arXiv subjects

Vaneet Aggarwal

Publications and source records attributed to Vaneet Aggarwal.

At least 19 recordsLinked to original sources

arXiv2026

Tilted Convexity and an Exact Offline-Online Separation

Convexity has three equivalent descriptions: support by affine functions, monotonicity of the subdifferential, and increasing secant slopes. We weaken all three by one distortion with parameter $γ\in(0,1]$, which multiplies positive changes by $γ$ and divides negative ones by $γ$. The resulting classes are the convex functions at $γ=1$ and, for $γ<1$, are distinct and nested, with the parameter dropping by at most a square along the chain. For proper lower semicontinuous functions in these classes, the effective domain is convex, local minimizers are global, and the function is locally Lipschitz on the relative interior of its domain. The classes separate offline from online optimization over compact convex sets with efficient projection, with access to function values and bounded support vectors. A smooth member of any class can be minimized to additive accuracy $\varepsilon$ with $\mathrm{poly}(1/\varepsilon)$ first-order queries and projections. Online, with nonpositive losses and $α$-regret measured against $α$ times the best fixed point, a polynomial-time algorithm achieves $γ$-regret $O(\sqrt T)$ on the largest class. For every fixed, effectively specified $γ\in(0,1)$ and every $α\in(γ,1]$, no randomized polynomial-time algorithm has sublinear expected $α$-regret unless $\mathbf{NP}=\mathbf{RP}$, already for smooth concave polynomial losses in the smallest class, each minimized by one linear optimization. Thus, unless $\mathbf{NP}=\mathbf{RP}$, $γ$ is the exact online coefficient of each class, while the offline coefficient is one.

cs.LG↗

arXiv2026

Improved Sample Complexity Analysis For First Order Bilevel Reinforcement Learning

Bilevel reinforcement learning optimizes an upper-level objective through the policy response to a lower-level reinforcement learning problem. Such problems arise when learning a reward model while adapting a policy to that reward. However, a gap remains between sample-complexity guarantees and practical actor--critic training. Tabular analyses restrict applicability to finite spaces, while continuous-state analyses often impose a uniform Polyak--Łojasiewicz (PL) condition. We show that this condition need not hold even for smooth positive policies with exactly representable values. We develop a single-loop, fully first-order method for continuous states and finite actions with specified multilayer neural actors and a persistently trained nonlinear critic built from fixed features. Under explicit initialization, regularity, feature-excitation, and value-realizability conditions, fixed reference-KL regularization yields local curvature, and we prove containment of the coupled iterates. Shared trajectories control sampling noise, while paired critic analysis controls normalized critic-error differences as the response perturbation shrinks. A joint descent and tracking argument establishes an improved $\widetilde{\mathcal O}(\varepsilon^{-2})$ sample complexity for $O(\varepsilon)$ expected squared gradient norm of the upper objective induced by the selected regularized policy response, including critic training.

cs.LG↗

arXiv2026

Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching

Flow matching has emerged as a promising alternative to diffusion-based generative modeling, offering a continuous-time ordinary differential equation formulation with fast sampling and a simple regression-based training objective. Despite its empirical success, some theoretical analyses idealize the learning procedure by assuming that the velocity field is obtained by exact empirical risk minimization. This assumption does not reflect the practical finite-dataset setting, where the empirical flow-matching loss is only approximately minimized by a finite number of stochastic-gradient updates. We develop an empirical-risk-minimizer-free analysis for flow matching that explicitly accounts for both finite-sample statistical error and finite-time empirical optimization error. Our analysis considers a finite dataset of $N$ flow-matching samples and a learned parameter $θ_T$ obtained after $T$ stochastic gradient updates on the empirical objective. Under regularity assumptions on the interpolation path, sub-exponential source and target distributions, approximation by a bounded-parameter smooth neural velocity class, empirical-loss optimization, and uniform flow stability, we prove a guarantee that holds with high probability over the finite training dataset and in expectation over the stochastic-gradient randomness. The resulting bound separates approximation error, statistical error from the finite dataset, tail error from unbounded source and target distributions, and empirical optimization error from approximate minimization of the finite-sample objective.

