While forward reasoning (i.e. find the answer given the question) has been explored extensively in the recent literature, backward reasoning is relatively unexplored. We examine the backward reasoning capabilities of LLMs on Math Word Problems (MWPs): given a mathematical question and its answer, with some details omitted from the question, can LLMs effectively retrieve the missing information? In this paper, we formally define the backward reasoning task on math word problems and modify three datasets to evaluate this task: GSM8k, SVAMP and MultiArith. Our findings show a significant drop in the accuracy of models on backward reasoning compared to forward reasoning across four SOTA LLMs (GPT4, GPT3.5, PaLM-2, and LLaMa-2). Utilizing the specific format of this task, we propose three novel techniques that improve performance: Rephrase reformulates the given problem into a forward reasoning problem, PAL-Tools combines the idea of Program-Aided LLMs to produce a set of equations that can be solved by an external solver, and Check your Work exploits the availability of natural verifier of high accuracy in the forward direction, interleaving solving and verification steps. Finally, realizing that each of our base methods correctly solves a different set of problems, we propose a novel Bayesian formulation for creating an ensemble over these base methods aided by a verifier to further boost the accuracy by a significant margin. Extensive experimentation demonstrates that our techniques successively improve the performance of LLMs on the backward reasoning task, with the final ensemble-based method resulting in a substantial performance gain compared to the raw LLMs with standard prompting techniques such as chain-of-thought.

Model-based reinforcement learning seeks to simultaneously learn the dynamics of an unknown stochastic environment and synthesise an optimal policy for acting in it. Ensuring the safety and robustness of sequential decisions made through a policy in such an environment is a key challenge for policies intended for safety-critical scenarios. In this work, we investigate two complementary problems: first, computing reach-avoid probabilities for iterative predictions made with dynamical models, with dynamics described by Bayesian neural network (BNN); second, synthesising control policies that are optimal with respect to a given reach-avoid specification (reaching a "target" state, while avoiding a set of "unsafe" states) and a learned BNN model. Our solution leverages interval propagation and backward recursion techniques to compute lower bounds for the probability that a policy's sequence of actions leads to satisfying the reach-avoid specification. Such computed lower bounds provide safety certification for the given policy and BNN model. We then introduce control synthesis algorithms to derive policies maximizing said lower bounds on the safety probability. We demonstrate the effectiveness of our method on a series of control benchmarks characterized by learned BNN dynamics models. On our most challenging benchmark, compared to purely data-driven policies the optimal synthesis algorithm is able to provide more than a four-fold increase in the number of certifiable states and more than a three-fold increase in the average guaranteed reach-avoid probability.

Bayesian personalized federated learning (BPFL) addresses challenges in existing personalized FL (PFL). BPFL aims to quantify the uncertainty and heterogeneity within and across clients towards uncertainty representations by addressing the statistical heterogeneity of client data. In PFL, some recent preliminary work proposes to decompose hidden neural representations into shared and local components and demonstrates interesting results. However, most of them do not address client uncertainty and heterogeneity in FL systems, while appropriately decoupling neural representations is challenging and often ad hoc. In this paper, we make the first attempt to introduce a general BPFL framework to decompose and jointly learn shared and personalized uncertainty representations on statistically heterogeneous client data over time. A Bayesian federated neural network BPFed instantiates BPFL by jointly learning cross-client shared uncertainty and client-specific personalized uncertainty over statistically heterogeneous and randomly participating clients. We further involve continual updating of prior distribution in BPFed to speed up the convergence and avoid catastrophic forgetting. Theoretical analysis and guarantees are provided in addition to the experimental evaluation of BPFed against the diversified baselines.

There has been an increasing interest in enhancing the fairness of machine learning (ML). Despite the growing number of fairness-improving methods, we lack a systematic understanding of the trade-offs among factors considered in the ML pipeline when fairness-improving methods are applied. This understanding is essential for developers to make informed decisions regarding the provision of fair ML services. Nonetheless, it is extremely difficult to analyze the trade-offs when there are multiple fairness parameters and other crucial metrics involved, coupled, and even in conflict with one another. This paper uses causality analysis as a principled method for analyzing trade-offs between fairness parameters and other crucial metrics in ML pipelines. To ractically and effectively conduct causality analysis, we propose a set of domain-specific optimizations to facilitate accurate causal discovery and a unified, novel interface for trade-off analysis based on well-established causal inference methods. We conduct a comprehensive empirical study using three real-world datasets on a collection of widelyused fairness-improving techniques. Our study obtains actionable suggestions for users and developers of fair ML. We further demonstrate the versatile usage of our approach in selecting the optimal fairness-improving method, paving the way for more ethical and socially responsible AI technologies.

Training reinforcement learning (RL) agents using scalar reward signals is often infeasible when an environment has sparse and non-Markovian rewards. Moreover, handcrafting these reward functions before training is prone to misspecification, especially when the environment's dynamics are only partially known. This paper proposes a novel pipeline for learning non-Markovian task specifications as succinct finite-state `task automata' from episodes of agent experience within unknown environments. We leverage two key algorithmic insights. First, we learn a product MDP, a model composed of the specification's automaton and the environment's MDP (both initially unknown), by treating the product MDP as a partially observable MDP and using the well-known Baum-Welch algorithm for learning hidden Markov models. Second, we propose a novel method for distilling the task automaton (assumed to be a deterministic finite automaton) from the learnt product MDP. Our learnt task automaton enables the decomposition of a task into its constituent sub-tasks, which improves the rate at which an RL agent can later synthesise an optimal policy. It also provides an interpretable encoding of high-level environmental and task features, so a human can readily verify that the agent has learnt coherent tasks with no misspecifications. In addition, we take steps towards ensuring that the learnt automaton is environment-agnostic, making it well-suited for use in transfer learning. Finally, we provide experimental results compared with two baselines to illustrate our algorithm's performance in different environments and tasks.

