Recent empirical evidence indicates that many machine learning applications involve heavy-tailed gradient noise, which challenges the standard assumptions of bounded variance in stochastic optimization. Gradient clipping has emerged as a popular tool to handle this heavy-tailed noise, as it achieves good performance both theoretically and practically. However, our current theoretical understanding of non-convex gradient clipping has three main shortcomings. First, the theory hinges on large, increasing clipping thresholds, which are in stark contrast to the small constant clipping thresholds employed in practice. Second, clipping thresholds require knowledge of problem-dependent parameters to guarantee convergence. Lastly, even with this knowledge, current sample complexity upper bounds for the method are sub-optimal in nearly all parameters. To address these issues and motivated by practical observations, we make the connection of gradient clipping to its close relative — Normalized SGD (NSGD) — and study its convergence properties. First, we establish a parameter-free sample complexity for NSGD of $\mathcal{O}\left(\varepsilon^{-\frac{2p}{p-1}}\right)$ to find an $\varepsilon$-stationary point, only assuming a finite $p$-th central moment of the noise, $p\in(1,2]$. Furthermore, we prove the tightness of this result, by providing a matching algorithm-specific lower bound. In the setting where all problem parameters are known, we show this complexity is improved to $\mathcal{O}\left(\varepsilon^{-\frac{3p-2}{p-1}}\right)$, matching the previously known lower bound for all first-order methods in all problem dependent parameters. Finally, we establish high-probability convergence of NSGD with a mild logarithmic dependence on the failure probability. Our work complements the studies of gradient clipping under heavy-tailed noise, improving the sample complexities of existing algorithms and offering an alternative mechanism to achieve high-probability convergence.

Incentive design is a popular framework for guiding agents’ learning dynamics towards desired outcomes by providing additional payments beyond intrinsic rewards. However, most existing works focus on a finite, small set of agents or assume complete knowledge of the game, limiting their applicability to real-world scenarios involving large populations and model uncertainty. To address this gap, we study the design of steering rewards in Mean-Field Games (MFGs) with density-independent transitions, where both the transition dynamics and intrinsic reward functions are unknown. This setting presents non-trivial challenges, as the mediator must incentivize the agents to explore for its model learning under uncertainty, while simultaneously steer them to converge to desired behaviors without incurring excessive incentive payments. Assuming agents exhibit no(-adaptive) regret behaviors, we contribute novel optimistic exploration algorithms. Theoretically, we establish sub-linear regret guarantees for the cumulative gaps between the agents’ behaviors and the desired ones. In terms of the steering cost, we demonstrate that our total incentive payments incur only sub-linear excess, competing with a baseline steering strategy that stabilizes the target policy as an equilibrium. Our work presents an effective framework for steering agents behaviors in large-population systems under uncertainty.

Designing incentives for an adapting population is a ubiquitous problem in a wide array of economic applications and beyond. In this work, we study how to design additional rewards to steer multi-agent systems towards desired policies \emph{without} prior knowledge of the agents' underlying learning dynamics. Motivated by the limitation of existing works, we consider a new and general category of learning dynamics called \emph{Markovian agents}. We introduce a model-based non-episodic Reinforcement Learning (RL) formulation for our steering problem. Importantly, we focus on learning a \emph{history-dependent} steering strategy to handle the inherent model uncertainty about the agents' learning dynamics. We introduce a novel objective function to encode the desiderata of achieving a good steering outcome with reasonable cost. Theoretically, we identify conditions for the existence of steering strategies to guide agents to the desired policies. Complementing our theoretical contributions, we provide empirical algorithms to approximately solve our objective, which effectively tackles the challenge in learning history-dependent strategies. We demonstrate the efficacy of our algorithms through empirical evaluations.

This work revisits the classical low-rank matrix factorization problem and unveils the critical role of initialization in shaping convergence rates for such nonconvex and nonsmooth optimization. We introduce Nystrom initialization, which significantly improves the global convergence of Scaled Gradient Descent (ScaledGD) in both symmetric and asymmetric matrix factorization tasks. Specifically, we prove that ScaledGD with Nystrom initialization achieves quadratic convergence in cases where only linear rates were previously known. Furthermore, we extend this initialization to low-rank adapters (LoRA) commonly used for finetuning foundation models. Our approach, NoRA, i.e., LoRA with Nystrom initialization, demonstrates superior performance across various downstream tasks and model scales, from 1B to 7B parameters, in large language and diffusion models.

Sample efficiency is critical for online Reinforcement Learning from Human Feedback (RLHF). While existing works investigate sample-efficient online exploration strategies, the potential of utilizing misspecified yet relevant reward models to accelerate learning remains underexplored. This paper studies how to transfer knowledge from those imperfect reward models in online RLHF. We start by identifying a novel property due to KL-regularization in the RLHF objective: a policy’s coverability of the optimal policy is captured by its sub-optimality. Building on this insight, we propose novel transfer learning principles and a theoretical algorithm—Transfer Policy Optimization (TPO)—with provable benefits compared to standard online learning. Empirically, inspired by our theoretical findings, we develop a win-rate-based transfer policy selection strategy with improved computational efficiency. Moreover, our empirical transfer learning technique is modular and can be integrated with various policy optimization methods, such as DPO, IPO and XPO, to further enhance their performance. We validate the effectiveness of our method through experiments on summarization tasks.

Exploration is critical for solving real-world decision-making problems such as scientific discovery, where the objective is to generate truly novel designs rather than mimic existing data distributions. In this work, we address the challenge of leveraging the representational power of generative models for exploration without relying on explicit uncertainty quantification. We introduce a novel framework that casts exploration as entropy maximization over the approximate data manifold implicitly defined by a pre-trained diffusion model. Then, we present a novel principle for exploration based on density estimation, a problem well-known to be challenging in practice. To overcome this issue and render this method truly scalable, we leverage a fundamental connection between the entropy of the density induced by a diffusion model and its score function. Building on this, we develop an algorithm based on mirror descent that solves the exploration problem as sequential fine-tuning of a pre-trained diffusion model. We prove its convergence to the optimal exploratory diffusion model under realistic assumptions by leveraging recent understanding of mirror flows. Finally, we empirically evaluate our approach on both synthetic and high-dimensional text-to-image diffusion, demonstrating promising results.

Designing incentives for a multi-agent system to induce a desirable Nash equilibrium is both a crucial and challenging problem appearing in many decision-making domains, especially for a large number of agents $N$. Under the exchangeability assumption, we formalize this incentive design (ID) problem as a parameterized mean-field game (PMFG), aiming to reduce complexity via an infinite-population limit. We first show that when dynamics and rewards are Lipschitz, the finite-$N$ ID objective is approximated by the PMFG at rate $\mathcal{O}(\frac{1}{\sqrt{N}})$. Moreover, beyond the Lipschitz-continuous setting, we prove the same $\mathcal{O}(\frac{1}{\sqrt{N}})$ decay for the important special case of sequential auctions, despite discontinuities in dynamics, through a tailored auction-specific analysis. Built on our novel approximation results, we further introduce our Adjoint Mean-Field Incentive Design (AMID) algorithm, which uses explicit differentiation of iterated equilibrium operators to compute gradients efficiently. By uniting approximation bounds with optimization guarantees, AMID delivers a powerful, scalable algorithmic tool for many-agent (large $N$) ID. Across diverse auction settings, the proposed AMID method substantially increases revenue over first-price formats and outperforms existing benchmark methods.

