Semih Cayci

Semih Cayci, Niao He, R. Srikant

Natural actor-critic (NAC) and its variants, equipped with the representation power of neural networks, have demonstrated impressive empirical success in solving Markov decision problems with large state spaces. In this paper, we present a finite-time analysis of NAC with neural network approximation, and identify the roles of neural networks, regularization and optimization techniques (e.g., gradient clipping and averaging) to achieve provably good performance in terms of sample complexity, iteration complexity and overparametrization bounds for the actor and the critic. In particular, we prove that (i) entropy regularization and averaging ensure stability by providing sufficient exploration to avoid near-deterministic and strictly suboptimal policies and (ii) regularization leads to sharp sample complexity and network width bounds in the regularized MDPs, yielding a favorable bias-variance tradeoff in policy optimization. In the process, we identify the importance of uniform approximation power of the actor neural network to achieve global optimality in policy optimization due to distributional shift.

Batuhan Yardim, Semih Cayci, Matthieu Geist et al.

Mean-field games have been used as a theoretical tool to obtain an approximate Nash equilibrium for symmetric and anonymous $N$-player games. However, limiting applicability, existing theoretical results assume variations of a "population generative model", which allows arbitrary modifications of the population distribution by the learning algorithm. Moreover, learning algorithms typically work on abstract simulators with population instead of the $N$-player game. Instead, we show that $N$ agents running policy mirror ascent converge to the Nash equilibrium of the regularized game within $\widetilde{\mathcal{O}}(\varepsilon^{-2})$ samples from a single sample trajectory without a population generative model, up to a standard $\mathcal{O}(\frac{1}{\sqrt{N}})$ error due to the mean field. Taking a divergent approach from the literature, instead of working with the best-response map we first show that a policy mirror ascent map can be used to construct a contractive operator having the Nash equilibrium as its fixed point. We analyze single-path TD learning for $N$-agent games, proving sample complexity guarantees by only using a sample path from the $N$-agent simulator without a population generative model. Furthermore, we demonstrate that our methodology allows for independent learning by $N$ agents with finite sample guarantees.

Semih Cayci, Atilla Eryilmaz

In a broad class of reinforcement learning applications, stochastic rewards have heavy-tailed distributions, which lead to infinite second-order moments for stochastic (semi)gradients in policy evaluation and direct policy optimization. In such instances, the existing RL methods may fail miserably due to frequent statistical outliers. In this work, we establish that temporal difference (TD) learning with a dynamic gradient clipping mechanism, and correspondingly operated natural actor-critic (NAC), can be provably robustified against heavy-tailed reward distributions. It is shown in the framework of linear function approximation that a favorable tradeoff between bias and variability of the stochastic gradients can be achieved with this dynamic gradient clipping mechanism. In particular, we prove that robust versions of TD learning achieve sample complexities of order $\mathcal{O}(\varepsilon^{-\frac{1}{p}})$ and $\mathcal{O}(\varepsilon^{-1-\frac{1}{p}})$ with and without the full-rank assumption on the feature matrix, respectively, under heavy-tailed rewards with finite moments of order $(1+p)$ for some $p\in(0,1]$, both in expectation and with high probability. We show that a robust variant of NAC based on Robust TD learning achieves $\tilde{\mathcal{O}}(\varepsilon^{-4-\frac{2}{p}})$ sample complexity. We corroborate our theoretical results with numerical experiments.

Enes Arda, Semih Cayci, Atilla Eryilmaz

Understanding why Transformers perform so well remains challenging due to their non-convex optimization landscape. In this work, we analyze a shallow Transformer with $m$ independent heads trained by projected gradient descent in the kernel regime. Our analysis reveals two main findings: (i) the width required for nonasymptotic guarantees scales only logarithmically with the sample size $n$, and (ii) the optimization error is independent of the sequence length $T$. This contrasts sharply with recurrent architectures, where the optimization error can grow exponentially with $T$. The trade-off is memory: to keep the full context, the Transformer's memory requirement grows with the sequence length. We validate our theoretical results numerically in a teacher-student setting and confirm the predicted scaling laws for Transformers.

