Lixue Cheng, P. Bernát Szabó, Zeno Schätzle et al. · microsoft-research
Variational ab-initio methods in quantum chemistry stand out among other methods in providing direct access to the wave function. This allows in principle straightforward extraction of any other observable of interest, besides the energy, but in practice this extraction is often technically difficult and computationally impractical. Here, we consider the electron density as a central observable in quantum chemistry and introduce a novel method to obtain accurate densities from real-space many-electron wave functions by representing the density with a neural network that captures known asymptotic properties and is trained from the wave function by score matching and noise-contrastive estimation. We use variational quantum Monte Carlo with deep-learning ansätze (deep QMC) to obtain highly accurate wave functions free of basis set errors, and from them, using our novel method, correspondingly accurate electron densities, which we demonstrate by calculating dipole moments, nuclear forces, contact densities, and other density-based properties.
Coarse-grained (CG) molecular simulations have become a standard tool to study molecular processes on time- and length-scales inaccessible to all-atom simulations. Parameterizing CG force fields to match all-atom simulations has mainly relied on force-matching or relative entropy minimization, which require many samples from costly simulations with all-atom or CG resolutions, respectively. Here we present flow-matching, a new training method for CG force fields that combines the advantages of both methods by leveraging normalizing flows, a generative deep learning method. Flow-matching first trains a normalizing flow to represent the CG probability density, which is equivalent to minimizing the relative entropy without requiring iterative CG simulations. Subsequently, the flow generates samples and forces according to the learned distribution in order to train the desired CG free energy model via force matching. Even without requiring forces from the all-atom simulations, flow-matching outperforms classical force-matching by an order of magnitude in terms of data efficiency, and produces CG models that can capture the folding and unfolding transitions of small proteins.
Jonas Köhler, Michele Invernizzi, Pim de Haan et al.
Normalizing flows (NF) are a class of powerful generative models that have gained popularity in recent years due to their ability to model complex distributions with high flexibility and expressiveness. In this work, we introduce a new type of normalizing flow that is tailored for modeling positions and orientations of multiple objects in three-dimensional space, such as molecules in a crystal. Our approach is based on two key ideas: first, we define smooth and expressive flows on the group of unit quaternions, which allows us to capture the continuous rotational motion of rigid bodies; second, we use the double cover property of unit quaternions to define a proper density on the rotation group. This ensures that our model can be trained using standard likelihood-based methods or variational inference with respect to a thermodynamic target density. We evaluate the method by training Boltzmann generators for two molecular examples, namely the multi-modal density of a tetrahedral system in an external field and the ice XI phase in the TIP4P water model. Our flows can be combined with flows operating on the internal degrees of freedom of molecules and constitute an important step towards the modeling of distributions of many interacting molecules.
Adam Polyak, Amit Zohar, Andrew Brown et al. · meta-ai
We present Movie Gen, a cast of foundation models that generates high-quality, 1080p HD videos with different aspect ratios and synchronized audio. We also show additional capabilities such as precise instruction-based video editing and generation of personalized videos based on a user's image. Our models set a new state-of-the-art on multiple tasks: text-to-video synthesis, video personalization, video editing, video-to-audio generation, and text-to-audio generation. Our largest video generation model is a 30B parameter transformer trained with a maximum context length of 73K video tokens, corresponding to a generated video of 16 seconds at 16 frames-per-second. We show multiple technical innovations and simplifications on the architecture, latent spaces, training objectives and recipes, data curation, evaluation protocols, parallelization techniques, and inference optimizations that allow us to reap the benefits of scaling pre-training data, model size, and training compute for training large scale media generation models. We hope this paper helps the research community to accelerate progress and innovation in media generation models. All videos from this paper are available at https://go.fb.me/MovieGenResearchVideos.
Yeongmin Kim, Sotiris Anagnostidis, Yuming Du et al.
