arXiv subjects

Caleb Ju

Publications and source records attributed to Caleb Ju.

12 recordsLinked to original sources

arXiv2026

Actor-Accelerated Policy Dual Averaging for Reinforcement Learning in Continuous Action Spaces

Policy Dual Averaging (PDA) offers a principled Policy Mirror Descent (PMD) framework that more naturally admits value function approximation than standard PMD, enabling the use of approximate advantage (or Q-) functions while retaining strong convergence guarantees. However, applying PDA in continuous state and action spaces remains computationally challenging, since action selection involves solving an optimization sub-problem at each decision step. In this paper, we propose \textit{actor-accelerated PDA}, which uses a learned policy network to approximate the solution of the optimization sub-problems, yielding faster runtimes while maintaining convergence guarantees. We provide a theoretical analysis that quantifies how actor approximation error impacts the convergence of PDA under suitable assumptions. We then evaluate its performance on several benchmarks in robotics, control, and operations research problems. Actor-accelerated PDA achieves superior performance compared to popular on-policy baselines such as Proximal Policy Optimization (PPO). Overall, our results bridge the gap between the theoretical advantages of PDA and its practical deployment in continuous-action problems with function approximation.

cs.LG↗

arXiv2025

Auto-exploration for online reinforcement learning

The exploration-exploitation dilemma in reinforcement learning (RL) is a fundamental challenge to efficient RL algorithms. Existing algorithms for finite state and action discounted RL problems address this by assuming sufficient exploration over both state and action spaces. However, this yields non-implementable algorithms and sub-optimal performance. To resolve these limitations, we introduce a new class of methods with auto-exploration, or methods that automatically explore both state and action spaces. Auto-exploration can be applied in both the tabular and linear function approximation setting. Under algorithm-independent assumptions on the existence of an exploring optimal policy, both settings attain $O(\epsilon^{-2})$ sample complexity to solve to $\epsilon$ error. These complexities are novel since they avoid algorithm-dependent parameters seen in prior works, which may be arbitrarily large. The methods are also simple to implement because they are parameter-free. We achieve these results by integrating auto-exploration into policy mirror descent to avoid the (unknown) stationary distribution seen in prior art. In the tabular setting, we introduce a dynamic exploration time with a data-driven stopping time, while for linear function approximation we propose a new sampling distribution based on the discounted visitation distribution that covers a more general class of Markov chains.

cs.LG↗

arXiv2025

Preconditioning via Randomized Range Deflation (RandRAND)

We introduce RandRAND, a new class of randomized preconditioning methods for large-scale linear systems. RandRAND deflates the spectrum via efficient orthogonal projections onto random subspaces, without computing eigenpairs or low-rank approximations. This leads to advantages in computational cost and numerical stability. We establish rigorous condition number bounds that depend only weakly on the problem size and that reduce to a small constant when the dimension of the deflated subspace is comparable to the effective spectral dimension. RandRAND can be employed without explicit operations with the deflation basis, enabling the effective use of fast randomized transforms. In this setting, the costly explicit basis orthogonalization is bypassed by using fast randomized Q-less QR factorizations or iterative methods for computing orthogonal projections. These strategies balance the cost of constructing RandRAND preconditioners and applying them within linear solvers, and can ensure robustness to rounding errors.

math.NA↗

arXiv2024

Learning a local trading strategy: deep reinforcement learning for grid-scale renewable energy integration

Variable renewable generation increases the challenge of balancing power supply and demand. Grid-scale batteries co-located with generation can help mitigate this misalignment. This paper explores the use of reinforcement learning (RL) for operating grid-scale batteries co-located with solar power. Our results show RL achieves an average of 61% (and up to 96%) of the approximate theoretical optimal (non-causal) operation, outperforming advanced control methods on average. Our findings suggest RL may be preferred when future signals are hard to predict. Moreover, RL has two significant advantages compared to simpler rules-based control: (1) that solar energy is more effectively shifted towards high demand periods, and (2) increased diversity of battery dispatch across different locations, reducing potential ramping issues caused by super-position of many similar actions.