Semih Cayci

An important question in deep learning is how higher-order optimization methods affect generalization. In this work, we analyze a stochastic Gauss-Newton (SGN) method with Levenberg-Marquardt damping and mini-batch sampling for training overparameterized deep neural networks with smooth activations in a regression setting. Our theoretical contributions are twofold. First, we establish finite-time convergence bounds via a variable-metric analysis in parameter space, with explicit dependencies on the batch size, network width and depth. Second, we derive non-asymptotic generalization bounds for SGN using uniform stability in the overparameterized regime, characterizing the impact of curvature, batch size, and overparameterization on generalization performance. Our theoretical results identify a favorable generalization regime for SGN in which a larger minimum eigenvalue of the Gauss-Newton matrix along the optimization path yields tighter stability bounds.

Semih Cayci, Atilla Eryilmaz

We analyze recurrent neural networks with diagonal hidden-to-hidden weight matrices, trained with gradient descent in the supervised learning setting, and prove that gradient descent can achieve optimality \emph{without} massive overparameterization. Our in-depth nonasymptotic analysis (i) provides improved bounds on the network size $m$ in terms of the sequence length $T$, sample size $n$ and ambient dimension $d$, and (ii) identifies the significant impact of long-term dependencies in the dynamical system on the convergence and network width bounds characterized by a cutoff point that depends on the Lipschitz continuity of the activation function. Remarkably, this analysis reveals that an appropriately-initialized recurrent neural network trained with $n$ samples can achieve optimality with a network size $m$ that scales only logarithmically with $n$. This sharply contrasts with the prior works that require high-order polynomial dependency of $m$ on $n$ to establish strong regularity conditions. Our results are based on an explicit characterization of the class of dynamical systems that can be approximated and learned by recurrent neural networks via norm-constrained transportation mappings, and establishing local smoothness properties of the hidden state with respect to the learnable parameters.

Semih Cayci

We analyze the convergence of Gauss-Newton dynamics for training neural networks with smooth activation functions. In the underparameterized regime, the Gauss-Newton gradient flow induces a Riemannian gradient flow on a low-dimensional, smooth, embedded submanifold of the Euclidean output space. Using tools from Riemannian optimization, we prove \emph{last-iterate} convergence of the Riemannian gradient flow to the optimal in-class predictor at an \emph{exponential rate} that is independent of the conditioning of the Gram matrix, \emph{without} requiring explicit regularization. We further characterize the critical impacts of the neural network scaling factor and the initialization on the convergence behavior. In the overparameterized regime, we show that the Levenberg-Marquardt dynamics with an appropriately chosen damping schedule yields fast convergence rate despite potentially ill-conditioned neural tangent kernel matrices, analogous to the underparameterized regime. These findings demonstrate the potential of Gauss-Newton methods for efficiently optimizing neural networks in the near-initialization regime, particularly in ill-conditioned problems where kernel and Gram matrices have small singular values.

Noah Oberweis, Semih Cayci

Continuous-time models provide important insights into the training dynamics of optimization algorithms in deep learning. In this work, we establish a non-asymptotic convergence analysis of stochastic gradient Langevin dynamics (SGLD), which is an Itô stochastic differential equation (SDE) approximation of stochastic gradient descent in continuous time, in the lazy training regime. We show that, under regularity conditions on the Hessian of the loss function, SGLD with multiplicative and state-dependent noise (i) yields a non-degenerate kernel throughout the training process with high probability, and (ii) achieves exponential convergence to the empirical risk minimizer in expectation, and we establish finite-time and finite-width bounds on the optimality gap. We corroborate our theoretical findings with numerical examples in the regression setting.