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arXiv2026

Finite-Time Online Sample Complexity for Conditional Diffusion Models

We study the finite-time training and sampling error of conditional diffusion models trained by online stochastic-gradient denoising score matching. Rather than assuming access to an exact empirical or population risk minimizer, we analyze the score network produced after a finite number of stochastic-gradient updates under an explicit localized Polyak--Łojasiewicz (PL) condition, stochastic-gradient regularity, and a trajectory-containment assumption. Our analysis separates the network approximation complexity from the online training-sample budget and propagates both errors through the reverse diffusion process. Under anisotropic smoothness, a joint intrinsic-dimensional structure for the data and conditioning variables, sub-exponential tails, and explicit approximation-class compatibility, we obtain a conditional score approximation error of order \[ \widetilde{\mathcal O}\!\left(M^{-η_{\mathrm{app}}}\right), \qquad η_{\mathrm{app}} = \frac{2α_x} {2α_x+d_x'+(α_x/α_y)d_y'}, \] where $M$ denotes the approximation complexity, $d_x'$ and $d_y'$ are the intrinsic response and conditioning dimensions, and $α_x,α_y$ are their corresponding smoothness exponents. For $N=RB$ online denoising samples used by SGD, the finite-time optimization error is controlled explicitly in terms of the PL, smoothness, gradient-noise, and initialization constants. Combining these ingredients with reverse-process stability yields a total-variation bound that decomposes into terminal-initialization, time-discretization, approximation/localization, and finite-time optimization errors.

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arXiv2026

Sample-Efficient Reinforcement Learning with General Utilities via Neural Dual Critics

Reinforcement learning with general utilities optimizes nonlinear functionals of policy-induced occupancy measures. We develop a natural actor-critic method for concave RLGU objectives using an exact variational dual representation, whose existence follows from concavity and differentiability, with explicit constructions for a broad class of separable occupancy utilities. Rather than estimating the occupancy measure and then forming the policy-gradient pseudo-reward, our method learns this pseudo-reward directly as a function of the state-action pair using a neural critic. The critic is trained by projected stochastic updates, while multilevel Monte Carlo combines outputs across training horizons to control its bias at logarithmic expected cost. Our analysis separates conditional actor bias from estimator second moments and accounts explicitly for dual optimization, critic approximation, and neural linearization errors. Under policy-transfer, dual-stability, and local critic-regularity conditions, the method achieves $\widetilde{O}(T^{-1/2})$ global convergence above policy, critic-class, and finite-width approximation errors using $\widetilde{O}(T)$ expected environment transitions and reference samples. For a fixed critic representation, this yields $\widetilde{O}(ε^{-2})$ sampling complexity above the approximation floor, covering infinite state-action spaces and occupancy utilities beyond fixed finite vectors of returns.

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arXiv2026

Geometry-Dependent Approximation for Non-Monotone $k$-Submodular Maximization

We study the maximization of a nonnegative $k$-submodular function, not assumed monotone, in which each of $n$ elements receives one of $k\ge2$ labels or stays unselected and the selected elements must satisfy a constraint. For a compact convex down-closed region $P\subseteq[0,1]^n$ of fractional selection vectors, the diagonal level $ζ(P)=\max\{t\in[0,1]:t(1,\ldots,1)\in P\}$ is the largest selection probability that can be given to all elements at once; it equals $r/n$ for a uniform matroid of rank $r$, and $ζ(P)=1$ means that the constraint is vacuous. We give approximation coefficients that increase continuously with $ζ$. They come from one inequality for the multilinear extension of the objective: the algorithm computes an action without querying the objective, and objective feedback gives a comparator-independent update direction. For $k\ge3$, feasibility of the action reduces to three polynomial inequalities in one variable; for $k=2$ the profile is closed form. At $ζ=0$ the coefficients are $0.4456\ldots$ for $k=2$ and $0.4541\ldots$ for every $k\ge3$. For every finite $k\ge3$ they reach $1/2$ above an explicit level, which is $0.4026\ldots$ for $k=3,4,5$, and at $ζ=1$ they reach $1/2$ for $k=2$, $(\sqrt{17}-3)/2$ for $k=3,4$, and $k/(2k-1)$ for $k\ge5$. Online linear optimization gives $O(\sqrt T)$ approximate regret, with gradient feedback or with value queries made after each action. The coefficients carry over to matroids, with expected and high-probability guarantees, and to any fixed number of knapsacks: the zero-level coefficients hold on every instance, and the larger ones when items are small or when prefix enumeration keeps the diagonal level on one branch. For one budget the enumeration exponent is $O(1/\varepsilon)$.