We present a new algorithm for amortized inference in sparse probabilistic graphical models (PGMs), which we call $\Delta$-amortized inference ($\Delta$-AI). Our approach is based on the observation that when the sampling of variables in a PGM is seen as a sequence of actions taken by an agent, sparsity of the PGM enables local credit assignment in the agent's policy learning objective. This yields a local constraint that can be turned into a local loss in the style of generative flow networks (GFlowNets) that enables off-policy training but avoids the need to instantiate all the random variables for each parameter update, thus speeding up training considerably. The $\Delta$-AI objective matches the conditional distribution of a variable given its Markov blanket in a tractable learned sampler, which has the structure of a Bayesian network, with the same conditional distribution under the target PGM. As such, the trained sampler recovers marginals and conditional distributions of interest and enables inference of partial subsets of variables. We illustrate $\Delta$-AI's effectiveness for sampling from synthetic PGMs and training latent variable models with sparse factor structure.

In estimation theory, the Kushner equation provides the evolution of the probability density of the state of a dynamical system given continuous-time observations. Building upon our recent work, we propose a new way to approximate the solution of the Kushner equation through tractable variational Gaussian approximations of two proximal losses associated with the propagation and Bayesian update of the probability density. The first is a proximal loss based on the Wasserstein metric and the second is a proximal loss based on the Fisher metric. The solution to this last proximal loss is given by implicit updates on the mean and covariance that we proposed earlier. These two variational updates can be fused and shown to satisfy a set of stochastic differential equations on the Gaussian's mean and covariance matrix. This Gaussian flow is consistent with the Kalman-Bucy and Riccati flows in the linear case and generalize them in the nonlinear one.

Point processes offer a versatile framework for sequential event modeling. However, the computational challenges and constrained representational power of the existing point process models have impeded their potential for wider applications. This limitation becomes especially pronounced when dealing with event data that is associated with multi-dimensional or high-dimensional marks such as texts or images. To address this challenge, this study proposes a novel event generative framework for modeling point processes with high-dimensional marks. We aim to capture the distribution of events without explicitly specifying the conditional intensity or probability density function. Instead, we use a conditional generator that takes the history of events as input and generates the high-quality subsequent event that is likely to occur given the prior observations. The proposed framework offers a host of benefits, including considerable representational power to capture intricate dynamics in multi- or even high-dimensional event space, as well as exceptional efficiency in learning the model and generating samples. Our numerical results demonstrate superior performance compared to other state-of-the-art baselines.

In this paper, we introduce an innovative approach for addressing Bayesian inverse problems through the utilization of physics-informed invertible neural networks (PI-INN). The PI-INN framework encompasses two sub-networks: an invertible neural network (INN) and a neural basis network (NB-Net). The primary role of the NB-Net lies in modeling the spatial basis functions characterizing the solution to the forward problem dictated by the underlying partial differential equation. Simultaneously, the INN is designed to partition the parameter vector linked to the input physical field into two distinct components: the expansion coefficients representing the forward problem solution and the Gaussian latent noise. If the forward mapping is precisely estimated, and the statistical independence between expansion coefficients and latent noise is well-maintained, the PI-INN offers a precise and efficient generative model for Bayesian inverse problems, yielding tractable posterior density estimates. As a particular physics-informed deep learning model, the primary training challenge for PI-INN centers on enforcing the independence constraint, which we tackle by introducing a novel independence loss based on estimated density. We support the efficacy and precision of the proposed PI-INN through a series of numerical experiments, including inverse kinematics, 1-dimensional and 2-dimensional diffusion equations, and seismic traveltime tomography. Specifically, our experimental results showcase the superior performance of the proposed independence loss in comparison to the commonly used but computationally demanding kernel-based maximum mean discrepancy loss.

Recent work has considered trust-aware decision making for human-robot collaboration (HRC) with a focus on model learning. In this paper, we are interested in enabling the HRC system to complete complex tasks specified using temporal logic that involve human trust. Since human trust in robots is not observable, we adopt the widely used partially observable Markov decision process (POMDP) framework for modelling the interactions between humans and robots. To specify the desired behaviour, we propose to use syntactically co-safe linear distribution temporal logic (scLDTL), a logic that is defined over predicates of states as well as belief states of partially observable systems. The incorporation of belief predicates in scLDTL enhances its expressiveness while simultaneously introducing added complexity. This also presents a new challenge as the belief predicates must be evaluated over the continuous (infinite) belief space. To address this challenge, we present an algorithm for solving the optimal policy synthesis problem. First, we enhance the belief MDP (derived by reformulating the POMDP) with a probabilistic labelling function. Then a product belief MDP is constructed between the probabilistically labelled belief MDP and the automaton translation of the scLDTL formula. Finally, we show that the optimal policy can be obtained by leveraging existing point-based value iteration algorithms with essential modifications. Human subject experiments with 21 participants on a driving simulator demonstrate the effectiveness of the proposed approach.