Mirror Descent

Regularized self-play -- the family behind DeepNash's Stratego play -- drives a two-player zero-sum policy to a Nash equilibrium by best-responding to a slowly moving, entropy-regularized reference policy $ρ$. When the game has a polytope of value-equivalent equilibria, the regularizer silently breaks the tie: with a uniform reference it selects the maximum-entropy member, the I-projection of $ρ$ onto the Nash set. Can the reference be used to choose the equilibrium on purpose? On five exactly solvable games plus a 2-D polytope, with exact best responses and equivalence tests over independent seeds, anchoring the reference at a target member and refining steers self-play to that member with mean coordinate error 0.007 at median exploitability $5\times10^{-5}$, TOST-equivalent to the request within $\pm0.05$; the anchoring persists through refinement and follows the reference, not the initialization. Selection follows the reach-weighted I-projection (slope 0.969 [0.950, 0.987]). We report with equal emphasis where the story breaks: fixed off-manifold references cost 0.08-0.25 exploitability; stiff or flat families require a smaller mirror step, set by a pre-registered rule; boundary targets undershoot; curvature predicts where boundary saturation bites (rank correlation 0.90, p=0.037) while interior precision is curvature-independent. Table and MLP steering maps are equivalent within $\pm0.03$ at every target (30 seeds); matched control arms show attention's robust signature is excess seed variance, any systematic shift bounded at 0.018 and not significant. Against a best response the selection-robustness trade-off is degenerate: steering matters only against fixed, non-equilibrium opponents. The recipe -- anchor the reference at the desired member and refine -- reinterprets the KL anchor of RLHF-style RL as a selection knob, not only a stability leash.

Reinforcement learning with verifiable rewards (RLVR) improves the reasoning capabilities of large language models (LLMs). We introduce Bellman Policy Optimization (BPO), a critic-free method derived from Policy Mirror Descent (PMD). For autoregressive generation with terminal rewards, BPO uses the Bellman equations to reformulate PMD as a trajectory-level objective. The reformulation avoids estimating state values at intermediate states. We prove that it has the same unique optimal solution as the original PMD objective. We derive the practical BPO loss by approximating this objective. Its mismatch-correction weight is a smoothed ratio of complementary token probabilities. Experiments on mathematical reasoning benchmarks demonstrate the effectiveness of BPO.

Online gradient descent is usually studied through external regret, where the learner competes with fixed alternatives. Recent work shows that first-order methods control richer action-dependent deviations. We ask for a geometric characterization of the deviations with respect to which online gradient descent, mirror descent, and follow-the-regularized-leader (FTRL) achieve no regret. We identify exactness as the common principle. Exactness means that the relevant displacement field is generated by a scalar potential, or equivalently that the associated one-form is exact in the geometry used by the algorithm. This geometry depends on the algorithm. For gradient descent it is Euclidean geometry, for mirror descent it is the geometry induced by the regularizer, and for FTRL it is the cumulative dual state. Under mild regularity conditions, exactness yields sublinear regret, while nonzero circulation provides the complementary obstruction and leads to linear regret. This gives a unified geometric framework for understanding the deviation classes controlled by these algorithms and reveals that different first-order methods can control genuinely different classes of deviations. These deviation classes have direct consequences for learning, particularly in games. We study the equilibrium notions induced by exact-form deviations and introduce conservative correlated equilibrium, reflecting both the conservative geometry of the underlying displacement fields and the restricted family of deviations available to the players. We characterize its relation to correlated equilibrium, determine when the resulting equilibrium notions coincide and when they separate, and show how these relationships depend on the geometry and the learning algorithm. Overall, this work gives a unified geometric account of what first-order online learning algorithms are no-regret with respect to, beyond fixed comparators.

Preference-based fine-tuning methods such as RLHF and DPO require substantial compute and large preference datasets. They also need direct access to the model parameters which are not provided by many state-of-the art models. Inference-time alignment offers a cost-effective alternative without updating model parameters. However, existing inference-time methods rely on a scalar reward model derived under a Bradley-Terry assumption, which cannot represent general preferences. Following recent work on fine-tuning with generalized preferences, in this work, we initiate the study of inference-time alignment under general preferences. We formulate the problem as obtaining a Nash equilibrium of a two-player zero-sum game between policies. We propose two algorithms: Best-of-Nash (BoN) and Nash Mirror Descent (NMD). We prove that both algorithms achieve a duality gap that matches the problem lower bound. Empirically, we implement the two methods on three datasets, which shows that our methods substantially outperform the base policy, converging to the performance of the fine-tuned models. Moreover, our results show that NMD remains robust across the regularization parameter.

