Beren Millidge

Hassan Shapourian, Kasra Hejazi, Olabode M. Sule et al.

We present Zamba2-VL, a suite of vision-language models built on Zamba2, a hybrid language-model architecture combining Mamba2 state-space layers with a small number of shared transformer blocks. Across a broad range of image understanding, reasoning, OCR, grounding, and counting benchmarks, Zamba2-VL is competitive with leading Transformer-based open-weight VLMs of comparable scale, including the Molmo2, Qwen3-VL, and InternVL3.5 families, and substantially outperforms prior SSM-based and hybrid VLMs such as VL-Mamba, Cobra, and mmMamba. Inheriting the near-linear prefill compute and small, near-constant recurrent state of its Zamba2 backbone, Zamba2-VL delivers roughly an order of magnitude lower time-to-first-token (TTFT) than these Transformer baselines at matched parameter scale, with the efficiency gap most pronounced at the smaller 1.2B and 2.7B scales most relevant to on-device and edge deployment. We release three models -- 1.2B, 2.7B, and 7B -- together with inference code at https://huggingface.co/collections/Zyphra/zamba2-vl.

Tommaso Salvatori, Yuhang Song, Yordan Yordanov et al. · oxford

Predictive coding networks are neuroscience-inspired models with roots in both Bayesian statistics and neuroscience. Training such models, however, is quite inefficient and unstable. In this work, we show how by simply changing the temporal scheduling of the update rule for the synaptic weights leads to an algorithm that is much more efficient and stable than the original one, and has theoretical guarantees in terms of convergence. The proposed algorithm, that we call incremental predictive coding (iPC) is also more biologically plausible than the original one, as it it fully automatic. In an extensive set of experiments, we show that iPC constantly performs better than the original formulation on a large number of benchmarks for image classification, as well as for the training of both conditional and masked language models, in terms of test accuracy, efficiency, and convergence with respect to a large set of hyperparameters.

Luca Pinchetti, Tommaso Salvatori, Yordan Yordanov et al. · oxford

A large amount of recent research has the far-reaching goal of finding training methods for deep neural networks that can serve as alternatives to backpropagation (BP). A prominent example is predictive coding (PC), which is a neuroscience-inspired method that performs inference on hierarchical Gaussian generative models. These methods, however, fail to keep up with modern neural networks, as they are unable to replicate the dynamics of complex layers and activation functions. In this work, we solve this problem by generalizing PC to arbitrary probability distributions, enabling the training of architectures, such as transformers, that are hard to approximate with only Gaussian assumptions. We perform three experimental analyses. First, we study the gap between our method and the standard formulation of PC on multiple toy examples. Second, we test the reconstruction quality on variational autoencoders, where our method reaches the same reconstruction quality as BP. Third, we show that our method allows us to train transformer networks and achieve a performance comparable with BP on conditional language models. More broadly, this method allows neuroscience-inspired learning to be applied to multiple domains, since the internal distributions can be flexibly adapted to the data, tasks, and architectures used.

Vasudev Shyam, Jonathan Pilault, Emily Shepperd et al.

Our formulation reveals that the reduction across the sequence axis can be efficiently computed in parallel through a tree reduction. Our algorithm, called Tree Attention, for parallelizing exact attention computation across multiple GPUs enables cross-device decoding to be performed asymptotically faster (up to 8x faster in our experiments) than state-of-the-art approaches such as Ring Attention, while also requiring significantly less communication volume and incurring 2x less peak memory. We demonstrate that Tree Attention speeds up decoding up to 4x on Llama 3.1-8B and can be applied to a variety of hardware and networking setups such as H100 DGX nodes, AMD MI300x nodes, and PCIe connected NVIDIA RTX 4090s. Our code is publicly available here: https://github.com/Zyphra/tree_attention

Karl J Friston, Maxwell J D Ramstead, Alex B Kiefer et al.