Diffusion models with transformer architectures have demonstrated promising capabilities in generating high-fidelity images and scalability for high resolution. However, iterative sampling process required for synthesis is very resource-intensive. A line of work has focused on distilling solutions to probability flow ODEs into few-step student models. Nevertheless, existing methods have been limited by their reliance on the most recent denoised samples as input, rendering them susceptible to exposure bias. To address this limitation, we propose AutoRegressive Distillation (ARD), a novel approach that leverages the historical trajectory of the ODE to predict future steps. ARD offers two key benefits: 1) it mitigates exposure bias by utilizing a predicted historical trajectory that is less susceptible to accumulated errors, and 2) it leverages the previous history of the ODE trajectory as a more effective source of coarse-grained information. ARD modifies the teacher transformer architecture by adding token-wise time embedding to mark each input from the trajectory history and employs a block-wise causal attention mask for training. Furthermore, incorporating historical inputs only in lower transformer layers enhances performance and efficiency. We validate the effectiveness of ARD in a class-conditioned generation on ImageNet and T2I synthesis. Our model achieves a $5\times$ reduction in FID degradation compared to the baseline methods while requiring only 1.1\% extra FLOPs on ImageNet-256. Moreover, ARD reaches FID of 1.84 on ImageNet-256 in merely 4 steps and outperforms the publicly available 1024p text-to-image distilled models in prompt adherence score with a minimal drop in FID compared to the teacher. Project page: https://github.com/alsdudrla10/ARD.
Felix Wimbauer, Bichen Wu, Edgar Schoenfeld et al.
Diffusion models have recently revolutionized the field of image synthesis due to their ability to generate photorealistic images. However, one of the major drawbacks of diffusion models is that the image generation process is costly. A large image-to-image network has to be applied many times to iteratively refine an image from random noise. While many recent works propose techniques to reduce the number of required steps, they generally treat the underlying denoising network as a black box. In this work, we investigate the behavior of the layers within the network and find that 1) the layers' output changes smoothly over time, 2) the layers show distinct patterns of change, and 3) the change from step to step is often very small. We hypothesize that many layer computations in the denoising network are redundant. Leveraging this, we introduce block caching, in which we reuse outputs from layer blocks of previous steps to speed up inference. Furthermore, we propose a technique to automatically determine caching schedules based on each block's changes over timesteps. In our experiments, we show through FID, human evaluation and qualitative analysis that Block Caching allows to generate images with higher visual quality at the same computational cost. We demonstrate this for different state-of-the-art models (LDM and EMU) and solvers (DDIM and DPM).
Recent advances in differentiable rendering have sparked an interest in learning generative models of textured 3D meshes from image collections. These models natively disentangle pose and appearance, enable downstream applications in computer graphics, and improve the ability of generative models to understand the concept of image formation. Although there has been prior work on learning such models from collections of 2D images, these approaches require a delicate pose estimation step that exploits annotated keypoints, thereby restricting their applicability to a few specific datasets. In this work, we propose a GAN framework for generating textured triangle meshes without relying on such annotations. We show that the performance of our approach is on par with prior work that relies on ground-truth keypoints, and more importantly, we demonstrate the generality of our method by setting new baselines on a larger set of categories from ImageNet - for which keypoints are not available - without any class-specific hyperparameter tuning. We release our code at https://github.com/dariopavllo/textured-3d-gan
Jonas Kohler, Albert Pumarola, Edgar Schönfeld et al.
Diffusion models are a powerful generative framework, but come with expensive inference. Existing acceleration methods often compromise image quality or fail under complex conditioning when operating in an extremely low-step regime. In this work, we propose a novel distillation framework tailored to enable high-fidelity, diverse sample generation using just one to three steps. Our approach comprises three key components: (i) Backward Distillation, which mitigates training-inference discrepancies by calibrating the student on its own backward trajectory; (ii) Shifted Reconstruction Loss that dynamically adapts knowledge transfer based on the current time step; and (iii) Noise Correction, an inference-time technique that enhances sample quality by addressing singularities in noise prediction. Through extensive experiments, we demonstrate that our method outperforms existing competitors in quantitative metrics and human evaluations. Remarkably, it achieves performance comparable to the teacher model using only three denoising steps, enabling efficient high-quality generation.