In this work, we study radically uncoupled learning in discounted general-sum Markov games. Assuming ``$\mathsf{ETH}$ for $\mathsf{PPAD}$", we show that, for every fixed discount factor, there is no polynomial-time algorithm for computing inverse-polynomially accurate coarse correlated equilibria in discounted general-sum Markov games when players learn independently in decentralized settings. Complementing this hardness result, we provide what appears to be the first \emph{radically uncoupled} algorithm with sub-exponential convergence guarantees to coarse correlated equilibria in discounted general-sum Markov games without imposing any structural restrictions on the game. Our algorithm is a \emph{layered} variant of optimistic mirror descent with an increasing step-size schedule tailored to the multi-agent setting. Finally, we develop both full-feedback and partial feedback versions of the aforementioned algorithm and establish sub-exponential convergence guarantees for each case.

A score-driven filter multiplies its scaled log-likelihood score by a scale parameter. We call this coefficient the gain and learn it online. Given the current state and realised scaled score, each admissible gain selects a reachable next state and predictive density. A scalar gain moves along a line; diagonal gains control coordinatewise transmission and may change direction. We evaluate gain selection using a one-step predictive Kullback-Leibler objective. In the scalar unscaled case, the negative consecutive-score product is a stochastic gradient; the positive product used in accelerated recursions is a descent direction. Positive scalar score scaling changes only the effective learning rate. Strictly increasing, continuously differentiable gain links with positive derivative induce mirror-descent geometry, while persistence adds a Bregman pull towards a reference gain. Under convexity, compactness, integrability, and schedule conditions, projected and discounted mirror updates satisfy dynamic-regret bounds relative to time-varying, current-information comparators. Simulations isolate score scaling, link geometry, persistence, and coordinatewise gains. Across twelve equity indices, the bounded discounted-logistic gain records a lower out-of-sample mean negative log score than the constant gain in eleven markets, although market-level evidence is mixed. It also avoids the extreme transients of the numerically capped exponential-link benchmark.

Sequence convergence to a boundary Karush--Kuhn--Tucker (KKT) point has long remained unclear for nonconvex mirror descent with Legendre kernels. The difficulty arises from the blow-up of the gradient of the Legendre kernel at the boundary. Recent work~\cite{dingtoh2026nonkkt} shows that mirror descent can accumulate at non-KKT boundary points despite decreasing objective values, precluding a convergence guarantee to KKT points in general. Despite this negative result, mirror descent remains effective in many real applications. Motivated by this contrast, we address the boundary difficulty directly and establish KKT convergence of mirror descent for a broad class of structured nonconvex problems. We analyze mirror descent in reparameterized variables, where the Hessian metric is flattened and remains nondegenerate as the boundary is approached. Under extension and definability conditions jointly coupling the objective, the Legendre kernel, and the feasible region, the reparameterized sequence has finite length and converges, thereby recovering convergence to a KKT point of the original sequence. Our general framework applies to some concrete instances: Shannon entropy, Fermi--Dirac entropy, and power kernels on polyhedron.

We propose a deep learning framework for image restoration from images degraded by both multiplicative Gamma noise and blur. Unlike conventional deep equilibrium (DEQ) models that rely on implicit neural regularization, the proposed method learns an explicit and interpretable regularizer parameterized by geometric priors associated with surface area and mean curvature. To minimize the resulting variational model, we develop a mirror descent algorithm tailored to the commonly used Gamma-noise fidelity terms. Leveraging the Kurdyka-Lojasiewicz property for functions defined in $o$-minimal structures, we establish the global convergence of the generated iterates to a critical point. Experimental results on both grayscale and color image restoration demonstrate that the proposed method consistently outperforms representative model-based approaches while achieving performance comparable to state-of-the-art DEQ models based on implicit regularization, despite requiring substantially fewer trainable parameters.

For mirror descent generated by a Legendre kernel, perhaps one of the most basic question in optimization is this: must every accumulation point of a bounded mirror descent sequence be Karush--Kuhn--Tucker (KKT) stationary under proper stepsizes? We show that the answer is no. A longstanding obstacle to resolving this question is the boundary blow-up of the Legendre gradient: it keeps every mirror step in the interior, while at a boundary limit, the inverse entropy metric vanishes on active coordinates and can erase the dual-feasibility in the KKT system. We construct $C^\infty$ objectives and bounded sequences generated by the Shannon-entropic mirror descent on the nonnegative orthant $\R_+^n$, for every $n\geq 3$, and on the probability simplex $Δ_n$, for every $n\geq 4$, such that, in each case, the set of accumulation points is a smooth boundary circle containing a nonempty relatively open arc of non-KKT points. The steps satisfy $α_k\asymp k^{-β}$ with $β\in(1/2,1)$, the objective values are nonincreasing, and the objectives are entropy-relatively smooth. Hence the pathology stems from the degeneracy of the Bregman geometry at the boundary, rather than from failure of descent, or improper stepsizes. To the best of our knowledge, these provide the first counterexamples to KKT accumulation for bounded mirror descent sequences with nonincreasing objective values.

