A Theoretical Analysis of Soft-Label vs Hard-Label Training in Neural Networks

Knowledge distillation, where a small student model learns from a pre-trained large teacher model, has achieved substantial empirical success since the seminal work of \citep{hinton2015distilling}. Despite prior theoretical studies exploring the benefits of knowledge distillation, an important question remains unanswered: why does soft-label training from the teacher require significantly fewer neurons than directly training a small neural network with hard labels? To address this, we first present motivating experimental results using simple neural network models on a binary classification problem. These results demonstrate that soft-label training consistently outperforms hard-label training in accuracy, with the performance gap becoming more pronounced as the dataset becomes increasingly difficult to classify. We then substantiate these observations with a theoretical contribution based on two-layer neural network models. Specifically, we show that soft-label training using gradient descent requires only $O\left(\frac{1}{\gamma^2 \epsilon}\right)$ neurons to achieve a classification loss averaged over epochs smaller than some $\epsilon > 0$, where $\gamma$ is the separation margin of the limiting kernel. In contrast, hard-label training requires $O\left(\frac{1}{\gamma^4} \cdot \ln\left(\frac{1}{\epsilon}\right)\right)$ neurons, as derived from an adapted version of the gradient descent analysis in \citep{ji2020polylogarithmic}. This implies that when $\gamma \leq \epsilon$, i.e., when the dataset is challenging to classify, the neuron requirement for soft-label training can be significantly lower than that for hard-label training. Finally, we present experimental results on deep neural networks, further validating these theoretical findings.

Decentralized and Uncoordinated Learning of Stable Matchings: A Game-Theoretic Approach

We consider the problem of learning stable matchings in a fully decentralized and uncoordinated manner. In this problem, there are $n$ men and $n$ women, each having preference over the other side. It is assumed that women know their preferences over men, but men are not aware of their preferences over women, and they only learn them if they propose and successfully get matched to women. A matching is called stable if no man and woman prefer each other over their current matches. When all the preferences are known a priori, the celebrated Deferred-Acceptance algorithm proposed by Gale and Shapley provides a decentralized and uncoordinated algorithm to obtain a stable matching. However, when the preferences are unknown, developing such an algorithm faces major challenges due to a lack of coordination. We achieve this goal by making a connection between stable matchings and learning Nash equilibria (NE) in noncooperative games. First, we provide a complete information game formulation for the stable matching problem with known preferences such that its set of pure NE coincides with the set of stable matchings, while its mixed NE can be rounded in a decentralized manner to a stable matching. Relying on such a game-theoretic formulation, we show that for hierarchical markets, adopting the exponential weight (EXP) learning algorithm for the stable matching game achieves logarithmic regret with polynomial dependence on the number of players, thus answering a question posed in previous literature. Moreover, we show that the same EXP learning algorithm converges locally and exponentially fast to a stable matching in general matching markets. We complement this result by introducing another decentralized and uncoordinated learning algorithm that globally converges to a stable matching with arbitrarily high probability, leveraging the weak acyclicity property of the stable matching game.

Performance of NPG in Countable State-Space Average-Cost RL

We consider policy optimization methods in reinforcement learning settings where the state space is arbitrarily large, or even countably infinite. The motivation arises from control problems in communication networks, matching markets, and other queueing systems. We consider Natural Policy Gradient (NPG), which is a popular algorithm for finite state spaces. Under reasonable assumptions, we derive a performance bound for NPG that is independent of the size of the state space, provided the error in policy evaluation is within a factor of the true value function. We obtain this result by establishing new policy-independent bounds on the solution to Poisson's equation, i.e., the relative value function, and by combining these bounds with previously known connections between MDPs and learning from experts.

On the Global Convergence of Policy Gradient in Average Reward Markov Decision Processes

We present the first finite time global convergence analysis of policy gradient in the context of infinite horizon average reward Markov decision processes (MDPs). Specifically, we focus on ergodic tabular MDPs with finite state and action spaces. Our analysis shows that the policy gradient iterates converge to the optimal policy at a sublinear rate of $O\left({\frac{1}{T}}\right),$ which translates to $O\left({\log(T)}\right)$ regret, where $T$ represents the number of iterations. Prior work on performance bounds for discounted reward MDPs cannot be extended to average reward MDPs because the bounds grow proportional to the fifth power of the effective horizon. Thus, our primary contribution is in proving that the policy gradient algorithm converges for average-reward MDPs and in obtaining finite-time performance guarantees. In contrast to the existing discounted reward performance bounds, our performance bounds have an explicit dependence on constants that capture the complexity of the underlying MDP. Motivated by this observation, we reexamine and improve the existing performance bounds for discounted reward MDPs. We also present simulations to empirically evaluate the performance of average reward policy gradient algorithm.

