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 $ φ$-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.

Neural Information Processing Systems http://nips.cc/

In this work, we revisit the generalization error of stochastic mirror descent for quadratically bounded losses studied in Telgarsky (2022). Quadratically bounded losses is a broad class of loss functions, capturing both Lipschitz and smooth functions, for both regression and classification problems. We study the high probability generalization for this class of losses on linear predictors in both realizable and non-realizable cases when the data are sampled IID or from a Markov chain. The prior work relies on an intricate coupling argument between the iterates of the original problem and those projected onto a bounded domain. This approach enables blackbox application of concentration inequalities, but also leads to suboptimal guarantees due in part to the use of a union bound across all iterations.

In this paper, we examine a class of non-convex stochastic optimization problems which we call variationally coherent, and which properly includes pseudo-/quasiconvex and star-convex optimization problems. To solve such problems, we focus on the widely used stochastic mirror descent (SMD) family of algorithms (which contains stochastic gradient descent as a special case), and we show that the last iterate of SMD converges to the problem's solution set with probability 1. This result contributes to the landscape of non-convex stochastic optimization by clarifying that neither pseudo-/quasi-convexity nor star-convexity is essential for (almost sure) global convergence; rather, variational coherence, a much weaker requirement, suffices. Characterization of convergence rates for the subclass of strongly variationally coherent optimization problems as well as simulation results are also presented.

The question of what we can guarantee beyond these settings remains open.

Adding experimental results as they promised to R3 would be valuable as well 3) As R2 pointed out, the intuition behind the analysis is not always clear. Given the rather convincing answers in the rebuttal, I think the authors can easily improve this aspect in the revised version.

In this work, we revisit the generalization error of stochastic mirror descent for quadratically bounded losses studied in Telgarsky (2022). Quadratically bounded losses is a broad class of loss functions, capturing both Lipschitz and smooth functions, for both regression and classification problems. We study the high probability generalization for this class of losses on linear predictors in both realizable and non-realizable cases when the data are sampled IID or from a Markov chain. The prior work relies on an intricate coupling argument between the iterates of the original problem and those projected onto a bounded domain. This approach enables blackbox application of concentration inequalities, but also leads to suboptimal guarantees due in part to the use of a union bound across all iterations.

We strengthen recent connections between prediction markets and learning by showing that a natural class of market makers can be understood as performing stochastic mirror descent when trader demands are sequentially drawn from a fixed distribution. This provides new insights into how market prices (and price paths) may be interpreted as a summary of the market's belief distribution by relating them to the optimization problem being solved. In particular, we show that under certain conditions the stationary point of the stochastic process of prices generated by the market is equal to the market's Walrasian equilibrium of classic market analysis. Together, these results suggest how traditional market making mechanisms might be replaced with general purpose learning algorithms while still retaining guarantees about their behaviour.

In this paper, we examine a class of non-convex stochastic optimization problems which we call variationally coherent, and which properly includes pseudo-/quasiconvex and star-convex optimization problems. To solve such problems, we focus on the widely used stochastic mirror descent (SMD) family of algorithms (which contains stochastic gradient descent as a special case), and we show that the last iterate of SMD converges to the problem's solution set with probability 1. This result contributes to the landscape of non-convex stochastic optimization by clarifying that neither pseudo-/quasi-convexity nor star-convexity is essential for (almost sure) global convergence; rather, variational coherence, a much weaker requirement, suffices. Characterization of convergence rates for the subclass of strongly variationally coherent optimization problems as well as simulation results are also presented.

The stochastic mirror descent (SMD) algorithm is a general class of training algorithms, which includes the celebrated stochastic gradient descent (SGD), as a special case. It utilizes a mirror potential to influence the implicit bias of the training algorithm. In this paper we explore the performance of the SMD iterates on mean-field ensemble models. Our results generalize earlier ones obtained for SGD on such models. The evolution of the distribution of parameters is mapped to a continuous time process in the space of probability distributions. Our main result gives a nonlinear partial differential equation to which the continuous time process converges in the asymptotic regime of large networks. The impact of the mirror potential appears through a multiplicative term that is equal to the inverse of its Hessian and which can be interpreted as defining a gradient flow over an appropriately defined Riemannian manifold. We provide numerical simulations which allow us to study and characterize the effect of the mirror potential on the performance of networks trained with SMD for some binary classification problems.