Reward-agnostic Fine-tuning: Provable Statistical Benefits of Hybrid Reinforcement Learning

This paper studies tabular reinforcement learning (RL) in the hybrid setting, which assumes access to both an offline dataset and online interactions with the unknown environment. A central question boils down to how to efficiently utilize online data collection to strengthen and complement the offline dataset and enable effective policy fine-tuning. Leveraging recent advances in reward-agnostic exploration and model-based offline RL, we design a three-stage hybrid RL algorithm that beats the best of both worlds -- pure offline RL and pure online RL -- in terms of sample complexities. The proposed algorithm does not require any reward information during data collection. Our theory is developed based on a new notion called single-policy partial concentrability, which captures the trade-off between distribution mismatch and miscoverage and guides the interplay between offline and online data.

Provable Guarantees for Nonlinear Feature Learning in Three-Layer Neural Networks

One of the central questions in the theory of deep learning is to understand how neural networks learn hierarchical features. The ability of deep networks to extract salient features is crucial to both their outstanding generalization ability and the modern deep learning paradigm of pretraining and finetuneing. However, this feature learning process remains poorly understood from a theoretical perspective, with existing analyses largely restricted to two-layer networks. In this work we show that three-layer neural networks have provably richer feature learning capabilities than two-layer networks. We analyze the features learned by a three-layer network trained with layer-wise gradient descent, and present a general purpose theorem which upper bounds the sample complexity and width needed to achieve low test error when the target has specific hierarchical structure. We instantiate our framework in specific statistical learning settings -- single-index models and functions of quadratic features -- and show that in the latter setting three-layer networks obtain a sample complexity improvement over all existing guarantees for two-layer networks. Crucially, this sample complexity improvement relies on the ability of three-layer networks to efficiently learn nonlinear features. We then establish a concrete optimization-based depth separation by constructing a function which is efficiently learnable via gradient descent on a three-layer network, yet cannot be learned efficiently by a two-layer network. Our work makes progress towards understanding the provable benefit of three-layer neural networks over two-layer networks in the feature learning regime.

Local Optimization Achieves Global Optimality in Multi-Agent Reinforcement Learning

Policy optimization methods with function approximation are widely used in multi-agent reinforcement learning. However, it remains elusive how to design such algorithms with statistical guarantees. Leveraging a multi-agent performance difference lemma that characterizes the landscape of multi-agent policy optimization, we find that the localized action value function serves as an ideal descent direction for each local policy. Motivated by the observation, we present a multi-agent PPO algorithm in which the local policy of each agent is updated similarly to vanilla PPO. We prove that with standard regularity conditions on the Markov game and problem-dependent quantities, our algorithm converges to the globally optimal policy at a sublinear rate. We extend our algorithm to the off-policy setting and introduce pessimism to policy evaluation, which aligns with experiments. To our knowledge, this is the first provably convergent multi-agent PPO algorithm in cooperative Markov games.

Can We Find Nash Equilibria at a Linear Rate in Markov Games?

We study decentralized learning in two-player zero-sum discounted Markov games where the goal is to design a policy optimization algorithm for either agent satisfying two properties. First, the player does not need to know the policy of the opponent to update its policy. Second, when both players adopt the algorithm, their joint policy converges to a Nash equilibrium of the game. To this end, we construct a meta algorithm, dubbed as $\texttt{Homotopy-PO}$, which provably finds a Nash equilibrium at a global linear rate. In particular, $\texttt{Homotopy-PO}$ interweaves two base algorithms $\texttt{Local-Fast}$ and $\texttt{Global-Slow}$ via homotopy continuation. $\texttt{Local-Fast}$ is an algorithm that enjoys local linear convergence while $\texttt{Global-Slow}$ is an algorithm that converges globally but at a slower sublinear rate. By switching between these two base algorithms, $\texttt{Global-Slow}$ essentially serves as a ``guide'' which identifies a benign neighborhood where $\texttt{Local-Fast}$ enjoys fast convergence. However, since the exact size of such a neighborhood is unknown, we apply a doubling trick to switch between these two base algorithms. The switching scheme is delicately designed so that the aggregated performance of the algorithm is driven by $\texttt{Local-Fast}$. Furthermore, we prove that $\texttt{Local-Fast}$ and $\texttt{Global-Slow}$ can both be instantiated by variants of optimistic gradient descent/ascent (OGDA) method, which is of independent interest.