We show that low-rank adaptation of large-scale models suffers from a low stable rank that is well below the linear algebraic rank of the subspace, degrading fine-tuning performance. To mitigate the underutilization of the allocated subspace, we propose PoLAR, a parameterization inspired by the polar decomposition that factorizes the low-rank update into two direction matrices constrained to Stiefel manifolds and an unconstrained scale matrix. Our theory shows that PoLAR yields an exponentially faster convergence rate on a canonical low-rank adaptation problem. Pairing the parameterization with Riemannian optimization leads to consistent gains on three different benchmarks testing general language understanding, commonsense reasoning, and mathematical problem solving with base model sizes ranging from 350M to 27B.

Adapting large-scale foundational flow and diffusion generative models to optimize task-specific objectives while preserving prior information is crucial for real-world applications such as molecular design, protein docking, and creative image generation. Existing principled fine-tuning methods aim to maximize the expected reward of generated samples, while retaining knowledge from the pre-trained model via KL-divergence regularization. In this work, we tackle the significantly more general problem of optimizing general utilities beyond average rewards, including risk-averse and novelty-seeking reward maximization, diversity measures for exploration, and experiment design objectives among others. Likewise, we consider more general ways to preserve prior information beyond KL-divergence, such as optimal transport distances and Rényi divergences. To this end, we introduce Flow Density Control (FDC), a simple algorithm that reduces this complex problem to a specific sequence of simpler fine-tuning tasks, each solvable via scalable established methods. We derive convergence guarantees for the proposed scheme under realistic assumptions by leveraging recent understanding of mirror flows. Finally, we validate our method on illustrative settings, text-to-image, and molecular design tasks, showing that it can steer pre-trained generative models to optimize objectives and solve practically relevant tasks beyond the reach of current fine-tuning schemes.

Stochastic Natural Gradient Variational Inference (NGVI) is a widely used method for approximating posterior distribution in probabilistic models. Despite its empirical success and foundational role in variational inference, its theoretical underpinnings remain limited, particularly in the case of non-conjugate likelihoods. While NGVI has been shown to be a special instance of Stochastic Mirror Descent, and recent work has provided convergence guarantees using relative smoothness and strong convexity for conjugate models, these results do not extend to the non-conjugate setting, where the variational loss becomes non-convex and harder to analyze. In this work, we focus on mean-field parameterization and advance the theoretical understanding of NGVI in three key directions. First, we derive sufficient conditions under which the variational loss satisfies relative smoothness with respect to a suitable mirror map. Second, leveraging this structure, we propose a modified NGVI algorithm incorporating non-Euclidean projections and prove its global non-asymptotic convergence to a stationary point. Finally, under additional structural assumptions about the likelihood, we uncover hidden convexity properties of the variational loss and establish fast global convergence of NGVI to a global optimum. These results provide new insights into the geometry and convergence behavior of NGVI in challenging inference settings.

Contrastive Language-Image Pretraining (CLIP) has demonstrated strong zero-shot performance across diverse downstream text-image tasks. Existing CLIP methods typically optimize a contrastive objective using negative samples drawn from each minibatch. To achieve robust representation learning, these methods require extremely large batch sizes and escalate computational demands to hundreds or even thousands of GPUs. Prior approaches to mitigate this issue often compromise downstream performance, prolong training duration, or face scalability challenges with very large datasets. To overcome these limitations, we propose AmorLIP, an efficient CLIP pretraining framework that amortizes expensive computations involved in contrastive learning through lightweight neural networks, which substantially improves training efficiency and performance. Leveraging insights from a spectral factorization of energy-based models, we introduce novel amortization objectives along with practical techniques to improve training stability. Extensive experiments across 38 downstream tasks demonstrate the superior zero-shot classification and retrieval capabilities of AmorLIP, consistently outperforming standard CLIP baselines with substantial relative improvements of up to 12.24%.

Zeroth-order methods are extensively used in machine learning applications where gradients are infeasible or expensive to compute, such as black-box attacks, reinforcement learning, and language model fine-tuning. Existing optimization theory focuses on convergence to an arbitrary stationary point, but less is known on the implicit regularization that provides a fine-grained characterization on which particular solutions are finally reached. We show that zeroth-order optimization with the standard two-point estimator favors solutions with small trace of Hessian, which is widely used in previous work to distinguish between sharp and flat minima. We further provide convergence rates of zeroth-order optimization to approximate flat minima for convex and sufficiently smooth functions, where flat minima are defined as the minimizers that achieve the smallest trace of Hessian among all optimal solutions. Experiments on binary classification tasks with convex losses and language model fine-tuning support our theoretical findings.

Policy gradient (PG) is widely used in reinforcement learning due to its scalability and good performance. In recent years, several variance-reduced PG methods have been proposed with a theoretical guarantee of converging to an approximate first-order stationary point (FOSP) with the sample complexity of $O(\epsilon^{-3})$. However, FOSPs could be bad local optima or saddle points. Moreover, these algorithms often use importance sampling (IS) weights which could impair the statistical effectiveness of variance reduction. In this paper, we propose a variance-reduced second-order method that uses second-order information in the form of Hessian vector products (HVP) and converges to an approximate second-order stationary point (SOSP) with sample complexity of $\tilde{O}(\epsilon^{-3})$. This rate improves the best-known sample complexity for achieving approximate SOSPs by a factor of $O(\epsilon^{-0.5})$. Moreover, the proposed variance reduction technique bypasses IS weights by using HVP terms. Our experimental results show that the proposed algorithm outperforms the state of the art and is more robust to changes in random seeds.

Dynamic mechanism design is a challenging extension to ordinary mechanism design in which the mechanism designer must make a sequence of decisions over time in the face of possibly untruthful reports of participating agents. Optimizing dynamic mechanisms for welfare is relatively well understood. However, there has been less work on optimizing for other goals (e.g., revenue), and without restrictive assumptions on valuations, it is remarkably challenging to characterize good mechanisms. Instead, we turn to automated mechanism design to find mechanisms with good performance in specific problem instances. We extend the class of affine maximizer mechanisms to MDPs where agents may untruthfully report their rewards. This extension results in a challenging bilevel optimization problem in which the upper problem involves choosing optimal mechanism parameters, and the lower problem involves solving the resulting MDP. Our approach can find truthful dynamic mechanisms that achieve strong performance on goals other than welfare, and can be applied to essentially any problem setting---without restrictions on valuations---for which RL can learn optimal policies.

This paper revisits the convergence of Stochastic Mirror Descent (SMD) in the contemporary nonconvex optimization setting. Existing results for batch-free nonconvex SMD restrict the choice of the distance generating function (DGF) to be differentiable with Lipschitz continuous gradients, thereby excluding important setups such as Shannon entropy. In this work, we present a new convergence analysis of nonconvex SMD supporting general DGF, that overcomes the above limitations and relies solely on the standard assumptions. Moreover, our convergence is established with respect to the Bregman Forward-Backward envelope, which is a stronger measure than the commonly used squared norm of gradient mapping. We further extend our results to guarantee high probability convergence under sub-Gaussian noise and global convergence under the generalized Bregman Proximal Polyak-{Ł}ojasiewicz condition. Additionally, we illustrate the advantages of our improved SMD theory in various nonconvex machine learning tasks by harnessing nonsmooth DGFs. Notably, in the context of nonconvex differentially private (DP) learning, our theory yields a simple algorithm with a (nearly) dimension-independent utility bound. For the problem of training linear neural networks, we develop provably convergent stochastic algorithms.