Gregor Bachmann, Sotiris Anagnostidis, Albert Pumarola et al.
The performance of large language models (LLMs) is closely linked to their underlying size, leading to ever-growing networks and hence slower inference. Speculative decoding has been proposed as a technique to accelerate autoregressive generation, leveraging a fast draft model to propose candidate tokens, which are then verified in parallel based on their likelihood under the target model. While this approach guarantees to reproduce the target output, it incurs a substantial penalty: many high-quality draft tokens are rejected, even when they represent objectively valid continuations. Indeed, we show that even powerful draft models such as GPT-4o, as well as human text cannot achieve high acceptance rates under the standard verification scheme. This severely limits the speedup potential of current speculative decoding methods, as an early rejection becomes overwhelmingly likely when solely relying on alignment of draft and target. We thus ask the following question: Can we adapt verification to recognize correct, but non-aligned replies? To this end, we draw inspiration from the LLM-as-a-judge framework, which demonstrated that LLMs are able to rate answers in a versatile way. We carefully design a dataset to elicit the same capability in the target model by training a compact module on top of the embeddings to produce ``judgements" of the current continuation. We showcase our strategy on the Llama-3.1 family, where our 8b/405B-Judge achieves a speedup of 9x over Llama-405B, while maintaining its quality on a large range of benchmarks. These benefits remain present even in optimized inference frameworks, where our method reaches up to 141 tokens/s for 8B/70B-Judge and 129 tokens/s for 8B/405B on 2 and 8 H100s respectively.
Angela Castillo, Jonas Kohler, Juan C. Pérez et al.
This paper presents a comprehensive study on the role of Classifier-Free Guidance (CFG) in text-conditioned diffusion models from the perspective of inference efficiency. In particular, we relax the default choice of applying CFG in all diffusion steps and instead search for efficient guidance policies. We formulate the discovery of such policies in the differentiable Neural Architecture Search framework. Our findings suggest that the denoising steps proposed by CFG become increasingly aligned with simple conditional steps, which renders the extra neural network evaluation of CFG redundant, especially in the second half of the denoising process. Building upon this insight, we propose "Adaptive Guidance" (AG), an efficient variant of CFG, that adaptively omits network evaluations when the denoising process displays convergence. Our experiments demonstrate that AG preserves CFG's image quality while reducing computation by 25%. Thus, AG constitutes a plug-and-play alternative to Guidance Distillation, achieving 50% of the speed-ups of the latter while being training-free and retaining the capacity to handle negative prompts. Finally, we uncover further redundancies of CFG in the first half of the diffusion process, showing that entire neural function evaluations can be replaced by simple affine transformations of past score estimates. This method, termed LinearAG, offers even cheaper inference at the cost of deviating from the baseline model. Our findings provide insights into the efficiency of the conditional denoising process that contribute to more practical and swift deployment of text-conditioned diffusion models.
Adam Foster, Zeno Schätzle, P. Bernát Szabó et al. · microsoft-research
Reliable description of bond breaking remains a major challenge for quantum chemistry due to the multireferential character of the electronic structure in dissociating species. Multireferential methods in particular suffer from large computational cost, which under the normal paradigm has to be paid anew for each system at a full price, ignoring commonalities in electronic structure across molecules. Quantum Monte Carlo with deep neural networks (deep QMC) uniquely offers to exploit such commonalities by pretraining transferable wavefunction models, but all such attempts were so far limited in scope. Here, we bring this new paradigm to fruition with Orbformer, a novel transferable wavefunction model pretrained on 22,000 equilibrium and dissociating structures that can be fine-tuned on unseen molecules reaching an accuracy-cost ratio rivalling classical multireferential methods. On established benchmarks as well as more challenging bond dissociations and Diels-Alder reactions, Orbformer is the only method that consistently converges to chemical accuracy (1 kcal/mol). This work turns the idea of amortizing the cost of solving the Schrödinger equation over many molecules into a practical approach in quantum chemistry.
Jaskirat Singh, Junshen Kevin Chen, Jonas Kohler et al.