cs.LG↗

arXiv2024

Strongly-polynomial time and validation analysis of policy gradient methods

This paper proposes a novel termination criterion, termed the advantage gap function, for finite state and action Markov decision processes (MDP) and reinforcement learning (RL). By incorporating this advantage gap function into the design of step size rules and deriving a new linear rate of convergence that is independent of the stationary state distribution of the optimal policy, we demonstrate that policy gradient methods can solve MDPs in strongly-polynomial time. To the best of our knowledge, this is the first time that such strong convergence properties have been established for policy gradient methods. Moreover, in the stochastic setting, where only stochastic estimates of policy gradients are available, we show that the advantage gap function provides close approximations of the optimality gap for each individual state and exhibits a sublinear rate of convergence at every state. The advantage gap function can be easily estimated in the stochastic case, and when coupled with easily computable upper bounds on policy values, they provide a convenient way to validate the solutions generated by policy gradient methods. Therefore, our developments offer a principled and computable measure of optimality for RL, whereas current practice tends to rely on algorithm-to-algorithm or baselines comparisons with no certificate of optimality.

cs.LG↗

arXiv2023

Efficient parallel implementation of the multiplicative weight update method for graph-based linear programs

Positive linear programs (LPs) model many graph and operations research problems. One can solve for a $(1+\epsilon)$-approximation for positive LPs, for any selected $\epsilon$, in polylogarithmic depth and near-linear work via variations of the multiplicative weight update (MWU) method. Despite extensive theoretical work on these algorithms through the decades, their empirical performance is not well understood. In this work, we implement and test an efficient parallel algorithm for solving positive LP relaxations, and apply it to graph problems such as densest subgraph, bipartite matching, vertex cover and dominating set. We accelerate the algorithm via a new step size search heuristic. Our implementation uses sparse linear algebra optimization techniques such as fusion of vector operations and use of sparse format. Furthermore, we devise an implicit representation for graph incidence constraints. We demonstrate the parallel scalability with the use of threading OpenMP and MPI on the Stampede2 supercomputer. We compare this implementation with exact libraries and specialized libraries for the above problems in order to evaluate MWU's practical standing for both accuracy and performance among other methods. Our results show this implementation is faster than general purpose LP solvers (IBM CPLEX, Gurobi) in all of our experiments, and in some instances, outperforms state-of-the-art specialized parallel graph algorithms.

cs.DC↗

arXiv2023

Dual dynamic programming for stochastic programs over an infinite horizon

We consider solving stochastic programs over an infinite horizon. By leveraging the stationarity of the problem, we develop a novel continually-exploring infinite-horizon explorative dual dynamic programming (CE-Inf-EDDP) algorithm. CE-Inf-EDDP builds upon the existing explorative dual dynamic programming, designed for the finite-horizon problem, by specializing it for the infinite-horizon, stationary case. By incorporating cut sharing, frequent cutting-plane model updates, and a new adaptive search point selection strategy, CE-Inf-EDDP provides state-of-the-art iteration complexity while offering encouraging numerical performance. In the newsvendor and hydrothermal planning problem, CE-Inf-EDDP can reduce the runtime of each iteration by up to one to two orders of magnitude compared to prior methods while maintaining similar solution quality. As a result, the final solution quality and its guarantees can be much improved over the same runtime. For example, in the hydrothermal planning problem, CE-Inf-EDDP attains about an order of magnitude improvement in the relative optimality gap compared to existing methods.

math.OC↗

arXiv2022

Policy Optimization over General State and Action Spaces

Reinforcement learning (RL) problems over general state and action spaces are notoriously challenging. In contrast to the tableau setting, one can not enumerate all the states and then iteratively update the policies for each state. This prevents the application of many well-studied RL methods especially those with provable convergence guarantees. In this paper, we first present a substantial generalization of the recently developed policy mirror descent method to deal with general state and action spaces. We introduce new approaches to incorporate function approximation into this method, so that we do not need to use explicit policy parameterization at all. Moreover, we present a novel policy dual averaging method for which possibly simpler function approximation techniques can be applied. We establish linear convergence rate to global optimality or sublinear convergence to stationarity for these methods applied to solve different classes of RL problems under exact policy evaluation. We then define proper notions of the approximation errors for policy evaluation and investigate their impact on the convergence of these methods applied to general-state RL problems with either finite-action or continuous-action spaces. To the best of our knowledge, the development of these algorithmic frameworks as well as their convergence analysis appear to be new in the literature. Preliminary numerical results demonstrate the robustness of the aforementioned methods and show they can be competitive with state-of-the-art RL algorithms.

cs.LG↗

arXiv2022

A model-free first-order method for linear quadratic regulator with $\tilde{O}(1/\varepsilon)$ sampling complexity

