arXiv subjects

Constantinos Daskalakis

Publications and source records attributed to Constantinos Daskalakis.

At least 37 records · Page 2Linked to original sources

arXiv2025

The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization

Motivated by applications in Optimization, Game Theory, and the training of Generative Adversarial Networks, the convergence properties of first order methods in min-max problems have received extensive study. It has been recognized that they may cycle, and there is no good understanding of their limit points when they do not. When they converge, do they converge to local min-max solutions? We characterize the limit points of two basic first order methods, namely Gradient Descent/Ascent (GDA) and Optimistic Gradient Descent Ascent (OGDA). We show that both dynamics avoid unstable critical points for almost all initializations. Moreover, for small step sizes and under mild assumptions, the set of \{OGDA\}-stable critical points is a superset of \{GDA\}-stable critical points, which is a superset of local min-max solutions (strict in some cases). The connecting thread is that the behavior of these dynamics can be studied from a dynamical systems perspective.

math.OC↗

arXiv2025

Ambient Diffusion Omni: Training Good Models with Bad Data

We show how to use low-quality, synthetic, and out-of-distribution images to improve the quality of a diffusion model. Typically, diffusion models are trained on curated datasets that emerge from highly filtered data pools from the Web and other sources. We show that there is immense value in the lower-quality images that are often discarded. We present Ambient Diffusion Omni, a simple, principled framework to train diffusion models that can extract signal from all available images during training. Our framework exploits two properties of natural images -- spectral power law decay and locality. We first validate our framework by successfully training diffusion models with images synthetically corrupted by Gaussian blur, JPEG compression, and motion blur. We then use our framework to achieve state-of-the-art ImageNet FID, and we show significant improvements in both image quality and diversity for text-to-image generative modeling. The core insight is that noise dampens the initial skew between the desired high-quality distribution and the mixed distribution we actually observe. We provide rigorous theoretical justification for our approach by analyzing the trade-off between learning from biased data versus limited unbiased data across diffusion times.

cs.GR↗

arXiv2025

Learning High-dimensional Gaussians from Censored Data

We provide efficient algorithms for the problem of distribution learning from high-dimensional Gaussian data where in each sample, some of the variable values are missing. We suppose that the variables are missing not at random (MNAR). The missingness model, denoted by $S(y)$, is the function that maps any point $y$ in $R^d$ to the subsets of its coordinates that are seen. In this work, we assume that it is known. We study the following two settings: (i) Self-censoring: An observation $x$ is generated by first sampling the true value $y$ from a $d$-dimensional Gaussian $N(μ*, Σ*)$ with unknown $μ*$ and $Σ*$. For each coordinate $i$, there exists a set $S_i$ subseteq $R^d$ such that $x_i = y_i$ if and only if $y_i$ in $S_i$. Otherwise, $x_i$ is missing and takes a generic value (e.g., "?"). We design an algorithm that learns $N(μ*, Σ*)$ up to total variation (TV) distance epsilon, using $poly(d, 1/ε)$ samples, assuming only that each pair of coordinates is observed with sufficiently high probability. (ii) Linear thresholding: An observation $x$ is generated by first sampling $y$ from a $d$-dimensional Gaussian $N(μ*, Σ)$ with unknown $μ*$ and known $Σ$, and then applying the missingness model $S$ where $S(y) = {i in [d] : v_i^T y <= b_i}$ for some $v_1, ..., v_d$ in $R^d$ and $b_1, ..., b_d$ in $R$. We design an efficient mean estimation algorithm, assuming that none of the possible missingness patterns is very rare conditioned on the values of the observed coordinates and that any small subset of coordinates is observed with sufficiently high probability.

