Critical windows: non-asymptotic theory for feature emergence in diffusion models

We develop theory to understand an intriguing property of diffusion models for image generation that we term critical windows. Empirically, it has been observed that there are narrow time intervals in sampling during which particular features of the final image emerge, e.g. the image class or background color (Ho et al., 2020b; Georgiev et al., 2023; Raya & Ambrogioni, 2023; Sclocchi et al., 2024; Biroli et al., 2024). While this is advantageous for interpretability as it implies one can localize properties of the generation to a small segment of the trajectory, it seems at odds with the continuous nature of the diffusion. We propose a formal framework for studying these windows and show that for data coming from a mixture of strongly log-concave densities, these windows can be provably bounded in terms of certain measures of inter- and intra-group separation. We also instantiate these bounds for concrete examples like well-conditioned Gaussian mixtures. Finally, we use our bounds to give a rigorous interpretation of diffusion models as hierarchical samplers that progressively "decide" output features over a discrete sequence of times. We validate our bounds with synthetic experiments. Additionally, preliminary experiments on Stable Diffusion suggest critical windows may serve as a useful tool for diagnosing fairness and privacy violations in real-world diffusion models.

An optimal tradeoff between entanglement and copy complexity for state tomography

There has been significant interest in understanding how practical constraints on contemporary quantum devices impact the complexity of quantum learning. For the classic question of tomography, recent work tightly characterized the copy complexity for any protocol that can only measure one copy of the unknown state at a time, showing it is polynomially worse than if one can make fully-entangled measurements. While we now have a fairly complete picture of the rates for such tasks in the near-term and fault-tolerant regimes, it remains poorly understood what the landscape in between looks like. In this work, we study tomography in the natural setting where one can make measurements of $t$ copies at a time. For sufficiently small $\epsilon$, we show that for any $t \le d^2$, $\widetilde{\Theta}(\frac{d^3}{\sqrt{t}\epsilon^2})$ copies are necessary and sufficient to learn an unknown $d$-dimensional state $\rho$ to trace distance $\epsilon$. This gives a smooth and optimal interpolation between the known rates for single-copy and fully-entangled measurements. To our knowledge, this is the first smooth entanglement-copy tradeoff known for any quantum learning task, and for tomography, no intermediate point on this curve was known, even at $t = 2$. An important obstacle is that unlike the optimal single-copy protocol, the optimal fully-entangled protocol is inherently biased and thus precludes naive batching approaches. Instead, we devise a novel two-stage procedure that uses Keyl's algorithm to refine a crude estimate for $\rho$ based on single-copy measurements. A key insight is to use Schur-Weyl sampling not to estimate the spectrum of $\rho$, but to estimate the deviation of $\rho$ from the maximally mixed state. When $\rho$ is far from the maximally mixed state, we devise a novel quantum splitting procedure that reduces to the case where $\rho$ is close to maximally mixed.

Provably learning a multi-head attention layer

The multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. Given a sequence length $k$, attention matrices $\mathbf{\Theta}_1,\ldots,\mathbf{\Theta}_m\in\mathbb{R}^{d\times d}$, and projection matrices $\mathbf{W}_1,\ldots,\mathbf{W}_m\in\mathbb{R}^{d\times d}$, the corresponding multi-head attention layer $F: \mathbb{R}^{k\times d}\to \mathbb{R}^{k\times d}$ transforms length-$k$ sequences of $d$-dimensional tokens $\mathbf{X}\in\mathbb{R}^{k\times d}$ via $F(\mathbf{X}) \triangleq \sum^m_{i=1} \mathrm{softmax}(\mathbf{X}\mathbf{\Theta}_i\mathbf{X}^\top)\mathbf{X}\mathbf{W}_i$. In this work, we initiate the study of provably learning a multi-head attention layer from random examples and give the first nontrivial upper and lower bounds for this problem: - Provided $\{\mathbf{W}_i, \mathbf{\Theta}_i\}$ satisfy certain non-degeneracy conditions, we give a $(dk)^{O(m^3)}$-time algorithm that learns $F$ to small error given random labeled examples drawn uniformly from $\{\pm 1\}^{k\times d}$. - We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable. We focus on Boolean $\mathbf{X}$ to mimic the discrete nature of tokens in large language models, though our techniques naturally extend to standard continuous settings, e.g. Gaussian. Our algorithm, which is centered around using examples to sculpt a convex body containing the unknown parameters, is a significant departure from existing provable algorithms for learning feedforward networks, which predominantly exploit algebraic and rotation invariance properties of the Gaussian distribution. In contrast, our analysis is more flexible as it primarily relies on various upper and lower tail bounds for the input distribution and "slices" thereof.