Training-free consistent text-to-image generation depicting the same subjects across different images is a topic of widespread recent interest. Existing works in this direction predominantly rely on cross-frame self-attention; which improves subject-consistency by allowing tokens in each frame to pay attention to tokens in other frames during self-attention computation. While useful for single subjects, we find that it struggles when scaling to multiple characters. In this work, we first analyze the reason for these limitations. Our exploration reveals that the primary-issue stems from self-attention-leakage, which is exacerbated when trying to ensure consistency across multiple-characters. This happens when tokens from one subject pay attention to other characters, causing them to appear like each other (e.g., a dog appearing like a duck). Motivated by these findings, we propose StoryBooth: a training-free approach for improving multi-character consistency. In particular, we first leverage multi-modal chain-of-thought reasoning and region-based generation to apriori localize the different subjects across the desired story outputs. The final outputs are then generated using a modified diffusion model which consists of two novel layers: 1) a bounded cross-frame self-attention layer for reducing inter-character attention leakage, and 2) token-merging layer for improving consistency of fine-grain subject details. Through both qualitative and quantitative results we find that the proposed approach surpasses prior state-of-the-art, exhibiting improved consistency across both multiple-characters and fine-grain subject details.
Sotiris Anagnostidis, Gregor Bachmann, Yeongmin Kim et al.
Despite their remarkable performance, modern Diffusion Transformers are hindered by substantial resource requirements during inference, stemming from the fixed and large amount of compute needed for each denoising step. In this work, we revisit the conventional static paradigm that allocates a fixed compute budget per denoising iteration and propose a dynamic strategy instead. Our simple and sample-efficient framework enables pre-trained DiT models to be converted into \emph{flexible} ones -- dubbed FlexiDiT -- allowing them to process inputs at varying compute budgets. We demonstrate how a single \emph{flexible} model can generate images without any drop in quality, while reducing the required FLOPs by more than $40$\% compared to their static counterparts, for both class-conditioned and text-conditioned image generation. Our method is general and agnostic to input and conditioning modalities. We show how our approach can be readily extended for video generation, where FlexiDiT models generate samples with up to $75$\% less compute without compromising performance.
Jonas Kohler, Nicolas Griffiths Sanchez, Luca Cavalli et al.
In this study, we propose two novel input processing paradigms for novel view synthesis (NVS) methods based on layered scene representations that significantly improve their runtime without compromising quality. Our approach identifies and mitigates the two most time-consuming aspects of traditional pipelines: building and processing the so-called plane sweep volume (PSV), which is a high-dimensional tensor of planar re-projections of the input camera views. In particular, we propose processing this tensor in parallel groups for improved compute efficiency as well as super-sampling adjacent input planes to generate denser, and hence more accurate scene representation. The proposed enhancements offer significant flexibility, allowing for a balance between performance and speed, thus making substantial steps toward real-time applications. Furthermore, they are very general in the sense that any PSV-based method can make use of them, including methods that employ multiplane images, multisphere images, and layered depth images. In a comprehensive set of experiments, we demonstrate that our proposed paradigms enable the design of an NVS method that achieves state-of-the-art on public benchmarks while being up to $50x$ faster than existing state-of-the-art methods. It also beats the current forerunner in terms of speed by over $3x$, while achieving significantly better rendering quality.
Jonas Kohler, Maarten C. Ottenhoff, Sophocles Goulis et al.
Speech Neuroprostheses have the potential to enable communication for people with dysarthria or anarthria. Recent advances have demonstrated high-quality text decoding and speech synthesis from electrocorticographic grids placed on the cortical surface. Here, we investigate a less invasive measurement modality in three participants, namely stereotactic EEG (sEEG) that provides sparse sampling from multiple brain regions, including subcortical regions. To evaluate whether sEEG can also be used to synthesize audio from neural recordings, we employ a recurrent encoder-decoder model based on modern deep learning methods. We find that speech can indeed be reconstructed with correlations up to 0.8 from these minimally invasive recordings, despite limited amounts of training data. In particular, the architecture we employ naturally picks up on the temporal nature of the data and thereby outperforms an existing benchmark based on non-regressive convolutional neural networks.