Layer-wise post-training quantization of large language models minimizes each layer's reconstruction error in isolation, allowing quantization errors to accumulate across depth and causing severe degradation in extreme low-bit regimes. We formulate quantization as a joint optimization over the discrete codes and scales of all layers, driven by two mechanisms: (i) cross-layer error compensation, which maintains the network-level accumulated error through the recursion e_{l+1} = A_l e_l + q_l, with a propagation operator A_l derived from the layer's input differential and a local quantization residual q_l evaluated at teacher features; and (ii) finite-sample feature-statistics matching, which aligns means, projected covariances, and centered empirical kernels between the full-precision and quantized networks under relative normalization. We prove that instantiating the propagation operator as a finite difference of the quantized network makes the recursion exact for arbitrary nonlinear layers, enabling an efficient forward-difference implementation. Binary weights are optimized via a mirror-descent parameterization u = tanh(beta*z) with annealed inverse temperature and group-wise log-scales. On Qwen2.5-1.5B with 1.125-bit group-binary weights, error compensation alone reaches a perplexity ratio of 9.56 +/- 0.15 over the FP16 teacher, outperforming logit distillation (14.09 +/- 0.53; 32 percent relative, more than 8 sigma over 3 seeds) and layer-local reconstruction by two orders of magnitude. The same objective transfers unchanged to 4-bit quantization (1.060 vs. 1.088 for layer-local). Out-of-domain evaluations (C4, CNN/DailyMail) show the advantage of error compensation grows off-domain, while statistics matching keeps feature-statistics discrepancy low off-domain (0.42-0.88 vs. 1.41-2.99 without it), revealing a complementary division of labor between the two mechanisms.

We study mirror flows generated by a convex quadratic loss and a general convex lower semicontinuous mirror potential. We show that, when initialized near the boundary of the domain of the mirror potential, their rescaled trajectories converge to a limiting mirror flow whose potential is the indicator function of the domain. In this limit, the primal variable minimizes the loss over a time-dependent hypothesis set: the subdifferential of the support function of the domain, evaluated at the dual variable. This characterization provides a general mechanism for incremental learning in mirror flows.

We propose Markov Chain from Human Feedback (MCHF), an elementary approach for aligning generative models from pairwise human preferences. Unlike Reinforcement Learning from Human Feedback (RLHF), which reduces comparisons to a scalar reward, and Nash Learning from Human Feedback (NLHF), which preserves pairwise utilities through a KL-regularized minimax optimization, MCHF uses pairwise preferences directly to define a transition mechanism over model outputs. Given a pairwise utility $U(x,y)$, which quantifies human preference for $y$ over $x$, and a reference probability distribution $μ_{\mathsf{ref}}$, we define a Markov kernel $\mathsf{P}(x, dy)\propto \exp(U(x,y))μ_{\mathsf{ref}}(dy)$, and take the Markov chain starting from $μ_{\mathsf{ref}}$ as an iterative alignment procedure. We show that MCHF converges geometrically fast to the stationary distribution, with a convergence rate governed by the seminorm $\|U\|_\oplus=\inf_{g,f\in L^\infty(μ_{\mathsf{ref}})}\|U-g\oplus f\|_\infty$, which quantifies the non-transitive structure of the pairwise utility. We further show that a mirror-descent algorithm for NLHF satisfies an analogous structure-adaptive convergence guarantee. Finally, through a perturbation analysis, we prove that when $\|U\|_\oplus$ is small, MCHF and NLHF agree up to first order around an RLHF solution, which yields a unified view of reward-based, game-theoretic, and Markovian approaches to alignment. In particular, for two natural algorithms that converge to the MCHF/NLHF equilibria, we show that the first step of MCHF and NLHF recovers the RLHF solution based on the column-sum reward $\hat{f}(y)=\int μ_{\mathsf{ref}}(dx) U(x, y)$, and starting from the second iteration, both algorithms incorporate the same linear functional of the residual $U-(-\hat f)\oplus \hat f$, which captures the non-transitive structure of the pairwise utility $U$.