We consider the classic stochastic linear quadratic regulator (LQR) problem under an infinite horizon average stage cost. By leveraging recent policy gradient methods from reinforcement learning, we obtain a first-order method that finds a stable feedback law whose objective function gap to the optima is at most $\varepsilon$ with high probability using $\tilde{O}(1/\varepsilon)$ samples, where $\tilde{O}$ hides polylogarithmic dependence on $\varepsilon$. Our proposed method seems to have the best dependence on $\varepsilon$ within the model-free literature without the assumption that all policies generated by the algorithm are stable almost surely, and it matches the best-known rate from the model-based literature, up to logarithmic factors. The improved dependence on $\varepsilon$ is achieved by showing the accuracy scales with the variance rather than the standard deviation of the gradient estimation error. Our developments that result in this improved sampling complexity fall in the category of actor-critic algorithms. The actor part involves a variational inequality formulation of the stochastic LQR problem, while in the critic part, we utilize a conditional stochastic primal-dual method and show that the algorithm has the optimal rate of convergence when paired with a shrinking multi-epoch scheme.

math.OC↗

arXiv2021

Implicit Regularization of Bregman Proximal Point Algorithm and Mirror Descent on Separable Data

Bregman proximal point algorithm (BPPA) has witnessed emerging machine learning applications, yet its theoretical understanding has been largely unexplored. We study the computational properties of BPPA through learning linear classifiers with separable data, and demonstrate provable algorithmic regularization of BPPA. For any BPPA instantiated with a fixed Bregman divergence, we provide a lower bound of the margin obtained by BPPA with respect to an arbitrarily chosen norm. The obtained margin lower bound differs from the maximal margin by a multiplicative factor, which inversely depends on the condition number of the distance-generating function measured in the dual norm. We show that the dependence on the condition number is tight, thus demonstrating the importance of divergence in affecting the quality of the learned classifiers. We then extend our findings to mirror descent, for which we establish similar connections between the margin and Bregman divergence, together with a non-asymptotic analysis. Numerical experiments on both synthetic and real-world datasets are provided to support our theoretical findings. To the best of our knowledge, the aforementioned findings appear to be new in the literature of algorithmic regularization.

cs.LG↗

arXiv2021

Communication lower bounds for nested bilinear algorithms via rank expansion of Kronecker products

We develop lower bounds on communication in the memory hierarchy or between processors for nested bilinear algorithms, such as Strassen's algorithm for matrix multiplication. We build on a previous framework that establishes communication lower bounds by use of the rank expansion, or the minimum rank of any fixed size subset of columns of a matrix, for each of the three matrices encoding a bilinear algorithm. This framework provides lower bounds for a class of dependency directed acyclic graphs (DAGs) corresponding to the execution of a given bilinear algorithm, in contrast to other approaches that yield bounds for specific DAGs. However, our lower bounds only apply to executions that do not compute the same DAG node multiple times. Two bilinear algorithms can be nested by taking Kronecker products between their encoding matrices. Our main result is a lower bound on the rank expansion of a matrix constructed by a Kronecker product derived from lower bounds on the rank expansion of the Kronecker product's operands. We apply the rank expansion lower bounds to obtain novel communication lower bounds for nested Toom-Cook convolution, Strassen's algorithm, and fast algorithms for contraction of partially symmetric tensors.

cs.DC↗

arXiv2019

Derivation and Analysis of Fast Bilinear Algorithms for Convolution

The prevalence of convolution in applications within signal processing, deep neural networks, and numerical solvers has motivated the development of numerous fast convolution algorithms. In many of these problems, convolution is performed on terabytes or petabytes of data, so even constant factors of improvement can significantly reduce the computation time. We leverage the formalism of bilinear algorithms to describe and analyze all of the most popular approaches. This unified lens permits us to study the relationship between different variants of convolution as well as to derive error bounds and analyze the cost of the various algorithms. We provide new derivations, which predominantly leverage matrix and tensor algebra, to describe the Winograd family of convolution algorithms as well as reductions between 1D and multidimensional convolution. We provide cost and error bounds as well as experimental numerical studies. Our experiments for two of these algorithms, the overlap-add approach and Winograd convolution algorithm with polynomials of degree greater than one, show that fast convolution algorithms can rival the accuracy of the fast Fourier transform (FFT) without using complex arithmetic. These algorithms can be used for convolution problems with multidimensional inputs or for filters larger than size of four, extending the state-of-the-art in Winograd-based convolution algorithms.

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