Independently-Normalized SGD for Generalized-Smooth Nonconvex Optimization

Recent studies have shown that many nonconvex machine learning problems meet a so-called generalized-smooth condition that extends beyond traditional smooth nonconvex optimization. However, the existing algorithms designed for generalized-smooth nonconvex optimization encounter significant limitations in both their design and convergence analysis. In this work, we first study deterministic generalized-smooth nonconvex optimization and analyze the convergence of normalized gradient descent under the generalized Polyak-Lojasiewicz condition. Our results provide a comprehensive understanding of the interplay between gradient normalization and function geometry. Then, for stochastic generalized-smooth nonconvex optimization, we propose an independently-normalized stochastic gradient descent algorithm, which leverages independent sampling, gradient normalization and clipping to achieve an $\mathcal{O}(\epsilon^{-4})$ sample complexity under relaxed assumptions. Experiments demonstrate the fast convergence of our algorithm.

Non-Asymptotic Analysis for Single-Loop Actor-Critic with Compatible Function Approximation

Actor-critic (AC) is a powerful method for learning an optimal policy in reinforcement learning, where the critic uses algorithms, e.g., temporal difference (TD) learning with function approximation, to evaluate the current policy and the actor updates the policy along an approximate gradient direction using information from the critic. This paper provides the \textit{tightest} non-asymptotic convergence bounds for both the AC and natural AC (NAC) algorithms. Specifically, existing studies show that AC converges to an $\epsilon+\varepsilon_{\text{critic}}$ neighborhood of stationary points with the best known sample complexity of $\mathcal{O}(\epsilon^{-2})$ (up to a log factor), and NAC converges to an $\epsilon+\varepsilon_{\text{critic}}+\sqrt{\varepsilon_{\text{actor}}}$ neighborhood of the global optimum with the best known sample complexity of $\mathcal{O}(\epsilon^{-3})$, where $\varepsilon_{\text{critic}}$ is the approximation error of the critic and $\varepsilon_{\text{actor}}$ is the approximation error induced by the insufficient expressive power of the parameterized policy class. This paper analyzes the convergence of both AC and NAC algorithms with compatible function approximation. Our analysis eliminates the term $\varepsilon_{\text{critic}}$ from the error bounds while still achieving the best known sample complexities. Moreover, we focus on the challenging single-loop setting with a single Markovian sample trajectory. Our major technical novelty lies in analyzing the stochastic bias due to policy-dependent and time-varying compatible function approximation in the critic, and handling the non-ergodicity of the MDP due to the single Markovian sample trajectory. Numerical results are also provided in the appendix.

On the Convergence of Multi-objective Optimization under Generalized Smoothness

Multi-objective optimization (MOO) is receiving more attention in various fields such as multi-task learning. Recent works provide some effective algorithms with theoretical analysis but they are limited by the standard $L$-smooth or bounded-gradient assumptions, which are typically unsatisfactory for neural networks, such as recurrent neural networks (RNNs) and transformers. In this paper, we study a more general and realistic class of $\ell$-smooth loss functions, where $\ell$ is a general non-decreasing function of gradient norm. We develop two novel single-loop algorithms for $\ell$-smooth MOO problems, Generalized Smooth Multi-objective Gradient descent (GSMGrad) and its stochastic variant, Stochastic Generalized Smooth Multi-objective Gradient descent (SGSMGrad), which approximate the conflict-avoidant (CA) direction that maximizes the minimum improvement among objectives. We provide a comprehensive convergence analysis of both algorithms and show that they converge to an $\epsilon$-accurate Pareto stationary point with a guaranteed $\epsilon$-level average CA distance (i.e., the gap between the updating direction and the CA direction) over all iterations, where totally $\mathcal{O}(\epsilon^{-2})$ and $\mathcal{O}(\epsilon^{-4})$ samples are needed for deterministic and stochastic settings, respectively. Our algorithms can also guarantee a tighter $\epsilon$-level CA distance in each iteration using more samples. Moreover, we propose a practical variant of GSMGrad named GSMGrad-FA using only constant-level time and space, while achieving the same performance guarantee as GSMGrad. Our experiments validate our theory and demonstrate the effectiveness of the proposed methods.