Normalizing flows are a promising tool for modeling probability distributions in physical systems. While state-of-the-art flows accurately approximate distributions and energies, applications in physics additionally require smooth energies to compute forces and higher-order derivatives. Furthermore, such densities are often defined on non-trivial topologies. A recent example are Boltzmann Generators for generating 3D-structures of peptides and small proteins. These generative models leverage the space of internal coordinates (dihedrals, angles, and bonds), which is a product of hypertori and compact intervals. In this work, we introduce a class of smooth mixture transformations working on both compact intervals and hypertori. Mixture transformations employ root-finding methods to invert them in practice, which has so far prevented bi-directional flow training. To this end, we show that parameter gradients and forces of such inverses can be computed from forward evaluations via the inverse function theorem. We demonstrate two advantages of such smooth flows: they allow training by force matching to simulation data and can be used as potentials in molecular dynamics simulations.
Ziyad Sheebaelhamd, Konstantinos Zisis, Athina Nisioti et al.
Multi-agent control problems constitute an interesting area of application for deep reinforcement learning models with continuous action spaces. Such real-world applications, however, typically come with critical safety constraints that must not be violated. In order to ensure safety, we enhance the well-known multi-agent deep deterministic policy gradient (MADDPG) framework by adding a safety layer to the deep policy network. In particular, we extend the idea of linearizing the single-step transition dynamics, as was done for single-agent systems in Safe DDPG (Dalal et al., 2018), to multi-agent settings. We additionally propose to circumvent infeasibility problems in the action correction step using soft constraints (Kerrigan & Maciejowski, 2000). Results from the theory of exact penalty functions can be used to guarantee constraint satisfaction of the soft constraints under mild assumptions. We empirically find that the soft formulation achieves a dramatic decrease in constraint violations, making safety available even during the learning procedure.
Søren Ager Meldgaard, Jonas Köhler, Henrik Lund Mortensen et al.
Chemical space is routinely explored by machine learning methods to discover interesting molecules, before time-consuming experimental synthesizing is attempted. However, these methods often rely on a graph representation, ignoring 3D information necessary for determining the stability of the molecules. We propose a reinforcement learning approach for generating molecules in cartesian coordinates allowing for quantum chemical prediction of the stability. To improve sample-efficiency we learn basic chemical rules from imitation learning on the GDB-11 database to create an initial model applicable for all stoichiometries. We then deploy multiple copies of the model conditioned on a specific stoichiometry in a reinforcement learning setting. The models correctly identify low energy molecules in the database and produce novel isomers not found in the training set. Finally, we apply the model to larger molecules to show how reinforcement learning further refines the imitation learning model in domains far from the training data.
Antonio Orvieto, Jonas Kohler, Dario Pavllo et al.
This paper revisits the so-called vanishing gradient phenomenon, which commonly occurs in deep randomly initialized neural networks. Leveraging an in-depth analysis of neural chains, we first show that vanishing gradients cannot be circumvented when the network width scales with less than O(depth), even when initialized with the popular Xavier and He initializations. Second, we extend the analysis to second-order derivatives and show that random i.i.d. initialization also gives rise to Hessian matrices with eigenspectra that vanish as networks grow in depth. Whenever this happens, optimizers are initialized in a very flat, saddle point-like plateau, which is particularly hard to escape with stochastic gradient descent (SGD) as its escaping time is inversely related to curvature. We believe that this observation is crucial for fully understanding (a) historical difficulties of training deep nets with vanilla SGD, (b) the success of adaptive gradient methods (which naturally adapt to curvature and thus quickly escape flat plateaus) and (c) the effectiveness of modern architectural components like residual connections and normalization layers.
Adrian Hoffmann, Claudio Fanconi, Rahul Rade et al.