Low-rank optimal transport (OT) mitigates the quadratic scaling of classical solvers, yet existing approaches rely heavily on first-order mirror-descent updates that require careful hyperparameter tuning and ignore the optimization landscape's curvature. To address these limitations, we propose a unified Riemannian geometric framework for low-rank OT, modeling balanced and unbalanced rank-$r$ positive factored couplings as novel smooth embedded submanifolds of the positive orthant. By equipping these manifolds with the Fisher-Rao product metric, we derive tractable formulations for Riemannian projectors, retractions, and Hessian-vector products. Our cost-agnostic framework seamlessly extends to linear OT, Gromov-Wasserstein (GW), fused GW, and their unbalanced counterparts. For balanced OT, our geometric ingredients are computed via efficient conjugate-gradient and iterative Bregman updates. For the unbalanced OT, our operations elegantly reduce to closed-form scalings, completely eliminating inner iterative loops. In both regimes, per-iteration complexity scales linearly with dataset size, and we provide a rank-sufficiency certificate for global optimality verification. Extensive experiments across a range of problem sizes demonstrate that our regularization-free first- and second-order solvers achieve faster convergence and superior performance over existing state-of-the-art low-rank OT solvers.

Mirror Descent (MD) extends Gradient Descent (GD) beyond Euclidean geometry and has recently reappeared as a lens for KL-regularized policy optimization in reinforcement learning and LLM post-training. This raises a basic robustness question, crucial to reproducibility and reliability: how sensitive are MD dynamics to their inputs? We focus on initialization, often itself a pretrained or previously aligned model. Quadratic-regularized MD, including GD and Mahalanobis geometries, is well-known to be stable for convex smooth objectives. We show a sharp contrast: once the regularizer is non-quadratic, MD can be exponentially more sensitive to initialization than GD, even with a well-conditioned regularizer in Euclidean norm. We give a three-dimensional construction with a convex, smooth objective and a strongly convex, smooth, well-conditioned regularizer where an initial $\varepsilon$ perturbation is quickly amplified to $\min\{\text{polylog}^{-1}(1/\varepsilon), \varepsilon e^{Ω(ηT)}\}$ after $T$ iterations of MD with step size $η$. For canonical KL-regularized MD on the simplex, we show that even linear objectives can amplify an initial $\varepsilon$ perturbation exponentially fast in high-dimensional or near-boundary regimes. Finally, we show that adding a Bregman regularization term toward an anchor point can stabilize the dynamics while largely preserving the optimization guarantees, and that the choice of anchor is crucial: anchoring at the initialization only partially mitigates the instability, whereas anchoring at a fixed point yields a more stable mechanism.

We study the robustness of stochastic mirror descent (SMD) under heavy-tailed noise, focusing on whether the method retains its convergence guarantees when run with infinite-variance stochastic gradient input. To address this question in a principled manner, we begin by introducing a continuous-time model of SMD as a stochastic differential equation (SDE) driven by a centered Lévy noise process with finite $p$-th order moments, $1 < p \leq 2$. This scheme -- which we call the Lévy mirror flow (LMF) -- arises naturally as the scaling limit of SMD in the presence of heavy-tailed noise. In particular, when $p < 2$ -- the heavy noise regime -- the trajectories of LMF generically exhibit jump discontinuities of arbitrary magnitude which, if frequent enough, lead to infinite variance. Nonetheless, despite this highly singular behavior, we show that LMF attains $ε$-optimality within $\mathcal{O}(ε^{-p/(p-1)})$ time in the convex case, and within $\mathcal{\tilde O}(ε^{-1/(p-1)})$ time for (relatively) strongly convex objectives. These guarantees provide a transparent characterization of the impact of frequent long jumps on the convergence of the process, and percolate to a series of matching discrete-time guarantees for several variants of SMD under heavy-tailed noise.

Maintaining predictive accuracy in non-stationary environments requires online model selection to adapt autonomously to unknown distribution shifts. However, existing tuning-free algorithms face a fundamental trade-off between robustness and agility. Specifically, to ensure dynamic regret bounds, they must restrict learning rates to small constants (e.g., $O(1)$). This restriction inevitably causes significant adaptation lag during abrupt changes. To resolve this, we propose a novel optimistic online mirror descent that utilizes safeguarded large learning rates up to $Θ(T)$, where $T$ is the number of rounds. Our key technical contribution is a post-hoc penalty mechanism that dynamically monitors unstable updates and excludes learning rates incurring excessive regret, eliminating the need for restrictive a priori constraints. We show that the cumulative penalty remains $O(\log T)$, allowing our algorithm to match near-optimal worst-case guarantees while achieving superior rates in benign cases. Empirical evaluations on three synthetic and eleven diverse real-world datasets demonstrate that our approach reduces the adaptation lag from hundreds of rounds to a few rounds, consistently outperforming tuning-free baselines.

