In performative prediction, a predictive model impacts the distribution that generates future data, a phenomenon that is being ignored in classical supervised learning. In this closed-loop setting, the natural measure of performance named performative risk ($\mathrm{PR}$), captures the expected loss incurred by a predictive model \emph{after} deployment. The core difficulty of using the performative risk as an optimization objective is that the data distribution itself depends on the model parameters. This dependence is governed by the environment and not under the control of the learner. As a consequence, even the choice of a convex loss function can result in a highly non-convex $\mathrm{PR}$ minimization problem. Prior work has identified a pair of general conditions on the loss and the mapping from model parameters to distributions that implies the convexity of the performative risk. In this paper, we relax these assumptions and focus on obtaining weaker notions of convexity, without sacrificing the amenability of the $\mathrm{PR}$ minimization problem for iterative optimization methods.

Policy Optimization (PO) algorithms have been proven particularly suited to handle the high-dimensionality of real-world continuous control tasks. In this context, Trust Region Policy Optimization methods represent a popular approach to stabilize the policy updates. These usually rely on the Kullback-Leibler (KL) divergence to limit the change in the policy. The Wasserstein distance represents a natural alternative, in place of the KL divergence, to define trust regions or to regularize the objective function. However, state-of-the-art works either resort to its approximations or do not provide an algorithm for continuous state-action spaces, reducing the applicability of the method. In this paper, we explore optimal transport discrepancies (which include the Wasserstein distance) to define trust regions, and we propose a novel algorithm - Optimal Transport Trust Region Policy Optimization (OT-TRPO) - for continuous state-action spaces. We circumvent the infinite-dimensional optimization problem for PO by providing a one-dimensional dual reformulation for which strong duality holds. We then analytically derive the optimal policy update given the solution of the dual problem. This way, we bypass the computation of optimal transport costs and of optimal transport maps, which we implicitly characterize by solving the dual formulation. Finally, we provide an experimental evaluation of our approach across various control tasks. Our results show that optimal transport discrepancies can offer an advantage over state-of-the-art approaches.

The prevalent fully-connected tensor network (FCTN) has achieved excellent success to compress data. However, the FCTN decomposition suffers from slow computational speed when facing higher-order and large-scale data. Naturally, there arises an interesting question: can a new model be proposed that decomposes the tensor into smaller ones and speeds up the computation of the algorithm? This work gives a positive answer by formulating a novel higher-order tensor decomposition model that utilizes latent matrices based on the tensor network structure, which can decompose a tensor into smaller-scale data than the FCTN decomposition, hence we named it Latent Matrices for Tensor Network Decomposition (LMTN). Furthermore, three optimization algorithms, LMTN-PAM, LMTN-SVD and LMTN-AR, have been developed and applied to the tensor-completion task. In addition, we provide proofs of theoretical convergence and complexity analysis for these algorithms. Experimental results show that our algorithm has the effectiveness in both deep learning dataset compression and higher-order tensor completion, and that our LMTN-SVD algorithm is 3-6 times faster than the FCTN-PAM algorithm and only a 1.8 points accuracy drop.

The problem of constrained Markov decision process is considered. An agent aims to maximize the expected accumulated discounted reward subject to multiple constraints on its costs (the number of constraints is relatively small). A new dual approach is proposed with the integration of two ingredients: entropy regularized policy optimizer and Vaidya's dual optimizer, both of which are critical to achieve faster convergence. The finite-time error bound of the proposed approach is provided. Despite the challenge of the nonconcave objective subject to nonconcave constraints, the proposed approach is shown to converge (with linear rate) to the global optimum. The complexity expressed in terms of the optimality gap and the constraint violation significantly improves upon the existing primal-dual approaches.

When robots enter everyday human environments, they need to understand their tasks and how they should perform those tasks. To encode these, reward functions, which specify the objective of a robot, are employed. However, designing reward functions can be extremely challenging for complex tasks and environments. A promising approach is to learn reward functions from humans. Recently, several robot learning works embrace this approach and leverage human demonstrations to learn the reward functions. Known as inverse reinforcement learning, this approach relies on a fundamental assumption: humans can provide near-optimal demonstrations to the robot. Unfortunately, this is rarely the case: human demonstrations to the robot are often suboptimal due to various reasons, e.g., difficulty of teleoperation, robot having high degrees of freedom, or humans' cognitive limitations. This thesis is an attempt towards learning reward functions from human users by using other, more reliable data modalities. Specifically, we study how reward functions can be learned using comparative feedback, in which the human user compares multiple robot trajectories instead of (or in addition to) providing demonstrations. To this end, we first propose various forms of comparative feedback, e.g., pairwise comparisons, best-of-many choices, rankings, scaled comparisons; and describe how a robot can use these various forms of human feedback to infer a reward function, which may be parametric or non-parametric. Next, we propose active learning techniques to enable the robot to ask for comparison feedback that optimizes for the expected information that will be gained from that user feedback. Finally, we demonstrate the applicability of our methods in a wide variety of domains, ranging from autonomous driving simulations to home robotics, from standard reinforcement learning benchmarks to lower-body exoskeletons.