Deep neural networks that yield human interpretable decisions by architectural design have lately become an increasingly popular alternative to post hoc interpretation of traditional black-box models. Among these networks, the arguably most widespread approach is so-called prototype learning, where similarities to learned latent prototypes serve as the basis of classifying an unseen data point. In this work, we point to an important shortcoming of such approaches. Namely, there is a semantic gap between similarity in latent space and similarity in input space, which can corrupt interpretability. We design two experiments that exemplify this issue on the so-called ProtoPNet. Specifically, we find that this network's interpretability mechanism can be led astray by intentionally crafted or even JPEG compression artefacts, which can produce incomprehensible decisions. We argue that practitioners ought to have this shortcoming in mind when deploying prototype-based models in practice.
Nikolaos Tselepidis, Jonas Kohler, Antonio Orvieto
In the context of deep learning, many optimization methods use gradient covariance information in order to accelerate the convergence of Stochastic Gradient Descent. In particular, starting with Adagrad, a seemingly endless line of research advocates the use of diagonal approximations of the so-called empirical Fisher matrix in stochastic gradient-based algorithms, with the most prominent one arguably being Adam. However, in recent years, several works cast doubt on the theoretical basis of preconditioning with the empirical Fisher matrix, and it has been shown that more sophisticated approximations of the actual Fisher matrix more closely resemble the theoretically well-motivated Natural Gradient Descent. One particularly successful variant of such methods is the so-called K-FAC optimizer, which uses a Kronecker-factored block-diagonal Fisher approximation as preconditioner. In this work, drawing inspiration from two-level domain decomposition methods used as preconditioners in the field of scientific computing, we extend K-FAC by enriching it with off-diagonal (i.e. global) curvature information in a computationally efficient way. We achieve this by adding a coarse-space correction term to the preconditioner, which captures the global Fisher information matrix at a coarser scale. We present a small set of experimental results suggesting improved convergence behaviour of our proposed method.
Many types of neural network layers rely on matrix properties such as invertibility or orthogonality. Retaining such properties during optimization with gradient-based stochastic optimizers is a challenging task, which is usually addressed by either reparameterization of the affected parameters or by directly optimizing on the manifold. This work presents a novel approach for training invertible linear layers. In lieu of directly optimizing the network parameters, we train rank-one perturbations and add them to the actual weight matrices infrequently. This P$^{4}$Inv update allows keeping track of inverses and determinants without ever explicitly computing them. We show how such invertible blocks improve the mixing and thus the mode separation of the resulting normalizing flows. Furthermore, we outline how the P$^4$ concept can be utilized to retain properties other than invertibility.
Normalizing flows are exact-likelihood generative neural networks which approximately transform samples from a simple prior distribution to samples of the probability distribution of interest. Recent work showed that such generative models can be utilized in statistical mechanics to sample equilibrium states of many-body systems in physics and chemistry. To scale and generalize these results, it is essential that the natural symmetries in the probability density -- in physics defined by the invariances of the target potential -- are built into the flow. We provide a theoretical sufficient criterion showing that the distribution generated by \textit{equivariant} normalizing flows is invariant with respect to these symmetries by design. Furthermore, we propose building blocks for flows which preserve symmetries which are usually found in physical/chemical many-body particle systems. Using benchmark systems motivated from molecular physics, we demonstrate that those symmetry preserving flows can provide better generalization capabilities and sampling efficiency.
Hadi Daneshmand, Jonas Kohler, Francis Bach et al.
Randomly initialized neural networks are known to become harder to train with increasing depth, unless architectural enhancements like residual connections and batch normalization are used. We here investigate this phenomenon by revisiting the connection between random initialization in deep networks and spectral instabilities in products of random matrices. Given the rich literature on random matrices, it is not surprising to find that the rank of the intermediate representations in unnormalized networks collapses quickly with depth. In this work we highlight the fact that batch normalization is an effective strategy to avoid rank collapse for both linear and ReLU networks. Leveraging tools from Markov chain theory, we derive a meaningful lower rank bound in deep linear networks. Empirically, we also demonstrate that this rank robustness generalizes to ReLU nets. Finally, we conduct an extensive set of experiments on real-world data sets, which confirm that rank stability is indeed a crucial condition for training modern-day deep neural architectures.