We study nonstationary generalized linear bandits (GLBs), where the expected reward is modeled through a nonlinear link function with an unknown time-varying parameter. This framework encompasses a broad class of reward models, including linear, Bernoulli, and binomial rewards. Existing approaches are predominantly based on maximum-likelihood estimation (MLE), using sliding-window, restart, or discounting mechanisms to handle nonstationarity. Although these methods achieve statistically efficient regret guarantees, they generally require revisiting past observations at every round, which leads to computation and memory costs that grow with time; moreover, several of them rely on a non-convex projection step. In this paper, we propose DOMD-GLB, a new algorithm for nonstationary GLBs that utilizes discounted online mirror descent (DOMD) for parameter estimation, thereby incurring only $O(1)$ computation and memory costs per round. We prove dynamic regret bounds of order $\tilde{O} \big(c_μ^{-1/2} d^{3/4} P_T^{1/4} T^{3/4}\big)$ in drifting environments and $\tilde{O}\big(c_μ^{-1/3} d^{2/3} Γ_T^{1/3} T^{2/3}\big)$in piecewise-stationary environments, where $d$ denotes the feature dimension, $T$ the time horizon, $P_T$ the path length, $Γ_T$ the number of change points, and $c_μ$ a curvature parameter associated with the link function, while substantially improving computational efficiency over prior work. To the best of our knowledge, this is the first algorithm for nonstationary GLBs with per-round computation and memory costs independent of time.

We revisit the classical problem of estimating an unknown distribution from its samples by fitting a mixture model that minimizes cross-entropy loss. Framing the task as a stochastic convex optimization problem over the space of $M$-component mixture distributions, we propose a family of estimators derived from the stochastic mirror descent (SMD) algorithm. This optimization-based approach provides a principled and flexible framework that generalizes traditional estimators and proposes a variety of novel estimators through the choice of Bregman divergences. A key advantage of our method is that it scales efficiently with the number of candidate components $f_i$; that is, one can employ a large set of basis distributions in the mixture model without incurring significant computational overhead. This enables richer approximations and improved estimation accuracy. Moreover, in the case of categorical distribution (discrete outcomes) our estimators do not require a strict lower bound, in other words our framework does not require the precise knowledge of the support of the distribution. We demonstrate that, under mild conditions, the proposed $\varphi$-SMD estimators achieve near-optimal convergence rates in both Kullback-Leibler (KL) divergence and $\ell_2$-norm and offer practical benefits when computation is expensive. Our numerical analysis highlights improved performance guaranties over classical estimators, particularly in terms of sample efficiency and scalability.

Online off-policy reinforcement learning (RL) is shaped by two coupled choices: the policy class and the update rule. Gaussian policies are fast and have tractable entropy, but struggle with multimodal action distributions. Generative policies are more expressive, but often require iterative sampling or lack tractable entropy estimates. On the optimisation side, SAC-style soft policy improvement and mirror descent (MD) can be viewed as minimising different KL divergences: the former moves the policy towards a value-induced Boltzmann distribution, while the latter regularises each update against the previous policy. Combining entropy regularisation with an MD constraint is therefore attractive, as it supports exploration while stabilising policy improvement; however, the resulting target can be multimodal and is poorly matched by unimodal Gaussian policies. We propose Stochastic MeanFlow Policies (SMFP), a one-step generative policy class that maps Gaussian noise to actions through a MeanFlow transformation. This stochastic reparameterisation yields a tractable entropy surrogate and allows MeanFlow policies to be trained within off-policy mirror descent under a unified objective for exploratory yet stable improvement. Across seven MuJoCo benchmarks, SMFP improves over Gaussian and generative baselines while retaining single-step inference efficiency.

Gradient temporal-difference methods provide stable off-policy prediction with linear function approximation, but their practical performance is strongly affected by the geometry induced by the auxiliary-variable metric. Existing Mirror-Prox TD methods typically use the feature covariance metric, whereas hybrid TD methods suggest that behavior-policy transition information can provide a more informative update geometry. This paper proposes a behavior-induced Mirror-Prox temporal-difference method, called STHTD-MP, which replaces the covariance metric in the primal-dual saddle-point formulation with the symmetric part of the behavior-policy Bellman matrix. The method keeps a single learning rate for the primal and auxiliary variables and applies a Mirror-Prox prediction-correction step to the resulting hybrid saddle-point operator. We provide a formal convergence analysis for fixed-policy linear prediction under standard stochastic approximation assumptions: the behavior-induced metric is positive definite, the joint mean system is Hurwitz, boundedness follows from a Lyapunov argument, and the stochastic recursion converges by the ODE method. We further derive projected-oracle ergodic gap bounds and an exact mean-operator comparison with GTD2-MP based on the spectral radius of the deterministic Mirror-Prox error matrix. The analysis shows that STHTD-MP can have a smaller mean contraction factor than GTD2-MP when the behavior-induced metric improves the saddle-point geometry. Exact numerical mean-operator analysis on two-state, Random Walk, and Boyan Chain benchmarks supports this condition, while Baird's counterexample is identified as a singular boundary case where the strict assumptions fail.