cs.LG↗

arXiv2025

On Tractable $Φ$-Equilibria in Non-Concave Games

While Online Gradient Descent and other no-regret learning procedures are known to efficiently converge to a coarse correlated equilibrium in games where each agent's utility is concave in their own strategy, this is not the case when utilities are non-concave -- a common scenario in machine learning applications involving strategies parameterized by deep neural networks, or when agents' utilities are computed by neural networks, or both. Non-concave games introduce significant game-theoretic and optimization challenges: (i) Nash equilibria may not exist; (ii) local Nash equilibria, though they exist, are intractable; and (iii) mixed Nash, correlated, and coarse correlated equilibria generally have infinite support and are intractable. To sidestep these challenges, we revisit the classical solution concept of $Φ$-equilibria introduced by Greenwald and Jafari [2003], which is guaranteed to exist for an arbitrary set of strategy modifications $Φ$ even in non-concave games [Stolz and Lugosi, 2007]. However, the tractability of $Φ$-equilibria in such games remains elusive. In this paper, we initiate the study of tractable $Φ$-equilibria in non-concave games and examine several natural families of strategy modifications. We show that when $Φ$ is finite, there exists an efficient uncoupled learning algorithm that converges to the corresponding $Φ$-equilibria. Additionally, we explore cases where $Φ$ is infinite but consists of local modifications. We show that approximating local $Φ$-equilibria beyond the first-order stationary regime is computationally intractable. In contrast, within this regime, we show Online Gradient Descent efficiently converges to $Φ$-equilibria for several natural infinite families of modifications, including a new structural family of modifications inspired by the well-studied proximal operator.

cs.GT↗

arXiv2025

From External to Swap Regret 2.0: An Efficient Reduction and Oblivious Adversary for Large Action Spaces

We provide a novel reduction from swap-regret minimization to external-regret minimization, which improves upon the classical reductions of Blum-Mansour [BM07] and Stolz-Lugosi [SL05] in that it does not require finiteness of the space of actions. We show that, whenever there exists a no-external-regret algorithm for some hypothesis class, there must also exist a no-swap-regret algorithm for that same class. For the problem of learning with expert advice, our result implies that it is possible to guarantee that the swap regret is bounded by ε after $\log(N)^{O(1/ε)}$ rounds and with $O(N)$ per iteration complexity, where $N$ is the number of experts, while the classical reductions of Blum-Mansour and Stolz-Lugosi require $O(N/ε^2)$ rounds and at least $Ω(N^2)$ per iteration complexity. Our result comes with an associated lower bound, which -- in contrast to that in [BM07] -- holds for oblivious and $\ell_1$-constrained adversaries and learners that can employ distributions over experts, showing that the number of rounds must be $\tildeΩ(N/ε^2)$ or exponential in $1/ε$. Our reduction implies that, if no-regret learning is possible in some game, then this game must have approximate correlated equilibria, of arbitrarily good approximation. This strengthens the folklore implication of no-regret learning that approximate coarse correlated equilibria exist. Importantly, it provides a sufficient condition for the existence of correlated equilibrium which vastly extends the requirement that the action set is finite, thus answering a question left open by [DG22; Ass+23]. Moreover, it answers several outstanding questions about equilibrium computation and learning in games.

cs.LG↗

arXiv2025

Is Efficient PAC Learning Possible with an Oracle That Responds 'Yes' or 'No'?

The empirical risk minimization (ERM) principle has been highly impactful in machine learning, leading both to near-optimal theoretical guarantees for ERM-based learning algorithms as well as driving many of the recent empirical successes in deep learning. In this paper, we investigate the question of whether the ability to perform ERM, which computes a hypothesis minimizing empirical risk on a given dataset, is necessary for efficient learning: in particular, is there a weaker oracle than ERM which can nevertheless enable learnability? We answer this question affirmatively, showing that in the realizable setting of PAC learning for binary classification, a concept class can be learned using an oracle which only returns a single bit indicating whether a given dataset is realizable by some concept in the class. The sample complexity and oracle complexity of our algorithm depend polynomially on the VC dimension of the hypothesis class, thus showing that there is only a polynomial price to pay for use of our weaker oracle. Our results extend to the agnostic learning setting with a slight strengthening of the oracle, as well as to the partial concept, multiclass and real-valued learning settings. In the setting of partial concept classes, prior to our work no oracle-efficient algorithms were known, even with a standard ERM oracle. Thus, our results address a question of Alon et al. (2021) who asked whether there are algorithmic principles which enable efficient learnability in this setting.