Finite-Time Analysis for Conflict-Avoidant Multi-Task Reinforcement Learning

Multi-task reinforcement learning (MTRL) has shown great promise in many real-world applications. Existing MTRL algorithms often aim to learn a policy that optimizes individual objective functions simultaneously with a given prior preference (or weights) on different tasks. However, these methods often suffer from the issue of \textit{gradient conflict} such that the tasks with larger gradients dominate the update direction, resulting in a performance degeneration on other tasks. In this paper, we develop a novel dynamic weighting multi-task actor-critic algorithm (MTAC) under two options of sub-procedures named as CA and FC in task weight updates. MTAC-CA aims to find a conflict-avoidant (CA) update direction that maximizes the minimum value improvement among tasks, and MTAC-FC targets at a much faster convergence rate. We provide a comprehensive finite-time convergence analysis for both algorithms. We show that MTAC-CA can find a $\epsilon+\epsilon_{\text{app}}$-accurate Pareto stationary policy using $\mathcal{O}({\epsilon^{-5}})$ samples, while ensuring a small $\epsilon+\sqrt{\epsilon_{\text{app}}}$-level CA distance (defined as the distance to the CA direction), where $\epsilon_{\text{app}}$ is the function approximation error. The analysis also shows that MTAC-FC improves the sample complexity to $\mathcal{O}(\epsilon^{-3})$, but with a constant-level CA distance. Our experiments on MT10 demonstrate the improved performance of our algorithms over existing MTRL methods with fixed preference.

Constrained Reinforcement Learning Under Model Mismatch

Existing studies on constrained reinforcement learning (RL) may obtain a well-performing policy in the training environment. However, when deployed in a real environment, it may easily violate constraints that were originally satisfied during training because there might be model mismatch between the training and real environments. To address the above challenge, we formulate the problem as constrained RL under model uncertainty, where the goal is to learn a good policy that optimizes the reward and at the same time satisfy the constraint under model mismatch. We develop a Robust Constrained Policy Optimization (RCPO) algorithm, which is the first algorithm that applies to large/continuous state space and has theoretical guarantees on worst-case reward improvement and constraint violation at each iteration during the training. We demonstrate the effectiveness of our algorithm on a set of RL tasks with constraints.

Convergence Guarantees for RMSProp and Adam in Generalized-smooth Non-convex Optimization with Affine Noise Variance

This paper provides the first tight convergence analyses for RMSProp and Adam in non-convex optimization under the most relaxed assumptions of coordinate-wise generalized smoothness and affine noise variance. We first analyze RMSProp, which is a special case of Adam with adaptive learning rates but without first-order momentum. Specifically, to solve the challenges due to dependence among adaptive update, unbounded gradient estimate and Lipschitz constant, we demonstrate that the first-order term in the descent lemma converges and its denominator is upper bounded by a function of gradient norm. Based on this result, we show that RMSProp with proper hyperparameters converges to an $\epsilon$-stationary point with an iteration complexity of $\mathcal O(\epsilon^{-4})$. We then generalize our analysis to Adam, where the additional challenge is due to a mismatch between the gradient and first-order momentum. We develop a new upper bound on the first-order term in the descent lemma, which is also a function of the gradient norm. We show that Adam with proper hyperparameters converges to an $\epsilon$-stationary point with an iteration complexity of $\mathcal O(\epsilon^{-4})$. Our complexity results for both RMSProp and Adam match with the complexity lower bound established in \cite{arjevani2023lower}.

Large-Scale Non-convex Stochastic Constrained Distributionally Robust Optimization

Distributionally robust optimization (DRO) is a powerful framework for training robust models against data distribution shifts. This paper focuses on constrained DRO, which has an explicit characterization of the robustness level. Existing studies on constrained DRO mostly focus on convex loss function, and exclude the practical and challenging case with non-convex loss function, e.g., neural network. This paper develops a stochastic algorithm and its performance analysis for non-convex constrained DRO. The computational complexity of our stochastic algorithm at each iteration is independent of the overall dataset size, and thus is suitable for large-scale applications. We focus on the general Cressie-Read family divergence defined uncertainty set which includes $\chi^2$-divergences as a special case. We prove that our algorithm finds an $\epsilon$-stationary point with a computational complexity of $\mathcal O(\epsilon^{-3k_*-5})$, where $k_*$ is the parameter of the Cressie-Read divergence. The numerical results indicate that our method outperforms existing methods.} Our method also applies to the smoothed conditional value at risk (CVaR) DRO.