In this paper, we study the fundamental statistical efficiency of Reinforcement Learning in Mean-Field Control (MFC) and Mean-Field Game (MFG) with general model-based function approximation. We introduce a new concept called Mean-Field Model-Based Eluder Dimension (MF-MBED), which characterizes the inherent complexity of mean-field model classes. We show that a rich family of Mean-Field RL problems exhibits low MF-MBED. Additionally, we propose algorithms based on maximal likelihood estimation, which can return an $\epsilon$-optimal policy for MFC or an $\epsilon$-Nash Equilibrium policy for MFG. The overall sample complexity depends only polynomially on MF-MBED, which is potentially much lower than the size of state-action space. Compared with previous works, our results only require the minimal assumptions including realizability and Lipschitz continuity.

Tuning hyperparameters, such as the stepsize, presents a major challenge of training machine learning models. To address this challenge, numerous adaptive optimization algorithms have been developed that achieve near-optimal complexities, even when stepsizes are independent of problem-specific parameters, provided that the loss function is $L$-smooth. However, as the assumption is relaxed to the more realistic $(L_0, L_1)$-smoothness, all existing convergence results still necessitate tuning of the stepsize. In this study, we demonstrate that Normalized Stochastic Gradient Descent with Momentum (NSGD-M) can achieve a (nearly) rate-optimal complexity without prior knowledge of any problem parameter, though this comes at the cost of introducing an exponential term dependent on $L_1$ in the complexity. We further establish that this exponential term is inevitable to such schemes by introducing a theoretical framework of lower bounds tailored explicitly for parameter-agnostic algorithms. Interestingly, in deterministic settings, the exponential factor can be neutralized by employing Gradient Descent with a Backtracking Line Search. To the best of our knowledge, these findings represent the first parameter-agnostic convergence results under the generalized smoothness condition. Our empirical experiments further confirm our theoretical insights.

Constrained Markov games offer a formal mathematical framework for modeling multi-agent reinforcement learning problems where the behavior of the agents is subject to constraints. In this work, we focus on the recently introduced class of constrained Markov Potential Games. While centralized algorithms have been proposed for solving such constrained games, the design of converging independent learning algorithms tailored for the constrained setting remains an open question. We propose an independent policy gradient algorithm for learning approximate constrained Nash equilibria: Each agent observes their own actions and rewards, along with a shared state. Inspired by the optimization literature, our algorithm performs proximal-point-like updates augmented with a regularized constraint set. Each proximal step is solved inexactly using a stochastic switching gradient algorithm. Notably, our algorithm can be implemented independently without a centralized coordination mechanism requiring turn-based agent updates. Under some technical constraint qualification conditions, we establish convergence guarantees towards constrained approximate Nash equilibria. We perform simulations to illustrate our results.

This paper studies the generalization performance of algorithms for solving nonconvex-(strongly)-concave (NC-SC/NC-C) stochastic minimax optimization measured by the stationarity of primal functions. We first establish algorithm-agnostic generalization bounds via uniform convergence between the empirical minimax problem and the population minimax problem. The sample complexities for achieving $\epsilon$-generalization are $\tilde{\mathcal{O}}(d\kappa^2\epsilon^{-2})$ and $\tilde{\mathcal{O}}(d\epsilon^{-4})$ for NC-SC and NC-C settings, respectively, where $d$ is the dimension of the primal variable and $\kappa$ is the condition number. We further study the algorithm-dependent generalization bounds via stability arguments of algorithms. In particular, we introduce a novel stability notion for minimax problems and build a connection between stability and generalization. As a result, we establish algorithm-dependent generalization bounds for stochastic gradient descent ascent (SGDA) and the more general sampling-determined algorithms (SDA).

We study the sample complexity of reinforcement learning (RL) in Mean-Field Games (MFGs) with model-based function approximation that requires strategic exploration to find a Nash Equilibrium policy. We introduce the Partial Model-Based Eluder Dimension (P-MBED), a more effective notion to characterize the model class complexity. Notably, P-MBED measures the complexity of the single-agent model class converted from the given mean-field model class, and potentially, can be exponentially lower than the MBED proposed by Huang et al. (2024). We contribute a model elimination algorithm featuring a novel exploration strategy and establish sample complexity results polynomial w.r.t. P-MBED. Crucially, our results reveal that, under the basic realizability and Lipschitz continuity assumptions, learning Nash Equilibrium in MFGs is no more statistically challenging than solving a logarithmic number of single-agent RL problems. We further extend our results to Multi-Type MFGs, generalizing from conventional MFGs and involving multiple types of agents. This extension implies statistical tractability of a broader class of Markov Games through the efficacy of mean-field approximation. Finally, inspired by our theoretical algorithm, we present a heuristic approach with improved computational efficiency and empirically demonstrate its effectiveness.

Constrained Markov decision processes (CMDPs) are a common way to model safety constraints in reinforcement learning. State-of-the-art methods for efficiently solving CMDPs are based on primal-dual algorithms. For these algorithms, all currently known regret bounds allow for error cancellations — one can compensate for a constraint violation in one round with a strict constraint satisfaction in another. This makes the online learning process unsafe since it only guarantees safety for the final (mixture) policy but not during learning. As Efroni et al. (2020) pointed out, it is an open question whether primal-dual algorithms can provably achieve sublinear regret if we do not allow error cancellations. In this paper, we give the first affirmative answer. We first generalize a result on last-iterate convergence of regularized primal-dual schemes to CMDPs with multiple constraints. Building upon this insight, we propose a model-based primal-dual algorithm to learn in an unknown CMDP. We prove that our algorithm achieves sublinear regret without error cancellations.

The widespread practice of fine-tuning large language models (LLMs) on domain-specific data faces two major challenges in memory and privacy. First, as the size of LLMs continues to grow, the memory demands of gradient-based training methods via backpropagation become prohibitively high. Second, given the tendency of LLMs to memorize training data, it is important to protect potentially sensitive information in the fine-tuning data from being regurgitated. Zeroth-order methods, which rely solely on forward passes, substantially reduce memory consumption during training. However, directly combining them with standard differentially private gradient descent suffers more as model size grows. To bridge this gap, we introduce DPZero, a novel private zeroth-order algorithm with nearly dimension-independent rates. The memory efficiency of DPZero is demonstrated in privately fine-tuning RoBERTa and OPT on several downstream tasks. Our code is available at https://github.com/Liang137/DPZero.

In this paper, we show that applying adaptive methods directly to distributed minimax problems can result in non-convergence due to inconsistency in locally computed adaptive stepsizes. To address this challenge, we propose D-AdaST, a Distributed Adaptive minimax method with Stepsize Tracking. The key strategy is to employ an adaptive stepsize tracking protocol involving the transmission of two extra (scalar) variables. This protocol ensures the consistency among stepsizes of nodes, eliminating the steady-state error due to the lack of coordination of stepsizes among nodes that commonly exists in vanilla distributed adaptive methods, and thus guarantees exact convergence. For nonconvex-strongly-concave distributed minimax problems, we characterize the specific transient times that ensure time-scale separation of stepsizes and quasi-independence of networks, leading to a near-optimal convergence rate of $\tilde{\mathcal{O}} \left( \epsilon ^{-\left( 4+\delta \right)} \right)$ for any small $\delta > 0$, matching that of the centralized counterpart. To our best knowledge, D-AdaST is the *first* distributed adaptive method achieving near-optimal convergence without knowing any problem-dependent parameters for nonconvex minimax problems. Extensive experiments are conducted to validate our theoretical results.