cs.DS↗

arXiv2026

0.4633-Approximation for Online and Offline on-Monotone DR-Submodular Maximization

We obtain the approximation coefficient $α_0=54440750000/117486474169>0.463378$ for maximizing nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the unit cube, with queries allowed throughout the cube. This improves the previous best coefficient, $4/9$. A coordinatewise map, exponential on small coordinates and linear-fractional on large ones, turns the learner's state, a feasible point, into a feasible action whose value is at least $α_0$ times that of any feasible point, up to a linear term. The proof is analytical and reduces to one-dimensional inequalities. The linear term has an unbiased estimator from one noisy-gradient query, so online maximization reduces to linear learning. Under a uniform bound on the gradients, the expected approximate regret is $O(\sqrt T)$ with projected ascent and $O(T^{3/4})$ with one linear-maximization call per round; with one noisy value per round, it is $O(T^{3/4})$ and $O(T^{7/8})$. Online-to-batch conversion gives offline guarantees. With exact values and gradients, contracting the objective toward a feasible interior point yields an expected $(α_0-\varepsilon)$-approximation for every nonnegative $C^1$ DR-submodular objective, with oracle counts polynomial in $n$ and $\varepsilon^{-1}$ and no gradient or smoothness parameter. Sampling and standard rounding give $(α_0-\varepsilon)$-approximation for submodular set functions under one matroid or a fixed number of knapsack constraints.

cs.DS↗

arXiv2026

Geometry-Dependent Online and Offline Approximation for Non-Monotone DR-Submodular Maximization

We study online and offline maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed sets $K\subseteq[0,1]^d$ that contain a supplied diagonal point $ζ\mathbf{1}$. Our online algorithm earns at least $\underlineα(ζ)=\frac{1}{2}-\frac{(1-2ζ)_+^2}{2(3-2ζ)^2}$ times the reward of the best fixed feasible point, minus $O(\sqrt{T})$, using one possibly stochastic gradient query and one projection per round; $\underlineα(0)=4/9$. One post-decision noisy value query per round gives $O(T^{3/4})$ regret, and two exact value queries give $O(\sqrt{T})$. Conversely, no offline algorithm with polynomially many exact value-and-gradient queries attains a coefficient above an explicit $β_*\approx 0.4704$ at $ζ=0$, improving the bound $2(e^{-1/2}-e^{-1})\approx 0.478$ of Oveis Gharan and Vondrák (2011); for $ζ>0$, two families of hard instances give an upper bound that we reduce exactly to finitely many convex problems below $ζ=1/8$ and give in closed form above it. The bound $β_*$ also holds for nonnegative submodular set functions under a cardinality constraint, improving the previous $0.478$. Online-to-offline conversion gives $\underlineα(ζ)-\varepsilon$ offline, and the upper bound also limits online learners with polynomial query budgets and uniform $O(T^κ)$ regret, $κ<1$. Both coefficients equal $1/2$ for $ζ\ge 1/2$ and fall short of $1/2$ by order $(1/2-ζ)^2$ below. For quadratic objectives we attain $1/2$, and for cubic objectives $\max\{8/17,\,2\sqrt{3-4r}/(1+\sqrt{3-4r})^2\}$ with $r=\min\{ζ,1/2\}$. For submodular set functions, rounding the multilinear extension gives $\underlineα(ζ)-\varepsilon$ under a matroid whose polytope contains $ζ\mathbf{1}$ and $4/9-\varepsilon$ under a fixed number of knapsack constraints.