Sample Complexity Characterization for Linear Contextual MDPs

Contextual Markov decision processes (CMDPs) describe a class of reinforcement learning problems in which the transition kernels and reward functions can change over time with different MDPs indexed by a context variable. While CMDPs serve as an important framework to model many real-world applications with time-varying environments, they are largely unexplored from theoretical perspective. In this paper, we study CMDPs under two linear function approximation models: Model I with context-varying representations and common linear weights for all contexts; and Model II with common representations for all contexts and context-varying linear weights. For both models, we propose novel model-based algorithms and show that they enjoy guaranteed $\epsilon$-suboptimality gap with desired polynomial sample complexity. In particular, instantiating our result for the first model to the tabular CMDP improves the existing result by removing the reachability assumption. Our result for the second model is the first-known result for such a type of function approximation models. Comparison between our results for the two models further indicates that having context-varying features leads to much better sample efficiency than having common representations for all contexts under linear CMDPs.

Quickest Change Detection in Autoregressive Models

The problem of quickest change detection (QCD) in autoregressive (AR) models is investigated. A system is being monitored with sequentially observed samples. At some unknown time, a disturbance signal occurs and changes the distribution of the observations. The disturbance signal follows an AR model, which is dependent over time. Before the change, observations only consist of measurement noise, and are independent and identically distributed (i.i.d.). After the change, observations consist of the disturbance signal and the measurement noise, are dependent over time, which essentially follow a continuous-state hidden Markov model (HMM). The goal is to design a stopping time to detect the disturbance signal as quickly as possible subject to false alarm constraints. Existing approaches for general non-i.i.d. settings and discrete-state HMMs cannot be applied due to their high computational complexity and memory consumption, and they usually assume some asymptotic stability condition. In this paper, the asymptotic stability condition is firstly theoretically proved for the AR model by a novel design of forward variable and auxiliary Markov chain. A computationally efficient Ergodic CuSum algorithm that can be updated recursively is then constructed and is further shown to be asymptotically optimal. The data-driven setting where the disturbance signal parameters are unknown is further investigated, and an online and computationally efficient gradient ascent CuSum algorithm is designed. The algorithm is constructed by iteratively updating the estimate of the unknown parameters based on the maximum likelihood principle and the gradient ascent approach. The lower bound on its average running length to false alarm is also derived for practical false alarm control. Simulation results are provided to demonstrate the performance of the proposed algorithms.

Robust Multi-Agent Reinforcement Learning with State Uncertainty

In real-world multi-agent reinforcement learning (MARL) applications, agents may not have perfect state information (e.g., due to inaccurate measurement or malicious attacks), which challenges the robustness of agents' policies. Though robustness is getting important in MARL deployment, little prior work has studied state uncertainties in MARL, neither in problem formulation nor algorithm design. Motivated by this robustness issue and the lack of corresponding studies, we study the problem of MARL with state uncertainty in this work. We provide the first attempt to the theoretical and empirical analysis of this challenging problem. We first model the problem as a Markov Game with state perturbation adversaries (MG-SPA) by introducing a set of state perturbation adversaries into a Markov Game. We then introduce robust equilibrium (RE) as the solution concept of an MG-SPA. We conduct a fundamental analysis regarding MG-SPA such as giving conditions under which such a robust equilibrium exists. Then we propose a robust multi-agent Q-learning (RMAQ) algorithm to find such an equilibrium, with convergence guarantees. To handle high-dimensional state-action space, we design a robust multi-agent actor-critic (RMAAC) algorithm based on an analytical expression of the policy gradient derived in the paper. Our experiments show that the proposed RMAQ algorithm converges to the optimal value function; our RMAAC algorithm outperforms several MARL and robust MARL methods in multiple multi-agent environments when state uncertainty is present. The source code is public on \url{https://github.com/sihongho/robust_marl_with_state_uncertainty}.