cs.LG↗

arXiv2025

Fixed Point Computation: Beating Brute Force with Smoothed Analysis

We propose a new algorithm that finds an $\varepsilon$-approximate fixed point of a smooth function from the $n$-dimensional $\ell_2$ unit ball to itself. We use the general framework of finding approximate solutions to a variational inequality, a problem that subsumes fixed point computation and the computation of a Nash Equilibrium. The algorithm's runtime is bounded by $e^{O(n)}/\varepsilon$, under the smoothed-analysis framework. This is the first known algorithm in such a generality whose runtime is faster than $(1/\varepsilon)^{O(n)}$, which is a time that suffices for an exhaustive search. We complement this result with a lower bound of $e^{Ω(n)}$ on the query complexity for finding an $O(1)$-approximate fixed point on the unit ball, which holds even in the smoothed-analysis model, yet without the assumption that the function is smooth. Existing lower bounds are only known for the hypercube, and adapting them to the ball does not give non-trivial results even for finding $O(1/\sqrt{n})$-approximate fixed points.

cs.GT↗

arXiv2024

Efficient Learning and Computation of Linear Correlated Equilibrium in General Convex Games

We propose efficient no-regret learning dynamics and ellipsoid-based methods for computing linear correlated equilibria$\unicode{x2014}$a relaxation of correlated equilibria and a strengthening of coarse correlated equilibria$\unicode{x2014}$in general convex games. These are games where the number of pure strategies is potentially exponential in the natural representation of the game, such as extensive-form games. Our work identifies linear correlated equilibria as the tightest known notion of equilibrium that is computable in polynomial time and is efficiently learnable for general convex games. Our results are enabled by a generalization of the seminal framework of of Gordon et al. [2008] for $Φ$-regret minimization, providing extensions to this framework that can be used even when the set of deviations $Φ$ is intractable to separate/optimize over. Our polynomial-time algorithms are similarly enabled by extending the Ellipsoid-Against-Hope approach of Papadimitriou and Roughgarden [2008] and its generalization to games of non-polynomial type proposed by Farina and Pipis [2024a]. We provide an extension to these approaches when we do not have access to the separation oracles required by these works for the dual player.

cs.GT↗

arXiv2024

How much is a noisy image worth? Data Scaling Laws for Ambient Diffusion

The quality of generative models depends on the quality of the data they are trained on. Creating large-scale, high-quality datasets is often expensive and sometimes impossible, e.g. in certain scientific applications where there is no access to clean data due to physical or instrumentation constraints. Ambient Diffusion and related frameworks train diffusion models with solely corrupted data (which are usually cheaper to acquire) but ambient models significantly underperform models trained on clean data. We study this phenomenon at scale by training more than $80$ models on data with different corruption levels across three datasets ranging from $30,000$ to $\approx 1.3$M samples. We show that it is impossible, at these sample sizes, to match the performance of models trained on clean data when only training on noisy data. Yet, a combination of a small set of clean data (e.g.~$10\%$ of the total dataset) and a large set of highly noisy data suffices to reach the performance of models trained solely on similar-size datasets of clean data, and in particular to achieve near state-of-the-art performance. We provide theoretical evidence for our findings by developing novel sample complexity bounds for learning from Gaussian Mixtures with heterogeneous variances. Our theoretical model suggests that, for large enough datasets, the effective marginal utility of a noisy sample is exponentially worse than that of a clean sample. Providing a small set of clean samples can significantly reduce the sample size requirements for noisy data, as we also observe in our experiments.

cs.LG↗

arXiv2024

Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Data

Ambient diffusion is a recently proposed framework for training diffusion models using corrupted data. Both Ambient Diffusion and alternative SURE-based approaches for learning diffusion models from corrupted data resort to approximations which deteriorate performance. We present the first framework for training diffusion models that provably sample from the uncorrupted distribution given only noisy training data, solving an open problem in this space. Our key technical contribution is a method that uses a double application of Tweedie's formula and a consistency loss function that allows us to extend sampling at noise levels below the observed data noise. We also provide further evidence that diffusion models memorize from their training sets by identifying extremely corrupted images that are almost perfectly reconstructed, raising copyright and privacy concerns. Our method for training using corrupted samples can be used to mitigate this problem. We demonstrate this by fine-tuning Stable Diffusion XL to generate samples from a distribution using only noisy samples. Our framework reduces the amount of memorization of the fine-tuning dataset, while maintaining competitive performance.