Exploration-Driven Policy Optimization in RLHF: Theoretical Insights on Efficient Data Utilization

Reinforcement Learning from Human Feedback (RLHF) has achieved impressive empirical successes while relying on a small amount of human feedback. However, there is limited theoretical justification for this phenomenon. Additionally, most recent studies focus on value-based algorithms despite the recent empirical successes of policy-based algorithms. In this work, we consider an RLHF algorithm based on policy optimization (PO-RLHF). The algorithm is based on the popular Policy Cover-Policy Gradient (PC-PG) algorithm, which assumes knowledge of the reward function. In PO-RLHF, knowledge of the reward function is not assumed and the algorithm relies on trajectory-based comparison feedback to infer the reward function. We provide performance bounds for PO-RLHF with low query complexity, which provides insight into why a small amount of human feedback may be sufficient to get good performance with RLHF. A key novelty is our trajectory-level elliptical potential analysis technique used to infer reward function parameters when comparison queries rather than reward observations are used. We provide and analyze algorithms in two settings: linear and neural function approximation, PG-RLHF and NN-PG-RLHF, respectively.

Convergence for Natural Policy Gradient on Infinite-State Average-Reward Markov Decision Processes

Infinite-state Markov Decision Processes (MDPs) are essential in modeling and optimizing a wide variety of engineering problems. In the reinforcement learning (RL) context, a variety of algorithms have been developed to learn and optimize these MDPs. At the heart of many popular policy-gradient based learning algorithms, such as natural actor-critic, TRPO, and PPO, lies the Natural Policy Gradient (NPG) algorithm. Convergence results for these RL algorithms rest on convergence results for the NPG algorithm. However, all existing results on the convergence of the NPG algorithm are limited to finite-state settings. We prove the first convergence rate bound for the NPG algorithm for infinite-state average-reward MDPs, proving a $O(1/\sqrt{T})$ convergence rate, if the NPG algorithm is initialized with a good initial policy. Moreover, we show that in the context of a large class of queueing MDPs, the MaxWeight policy suffices to satisfy our initial-policy requirement and achieve a $O(1/\sqrt{T})$ convergence rate. Key to our result are state-dependent bounds on the relative value function achieved by the iterate policies of the NPG algorithm.

Rates of Convergence in the Central Limit Theorem for Markov Chains, with an Application to TD Learning

We prove a non-asymptotic central limit theorem for vector-valued martingale differences using Stein's method, and use Poisson's equation to extend the result to functions of Markov Chains. We then show that these results can be applied to establish a non-asymptotic central limit theorem for Temporal Difference (TD) learning with averaging.

Cascading Reinforcement Learning

Cascading bandits have gained popularity in recent years due to their applicability to recommendation systems and online advertising. In the cascading bandit model, at each timestep, an agent recommends an ordered subset of items (called an item list) from a pool of items, each associated with an unknown attraction probability. Then, the user examines the list, and clicks the first attractive item (if any), and after that, the agent receives a reward. The goal of the agent is to maximize the expected cumulative reward. However, the prior literature on cascading bandits ignores the influences of user states (e.g., historical behaviors) on recommendations and the change of states as the session proceeds. Motivated by this fact, we propose a generalized cascading RL framework, which considers the impact of user states and state transition into decisions. In cascading RL, we need to select items not only with large attraction probabilities but also leading to good successor states. This imposes a huge computational challenge due to the combinatorial action space. To tackle this challenge, we delve into the properties of value functions, and design an oracle BestPerm to efficiently find the optimal item list. Equipped with BestPerm, we develop two algorithms CascadingVI and CascadingBPI, which are both computationally-efficient and sample-efficient, and provide near-optimal regret and sample complexity guarantees. Furthermore, we present experiments to show the improved computational and sample efficiencies of our algorithms compared to straightforward adaptations of existing RL algorithms in practice.

Striking a Balance: An Optimal Mechanism Design for Heterogenous Differentially Private Data Acquisition for Logistic Regression

We investigate the problem of performing logistic regression on data collected from privacy-sensitive sellers. Since the data is private, sellers must be incentivized through payments to provide their data. Thus, the goal is to design a mechanism that optimizes a weighted combination of test loss, seller privacy, and payment, i.e., strikes a balance between multiple objectives of interest. We solve the problem by combining ideas from game theory, statistical learning theory, and differential privacy. The buyer's objective function can be highly non-convex. However, we show that, under certain conditions on the problem parameters, the problem can be convexified by using a change of variables. We also provide asymptotic results characterizing the buyer's test error and payments when the number of sellers becomes large. Finally, we demonstrate our ideas by applying them to a real healthcare data set.

Collaborative Multi-Agent Heterogeneous Multi-Armed Bandits

The study of collaborative multi-agent bandits has attracted significant attention recently. In light of this, we initiate the study of a new collaborative setting, consisting of $N$ agents such that each agent is learning one of $M$ stochastic multi-armed bandits to minimize their group cumulative regret. We develop decentralized algorithms which facilitate collaboration between the agents under two scenarios. We characterize the performance of these algorithms by deriving the per agent cumulative regret and group regret upper bounds. We also prove lower bounds for the group regret in this setting, which demonstrates the near-optimal behavior of the proposed algorithms.