We consider the general setting for partial matching of two or multiple graphs, in the sense that not necessarily all the nodes in one graph can find their correspondences in another graph and vice versa. We take a universe matching perspective to this ubiquitous problem, whereby each node is either matched into an anchor in a virtual universe graph or regarded as an outlier. Such a universe matching scheme enjoys a few important merits, which have not been adopted in existing learning-based graph matching (GM) literature. First, the subtle logic for inlier matching and outlier detection can be clearly modeled, which is otherwise less convenient to handle in the pairwise matching scheme. Second, it enables end-to-end learning especially for universe level affinity metric learning for inliers matching, and loss design for gathering outliers together. Third, the resulting matching model can easily handle new arriving graphs under online matching, or even the graphs coming from different categories of the training set. To our best knowledge, this is the first deep learning network that can cope with two-graph matching, multiple-graph matching, online matching, and mixture graph matching simultaneously. Extensive experimental results show the state-of-the-art performance of our method in these settings.

We study differentially private (DP) algorithms for smooth stochastic minimax optimization, with stochastic minimization as a byproduct. The holy grail of these settings is to guarantee the optimal trade-off between the privacy and the excess population loss, using an algorithm with a linear time-complexity in the number of training samples. We provide a general framework for solving differentially private stochastic minimax optimization (DP-SMO) problems, which enables the practitioners to bring their own base optimization algorithm and use it as a black-box to obtain the near-optimal privacy-loss trade-off. Our framework is inspired from the recently proposed Phased-ERM method [22] for nonsmooth differentially private stochastic convex optimization (DP-SCO), which exploits the stability of the empirical risk minimization (ERM) for the privacy guarantee. The flexibility of our approach enables us to sidestep the requirement that the base algorithm needs to have bounded sensitivity, and allows the use of sophisticated variance-reduced accelerated methods to achieve near-linear time-complexity. To the best of our knowledge, these are the first near-linear time algorithms with near-optimal guarantees on the population duality gap for smooth DP-SMO, when the objective is (strongly-)convex--(strongly-)concave. Additionally, based on our flexible framework, we enrich the family of near-linear time algorithms for smooth DP-SCO with the near-optimal privacy-loss trade-off.

In this work, we provide a fundamental unified convergence theorem used for deriving expected and almost sure convergence results for a series of stochastic optimization methods. Our unified theorem only requires to verify several representative conditions and is not tailored to any specific algorithm. As a direct application, we recover expected and almost sure convergence results of the stochastic gradient method (SGD) and random reshuffling (RR) under more general settings. Moreover, we establish new expected and almost sure convergence results for the stochastic proximal gradient method (prox-SGD) and stochastic model-based methods (SMM) for nonsmooth nonconvex optimization problems. These applications reveal that our unified theorem provides a plugin-type convergence analysis and strong convergence guarantees for a wide class of stochastic optimization methods.

Multi-aspect controllable text generation is a more challenging and practical task than single-aspect control. Existing methods achieve complex multi-aspect control by fusing multiple controllers learned from single-aspect, but suffer from attribute degeneration caused by the mutual interference of these controllers. To address this, we provide observations on attribute fusion from a distributional perspective and propose to directly search for the intersection areas of multiple attribute distributions as their combination for generation. Our method first estimates the attribute space with an autoencoder structure. Afterward, we iteratively approach the intersections by jointly minimizing distances to points representing different attributes. Finally, we map them to attribute-relevant sentences with a prefix-tuning-based decoder. Experiments on the three-aspect control task, including sentiment, topic, and detoxification aspects, reveal that our method outperforms several strong baselines on attribute relevance and text quality and achieves the SOTA. Further analysis also supplies some explanatory support for the effectiveness of our approach.

We study the problem of policy optimization (PO) with linear temporal logic (LTL) constraints. The language of LTL allows flexible description of tasks that may be unnatural to encode as a scalar cost function. We consider LTL-constrained PO as a systematic framework, decoupling task specification from policy selection, and as an alternative to the standard of cost shaping. With access to a generative model, we develop a model-based approach that enjoys a sample complexity analysis for guaranteeing both task satisfaction and cost optimality (through a reduction to a reachability problem). Empirically, our algorithm can achieve strong performance even in low-sample regimes.