Sharpness-aware minimization (SAM) improves generalization of various deep learning tasks. Motivated by popular architectures such as LoRA, we explore the implicit regularization of SAM for scale-invariant problems involving two groups of variables. Instead of focusing on commonly used sharpness, this work introduces a concept termed *balancedness*, defined as the difference between the squared norm of two variables. This allows us to depict richer global behaviors of SAM. In particular, our theoretical and empirical findings reveal that i) SAM promotes balancedness; and ii) the regularization on balancedness is *data-responsive* -- outliers have stronger impact. The latter coincides with empirical observations that SAM outperforms SGD in the presence of outliers. Leveraging the implicit regularization, we develop a resource-efficient SAM variant, balancedness-aware regularization (BAR), tailored for scale-invariant problems such as finetuning language models with LoRA. BAR saves 95% computational overhead of SAM, with enhanced test performance across various tasks on RoBERTa, GPT2, and OPT-1.3B.

In real-world decision-making, uncertainty is important yet difficult to handle. Stochastic dominance provides a theoretically sound approach to comparing uncertain quantities, but optimization with stochastic dominance constraints is often computationally expensive, which limits practical applicability. In this paper, we develop a simple yet efficient approach for the problem, Light Stochastic Dominance Solver (light-SD), by leveraging properties of the Lagrangian. We recast the inner optimization in the Lagrangian as a learning problem for surrogate approximation, which bypasses the intractability and leads to tractable updates or even closed-form solutions for gradient calculations. We prove convergence of the algorithm and test it empirically. The proposed light-SD demonstrates superior performance on several representative problems ranging from finance to supply chain management.

We study policy evaluation of offline contextual bandits subject to unobserved confounders. Sensitivity analysis methods are commonly used to estimate the policy value under the worst-case confounding over a given uncertainty set. However, existing work often resorts to some coarse relaxation of the uncertainty set for the sake of tractability, leading to overly conservative estimation of the policy value. In this paper, we propose a general estimator that provides a sharp lower bound of the policy value. It can be shown that our estimator contains the recently proposed sharp estimator by Dorn and Guo (2022) as a special case, and our method enables a novel extension of the classical marginal sensitivity model using f-divergence. To construct our estimator, we leverage the kernel method to obtain a tractable approximation to the conditional moment constraints, which traditional non-sharp estimators failed to take into account. In the theoretical analysis, we provide a condition for the choice of the kernel which guarantees no specification error that biases the lower bound estimation. Furthermore, we provide consistency guarantees of policy evaluation and learning. In the experiments with synthetic and real-world data, we demonstrate the effectiveness of the proposed method.

We consider the reinforcement learning (RL) problem with general utilities which consists in maximizing a function of the state-action occupancy measure. Beyond the standard cumulative reward RL setting, this problem includes as particular cases constrained RL, pure exploration and learning from demonstrations among others. For this problem, we propose a simpler single-loop parameter-free normalized policy gradient algorithm. Implementing a recursive momentum variance reduction mechanism, our algorithm achieves $\tilde{\mathcal{O}}(\epsilon^{-3})$ and $\tilde{\mathcal{O}}(\epsilon^{-2})$ sample complexities for $\epsilon$-first-order stationarity and $\epsilon$-global optimality respectively, under adequate assumptions. We further address the setting of large finite state action spaces via linear function approximation of the occupancy measure and show a $\tilde{\mathcal{O}}(\epsilon^{-4})$ sample complexity for a simple policy gradient method with a linear regression subroutine.

Recently, the impressive empirical success of policy gradient (PG) methods has catalyzed the development of their theoretical foundations. Despite the huge efforts directed at the design of efficient stochastic PG-type algorithms, the understanding of their convergence to a globally optimal policy is still limited. In this work, we develop improved global convergence guarantees for a general class of Fisher-non-degenerate parameterized policies which allows to address the case of continuous state action spaces. First, we propose a Normalized Policy Gradient method with Implicit Gradient Transport (N-PG-IGT) and derive a $\tilde{\mathcal{O}}(\varepsilon^{-2.5})$ sample complexity of this method for finding a global $\varepsilon$-optimal policy. Improving over the previously known $\tilde{\mathcal{O}}(\varepsilon^{-3})$ complexity, this algorithm does not require the use of importance sampling or second-order information and samples only one trajectory per iteration. Second, we further improve this complexity to $\tilde{ \mathcal{\mathcal{O}} }(\varepsilon^{-2})$ by considering a Hessian-Aided Recursive Policy Gradient ((N)-HARPG) algorithm enhanced with a correction based on a Hessian-vector product. Interestingly, both algorithms are $(i)$ simple and easy to implement: single-loop, do not require large batches of trajectories and sample at most two trajectories per iteration; $(ii)$ computationally and memory efficient: they do not require expensive subroutines at each iteration and can be implemented with memory linear in the dimension of parameters.

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.

In this paper, we study the Tiered Reinforcement Learning setting, a parallel transfer learning framework, where the goal is to transfer knowledge from the low-tier (source) task to the high-tier (target) task to reduce the exploration risk of the latter while solving the two tasks in parallel. Unlike previous work, we do not assume the low-tier and high-tier tasks share the same dynamics or reward functions, and focus on robust knowledge transfer without prior knowledge on the task similarity. We identify a natural and necessary condition called the ``Optimal Value Dominance'' for our objective. Under this condition, we propose novel online learning algorithms such that, for the high-tier task, it can achieve constant regret on partial states depending on the task similarity and retain near-optimal regret when the two tasks are dissimilar, while for the low-tier task, it can keep near-optimal without making sacrifice. Moreover, we further study the setting with multiple low-tier tasks, and propose a novel transfer source selection mechanism, which can ensemble the information from all low-tier tasks and allow provable benefits on a much larger state-action space.

We explore the problem of imitation learning (IL) in the context of mean-field games (MFGs), where the goal is to imitate the behavior of a population of agents following a Nash equilibrium policy according to some unknown payoff function. IL in MFGs presents new challenges compared to single-agent IL, particularly when both the reward function and the transition kernel depend on the population distribution. In this paper, departing from the existing literature on IL for MFGs, we introduce a new solution concept called the Nash imitation gap. Then we show that when only the reward depends on the population distribution, IL in MFGs can be reduced to single-agent IL with similar guarantees. However, when the dynamics is population-dependent, we provide a novel upper-bound that suggests IL is harder in this setting. To address this issue, we propose a new adversarial formulation where the reinforcement learning problem is replaced by a mean-field control (MFC) problem, suggesting progress in IL within MFGs may have to build upon MFC.

