Differentiable Tree Operations Promote Compositional Generalization

In the context of structure-to-structure transformation tasks, learning sequences of discrete symbolic operations poses significant challenges due to their non-differentiability. To facilitate the learning of these symbolic sequences, we introduce a differentiable tree interpreter that compiles high-level symbolic tree operations into subsymbolic matrix operations on tensors. We present a novel Differentiable Tree Machine (DTM) architecture that integrates our interpreter with an external memory and an agent that learns to sequentially select tree operations to execute the target transformation in an end-to-end manner. With respect to out-of-distribution compositional generalization on synthetic semantic parsing and language generation tasks, DTM achieves 100% while existing baselines such as Transformer, Tree Transformer, LSTM, and Tree2Tree LSTM achieve less than 30%. DTM remains highly interpretable in addition to its perfect performance.

GFlowNet-EM for learning compositional latent variable models

Latent variable models (LVMs) with discrete compositional latents are an important but challenging setting due to a combinatorially large number of possible configurations of the latents. A key tradeoff in modeling the posteriors over latents is between expressivity and tractable optimization. For algorithms based on expectation-maximization (EM), the E-step is often intractable without restrictive approximations to the posterior. We propose the use of GFlowNets, algorithms for sampling from an unnormalized density by learning a stochastic policy for sequential construction of samples, for this intractable E-step. By training GFlowNets to sample from the posterior over latents, we take advantage of their strengths as amortized variational inference algorithms for complex distributions over discrete structures. Our approach, GFlowNet-EM, enables the training of expressive LVMs with discrete compositional latents, as shown by experiments on non-context-free grammar induction and on images using discrete variational autoencoders (VAEs) without conditional independence enforced in the encoder.

GFlowNets and variational inference

This paper builds bridges between two families of probabilistic algorithms: (hierarchical) variational inference (VI), which is typically used to model distributions over continuous spaces, and generative flow networks (GFlowNets), which have been used for distributions over discrete structures such as graphs. We demonstrate that, in certain cases, VI algorithms are equivalent to special cases of GFlowNets in the sense of equality of expected gradients of their learning objectives. We then point out the differences between the two families and show how these differences emerge experimentally. Notably, GFlowNets, which borrow ideas from reinforcement learning, are more amenable than VI to off-policy training without the cost of high gradient variance induced by importance sampling. We argue that this property of GFlowNets can provide advantages for capturing diversity in multimodal target distributions.

Randomized Smoothing of All Shapes and Sizes

Randomized smoothing is a recently proposed defense against adversarial attacks that has achieved state-of-the-art provable robustness against $\ell_2$ perturbations. Soon after, a number of works devised new randomized smoothing schemes for other metrics, such as $\ell_1$ or $\ell_\infty$; however, for each geometry, substantial effort was needed to derive new robustness guarantees. This begs the question: can we find a general theory for randomized smoothing? In this work we propose a novel framework for devising and analyzing randomized smoothing schemes, and validate its effectiveness in practice. Our theoretical contributions are as follows: (1) We show that for an appropriate notion of "optimal", the optimal smoothing distributions for any "nice" norm have level sets given by the *Wulff Crystal* of that norm. (2) We propose two novel and complementary methods for deriving provably robust radii for any smoothing distribution. Finally, (3) we show fundamental limits to current randomized smoothing techniques via the theory of *Banach space cotypes*. By combining (1) and (2), we significantly improve the state-of-the-art certified accuracy in $\ell_1$ on standard datasets. On the other hand, using (3), we show that, without more information than label statistics under random input perturbations, randomized smoothing cannot achieve nontrivial certified accuracy against perturbations of $\ell_\infty$-norm $\Omega(1/\sqrt d)$, when the input dimension $d$ is large. We provide code in github.com/tonyduan/rs4a.