Futility and utility of a few ancillas for Pauli channel learning

In this paper we revisit one of the prototypical tasks for characterizing the structure of noise in quantum devices, estimating the eigenvalues of an $n$-qubit Pauli noise channel. Prior work (Chen et al., 2022) established exponential lower bounds for this task for algorithms with limited quantum memory. We first improve upon their lower bounds and show: (1) Any algorithm without quantum memory must make $\Omega(2^n/\epsilon^2)$ measurements to estimate each eigenvalue within error $\epsilon$. This is tight and implies the randomized benchmarking protocol is optimal, resolving an open question of (Flammia and Wallman, 2020). (2) Any algorithm with $\le k$ ancilla qubits of quantum memory must make $\Omega(2^{(n-k)/3})$ queries to the unknown channel. Crucially, unlike in (Chen et al., 2022), our bound holds even if arbitrary adaptive control and channel concatenation are allowed. In fact these lower bounds, like those of (Chen et al., 2022), hold even for the easier hypothesis testing problem of determining whether the underlying channel is completely depolarizing or has exactly one other nontrivial eigenvalue. Surprisingly, we show that: (3) With only $k=2$ ancilla qubits of quantum memory, there is an algorithm that solves this hypothesis testing task with high probability using a single measurement. Note that (3) does not contradict (2) as the protocol concatenates exponentially many queries to the channel before the measurement. This result suggests a novel mechanism by which channel concatenation and $O(1)$ qubits of quantum memory could work in tandem to yield striking speedups for quantum process learning that are not possible for quantum state learning.

A faster and simpler algorithm for learning shallow networks

We revisit the well-studied problem of learning a linear combination of $k$ ReLU activations given labeled examples drawn from the standard $d$-dimensional Gaussian measure. Chen et al. [CDG+23] recently gave the first algorithm for this problem to run in $\text{poly}(d,1/\varepsilon)$ time when $k = O(1)$, where $\varepsilon$ is the target error. More precisely, their algorithm runs in time $(d/\varepsilon)^{\mathrm{quasipoly}(k)}$ and learns over multiple stages. Here we show that a much simpler one-stage version of their algorithm suffices, and moreover its runtime is only $(d/\varepsilon)^{O(k^2)}$.

Learning Mixtures of Gaussians Using the DDPM Objective

Recent works have shown that diffusion models can learn essentially any distribution provided one can perform score estimation. Yet it remains poorly understood under what settings score estimation is possible, let alone when practical gradient-based algorithms for this task can provably succeed. In this work, we give the first provably efficient results along these lines for one of the most fundamental distribution families, Gaussian mixture models. We prove that gradient descent on the denoising diffusion probabilistic model (DDPM) objective can efficiently recover the ground truth parameters of the mixture model in the following two settings: 1) We show gradient descent with random initialization learns mixtures of two spherical Gaussians in $d$ dimensions with $1/\text{poly}(d)$-separated centers. 2) We show gradient descent with a warm start learns mixtures of $K$ spherical Gaussians with $\Omega(\sqrt{\log(\min(K,d))})$-separated centers. A key ingredient in our proofs is a new connection between score-based methods and two other approaches to distribution learning, the EM algorithm and spectral methods.

The probability flow ODE is provably fast

We provide the first polynomial-time convergence guarantees for the probability flow ODE implementation (together with a corrector step) of score-based generative modeling. Our analysis is carried out in the wake of recent results obtaining such guarantees for the SDE-based implementation (i.e., denoising diffusion probabilistic modeling or DDPM), but requires the development of novel techniques for studying deterministic dynamics without contractivity. Through the use of a specially chosen corrector step based on the underdamped Langevin diffusion, we obtain better dimension dependence than prior works on DDPM ($O(\sqrt{d})$ vs. $O(d)$, assuming smoothness of the data distribution), highlighting potential advantages of the ODE framework.

Learning Narrow One-Hidden-Layer ReLU Networks

We consider the well-studied problem of learning a linear combination of $k$ ReLU activations with respect to a Gaussian distribution on inputs in $d$ dimensions. We give the first polynomial-time algorithm that succeeds whenever $k$ is a constant. All prior polynomial-time learners require additional assumptions on the network, such as positive combining coefficients or the matrix of hidden weight vectors being well-conditioned. Our approach is based on analyzing random contractions of higher-order moment tensors. We use a multi-scale analysis to argue that sufficiently close neurons can be collapsed together, sidestepping the conditioning issues present in prior work. This allows us to design an iterative procedure to discover individual neurons.

Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers

We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. Several recent works have analyzed stochastic samplers using tools like Girsanov's theorem and a chain rule variant of the interpolation argument. Unfortunately, these techniques give vacuous bounds when applied to deterministic samplers. We give a new operational interpretation for deterministic sampling by showing that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs gradient ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current iterate. This perspective allows us to extend denoising diffusion implicit models to general, non-linear forward processes. We then develop the first polynomial convergence bounds for these samplers under mild conditions on the data distribution.

Learning to predict arbitrary quantum processes

We present an efficient machine learning (ML) algorithm for predicting any unknown quantum process $\mathcal{E}$ over $n$ qubits. For a wide range of distributions $\mathcal{D}$ on arbitrary $n$-qubit states, we show that this ML algorithm can learn to predict any local property of the output from the unknown process $\mathcal{E}$, with a small average error over input states drawn from $\mathcal{D}$. The ML algorithm is computationally efficient even when the unknown process is a quantum circuit with exponentially many gates. Our algorithm combines efficient procedures for learning properties of an unknown state and for learning a low-degree approximation to an unknown observable. The analysis hinges on proving new norm inequalities, including a quantum analogue of the classical Bohnenblust-Hille inequality, which we derive by giving an improved algorithm for optimizing local Hamiltonians. Overall, our results highlight the potential for ML models to predict the output of complex quantum dynamics much faster than the time needed to run the process itself.