cs.CV↗

arXiv2024

Smooth Nash Equilibria: Algorithms and Complexity

A fundamental shortcoming of the concept of Nash equilibrium is its computational intractability: approximating Nash equilibria in normal-form games is PPAD-hard. In this paper, inspired by the ideas of smoothed analysis, we introduce a relaxed variant of Nash equilibrium called $σ$-smooth Nash equilibrium, for a smoothness parameter $σ$. In a $σ$-smooth Nash equilibrium, players only need to achieve utility at least as high as their best deviation to a $σ$-smooth strategy, which is a distribution that does not put too much mass (as parametrized by $σ$) on any fixed action. We distinguish two variants of $σ$-smooth Nash equilibria: strong $σ$-smooth Nash equilibria, in which players are required to play $σ$-smooth strategies under equilibrium play, and weak $σ$-smooth Nash equilibria, where there is no such requirement. We show that both weak and strong $σ$-smooth Nash equilibria have superior computational properties to Nash equilibria: when $σ$ as well as an approximation parameter $ε$ and the number of players are all constants, there is a constant-time randomized algorithm to find a weak $ε$-approximate $σ$-smooth Nash equilibrium in normal-form games. In the same parameter regime, there is a polynomial-time deterministic algorithm to find a strong $ε$-approximate $σ$-smooth Nash equilibrium in a normal-form game. These results stand in contrast to the optimal algorithm for computing $ε$-approximate Nash equilibria, which cannot run in faster than quasipolynomial-time. We complement our upper bounds by showing that when either $σ$ or $ε$ is an inverse polynomial, finding a weak $ε$-approximate $σ$-smooth Nash equilibria becomes computationally intractable.

cs.GT↗

arXiv2024

Maximizing utility in multi-agent environments by anticipating the behavior of other learners

Learning algorithms are often used to make decisions in sequential decision-making environments. In multi-agent settings, the decisions of each agent can affect the utilities/losses of the other agents. Therefore, if an agent is good at anticipating the behavior of the other agents, in particular how they will make decisions in each round as a function of their experience that far, it could try to judiciously make its own decisions over the rounds of the interaction so as to influence the other agents to behave in a way that ultimately benefits its own utility. In this paper, we study repeated two-player games involving two types of agents: a learner, which employs an online learning algorithm to choose its strategy in each round; and an optimizer, which knows the learner's utility function and the learner's online learning algorithm. The optimizer wants to plan ahead to maximize its own utility, while taking into account the learner's behavior. We provide two results: a positive result for repeated zero-sum games and a negative result for repeated general-sum games. Our positive result is an algorithm for the optimizer, which exactly maximizes its utility against a learner that plays the Replicator Dynamics -- the continuous-time analogue of Multiplicative Weights Update (MWU). Additionally, we use this result to provide an algorithm for the optimizer against MWU, i.e.~for the discrete-time setting, which guarantees an average utility for the optimizer that is higher than the value of the one-shot game. Our negative result shows that, unless P=NP, there is no Fully Polynomial Time Approximation Scheme (FPTAS) for maximizing the utility of an optimizer against a learner that best-responds to the history in each round. Yet, this still leaves open the question of whether there exists a polynomial-time algorithm that optimizes the utility up to $o(T)$.

cs.GT↗

arXiv2024

A Lower Bound on Swap Regret in Extensive-Form Games

Recent simultaneous works by Peng and Rubinstein [2024] and Dagan et al. [2024] have demonstrated the existence of a no-swap-regret learning algorithm that can reach $ε$ average swap regret against an adversary in any extensive-form game within $m^{\tilde{\mathcal O}(1/ε)}$ rounds, where $m$ is the number of nodes in the game tree. However, the question of whether a $\mathrm{poly}(m, 1/ε)$-round algorithm could exist remained open. In this paper, we show a lower bound that precludes the existence of such an algorithm. In particular, we show that achieving average swap regret $ε$ against an oblivious adversary in general extensive-form games requires at least $\mathrm{exp}\left(Ω\left(\min\left\{m^{1/14}, ε^{-1/6}\right\}\right)\right)$ rounds.

