL algorithms, DAGGER. Specifically, if the class of policies is sufficiently expressive to contain the expert policy, we prove that DAGGER achieves constant regret. Unlike previous bounds that require the losses to be strongly-convex, our result only requires the weaker assumption that the losses be strongly-convex with respect to the policy's sufficient statistics (not its parameterization). In order to ensure convergence for a wider class of policies and losses, we augment DAGGER with an additional regularization term. In particular, we propose a variant of Follow-the-Regularized-Leader (FTRL) and its adaptive variant for OIL and develop a memory-efficient implementation, which matches the memory requirements of FTL. Assuming that the loss functions are smooth and convex with respect to the parameters of the policy, we also prove that FTRL achieves constant regret for any sufficiently expressive policy class, while retaining $O(\sqrt{T})$ regret in the worst-case. We demonstrate the effectiveness of these algorithms with experiments on synthetic and high-dimensional control tasks.

Near-Optimal Sample Complexity Bounds for Constrained MDPs

In contrast to the advances in characterizing the sample complexity for solving Markov decision processes (MDPs), the optimal statistical complexity for solving constrained MDPs (CMDPs) remains unknown. We resolve this question by providing minimax upper and lower bounds on the sample complexity for learning near-optimal policies in a discounted CMDP with access to a generative model (simulator). In particular, we design a model-based algorithm that addresses two settings: (i) relaxed feasibility, where small constraint violations are allowed, and (ii) strict feasibility, where the output policy is required to satisfy the constraint. For (i), we prove that our algorithm returns an $\epsilon$-optimal policy with probability $1 - \delta$, by making $\tilde{O}\left(\frac{S A \log(1/\delta)}{(1 - \gamma)^3 \epsilon^2}\right)$ queries to the generative model, thus matching the sample-complexity for unconstrained MDPs. For (ii), we show that the algorithm's sample complexity is upper-bounded by $\tilde{O} \left(\frac{S A \, \log(1/\delta)}{(1 - \gamma)^5 \, \epsilon^2 \zeta^2} \right)$ where $\zeta$ is the problem-dependent Slater constant that characterizes the size of the feasible region. Finally, we prove a matching lower-bound for the strict feasibility setting, thus obtaining the first near minimax optimal bounds for discounted CMDPs. Our results show that learning CMDPs is as easy as MDPs when small constraint violations are allowed, but inherently more difficult when we demand zero constraint violation.

Towards Painless Policy Optimization for Constrained MDPs

We study policy optimization in an infinite horizon, $\gamma$-discounted constrained Markov decision process (CMDP). Our objective is to return a policy that achieves large expected reward with a small constraint violation. We consider the online setting with linear function approximation and assume global access to the corresponding features. We propose a generic primal-dual framework that allows us to bound the reward sub-optimality and constraint violation for arbitrary algorithms in terms of their primal and dual regret on online linear optimization problems. We instantiate this framework to use coin-betting algorithms and propose the Coin Betting Politex (CBP) algorithm. Assuming that the action-value functions are $\varepsilon_b$-close to the span of the $d$-dimensional state-action features and no sampling errors, we prove that $T$ iterations of CBP result in an $O\left(\frac{1}{(1 - \gamma)^3 \sqrt{T}} + \frac{\varepsilon_b\sqrt{d}}{(1 - \gamma)^2} \right)$ reward sub-optimality and an $O\left(\frac{1}{(1 - \gamma)^2 \sqrt{T}} + \frac{\varepsilon_b \sqrt{d}}{1 - \gamma} \right)$ constraint violation. Importantly, unlike gradient descent-ascent and other recent methods, CBP does not require extensive hyperparameter tuning. Via experiments on synthetic and Cartpole environments, we demonstrate the effectiveness and robustness of CBP.