Johannes Müller, Semih Cayci

We study the error introduced by entropy regularization in infinite-horizon discrete discounted Markov decision processes. We show that this error decreases exponentially in the inverse regularization strength, both in a weighted KL-divergence and in value with a problem-specific exponent. This is in contrast to previously known estimates, of the order $O(τ)$, where $τ$ is the regularization strength. We provide a lower bound that matches our upper bound up to a polynomial term, thereby characterizing the exponential convergence rate for entropy regularization. Our proof relies on the observation that the solutions of entropy-regularized Markov decision processes solve a gradient flow of the unregularized reward with respect to a Riemannian metric common in natural policy gradient methods. This correspondence allows us to identify the limit of this gradient flow as the generalized maximum entropy optimal policy, thereby characterizing the implicit bias of this gradient flow, which corresponds to a time-continuous version of the natural policy gradient method. We use our improved error estimates to show that for entropy-regularized natural policy gradient methods, the overall error decays exponentially in the square root of the number of iterations, improving over existing sublinear guarantees. Finally, we extend our analysis to settings beyond the entropy. In particular, we characterize the implicit bias regarding general convex potentials and their resulting generalized natural policy gradients.

Semih Cayci, Niao He, R. Srikant

We consider the reinforcement learning problem for partially observed Markov decision processes (POMDPs) with large or even countably infinite state spaces, where the controller has access to only noisy observations of the underlying controlled Markov chain. We consider a natural actor-critic method that employs a finite internal memory for policy parameterization, and a multi-step temporal difference learning algorithm for policy evaluation. We establish, to the best of our knowledge, the first non-asymptotic global convergence of actor-critic methods for partially observed systems under function approximation. In particular, in addition to the function approximation and statistical errors that also arise in MDPs, we explicitly characterize the error due to the use of finite-state controllers. This additional error is stated in terms of the total variation distance between the traditional belief state in POMDPs and the posterior distribution of the hidden state when using a finite-state controller. Further, we show that this error can be made small in the case of sliding-block controllers by using larger block sizes.

Semih Cayci, Yilin Zheng, Atilla Eryilmaz

In a wide variety of applications including online advertising, contractual hiring, and wireless scheduling, the controller is constrained by a stringent budget constraint on the available resources, which are consumed in a random amount by each action, and a stochastic feasibility constraint that may impose important operational limitations on decision-making. In this work, we consider a general model to address such problems, where each action returns a random reward, cost, and penalty from an unknown joint distribution, and the decision-maker aims to maximize the total reward under a budget constraint $B$ on the total cost and a stochastic constraint on the time-average penalty. We propose a novel low-complexity algorithm based on Lyapunov optimization methodology, named ${\tt LyOn}$, and prove that for $K$ arms it achieves $O(\sqrt{K B\log B})$ regret and zero constraint-violation when $B$ is sufficiently large. The low computational cost and sharp performance bounds of ${\tt LyOn}$ suggest that Lyapunov-based algorithm design methodology can be effective in solving constrained bandit optimization problems.

Semih Cayci, Niao He, R. Srikant

Natural policy gradient (NPG) methods with entropy regularization achieve impressive empirical success in reinforcement learning problems with large state-action spaces. However, their convergence properties and the impact of entropy regularization remain elusive in the function approximation regime. In this paper, we establish finite-time convergence analyses of entropy-regularized NPG with linear function approximation under softmax parameterization. In particular, we prove that entropy-regularized NPG with averaging satisfies the \emph{persistence of excitation} condition, and achieves a fast convergence rate of $\tilde{O}(1/T)$ up to a function approximation error in regularized Markov decision processes. This convergence result does not require any a priori assumptions on the policies. Furthermore, under mild regularity conditions on the concentrability coefficient and basis vectors, we prove that entropy-regularized NPG exhibits \emph{linear convergence} up to a function approximation error.

Semih Cayci, Siddhartha Satpathi, Niao He et al.

In this paper, we study the dynamics of temporal difference learning with neural network-based value function approximation over a general state space, namely, \emph{Neural TD learning}. We consider two practically used algorithms, projection-free and max-norm regularized Neural TD learning, and establish the first convergence bounds for these algorithms. An interesting observation from our results is that max-norm regularization can dramatically improve the performance of TD learning algorithms, both in terms of sample complexity and overparameterization. In particular, we prove that max-norm regularization improves state-of-the-art sample complexity and overparameterization bounds. The results in this work rely on a novel Lyapunov drift analysis of the network parameters as a stopped and controlled random process.