The classical analysis of Stochastic Gradient Descent (SGD) with polynomially decaying stepsize $\eta_t = \eta/\sqrt{t}$ relies on well-tuned $\eta$ depending on problem parameters such as Lipschitz smoothness constant, which is often unknown in practice. In this work, we prove that SGD with arbitrary $\eta > 0$, referred to as untuned SGD, still attains an order-optimal convergence rate $\widetilde{\mathcal{O}}(T^{-1/4})$ in terms of gradient norm for minimizing smooth objectives. Unfortunately, it comes at the expense of a catastrophic exponential dependence on the smoothness constant, which we show is unavoidable for this scheme even in the noiseless setting. We then examine three families of adaptive methods — Normalized SGD (NSGD), AMSGrad, and AdaGrad — unveiling their power in preventing such exponential dependency in the absence of information about the smoothness parameter and boundedness of stochastic gradients. Our results provide theoretical justification for the advantage of adaptive methods over untuned SGD in alleviating the issue with large gradients.

Algorithmic reproducibility measures the deviation in outputs of machine learning algorithms upon minor changes in the training process. Previous work suggests that first-order methods would need to trade-off convergence rate (gradient complexity) for better reproducibility. In this work, we challenge this perception and demonstrate that both optimal reproducibility and near-optimal convergence guarantees can be achieved for smooth convex minimization and smooth convex-concave minimax problems under various error-prone oracle settings. Particularly, given the inexact initialization oracle, our regularization-based algorithms achieve the best of both worlds -- optimal reproducibility and near-optimal gradient complexity -- for minimization and minimax optimization. With the inexact gradient oracle, the near-optimal guarantees also hold for minimax optimization. Additionally, with the stochastic gradient oracle, we show that stochastic gradient descent ascent is optimal in terms of both reproducibility and gradient complexity. We believe our results contribute to an enhanced understanding of the reproducibility-convergence trade-off in the context of convex optimization.

We study the bilinearly coupled minimax problem: $\min_{x} \max_{y} f(x) + y^\top A x - h(y)$, where $f$ and $h$ are both strongly convex smooth functions and admit first-order gradient oracles. Surprisingly, no known first-order algorithms have hitherto achieved the lower complexity bound of $\Omega((\sqrt{\frac{L_x}{\mu_x}} + \frac{\|A\|}{\sqrt{\mu_x \mu_y}} + \sqrt{\frac{L_y}{\mu_y}}) \log(\frac1{\varepsilon}))$ for solving this problem up to an $\varepsilon$ primal-dual gap in the general parameter regime, where $L_x, L_y,\mu_x,\mu_y$ are the corresponding smoothness and strongly convexity constants. We close this gap by devising the first optimal algorithm, the Lifted Primal-Dual (LPD) method. Our method lifts the objective into an extended form that allows both the smooth terms and the bilinear term to be handled optimally and seamlessly with the same primal-dual framework. Besides optimality, our method yields a desirably simple single-loop algorithm that uses only one gradient oracle call per iteration. Moreover, when $f$ is just convex, the same algorithm applied to a smoothed objective achieves the nearly optimal iteration complexity. We also provide a direct single-loop algorithm, using the LPD method, that achieves the iteration complexity of $O(\sqrt{\frac{L_x}{\varepsilon}} + \frac{\|A\|}{\sqrt{\mu_y \varepsilon}} + \sqrt{\frac{L_y}{\varepsilon}})$. Numerical experiments on quadratic minimax problems and policy evaluation problems further demonstrate the fast convergence of our algorithm in practice.

Gradient descent ascent (GDA), the simplest single-loop algorithm for nonconvex minimax optimization, is widely used in practical applications such as generative adversarial networks (GANs) and adversarial training. Albeit its desirable simplicity, recent work shows inferior convergence rates of GDA in theory, even when assuming strong concavity of the objective in terms of one variable. This paper establishes new convergence results for two alternative single-loop algorithms – alternating GDA and smoothed GDA – under the mild assumption that the objective satisfies the Polyak-Lojasiewicz (PL) condition about one variable. We prove that, to find an $\epsilon$-stationary point, (i) alternating GDA and its stochastic variant (without mini batch) respectively require $O(\kappa^{2} \epsilon^{-2})$ and $O(\kappa^{4} \epsilon^{-4})$ iterations, while (ii) smoothed GDA and its stochastic variant (without mini batch) respectively require $O(\kappa \epsilon^{-2})$ and $O(\kappa^{2} \epsilon^{-4})$ iterations. The latter greatly improves over the vanilla GDA and gives the hitherto best known complexity results among single-loop algorithms under similar settings. We further showcase the empirical efficiency of these algorithms in training GANs and robust nonlinear regression.

We introduce algorithms based on natural actor-critic and analyze their sample complexity for solving two player zero-sum Markov games in the tabular case. Our results improve the best-known sample complexities of policy gradient/actor-critic methods for convergence to Nash equilibrium in the multi-agent setting. We use the error propagation scheme in approximate dynamic programming, recent advances for global convergence of policy gradient methods, temporal difference learning, and techniques from stochastic primal-dual optimization. Our algorithms feature two stages, requiring agents to agree on an etiquette before starting their interactions, which is feasible for instance in self-play. However, the agents only access to joint reward and joint next state and not to each other’s actions or policies. Our complexity results match the best-known results for global convergence of policy gradient algorithms for single agent RL. We provide numerical verification of our methods for a two player bandit environment and a two player game, Alesia. We observe improved empirical performance as compared to the recently proposed optimistic gradient descent-ascent variant for Markov games.

We study the complexity of finding the global solution to stochastic nonconvex optimization when the objective function satisfies global Kurdyka-{\L}ojasiewicz (KL) inequality and the queries from stochastic gradient oracles satisfy mild expected smoothness assumption. We first introduce a general framework to analyze Stochastic Gradient Descent (SGD) and its associated nonlinear dynamics under the setting. As a byproduct of our analysis, we obtain a sample complexity of $\mathcal{O}(\epsilon^{-(4-\alpha)/\alpha})$ for SGD when the objective satisfies the so called $\alpha$-P{\L} condition, where $\alpha$ is the degree of gradient domination. Furthermore, we show that a modified SGD with variance reduction and restarting (PAGER) achieves an improved sample complexity of $\mathcal{O}(\epsilon^{-2/\alpha})$ when the objective satisfies the average smoothness assumption. This leads to the first optimal algorithm for the important case of $\alpha=1$ which appears in applications such as policy optimization in reinforcement learning.

We study the performance of Stochastic Cubic Regularized Newton (SCRN) on a class of functions satisfying gradient dominance property with $1\le\alpha\le2$ which holds in a wide range of applications in machine learning and signal processing. This condition ensures that any first-order stationary point is a global optimum. We prove that the total sample complexity of SCRN in achieving $\epsilon$-global optimum is $\mathcal{O}(\epsilon^{-7/(2\alpha)+1})$ for $1\le\alpha< 3/2$ and $\mathcal{\tilde{O}}(\epsilon^{-2/(\alpha)})$ for $3/2\le\alpha\le 2$. SCRN improves the best-known sample complexity of stochastic gradient descent. Even under a weak version of gradient dominance property, which is applicable to policy-based reinforcement learning (RL), SCRN achieves the same improvement over stochastic policy gradient methods. Additionally, we show that the average sample complexity of SCRN can be reduced to ${\mathcal{O}}(\epsilon^{-2})$ for $\alpha=1$ using a variance reduction method with time-varying batch sizes. Experimental results in various RL settings showcase the remarkable performance of SCRN compared to first-order methods.