We study the problem of learning Nash equilibria in offline two-player zero-sum Markov games. While existing approaches often rely on explicit pessimism to address distribution shift, we show that KL regularization alone suffices to stabilize learning and guarantee convergence. We first introduce Regularized Offline Sequential Equilibrium (ROSE), a theoretical framework that achieves a fast $\widetilde{\mathcal{O}}(1/n)$ convergence rate under \textit{unilateral concentrability}, improving over the standard $\widetilde{\mathcal{O}}(1/\sqrt{n})$ rates in unregularized settings. We then propose Sequential Offline Self-play Mirror Descent (SOS-MD), a practical model-free algorithm based on least-squares value estimation and iterative self-play updates. We prove that the last iterate of SOS-MD attains the same $\widetilde{\mathcal{O}}(1/n)$ statistical rate up to a vanishing optimization error of order $\widetilde{\mathcal{O}}(1/\sqrt{T})$ in the number of self-play iterations $T$.

Preference-conditioned multi-objective reinforcement learning aims to learn a single policy that captures trade-offs across preferences, but under nonlinear scalarization the uniqueness and continuity of the preference-to-solution correspondence remain unclear. We study this problem in tabular multi-objective Markov decision processes (MDPs) using smooth Tchebycheff scalarization as a monotone utility. Under mild interior conditions on the preference set, we prove that each preference induces a unique Pareto-optimal return vector and that this vector depends Lipschitz-continuously on the preference, providing a principled foundation for preference sweeping toward dense Pareto-front coverage. To compute these targets, we formulate the problem over occupancy measures and derive Concave Mirror Descent Policy Iteration (CMDPI), which achieves an $O(1/k)$ objective-suboptimality rate. We further show that each update is equivalent to solving a Kullback-Leibler-regularized MDP with the previous policy as reference, yielding a policy-iteration interpretation and finite-iterate policy continuity across preferences. We instantiate the update as a deep actor-critic algorithm preserving previous-policy regularization. On eight MO-Gymnasium tasks, it achieves the best average hypervolume rank among recent baselines and strong expected-utility performance. Continuous-control experiments indicate gains beyond the discrete-action setting.

Online reinforcement learning with verifiable rewards (RLVR) turns checkable outcomes into a scalable training signal, but it keeps rollout generation, verifier scoring, and reference-policy evaluations on the optimization path. Static weighted supervised fine-tuning (SFT) on precomputed rollouts seems to remove this bottleneck, yet a weighted likelihood is not specified by rewards alone: its sampler and weights induce the policy being fit. This paper identifies the reference-sampled weighted-SFT objective whose induced policy equals the fixed-reference KL-regularized RLVR optimizer. The optimizer is the standard Boltzmann target policy, obtained by exponentially tilting the reference policy by verifier reward. Matching a weighted-SFT induced policy to this target forces density-ratio weights; in the reference-sampled subclass, this reduces uniquely, up to prompt scaling, to the prompt-normalized Boltzmann weight $\exp(r(x,y)/β)/Z(x)$. BOLT, a Boltzmann-Targeted SFT procedure, is the empirical estimator of this projection. The finite one-shot analysis separates the exact stored-support price $β\log(1/π^*(S_N\mid x))$ from partition estimation, effective-sample-size variance, generalization, optimization, and approximation errors. This decomposition explains why extra SFT epochs cannot repair missing reference-policy coverage and exposes the temperature--coverage--variance frontier. When coverage needs adaptive sampling, refreshed Boltzmann projections become KL policy mirror descent; finite inner solves enter as additive drift from the exact mirror step. Single-run Qwen experiments provide projection evidence for the target-matched weight, one-shot saturation, refreshed-sampler gains, and optimization-time savings, within the stated single-run scope.

Offline multi-agent reinforcement learning in general-sum settings is challenged by the distribution shift between logged datasets and target equilibrium policies. While standard methods rely on manual pessimistic penalties, we demonstrate that KL regularization suffices to stabilize learning and achieve equilibrium recovery. We propose General-sum Anchored Nash Equilibrium (GANE), which recovers regularized Nash equilibria at an accelerated statistical rate of $\widetilde{O}(1/n)$. For computational tractability, we develop General-sum Anchored Mirror Descent (GAMD), an iterative algorithm converging to a Coarse Correlated Equilibrium at the standard rate of $\widetilde{O}(1/\sqrt{n}+1/T)$. These results establish KL regularization as a standalone mechanism for pessimism-free offline learning that achieves equivalent or accelerated rates in multi-player general-sum games.