Provably Efficient Reinforcement Learning via Surprise Bound

Value function approximation is important in modern reinforcement learning (RL) problems especially when the state space is (infinitely) large. Despite the importance and wide applicability of value function approximation, its theoretical understanding is still not as sophisticated as its empirical success, especially in the context of general function approximation. In this paper, we propose a provably efficient RL algorithm (both computationally and statistically) with general value function approximations. We show that if the value functions can be approximated by a function class that satisfies the Bellman-completeness assumption, our algorithm achieves an $\widetilde{O}(\text{poly}(\iota H)\sqrt{T})$ regret bound where $\iota$ is the product of the surprise bound and log-covering numbers, $H$ is the planning horizon, $K$ is the number of episodes and $T = HK$ is the total number of steps the agent interacts with the environment. Our algorithm achieves reasonable regret bounds when applied to both the linear setting and the sparse high-dimensional linear setting. Moreover, our algorithm only needs to solve $O(H\log K)$ empirical risk minimization (ERM) problems, which is far more efficient than previous algorithms that need to solve ERM problems for $\Omega(HK)$ times.

Efficient displacement convex optimization with particle gradient descent

Particle gradient descent, which uses particles to represent a probability measure and performs gradient descent on particles in parallel, is widely used to optimize functions of probability measures. This paper considers particle gradient descent with a finite number of particles and establishes its theoretical guarantees to optimize functions that are \emph{displacement convex} in measures. Concretely, for Lipschitz displacement convex functions defined on probability over $\mathbb{R}^d$, we prove that $O(1/\epsilon^2)$ particles and $O(d/\epsilon^4)$ computations are sufficient to find the $\epsilon$-optimal solutions. We further provide improved complexity bounds for optimizing smooth displacement convex functions. We demonstrate the application of our results for function approximation with specific neural architectures with two-dimensional inputs.

Refined Value-Based Offline RL under Realizability and Partial Coverage

In offline reinforcement learning (RL) we have no opportunity to explore so we must make assumptions that the data is sufficient to guide picking a good policy, taking the form of assuming some coverage, realizability, Bellman completeness, and/or hard margin (gap). In this work we propose value-based algorithms for offline RL with PAC guarantees under just partial coverage, specifically, coverage of just a single comparator policy, and realizability of soft (entropy-regularized) Q-function of the single policy and a related function defined as a saddle point of certain minimax optimization problem. This offers refined and generally more lax conditions for offline RL. We further show an analogous result for vanilla Q-functions under a soft margin condition. To attain these guarantees, we leverage novel minimax learning algorithms to accurately estimate soft or vanilla Q-functions with $L^2$-convergence guarantees. Our algorithms' loss functions arise from casting the estimation problems as nonlinear convex optimization problems and Lagrangifying.

Looped Transformers as Programmable Computers

We present a framework for using transformer networks as universal computers by programming them with specific weights and placing them in a loop. Our input sequence acts as a punchcard, consisting of instructions and memory for data read/writes. We demonstrate that a constant number of encoder layers can emulate basic computing blocks, including embedding edit operations, non-linear functions, function calls, program counters, and conditional branches. Using these building blocks, we emulate a small instruction-set computer. This allows us to map iterative algorithms to programs that can be executed by a looped, 13-layer transformer. We show how this transformer, instructed by its input, can emulate a basic calculator, a basic linear algebra library, and in-context learning algorithms that employ backpropagation. Our work highlights the versatility of the attention mechanism, and demonstrates that even shallow transformers can execute full-fledged, general-purpose programs.

Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing

It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.

From Gradient Flow on Population Loss to Learning with Stochastic Gradient Descent

Stochastic Gradient Descent (SGD) has been the method of choice for learning large-scale non-convex models. While a general analysis of when SGD works has been elusive, there has been a lot of recent progress in understanding the convergence of Gradient Flow (GF) on the population loss, partly due to the simplicity that a continuous-time analysis buys us. An overarching theme of our paper is providing general conditions under which SGD converges, assuming that GF on the population loss converges. Our main tool to establish this connection is a general converse Lyapunov like theorem, which implies the existence of a Lyapunov potential under mild assumptions on the rates of convergence of GF. In fact, using these potentials, we show a one-to-one correspondence between rates of convergence of GF and geometrical properties of the underlying objective. When these potentials further satisfy certain self-bounding properties, we show that they can be used to provide a convergence guarantee for Gradient Descent (GD) and SGD (even when the paths of GF and GD/SGD are quite far apart). It turns out that these self-bounding assumptions are in a sense also necessary for GD/SGD to work. Using our framework, we provide a unified analysis for GD/SGD not only for classical settings like convex losses, or objectives that satisfy PL / KL properties, but also for more complex problems including Phase Retrieval and Matrix sq-root, and extending the results in the recent work of Chatterjee 2022.