Adaptive algorithms like AdaGrad and AMSGrad are successful in nonconvex optimization owing to their parameter-agnostic ability – requiring no a priori knowledge about problem-specific parameters nor tuning of learning rates. However, when it comes to nonconvex minimax optimization, direct extensions of such adaptive optimizers without proper time-scale separation may fail to work in practice. We provide such an example proving that the simple combination of Gradient Descent Ascent (GDA) with adaptive stepsizes can diverge if the primal-dual stepsize ratio is not carefully chosen; hence, a fortiori, such adaptive extensions are not parameter-agnostic. To address the issue, we formally introduce a Nested Adaptive framework, NeAda for short, that carries an inner loop for adaptively maximizing the dual variable with controllable stopping criteria and an outer loop for adaptively minimizing the primal variable. Such mechanism can be equipped with off-the-shelf adaptive optimizers and automatically balance the progress in the primal and dual variables. Theoretically, for nonconvex-strongly-concave minimax problems, we show that NeAda with AdaGrad stepsizes can achieve the near-optimal $\widetilde{O}(\epsilon^{-2})$ and $\widetilde{O}(\epsilon^{-4})$ gradient complexities respectively in the deterministic and stochastic settings, without prior information on the problem's smoothness and strong concavity parameters. To the best of our knowledge, this is the first algorithm that simultaneously achieves near-optimal convergence rates and parameter-agnostic adaptation in the nonconvex minimax setting. Numerically, we further illustrate the robustness of the NeAda family with experiments on simple test functions and a real-world application.

We study differentially private (DP) algorithms for smooth stochastic minimax optimization, with stochastic minimization as a byproduct. The holy grail of these settings is to guarantee the optimal trade-off between the privacy and the excess population loss, using an algorithm with a linear time-complexity in the number of training samples. We provide a general framework for solving differentially private stochastic minimax optimization (DP-SMO) problems, which enables the practitioners to bring their own base optimization algorithm and use it as a black-box to obtain the near-optimal privacy-loss trade-off. Our framework is inspired from the recently proposed Phased-ERM method [22] for nonsmooth differentially private stochastic convex optimization (DP-SCO), which exploits the stability of the empirical risk minimization (ERM) for the privacy guarantee. The flexibility of our approach enables us to sidestep the requirement that the base algorithm needs to have bounded sensitivity, and allows the use of sophisticated variance-reduced accelerated methods to achieve near-linear time-complexity. To the best of our knowledge, these are the first near-linear time algorithms with near-optimal guarantees on the population duality gap for smooth DP-SMO, when the objective is (strongly-)convex--(strongly-)concave. Additionally, based on our flexible framework, we enrich the family of near-linear time algorithms for smooth DP-SCO with the near-optimal privacy-loss trade-off.

We consider stochastic optimization when one only has access to biased stochastic oracles of the objective, and obtaining stochastic gradients with low biases comes at high costs. This setting captures a variety of optimization paradigms widely used in machine learning, such as conditional stochastic optimization, bilevel optimization, and distributionally robust optimization. We examine a family of multi-level Monte Carlo (MLMC) gradient methods that exploit a delicate trade-off among the bias, the variance, and the oracle cost. We provide a systematic study of their convergences and total computation complexities for strongly convex, convex, and nonconvex objectives, and demonstrate their superiority over the naive biased stochastic gradient method. Moreover, when applied to conditional stochastic optimization, the MLMC gradient methods significantly improve the best-known sample complexity in the literature.

This paper studies the complexity for finding approximate stationary points of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general and averaged smooth finite-sum settings. We establish nontrivial lower complexity bounds for the two settings, respectively. Our result reveals substantial gaps between these limits and best-known upper bounds in the literature. To close these gaps, we introduce a generic acceleration scheme that deploys existing gradient-based methods to solve a sequence of crafted strongly-convex-strongly-concave subproblems. In the general setting, the complexity of our proposed algorithm nearly matches the lower bound; in particular, it removes an additional poly-logarithmic dependence on accuracy present in previous works. In the averaged smooth finite-sum setting, our proposed algorithm improves over previous algorithms by providing a nearly-tight dependence on the condition number.

This paper develops a novel and unified framework to analyze the convergence of a large family of Q-learning algorithms from the switching system perspective. We show that the nonlinear ODE models associated with Q-learning and many of its variants can be naturally formulated as affine switching systems. Building on their asymptotic stability, we obtain a number of interesting results: (i) we provide a simple ODE analysis for the convergence of asynchronous Q-learning under relatively weak assumptions; (ii) we establish the first convergence analysis of the averaging Q-learning algorithm; and (iii) we derive a new sufficient condition for the convergence of Q-learning with linear function approximation.

Conditional stochastic optimization covers a variety of applications ranging from invariant learning and causal inference to meta-learning. However, constructing unbiased gradient estimators for such problems is challenging due to the composition structure. As an alternative, we propose a biased stochastic gradient descent (BSGD) algorithm and study the bias-variance tradeoff under different structural assumptions. We establish the sample complexities of BSGD for strongly convex, convex, and weakly convex objectives under smooth and non-smooth conditions. Our lower bound analysis shows that the sample complexities of BSGD cannot be improved for general convex objectives and nonconvex objectives except for smooth nonconvex objectives with Lipschitz continuous gradient estimator. For this special setting, we propose an accelerated algorithm called biased SpiderBoost (BSpiderBoost) that matches the lower bound complexity. We further conduct numerical experiments on invariant logistic regression and model-agnostic meta-learning to illustrate the performance of BSGD and BSpiderBoost.

In this paper, we establish a theoretical comparison between the asymptotic mean square errors of double Q-learning and Q-learning. Our result builds upon an analysis for linear stochastic approximation based on Lyapunov equations and applies to both tabular setting or with linear function approximation, provided that the optimal policy is unique and the algorithms converge. We show that the asymptotic mean-square error of Double Q-learning is exactly equal to that of Q-learning if Double Q-learning uses twice the learning rate of Q-learning and the output of Double Q-learning is the average of its two estimators. We also present some practical implications of this theoretical observation using simulations.

Nonconvex minimax problems appear frequently in emerging machine learning applications, such as generative adversarial networks and adversarial learning. Simple algorithms such as the gradient descent ascent (GDA) are the common practice for solving these nonconvex games and receive lots of empirical success. Yet, it is known that these vanilla GDA algorithms with constant stepsize can potentially diverge even in the convex setting. In this work, we show that for a subclass of nonconvex-nonconcave objectives satisfying a so-called two-sided Polyak-{\L}ojasiewicz inequality, the alternating gradient descent ascent (AGDA) algorithm converges globally at a linear rate and the stochastic AGDA achieves a sublinear rate. We further develop a variance reduced algorithm that attains a provably faster rate than AGDA when the problem has the finite-sum structure.

Macroscopic data aggregated from microscopic events are pervasive in machine learning, such as country-level COVID-19 infection statistics based on city-level data. Yet, many existing approaches for predicting macroscopic behavior only use aggregated data, leaving a large amount of fine-grained microscopic information unused. In this paper, we propose a principled optimization framework for macroscopic prediction by fitting microscopic models based on conditional stochastic optimization. The framework leverages both macroscopic and microscopic information, and adapts to individual microscopic models involved in the aggregation. In addition, we propose efficient learning algorithms with convergence guarantees. In our experiments, we show that the proposed learning framework clearly outperforms other plug-in supervised learning approaches in real-world applications, including the prediction of daily infections of COVID-19 and medicare claims.