We study the problem of learning in zero-sum matrix games with repeated play and bandit feedback. Specifically, we focus on developing uncoupled algorithms that guarantee, without communication between players, the convergence of the last-iterate to a Nash equilibrium. Although the non-bandit case has been studied extensively, this setting has only been explored recently, with a bound of $\mathcal{O}(T^{-1/8})$ on the exploitability gap. We show that, for uncoupled algorithms, guaranteeing convergence of the policy profiles to a Nash equilibrium is detrimental to the performance, with the best attainable rate being $Ω(T^{-1/4})$ in contrast to the usual $Ω(T^{-1/2})$ rate for convergence of the average iterates. We then propose two algorithms that achieve this optimal rate up to constant and logarithmic factors. The first algorithm leverages a straightforward trade-off between exploration and exploitation, while the second employs a regularization technique based on a two-step mirror descent approach.

We study the problem of learning minimax policies in zero-sum matrix games. Fiegel et al. (2025) recently showed that achieving last-iterate convergence in this setting is harder when the players are uncoupled, by proving a lower bound on the exploitability gap of Omega(t^{-1/4}). Some online mirror descent algorithms were proposed in the literature for this problem, but none have truly attained this rate yet. We show that the use of a log-barrier regularization, along with a dual-focused analysis, allows this O-tilde(t^{-1/4}) convergence with high-probability. We additionally extend our idea to the setting of extensive-form games, proving a bound with the same rate.

Value iteration-type methods have been extensively studied for computing a nearly optimal value function in reinforcement learning (RL). Under a generative sampling model, these methods can achieve sharper sample complexity than policy optimization approaches, particularly in their dependence on the discount factor. In practice, they are often employed for offline training. In this paper, we consider discounted Markov decision processes with state space S, action space A, discount factor $γ\in(0,1)$ and costs in $[0,1]$. We introduce a novel value optimization method, termed value mirror descent (VMD), which integrates mirror descent from convex optimization into the classical value iteration framework. In the deterministic setting with known transition kernels, we show that VMD converges linearly. For the stochastic setting with a generative model, we develop a stochastic variant, SVMD, which incorporates variance reduction commonly used in stochastic value iteration-type methods. For RL problems with general convex regularizers, SVMD attains a near-optimal sample complexity of $\tilde{O}(|S||A|(1-γ)^{-3}ε^{-2})$. Moreover, we establish that the Bregman divergence between the generated and optimal policies remains bounded throughout the iterations, even under the presence of model misspecification. This property is absent in existing stochastic value iteration-type methods but is important for enabling effective online (continual) learning following offline training. Under a strongly convex regularizer, SVMD achieves sample complexity of $\tilde{O}(|S||A|(1-γ)^{-5}ε^{-1})$, improving performance in the high-accuracy regime. Furthermore, we prove convergence of the generated policy to the optimal policy. Overall, the proposed method, its analysis, and the resulting guarantees, constitute new contributions to the RL and optimization literature.

This work uncovers variational principles behind symmetrizing the Bregman divergences induced by generic mirror maps over the cone of positive definite matrices. We show that computing the canonical means for this symmetrization can be posed as minimizing the desired symmetrized divergences over a set of mean functionals defined axiomatically to satisfy certain properties. For the forward symmetrization, we prove that the arithmetic mean over the primal space is canonical for any mirror map over the positive definite cone. For the reverse symmetrization, we show that the canonical mean is the arithmetic mean over the dual space, pulled back to the primal space. Applying this result to three common mirror maps used in practice, we show that the canonical means for reverse symmetrization, in those cases, turn out to be the arithmetic, log-Euclidean and harmonic means. Our results improve understanding of existing symmetrization practices in the literature, and can be seen as a navigational chart to help decide which mean to use when.

