Two commonly arising computational tasks in Bayesian learning are Optimization (Maximum A Posteriori estimation) and Sampling (from the posterior distribution). In the convex case these two problems are efficiently reducible to each other. Recent work (Ma et al. 2019) shows that in the non-convex case, sampling can sometimes be provably faster. We present a simpler and stronger separation. We then compare sampling and optimization in more detail and show that they are provably incomparable: there are families of continuous functions for which optimization is easy but sampling is NP-hard, and vice versa. Further, we show function families that exhibit a sharp phase transition in the computational complexity of sampling, as one varies the natural temperature parameter. Our results draw on a connection to analogous separations in the discrete setting which are well-studied.

A recent line of work has shown a qualitative equivalence between differentially private PAC learning and online learning: A concept class is privately learnable if and only if it is online learnable with a finite mistake bound. However, both directions of this equivalence incur significant losses in both sample and computational efficiency. Studying a special case of this connection, Gonen, Hazan, and Moran (NeurIPS 2019) showed that uniform or highly sample-efficient pure-private learners can be time-efficiently compiled into online learners. We show that, assuming the existence of one-way functions, such an efficient conversion is impossible even for general pure-private learners with polynomial sample complexity.

Recent work [Ma et al., 2019] shows that in the non-convex case, sampling

Recent work [Ma et al., 2019] shows that in the non-convex case, sampling

Summary and Contributions: This work shows that there is a class that is privately PAC learnable in polynomial time, but not efficiently learnable (assuming the existence of one-way functions) in the online setting, i.e., there is no polynomial time algorithm with a polynomial mistake bound. A line of recent work (focused on sample complexity) has demonstrated various equivalences between private PAC and online learning; this result focuses on the question of efficiency, proving that efficient private learnability does not imply efficient online learnability if one-way functions exist. The cryptographic assumption is standard for such results and is needed when ruling out general polynomial time learners. The class of functions that cannot be efficiently learned in the online model was given by [Blum 1994], which showed a separation between distribution free PAC learning and online learning. Much of the work toward separating the models of learning, which involves working out the cryptographic construction and giving an efficient PAC algorithm, was done there -- the technical contribution here is to give a private version of that algorithm.

The goal is to show that under some situations, one of these problems is easy and the other is hard. To show that optimization can be harder than sampling, the construction hides the solution of an NP-hard problem as a small bump in a mostly flat function. Thus, approximate sampling is easy (the distribution is mostly uniform), but optimization would result in solving an NP-hard problem. To show that sampling can be harder than optimization, the construction amplifies the number of solutions of an NP-hard problem and plants an additional simple solution, and then encodes this into a function that is flat in many places, but has bumps at every possible solution of the NP-hard problem. Optimization is as easy as finding the planted simple solution, but, intuitively, sampling requires finding many of the hard solutions.

A recent line of work has shown a qualitative equivalence between differentially private PAC learning and online learning: A concept class is privately learnable if and only if it is online learnable with a finite mistake bound. However, both directions of this equivalence incur significant losses in both sample and computational efficiency. Studying a special case of this connection, Gonen, Hazan, and Moran (NeurIPS 2019) showed that uniform or highly sample-efficient pure-private learners can be time-efficiently compiled into online learners. We show that, assuming the existence of one-way functions, such an efficient conversion is impossible even for general pure-private learners with polynomial sample complexity.

Two commonly arising computational tasks in Bayesian learning are Optimization (Maximum A Posteriori estimation) and Sampling (from the posterior distribution). In the convex case these two problems are efficiently reducible to each other. Recent work (Ma et al. 2019) shows that in the non-convex case, sampling can sometimes be provably faster. We present a simpler and stronger separation. We then compare sampling and optimization in more detail and show that they are provably incomparable: there are families of continuous functions for which optimization is easy but sampling is NP-hard, and vice versa.

In multimodal machine learning, multiple modalities of data (e.g., text and images) are combined to facilitate the learning of a better machine learning model, which remains applicable to a corresponding unimodal task (e.g., text generation). Recently, multimodal machine learning has enjoyed huge empirical success (e.g. GPT-4). Motivated to develop theoretical justification for this empirical success, Lu (NeurIPS '23, ALT '24) introduces a theory of multimodal learning, and considers possible separations between theoretical models of multimodal and unimodal learning. In particular, Lu (ALT '24) shows a computational separation, which is relevant to worst-case instances of the learning task. In this paper, we give a stronger average-case computational separation, where for "typical" instances of the learning task, unimodal learning is computationally hard, but multimodal learning is easy. We then question how "organic" the average-case separation is. Would it be encountered in practice? To this end, we prove that under natural conditions, any given computational separation between average-case unimodal and multimodal learning tasks implies a corresponding cryptographic key agreement protocol. We suggest to interpret this as evidence that very strong computational advantages of multimodal learning may arise infrequently in practice, since they exist only for the "pathological" case of inherently cryptographic distributions. However, this does not apply to possible (super-polynomial) statistical advantages.

Two commonly arising computational tasks in Bayesian learning are Optimization (Maximum A Posteriori estimation) and Sampling (from the posterior distribution). In the convex case these two problems are efficiently reducible to each other. Recent work (Ma et al. 2019) shows that in the non-convex case, sampling can sometimes be provably faster. We present a simpler and stronger separation. We then compare sampling and optimization in more detail and show that they are provably incomparable: there are families of continuous functions for which optimization is easy but sampling is NP-hard, and vice versa.