cs.LG↗

arXiv2026

Minimax-Optimal Predictive Flow Matching under Distribution Drift

Flow matching learns a generative velocity field by regression, but under distribution drift a field must be deployed before the current data arrive. We study this predictive problem for a fixed, possibly nonconvex, neural class, with dynamic regret against the best represented field of each round. We propose DyProd-FM, which refits flow matching on dyadic historical prefixes and selects among the fits with a sleeping Prod master. Without knowing any drift budget, it attains regret $\widetilde{O}(\min\{N_ω+1,1+\sqrt{TΩ}\})$ in the relative velocity risk variation $Ω$ and $\widetilde{O}(\min\{N_μ+1,1+T^{1/3}V_μ^{2/3}\})$ in the target-law variation $V_μ=\sum_t W_2(μ_{t+1},μ_t)$, up to approximation and fitting costs. Matching lower bounds for fixed classes show that both rates are optimal in $T$ and in the budgets up to logarithmic factors. Prior optimal dynamic-regret guarantees rely on a convex decision set with strongly convex or exp-concave losses, whereas our class may be nonconvex and no curvature in its parameters is assumed. The target-law rate rests on a second-order link between the two variations, which the base noise of flow matching provides.

cs.LG↗

arXiv2026

Hierarchical Multilevel Monte Carlo for Order-Optimal Neural Actor-Critic in Average-Reward CMDPs

Constrained Markov Decision Processes (CMDPs) provide a natural framework for reinforcement learning in safety-critical applications, where agents maximize long-term reward while satisfying long-term constraints. Although primal-dual actor-critic methods with linear critics are well understood, extending order-optimal convergence guarantees to neural critics in average-reward CMDPs has remained open. The main challenge is a fundamental bias-cost trade-off in neural critic estimation: under Neural Tangent Kernel (NTK) analysis, reducing critic bias substantially increases critic optimization cost, preventing order-optimal convergence in the primal-dual framework. We resolve this bottleneck by introducing a hierarchical Multilevel Monte Carlo (MLMC) neural critic that performs debiasing simultaneously across trajectory sampling and critic optimization. The resulting estimator attains the bias of a long critic optimization run with only logarithmic expected sample cost. Building on this estimator, we develop a primal-dual Natural Actor-Critic algorithm that achieves both an optimality gap and a constraint violation of order $\tilde{O}(T^{-1/2})$. This establishes the first order-optimal convergence guarantees for infinite-horizon average-reward CMDPs with general policy parameterization and neural critics, while eliminating the need to know the underlying mixing time. Our results are novel even in the unconstrained setting.

cs.LG↗

arXiv2026

Order-Optimal Sample Complexity of Rectified Flows

Recently, flow-based generative models have shown superior efficiency compared to diffusion models. In this paper, we study rectified flow models, which constrain transport trajectories to be linear from the base distribution to the data distribution. This structural restriction greatly accelerates sampling, often enabling high-quality generation with a single Euler step. Under standard assumptions on the neural network classes used to parameterize the velocity field and data distribution, we prove that rectified flows achieve sample complexity $\tilde{O}(\varepsilon^{-2})$. This improves on the best known $O(\varepsilon^{-4})$ bounds for flow matching model and matches the optimal rate for mean estimation. Our analysis exploits the particular structure of rectified flows: because the model is trained with a squared loss along linear paths, the associated hypothesis class admits a sharply controlled localized Rademacher complexity. This yields the improved, order-optimal sample complexity and provides a theoretical explanation for the strong empirical performance of rectified flow models.

cs.LG↗

arXiv2026

Decentralized Projection-free Online Upper-Linearizable Optimization with Applications to DR-Submodular Optimization

We introduce a novel framework for decentralized projection-free optimization, extending projection-free methods to a broader class of upper-linearizable functions. Our approach leverages decentralized optimization techniques with the flexibility of upper-linearizable function frameworks, effectively generalizing traditional DR-submodular function optimization. We obtain the regret of $O(T^{1-θ/2})$ with communication complexity of $O(T^θ)$ and number of linear optimization oracle calls of $O(T^{2θ})$ for decentralized upper-linearizable function optimization, for any $0\le θ\le 1$. This approach allows for the first results for monotone up-concave optimization with general convex constraints and non-monotone up-concave optimization with general convex constraints. Further, the above results for first order feedback are extended to zeroth order, semi-bandit, and bandit feedback.