The sampling of probability distributions specified up to a normalization constant is an important problem in both machine learning and statistical mechanics. While classical stochastic sampling methods such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics (LD) can suffer from slow mixing times there is a growing interest in using normalizing flows in order to learn the transformation of a simple prior distribution to the given target distribution. Here we propose a generalized and combined approach to sample target densities: Stochastic Normalizing Flows (SNF) -- an arbitrary sequence of deterministic invertible functions and stochastic sampling blocks. We show that stochasticity overcomes expressivity limitations of normalizing flows resulting from the invertibility constraint, whereas trainable transformations between sampling steps improve efficiency of pure MCMC/LD along the flow. By invoking ideas from non-equilibrium statistical mechanics we derive an efficient training procedure by which both the sampler's and the flow's parameters can be optimized end-to-end, and by which we can compute exact importance weights without having to marginalize out the randomness of the stochastic blocks. We illustrate the representational power, sampling efficiency and asymptotic correctness of SNFs on several benchmarks including applications to sampling molecular systems in equilibrium.
We present a stochastic optimization method that uses a fourth-order regularized model to find local minima of smooth and potentially non-convex objective functions with a finite-sum structure. This algorithm uses sub-sampled derivatives instead of exact quantities. The proposed approach is shown to find an $(ε_1,ε_2,ε_3)$-third-order critical point in at most $\bigO\left(\max\left(ε_1^{-4/3}, ε_2^{-2}, ε_3^{-4}\right)\right)$ iterations, thereby matching the rate of deterministic approaches. In order to prove this result, we derive a novel tensor concentration inequality for sums of tensors of any order that makes explicit use of the finite-sum structure of the objective function.
Developing a differentially private deep learning algorithm is challenging, due to the difficulty in analyzing the sensitivity of objective functions that are typically used to train deep neural networks. Many existing methods resort to the stochastic gradient descent algorithm and apply a pre-defined sensitivity to the gradients for privatizing weights. However, their slow convergence typically yields a high cumulative privacy loss. Here, we take a different route by employing the method of auxiliary coordinates, which allows us to independently update the weights per layer by optimizing a per-layer objective function. This objective function can be well approximated by a low-order Taylor's expansion, in which sensitivity analysis becomes tractable. We perturb the coefficients of the expansion for privacy, which we optimize using more advanced optimization routines than SGD for faster convergence. We empirically show that our algorithm provides a decent trained model quality under a modest privacy budget.
Flows are exact-likelihood generative neural networks that transform samples from a simple prior distribution to the samples of the probability distribution of interest. Boltzmann Generators (BG) combine flows and statistical mechanics to sample equilibrium states of strongly interacting many-body systems such as proteins with 1000 atoms. In order to scale and generalize these results, it is essential that the natural symmetries of the probability density - in physics defined by the invariances of the energy function - are built into the flow. Here we develop theoretical tools for constructing such equivariant flows and demonstrate that a BG that is equivariant with respect to rotations and particle permutations can generalize to sampling nontrivially new configurations where a nonequivariant BG cannot.
The choice of how to retain information about past gradients dramatically affects the convergence properties of state-of-the-art stochastic optimization methods, such as Heavy-ball, Nesterov's momentum, RMSprop and Adam. Building on this observation, we use stochastic differential equations (SDEs) to explicitly study the role of memory in gradient-based algorithms. We first derive a general continuous-time model that can incorporate arbitrary types of memory, for both deterministic and stochastic settings. We provide convergence guarantees for this SDE for weakly-quasi-convex and quadratically growing functions. We then demonstrate how to discretize this SDE to get a flexible discrete-time algorithm that can implement a board spectrum of memories ranging from short- to long-term. Not only does this algorithm increase the degrees of freedom in algorithmic choice for practitioners but it also comes with better stability properties than classical momentum in the convex stochastic setting. In particular, no iterate averaging is needed for convergence. Interestingly, our analysis also provides a novel interpretation of Nesterov's momentum as stable gradient amplification and highlights a possible reason for its unstable behavior in the (convex) stochastic setting. Furthermore, we discuss the use of long term memory for second-moment estimation in adaptive methods, such as Adam and RMSprop. Finally, we provide an extensive experimental study of the effect of different types of memory in both convex and nonconvex settings.