Semih Cayci, Atilla Eryilmaz, R. Srikant

Time-constrained decision processes have been ubiquitous in many fundamental applications in physics, biology and computer science. Recently, restart strategies have gained significant attention for boosting the efficiency of time-constrained processes by expediting the completion times. In this work, we investigate the bandit problem with controlled restarts for time-constrained decision processes, and develop provably good learning algorithms. In particular, we consider a bandit setting where each decision takes a random completion time, and yields a random and correlated reward at the end, with unknown values at the time of decision. The goal of the decision-maker is to maximize the expected total reward subject to a time constraint $τ$. As an additional control, we allow the decision-maker to interrupt an ongoing task and forgo its reward for a potentially more rewarding alternative. For this problem, we develop efficient online learning algorithms with $O(\log(τ))$ and $O(\sqrt{τ\log(τ)})$ regret in a finite and continuous action space of restart strategies, respectively. We demonstrate an applicability of our algorithm by using it to boost the performance of SAT solvers.

Semih Cayci, Swati Gupta, Atilla Eryilmaz

The theory of discrete-time online learning has been successfully applied in many problems that involve sequential decision-making under uncertainty. However, in many applications including contractual hiring in online freelancing platforms and server allocation in cloud computing systems, the outcome of each action is observed only after a random and action-dependent time. Furthermore, as a consequence of certain ethical and economic concerns, the controller may impose deadlines on the completion of each task, and require fairness across different groups in the allocation of total time budget $B$. In order to address these applications, we consider continuous-time online learning problem with fairness considerations, and present a novel framework based on continuous-time utility maximization. We show that this formulation recovers reward-maximizing, max-min fair and proportionally fair allocation rules across different groups as special cases. We characterize the optimal offline policy, which allocates the total time between different actions in an optimally fair way (as defined by the utility function), and impose deadlines to maximize time-efficiency. In the absence of any statistical knowledge, we propose a novel online learning algorithm based on dual ascent optimization for time averages, and prove that it achieves $\tilde{O}(B^{-1/2})$ regret bound.

Semih Cayci, Atilla Eryilmaz, R. Srikant

We consider a budget-constrained bandit problem where each arm pull incurs a random cost, and yields a random reward in return. The objective is to maximize the total expected reward under a budget constraint on the total cost. The model is general in the sense that it allows correlated and potentially heavy-tailed cost-reward pairs that can take on negative values as required by many applications. We show that if moments of order $(2+γ)$ for some $γ> 0$ exist for all cost-reward pairs, $O(\log B)$ regret is achievable for a budget $B>0$. In order to achieve tight regret bounds, we propose algorithms that exploit the correlation between the cost and reward of each arm by extracting the common information via linear minimum mean-square error estimation. We prove a regret lower bound for this problem, and show that the proposed algorithms achieve tight problem-dependent regret bounds, which are optimal up to a universal constant factor in the case of jointly Gaussian cost and reward pairs.

Semih Cayci, Atilla Eryilmaz

We study the problem of serving randomly arriving and delay-sensitive traffic over a multi-channel communication system with time-varying channel states and unknown statistics. This problem deviates from the classical exploration-exploitation setting in that the design and analysis must accommodate the dynamics of packet availability and urgency as well as the cost of each channel use at the time of decision. To that end, we have developed and investigated an index-based policy UCB-Deadline, which performs dynamic channel allocation decisions that incorporate these traffic requirements and costs. Under symmetric channel conditions, we have proved that the UCB-Deadline policy can achieve bounded regret in the likely case where the cost of using a channel is not too high to prevent all transmissions, and logarithmic regret otherwise. In this case, we show that UCB-Deadline is order-optimal. We also perform numerical investigations to validate the theoretical findings, and also compare the performance of the UCB-Deadline to another learning algorithm that we propose based on Thompson Sampling.