We consider the problem of heteroskedastic generalized linear bandits (GLBs) with adversarial corruptions, which subsumes heteroskedastic linear bandits and logistic/Poisson bandits, in the presence of adversarial corruptions. We propose HCW-GLB-OMD, which consists of two components: an online mirror descent (OMD)-based estimator and Hessian-based confidence weights to achieve corruption robustness. This is computationally efficient in that it only requires ${O}(1)$ space and time complexity per iteration. Under the self-concordance assumption on the link function, we show a regret bound of $\tilde{O}\left( d \sqrt{\sum_t g(τ_t) \dotμ_{t,\star}} + d^2 g_{\max} κ+ d (g_{\max} + κ) C \right)$, where $\dotμ_{t,\star}$ is the slope of $μ$ around the optimal arm at time $t$, $g(τ_t)$'s are potentially exogenously time-varying dispersions (e.g., $g(τ_t) = σ_t^2$ for heteroskedastic linear bandits, $g(τ_t) = 1$ for Bernoulli and Poisson), $g_{\max} = \max_{t \in [T]} g(τ_t)$ is the maximum dispersion, and $C \geq 0$ is the total corruption budget of the adversary. We complement this with a lower bound of $\tildeΩ(d \sqrt{\sum_t g(τ_t) \dotμ_{t,\star}} + d C)$, unifying previous problem-specific lower bounds. Thus, our algorithm achieves, up to a $κ$-factor in the corruption term, instance-wise minimax optimality simultaneously across various instances of heteroskedastic GLBs with adversarial corruptions.

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 $O(ε^{-2})$ sample complexity to solve to $ε$ 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.

In this paper, we study nonconvex constrained stochastic zeroth-order optimization problems with exact constraints and stochastic objective evaluations. To solve this class of problems, we propose a framework of mirror descent linearized augmented Lagrangian methods that employs two-point stochastic zeroth-order gradient estimators and exploits non-Euclidean mirror descent geometry. Under mild assumptions, we establish oracle complexity guarantees for finding an $ε$-KKT point parameterized by $p \geq 2$. Under Rademacher smoothing, our analysis reveals a trade-off between the variance of the zeroth-order gradient estimators and the smoothness of the mirror map. In the high-accuracy regime, the resulting effective oracle complexity is $\mathcal{O}(p d^{2/p}ε^{-3})$ for $p \in [2,2\ln d]$ and $\mathcal{O}(\ln d\,ε^{-3})$ for $p > 2\ln d$. These bounds reduce the dimension dependence in the leading term. When $p=2$, our method recovers the Euclidean setting with an oracle complexity of $\mathcal{O}(dε^{-3})$, improving the $ε$-dependence over existing methods. Furthermore, to eliminate initial near-feasibility requirements, we introduce a multi-stage scheme that finds an $ε$-KKT point within $\mathcal{O}(1+\log\log(e/ε))$ stages while maintaining the leading-order complexity. Numerical tests on QCQPs, black-box adversarial attacks, and fairness-constrained classification demonstrate the effectiveness of our proposed method.

In this book, I introduce the concepts of online learning through a modern view based on convex optimization. Here, online learning refers to the framework of regret minimization under worst-case assumptions. I attempted to unify all the literature as instantiations of Online Mirror Descent and Follow-the-Regularized-Leader (and their variants). I paid particular attention to the issue of tuning the parameters of the algorithms, through adaptive and parameter-free online learning algorithms. The bandit setting is also briefly discussed, touching on the problem of adversarial and stochastic multi-armed bandits. Building on fundamental algorithms and concepts, I also cover advanced topics, including black-box reductions, saddle-point optimization, sequential investment, and non-stationary forms of regret analysis. Finally, I conclude with a selection of applications of online learning to domains far from it, such as generalization theory and concentration inequalities. I attempted to maintain an informal, yet mathematically rigorous, tone throughout the book. Moreover, all the included proofs have been carefully chosen to be as simple and as short as possible. This also means that sometimes I have added one or two additional assumptions, just to simplify the proofs.

The performance of online mirror descent depends critically on the geometry induced by its mirror map, yet standard algorithms largely rely on two canonical choices: Euclidean and entropic geometry. We show that these two geometries can both be substantially suboptimal when loss gradients are sparse. We introduce a family of randomized block-norm mirror maps that interpolates between Euclidean and entropic geometries and adapts to intermediate sparsity structure. For several standard convex sets, including $\ell_p$ balls, ellipsoids, boxes, and Minkowski sums of norm balls, we prove polynomial-in-dimension improvements in regret bounds over the better of online projected gradient descent and exponentiated gradient. We further construct explicit online convex optimization instances for which these improvements are realized: on a simple polytope, an intermediate block geometry achieves a $\text{poly}(d)$ separation in regret from both Euclidean and entropic geometries in dimension $d$, while on the probability simplex we obtain a separation of order $\Omega(\sqrt{\log d}/\log\log d)$. Finally, we study geometry selection when sparsity is unknown. We show that naively alternating between mirror maps can incur linear regret, even though either mirror map alone has sublinear regret, and give a Hedge meta-algorithm that competes with the best mirror map in a finite portfolio. For random block geometries, this yields regret within an $O(\sqrt{\log\log d})$ factor of the best random uniform block norm chosen in hindsight.