Computing equilibrium states in condensed-matter many-body systems, such as solvated proteins, is a long-standing challenge. Lacking methods for generating statistically independent equilibrium samples in "one shot", vast computational effort is invested for simulating these system in small steps, e.g., using Molecular Dynamics. Combining deep learning and statistical mechanics, we here develop Boltzmann Generators, that are shown to generate unbiased one-shot equilibrium samples of representative condensed matter systems and proteins. Boltzmann Generators use neural networks to learn a coordinate transformation of the complex configurational equilibrium distribution to a distribution that can be easily sampled. Accurate computation of free energy differences and discovery of new configurations are demonstrated, providing a statistical mechanics tool that can avoid rare events during sampling without prior knowledge of reaction coordinates.
Jonas Kohler, Hadi Daneshmand, Aurelien Lucchi et al.
Normalization techniques such as Batch Normalization have been applied successfully for training deep neural networks. Yet, despite its apparent empirical benefits, the reasons behind the success of Batch Normalization are mostly hypothetical. We here aim to provide a more thorough theoretical understanding from a classical optimization perspective. Our main contribution towards this goal is the identification of various problem instances in the realm of machine learning where % -- under certain assumptions-- Batch Normalization can provably accelerate optimization. We argue that this acceleration is due to the fact that Batch Normalization splits the optimization task into optimizing length and direction of the parameters separately. This allows gradient-based methods to leverage a favourable global structure in the loss landscape that we prove to exist in Learning Halfspace problems and neural network training with Gaussian inputs. We thereby turn Batch Normalization from an effective practical heuristic into a provably converging algorithm for these settings. Furthermore, we substantiate our analysis with empirical evidence that suggests the validity of our theoretical results in a broader context.
Hadi Daneshmand, Jonas Kohler, Aurelien Lucchi et al.
We analyze the variance of stochastic gradients along negative curvature directions in certain non-convex machine learning models and show that stochastic gradients exhibit a strong component along these directions. Furthermore, we show that - contrary to the case of isotropic noise - this variance is proportional to the magnitude of the corresponding eigenvalues and not decreasing in the dimensionality. Based upon this observation we propose a new assumption under which we show that the injection of explicit, isotropic noise usually applied to make gradient descent escape saddle points can successfully be replaced by a simple SGD step. Additionally - and under the same condition - we derive the first convergence rate for plain SGD to a second-order stationary point in a number of iterations that is independent of the problem dimension.
The success of convolutional networks in learning problems involving planar signals such as images is due to their ability to exploit the translation symmetry of the data distribution through weight sharing. Many areas of science and egineering deal with signals with other symmetries, such as rotation invariant data on the sphere. Examples include climate and weather science, astrophysics, and chemistry. In this paper we present spherical convolutional networks. These networks use convolutions on the sphere and rotation group, which results in rotational weight sharing and rotation equivariance. Using a synthetic spherical MNIST dataset, we show that spherical convolutional networks are very effective at dealing with rotationally invariant classification problems.
We consider the minimization of non-convex functions that typically arise in machine learning. Specifically, we focus our attention on a variant of trust region methods known as cubic regularization. This approach is particularly attractive because it escapes strict saddle points and it provides stronger convergence guarantees than first- and second-order as well as classical trust region methods. However, it suffers from a high computational complexity that makes it impractical for large-scale learning. Here, we propose a novel method that uses sub-sampling to lower this computational cost. By the use of concentration inequalities we provide a sampling scheme that gives sufficiently accurate gradient and Hessian approximations to retain the strong global and local convergence guarantees of cubically regularized methods. To the best of our knowledge this is the first work that gives global convergence guarantees for a sub-sampled variant of cubic regularization on non-convex functions. Furthermore, we provide experimental results supporting our theory.