We introduce a generic \emph{two-loop} scheme for smooth minimax optimization with strongly-convex-concave objectives. Our approach applies the accelerated proximal point framework (or Catalyst) to the associated \emph{dual problem} and takes full advantage of existing gradient-based algorithms to solve a sequence of well-balanced strongly-convex-strongly-concave minimax problems. Despite its simplicity, this leads to a family of near-optimal algorithms with improved complexity over all existing methods designed for strongly-convex-concave minimax problems. Additionally, we obtain the first variance-reduced algorithms for this class of minimax problems with finite-sum structure and establish even faster convergence rate. Furthermore, when extended to the nonconvex-concave minimax optimization, our algorithm again achieves the state-of-the-art complexity for finding a stationary point. We carry out several numerical experiments showcasing the superiority of the Catalyst framework in practice.

We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization (QDSFM), which allows to model a number of learning tasks on graphs and hypergraphs. The problem exhibits close ties to decomposable submodular function minimization (DSFM) yet is much more challenging to solve. We approach the problem via a new dual strategy and formulate an objective that can be optimized through a number of double-loop algorithms. The outer-loop uses either random coordinate descent (RCD) or alternative projection (AP) methods, for both of which we prove linear convergence rates. The inner-loop computes projections onto cones generated by base polytopes of the submodular functions via the modified min-norm-point or Frank-Wolfe algorithms. We also describe two new applications of QDSFM: hypergraph-adapted PageRank and semi-supervised learning. The proposed hypergraph-based PageRank algorithm can be used for local hypergraph partitioning and comes with provable performance guarantees. For hypergraph-adapted semi-supervised learning, we provide numerical experiments demonstrating the efficiency of our QDSFM solvers and their significant improvements on prediction accuracy when compared to state-of-the-art methods.

We investigate penalized maximum log-likelihood estimation for exponential family distributions whose natural parameter resides in a reproducing kernel Hilbert space. Key to our approach is a novel technique, doubly dual embedding, that avoids computation of the partition function. This technique also allows the development of a flexible sampling strategy that amortizes the cost of Monte-Carlo sampling in the inference stage. The resulting estimator can be easily generalized to kernel conditional exponential families. We establish a connection between kernel exponential family estimation and MMD-GANs, revealing a new perspective for understanding GANs. Compared to the score matching based estimators, the proposed method improves both memory and time efficiency while enjoying stronger statistical properties, such as fully capturing smoothness in its statistical convergence rate while the score matching estimator appears to saturate. Finally, we show that the proposed estimator empirically outperforms state-of-the-art methods in both kernel exponential family estimation and its conditional extension.

The use of target networks has been a popular and key component of recent deep Q-learning algorithms for reinforcement learning, yet little is known from the theory side. In this work, we introduce a new family of target-based temporal difference (TD) learning algorithms that maintain two separate learning parameters {–} the target variable and online variable. We propose three members in the family, the averaging TD, double TD, and periodic TD, where the target variable is updated through an averaging, symmetric, or periodic fashion, respectively, mirroring those techniques used in deep Q-learning practice. We establish asymptotic convergence analyses for both averaging TD and double TD and a finite sample analysis for periodic TD. In addition, we provide some simulation results showing potentially superior convergence of these target-based TD algorithms compared to the standard TD-learning. While this work focuses on linear function approximation and policy evaluation setting, we consider this as a meaningful step towards the theoretical understanding of deep Q-learning variants with target networks.

We present an efficient algorithm for maximum likelihood estimation (MLE) of exponential family models, with a general parametrization of the energy function that includes neural networks. We exploit the primal-dual view of the MLE with a kinetics augmented model to obtain an estimate associated with an adversarial dual sampler. To represent this sampler, we introduce a novel neural architecture, dynamics embedding, that generalizes Hamiltonian Monte-Carlo (HMC). The proposed approach inherits the flexibility of HMC while enabling tractable entropy estimation for the augmented model. By learning both a dual sampler and the primal model simultaneously, and sharing parameters between them, we obviate the requirement to design a separate sampling procedure once the model has been trained, leading to more effective learning. We show that many existing estimators, such as contrastive divergence, pseudo/composite-likelihood, score matching, minimum Stein discrepancy estimator, non-local contrastive objectives, noise-contrastive estimation, and minimum probability flow, are special cases of the proposed approach, each expressed by a different (fixed) dual sampler. An empirical investigation shows that adapting the sampler during MLE can significantly improve on state-of-the-art estimators.

The nonparametric learning of positive-valued functions appears widely in machine learning, especially in the context of estimating intensity functions of point processes. Yet, existing approaches either require computing expensive projections or semidefinite relaxations, or lack convexity and theoretical guarantees after introducing nonlinear link functions. In this paper, we propose a novel algorithm, pseudo mirror descent, that performs efficient estimation of positive functions within a Hilbert space without expensive projections. The algorithm guarantees positivity by performing mirror descent with an appropriately selected Bregman divergence, and a pseudo-gradient is adopted to speed up the gradient evaluation procedure in practice. We analyze both asymptotic and nonasymptotic convergence of the algorithm. Through simulations, we show that pseudo mirror descent outperforms the state-of-the-art benchmarks for learning intensities of Poisson and multivariate Hawkes processes, in terms of both computational efficiency and accuracy.

Power control in wireless networks is a well-studied problem. However, recently it has been demonstrated that significant throughput gains can be achieved using data-driven online learning algorithms, supported by a cloud computing infrastructure. In this paper, we provide theoretical guarantees for such algorithms. In particular, we consider two variants of the problem: one which emphasizes long-term throughput and the other which emphasizes robust short-term throughput. The first problem reduces to solving a convex optimization problem with noisy, stochastic measurements while the second one is an online optimization problem where an adversary chooses the reward functions. We provide stochastic and online gradient descent methods customized for the power control problem and establish their convergence analysis. We show that in both cases, the total regret over a time horizon T grows sublinearly at rate O(√T) for suitable choices of algorithms and algorithm parameters.

When function approximation is used, solving the Bellman optimality equation with stability guarantees has remained a major open problem in reinforcement learning for decades. The fundamental difficulty is that the Bellman operator may become an expansion in general, resulting in oscillating and even divergent behavior of popular algorithms like Q-learning. In this paper, we revisit the Bellman equation, and reformulate it into a novel primal-dual optimization problem using Nesterov’s smoothing technique and the Legendre-Fenchel transformation. We then develop a new algorithm, called Smoothed Bellman Error Embedding, to solve this optimization problem where any differentiable function class may be used. We provide what we believe to be the first convergence guarantee for general nonlinear function approximation, and analyze the algorithm’s sample complexity. Empirically, our algorithm compares favorably to state-of-the-art baselines in several benchmark control problems.