math.OC↗

arXiv2026

Dec-BFTRL: Squre-Root Regret for Decentralized Online Upper-Linearizable Optimization under Separation Access with Application to Continuous Submodular Maximization

We study decentralized online optimization of upper-linearizable payoffs over an action set under efficient separation access, with applications to online continuous diminishing-return (DR) submodular maximization. We propose Decentralized Barrier Follow-the-Regularized-Leader (Dec-BFTRL), and evaluate each agent's played action against the average of all local objectives. Each agent maps an internal iterate to a feasible action through an approximate gauge projection, communicates only a cumulative surrogate-gradient dual state, and invokes the local HybridNewton procedure to approximately minimize its post-communication BFTRL potential. For every agent, we achieve expected network-aggregate regret of $\widetilde O(\sqrt{T})$. Over $T$ rounds, each agent uses $T$ neighbor-mixing steps and $\widetilde O(T)$ separation-oracle calls. We give wrapper instantiations covering four up-concave or DR-submodular maximization problems.

math.OC↗

arXiv2026

Parameter-Free Interval-Dynamic Regret under Heavy-Tailed Noise

We study online convex optimization with one unbiased stochastic subgradient per round and an unknown finite conditional $p$th noise moment, $1<p\le2$. For every fixed interval $I$ of length $n$ and comparator path with $Λ_I=1+P_I/D$, one learner achieves \[ E[Regret_I(u)]\le\min(GDn, C[GD\sqrt{n(Λ_I+\log^2(2T))} +σDn^{1/p}(Λ_I+\log^2(2T))^{(p-1)/p}]). \] The learner uses none of $G,σ,p,I,P_I$, and the constant is universal. Interval adaptation adds to comparator complexity, preserving the distinct mean-gradient and noise exponents. The analysis controls calibration in expectation and limits the cost of observation-scale changes. Its general theorem compares to distributions over predictably available experts with relative-entropy dependence on a nonuniform prior. A common prior favors long windows and long restart lengths. With the statistics supplied, the interval cost becomes $1+\log(T/n)$, including the optimal full-horizon static rate. A change-of-measure lower bound identifies the noise power of this logarithm for learners retaining a full-horizon optimal guarantee, under explicit conditions. Static comparisons and deterministic partitions follow from the same decisions.

cs.LG↗

arXiv2026

SQUARE: Structured Quantum Representation Adapters as Compact Quadratic Feature Maps for Frozen Language Models

Frozen language models (LMs) are increasingly used as fixed feature extractors for downstream reranking, scoring, and preference modeling, raising a practical question: how should a compact module represent interactions among features in a fixed low-dimensional bottleneck? Common linear and low-rank adapters remain linear at the adaptation module itself, whereas explicit second-order alternatives introduce pairwise interactions through direct parameterization or predefined factorizations. We propose SQUARE, a Structured QUAntum REpresentation adapter that amplitude-encodes the bottleneck vector, applies a parameterized quantum circuit, and measures the resulting state. We show that each basis-probability feature is exactly a normalized quadratic form in the bottleneck coordinates, while the additional Pauli-$Z$ readouts are signed linear combinations of these probabilities. The measured map can therefore parameterize interactions over $O(d^2)$ coordinate pairs through a small set of shared circuit parameters, where $d$ is the bottleneck dimension. It provides a structured parameterization within, rather than beyond, the classical normalized-quadratic feature class. In a disjoint same-pipeline evaluation over eight GLUE-derived controlled interaction tasks and five shared seeds, SQUARE achieves an average test accuracy of $0.7565$, compared with $0.7355$ for an affine normalized-quadratic predictor, $0.7271$ for the evaluated parameter-matched Givens mixing model, $0.6817$ for an MLP, and $0.6155$ for a frozen-circuit control. Under reduced supervision, it also shows consistent gains over the strongest evaluated classical comparator, with the same qualitative pattern across multiple frozen LM backbones. All circuit experiments use simulation, while the learned feature map can be evaluated exactly in batched PyTorch without quantum hardware.