cs.GT↗

arXiv2024

Near-Optimal Learning and Planning in Separated Latent MDPs

We study computational and statistical aspects of learning Latent Markov Decision Processes (LMDPs). In this model, the learner interacts with an MDP drawn at the beginning of each epoch from an unknown mixture of MDPs. To sidestep known impossibility results, we consider several notions of separation of the constituent MDPs. The main thrust of this paper is in establishing a nearly-sharp *statistical threshold* for the horizon length necessary for efficient learning. On the computational side, we show that under a weaker assumption of separability under the optimal policy, there is a quasi-polynomial algorithm with time complexity scaling in terms of the statistical threshold. We further show a near-matching time complexity lower bound under the exponential time hypothesis.

cs.LG↗

arXiv2023

Learning and Testing Latent-Tree Ising Models Efficiently

We provide time- and sample-efficient algorithms for learning and testing latent-tree Ising models, i.e. Ising models that may only be observed at their leaf nodes. On the learning side, we obtain efficient algorithms for learning a tree-structured Ising model whose leaf node distribution is close in Total Variation Distance, improving on the results of prior work. On the testing side, we provide an efficient algorithm with fewer samples for testing whether two latent-tree Ising models have leaf-node distributions that are close or far in Total Variation distance. We obtain our algorithms by showing novel localization results for the total variation distance between the leaf-node distributions of tree-structured Ising models, in terms of their marginals on pairs of leaves.

cs.LG↗

arXiv2023

Online Learning and Solving Infinite Games with an ERM Oracle

While ERM suffices to attain near-optimal generalization error in the stochastic learning setting, this is not known to be the case in the online learning setting, where algorithms for general concept classes rely on computationally inefficient oracles such as the Standard Optimal Algorithm (SOA). In this work, we propose an algorithm for online binary classification setting that relies solely on ERM oracle calls, and show that it has finite regret in the realizable setting and sublinearly growing regret in the agnostic setting. We bound the regret in terms of the Littlestone and threshold dimensions of the underlying concept class. We obtain similar results for nonparametric games, where the ERM oracle can be interpreted as a best response oracle, finding the best response of a player to a given history of play of the other players. In this setting, we provide learning algorithms that only rely on best response oracles and converge to approximate-minimax equilibria in two-player zero-sum games and approximate coarse correlated equilibria in multi-player general-sum games, as long as the game has a bounded fat-threshold dimension. Our algorithms apply to both binary-valued and real-valued games and can be viewed as providing justification for the wide use of double oracle and multiple oracle algorithms in the practice of solving large games.

cs.LG↗

arXiv2023

Consistent Diffusion Models: Mitigating Sampling Drift by Learning to be Consistent

Imperfect score-matching leads to a shift between the training and the sampling distribution of diffusion models. Due to the recursive nature of the generation process, errors in previous steps yield sampling iterates that drift away from the training distribution. Yet, the standard training objective via Denoising Score Matching (DSM) is only designed to optimize over non-drifted data. To train on drifted data, we propose to enforce a \emph{consistency} property which states that predictions of the model on its own generated data are consistent across time. Theoretically, we show that if the score is learned perfectly on some non-drifted points (via DSM) and if the consistency property is enforced everywhere, then the score is learned accurately everywhere. Empirically we show that our novel training objective yields state-of-the-art results for conditional and unconditional generation in CIFAR-10 and baseline improvements in AFHQ and FFHQ. We open-source our code and models: https://github.com/giannisdaras/cdm

cs.LG↗

arXiv2023

SDYN-GANs: Adversarial Learning Methods for Multistep Generative Models for General Order Stochastic Dynamics

We introduce adversarial learning methods for data-driven generative modeling of the dynamics of $n^{th}$-order stochastic systems. Our approach builds on Generative Adversarial Networks (GANs) with generative model classes based on stable $m$-step stochastic numerical integrators. We introduce different formulations and training methods for learning models of stochastic dynamics based on observation of trajectory samples. We develop approaches using discriminators based on Maximum Mean Discrepancy (MMD), training protocols using conditional and marginal distributions, and methods for learning dynamic responses over different time-scales. We show how our approaches can be used for modeling physical systems to learn force-laws, damping coefficients, and noise-related parameters. The adversarial learning approaches provide methods for obtaining stable generative models for dynamic tasks including long-time prediction and developing simulations for stochastic systems.

cs.LG↗