Variational inference plays a vital role in learning graphical models, especially on large-scale datasets. Much of its success depends on a proper choice of auxiliary distribution class for posterior approximation. However, how to pursue an auxiliary distribution class that achieves both good approximation ability and computation efficiency remains a core challenge. In this paper, we proposed coupled variational Bayes which exploits the primal-dual view of the ELBO with the variational distribution class generated by an optimization procedure, which is termed optimization embedding. This flexible function class couples the variational distribution with the original parameters in the graphical models, allowing end-to-end learning of the graphical models by back-propagation through the variational distribution. Theoretically, we establish an interesting connection to gradient flow and demonstrate the extreme flexibility of this implicit distribution family in the limit sense. Empirically, we demonstrate the effectiveness of the proposed method on multiple graphical models with either continuous or discrete latent variables comparing to state-of-the-art methods.

We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization. The problem is closely related to decomposable submodular function minimization and arises in many learning on graphs and hypergraphs settings, such as graph-based semi-supervised learning and PageRank. We approach the problem via a new dual strategy and describe an objective that may be optimized via random coordinate descent (RCD) methods and projections onto cones. We also establish the linear convergence rate of the RCD algorithm and develop efficient projection algorithms with provable performance guarantees. Numerical experiments in semi-supervised learning on hypergraphs confirm the efficiency of the proposed algorithm and demonstrate the significant improvements in prediction accuracy with respect to state-of-the-art methods.

Approximate Bayesian computation (ABC) is an important methodology for Bayesian inference when the likelihood function is intractable. Sampling-based ABC algorithms such as rejection- and K2-ABC are inefficient when the parameters have high dimensions, while the regression-based algorithms such as K- and DR-ABC are hard to scale. In this paper, we introduce an optimization-based ABC framework that addresses these deficiencies. Leveraging a generative model for posterior and joint distribution matching, we show that ABC can be framed as saddle point problems, whose objectives can be accessed directly with samples. We present the predictive ABC algorithm (P-ABC), and provide a probabilistically approximately correct (PAC) bound that guarantees its learning consistency. Numerical experiment shows that P-ABC outperforms both K2- and DR-ABC significantly.

Many machine learning tasks, such as learning with invariance and policy evaluation in reinforcement learning, can be characterized as problems of learning from conditional distributions. In such problems, each sample x itself is associated with a conditional distribution $p(z|x)$ represented by samples $z_i_i=1^M$, and the goal is to learn a function f that links these conditional distributions to target values y. These problems become very challenging when we only have limited samples or in the extreme case only one sample from each conditional distribution. Commonly used approaches either assume that z is independent of x, or require an overwhelmingly large set of samples from each conditional distribution. To address these challenges, we propose a novel approach which employs a new min-max reformulation of the learning from conditional distribution problem. With such new reformulation, we only need to deal with the joint distribution p(z,x). We also design an efficient learning algorithm, Embedding-SGD, and establish theoretical sample complexity for such problems. Finally, our numerical experiments, on both synthetic and real-world datasets, show that the proposed approach can significantly improve over existing algorithms.

Learning-based binary hashing has become a powerful paradigm for fast search and retrieval in massive databases. However, due to the requirement of discrete outputs for the hash functions, learning such functions is known to be very challenging. In addition, the objective functions adopted by existing hashing techniques are mostly chosen heuristically. In this paper, we propose a novel generative approach to learn hash functions through Minimum Description Length principle such that the learned hash codes maximally compress the dataset and can also be used to regenerate the inputs. We also develop an efficient learning algorithm based on the stochastic distributional gradient, which avoids the notorious difficulty caused by binary output constraints, to jointly optimize the parameters of the hash function and the associated generative model. Extensive experiments on a variety of large-scale datasets show that the proposed method achieves better retrieval results than the existing state-of-the-art methods.

We develop a nonparametric and online learning algorithm that estimates the triggering functions of a multivariate Hawkes process (MHP). The approach we take approximates the triggering function $f_{i,j}(t)$ by functions in a reproducing kernel Hilbert space (RKHS), and maximizes a time-discretized version of the log-likelihood, with Tikhonov regularization. Theoretically, our algorithm achieves an $\calO(\log T)$ regret bound. Numerical results show that our algorithm offers a competing performance to that of the nonparametric batch learning algorithm, with a run time comparable to the parametric online learning algorithm.

Bayesian methods are appealing in their flexibility in modeling complex data and ability in capturing uncertainty in parameters. However, when Bayes’ rule does not result in tractable closed-form, most approximate inference algorithms lack either scalability or rigorous guarantees. To tackle this challenge, we propose a simple yet provable algorithm, Particle Mirror Descent (PMD), to iteratively approximate the posterior density. PMD is inspired by stochastic functional mirror descent where one descends in the density space using a small batch of data points at each iteration, and by particle filtering where one uses samples to approximate a function. We prove result of the first kind that, with m particles, PMD provides a posterior density estimator that converges in terms of KL-divergence to the true posterior in rate O(1/\sqrtm). We demonstrate competitive empirical performances of PMD compared to several approximate inference algorithms in mixture models, logistic regression, sparse Gaussian processes and latent Dirichlet allocation on large scale datasets.

By making personalized suggestions, a recommender system is playing a crucial role in improving the engagement of users in modern web-services. However, most recommendation algorithms do not explicitly take into account the temporal behavior and the recurrent activities of users. Two central but less explored questions are how to recommend the most desirable item \emph{at the right moment}, and how to predict \emph{the next returning time} of a user to a service. To address these questions, we propose a novel framework which connects self-exciting point processes and low-rank models to capture the recurrent temporal patterns in a large collection of user-item consumption pairs. We show that the parameters of the model can be estimated via a convex optimization, and furthermore, we develop an efficient algorithm that maintains $O(1 / \epsilon)$ convergence rate, scales up to problems with millions of user-item pairs and thousands of millions of temporal events. Compared to other state-of-the-arts in both synthetic and real datasets, our model achieves superb predictive performance in the two time-sensitive recommendation questions. Finally, we point out that our formulation can incorporate other extra context information of users, such as profile, textual and spatial features.

We propose a new first-order optimization algorithm to solve high-dimensional non-smooth composite minimization problems. Typical examples of such problems have an objective that decomposes into a non-smooth empirical risk part and a non-smooth regularization penalty. The proposed algorithm, called Semi-Proximal Mirror-Prox, leverages the saddle point representation of one part of the objective while handling the other part of the objective via linear minimization over the domain. The algorithm stands in contrast with more classical proximal gradient algorithms with smoothing, which require the computation of proximal operators at each iteration and can therefore be impractical for high-dimensional problems. We establish the theoretical convergence rate of Semi-Proximal Mirror-Prox, which exhibits the optimal complexity bounds for the number of calls to linear minimization oracle. We present promising experimental results showing the interest of the approach in comparison to competing methods.

The Alternating Direction Method of Multipliers (ADMM) has received lots of attention recently due to the tremendous demand from large-scale and data-distributed machine learning applications. In this paper, we present a stochastic setting for optimization problems with non-smooth composite objective functions. To solve this problem, we propose a stochastic ADMM algorithm. Our algorithm applies to a more general class of convex and nonsmooth objective functions, beyond the smooth and separable least squares loss used in lasso. We also demonstrate the rates of convergence for our algorithm under various structural assumptions of the stochastic function: O(1/\sqrtt) for convex functions and O(\log t/t) for strongly convex functions. Compared to previous literature, we establish the convergence rate of ADMM for convex problems in terms of both the objective value and the feasibility violation. A novel application named Graph-Guided SVM is proposed to demonstrate the usefulness of our algorithm.