quant-ph↗

arXiv2026

Online Convex Optimization with Dueling Feedback

Noisy binary comparison between two candidates is a common interface between human and learning systems, especially in modern large language model (LLM) post-training alignment. We study online convex optimization with dueling (pairwise comparison) feedback, where the learner observes only a binary preference between two queried points. We consider adversarial sequences of convex losses and measure regret with the loss at both queried points, under a comparison link with a known nonzero slope at the origin. We propose a simple reduction that converts dueling feedback into approximate gradients, enabling the use of standard first-order methods. We show that regret guarantees transfer under this reduction, yielding $\mathcal O(T^{3/4})$ static and adaptive regret, and $\mathcal O(T^{3/4}\sqrt{1+P_T/D})$ dynamic regret with unknown comparator path length $P_T$. For strongly convex losses, the static and adaptive bounds improve to $\widetilde{\mathcal O}(T^{2/3})$. For smooth losses, we presents unified dueling ellipsoidal FTRL, and proves $\widetilde{\mathcal O}(T^{2/3})$ static regret, which improves to $\widetilde{\mathcal O}(\sqrt T)$ under additional strong convexity.

cs.LG↗

arXiv2026

Fourier-Geometric Circuit Design for Gate and Entanglement Placement in Quantum Neural Networks

The output of a parameterized quantum circuit (PQC) can be expressed as a finite Fourier series whose accessible frequencies are fixed by the data-encoding gates. While the encoder determines which frequencies can appear, the corresponding Fourier coefficients depend on how the trainable and entangling gates are arranged. Although existing studies provide metrics for characterizing how gate structure affects Fourier coefficients, they do not translate these analyses into an explicit design criterion specifying how gates should be arranged to make a target coefficient reachable. In this paper, we provide such a criterion. Using the adjoint action of the encoding generator, we decompose operator space into two-dimensional invariant planes indexed by frequency and show that each Fourier coefficient is exactly a sum of bilinear projections of the effective state and observable onto the planes at that frequency. Because, for Pauli encodings, high-frequency planes are spanned by mixed multi-qubit Pauli strings, a target coefficient can contribute to the output only when local rotations and entangling layers are arranged so that both the effective state and observable acquire support on one of its planes. For Pauli readouts and commuting two-qubit entanglers, this yields a circuit design rule that specifies, for a given interaction graph, a placement of local rotations and entangling layers that makes a target Fourier coefficient reachable. Using a single encoding layer, we validate the proposed design through a placement ablation, regression tasks from PDEBench and physics-informed Maxwell field modeling, and further demonstrate its robustness to moderate simulated gate noise and damping.

quant-ph↗

arXiv2026

CARVE: Breaking Data Barriers in Chip Placement by Harnessing Reusable Expertise

Pretrained macro-placement policies can reduce repeated optimization across circuits, but deployment often exposes them to unfamiliar designs when the original training data are unavailable. Repeatedly fine-tuning a single serving model can overwrite earlier improvements, while simply saving checkpoints does not determine where they can be reliably reused. We introduce Continual Adaptation through the Reuse of Validated Expertise (CARVE), a framework that represents accumulated expertise as a frozen base policy, immutable specialists, and task-specific credentials obtained through local validation. For a new task, CARVE first checks existing specialists and trains a new specialist from the frozen base only when none qualifies. Under fixed task distributions and validation rules that control cumulative error, we establish expected-performance guarantees for repeated reuse. For bounded losses, we also derive matching worst-case bounds on the local samples needed for reliable reuse. In macro placement, a reuse-first follow-up reduces recorded training time by 58.5% (9.66 to 4.01 hours), while mean HPWL gain changes only from 8.41% to 7.86%. In a simulated receiving deployment, imported specialists are reused on six of seven new IBM circuits with no receiver-side training, achieving a 5.76% mean HPWL gain. Navigation studies provide complementary evidence on repair retention and repeated adaptation.

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