This white paper lays out a vision of research and development in the field of artificial intelligence for the next decade (and beyond). Its denouement is a cyber-physical ecosystem of natural and synthetic sense-making, in which humans are integral participants -- what we call ''shared intelligence''. This vision is premised on active inference, a formulation of adaptive behavior that can be read as a physics of intelligence, and which inherits from the physics of self-organization. In this context, we understand intelligence as the capacity to accumulate evidence for a generative model of one's sensed world -- also known as self-evidencing. Formally, this corresponds to maximizing (Bayesian) model evidence, via belief updating over several scales: i.e., inference, learning, and model selection. Operationally, this self-evidencing can be realized via (variational) message passing or belief propagation on a factor graph. Crucially, active inference foregrounds an existential imperative of intelligent systems; namely, curiosity or the resolution of uncertainty. This same imperative underwrites belief sharing in ensembles of agents, in which certain aspects (i.e., factors) of each agent's generative world model provide a common ground or frame of reference. Active inference plays a foundational role in this ecology of belief sharing -- leading to a formal account of collective intelligence that rests on shared narratives and goals. We also consider the kinds of communication protocols that must be developed to enable such an ecosystem of intelligences and motivate the development of a shared hyper-spatial modeling language and transaction protocol, as a first -- and key -- step towards such an ecology.

Alexander Tschantz, Beren Millidge, Anil K Seth et al.

Predictive coding is an influential model of cortical neural activity. It proposes that perceptual beliefs are furnished by sequentially minimising "prediction errors" - the differences between predicted and observed data. Implicit in this proposal is the idea that perception requires multiple cycles of neural activity. This is at odds with evidence that several aspects of visual perception - including complex forms of object recognition - arise from an initial "feedforward sweep" that occurs on fast timescales which preclude substantial recurrent activity. Here, we propose that the feedforward sweep can be understood as performing amortized inference and recurrent processing can be understood as performing iterative inference. We propose a hybrid predictive coding network that combines both iterative and amortized inference in a principled manner by describing both in terms of a dual optimization of a single objective function. We show that the resulting scheme can be implemented in a biologically plausible neural architecture that approximates Bayesian inference utilising local Hebbian update rules. We demonstrate that our hybrid predictive coding model combines the benefits of both amortized and iterative inference -- obtaining rapid and computationally cheap perceptual inference for familiar data while maintaining the context-sensitivity, precision, and sample efficiency of iterative inference schemes. Moreover, we show how our model is inherently sensitive to its uncertainty and adaptively balances iterative and amortized inference to obtain accurate beliefs using minimum computational expense. Hybrid predictive coding offers a new perspective on the functional relevance of the feedforward and recurrent activity observed during visual perception and offers novel insights into distinct aspects of visual phenomenology.

Beren Millidge, Yuhang Song, Tommaso Salvatori et al.

How the brain performs credit assignment is a fundamental unsolved problem in neuroscience. Many `biologically plausible' algorithms have been proposed, which compute gradients that approximate those computed by backpropagation (BP), and which operate in ways that more closely satisfy the constraints imposed by neural circuitry. Many such algorithms utilize the framework of energy-based models (EBMs), in which all free variables in the model are optimized to minimize a global energy function. However, in the literature, these algorithms exist in isolation and no unified theory exists linking them together. Here, we provide a comprehensive theory of the conditions under which EBMs can approximate BP, which lets us unify many of the BP approximation results in the literature (namely, predictive coding, equilibrium propagation, and contrastive Hebbian learning) and demonstrate that their approximation to BP arises from a simple and general mathematical property of EBMs at free-phase equilibrium. This property can then be exploited in different ways with different energy functions, and these specific choices yield a family of BP-approximating algorithms, which both includes the known results in the literature and can be used to derive new ones.

Beren Millidge, Yuhang Song, Tommaso Salvatori et al.

Predictive coding (PC) is an influential theory in computational neuroscience, which argues that the cortex forms unsupervised world models by implementing a hierarchical process of prediction error minimization. PC networks (PCNs) are trained in two phases. First, neural activities are updated to optimize the network's response to external stimuli. Second, synaptic weights are updated to consolidate this change in activity -- an algorithm called \emph{prospective configuration}. While previous work has shown how in various limits, PCNs can be found to approximate backpropagation (BP), recent work has demonstrated that PCNs operating in this standard regime, which does not approximate BP, nevertheless obtain competitive training and generalization performance to BP-trained networks while outperforming them on tasks such as online, few-shot, and continual learning, where brains are known to excel. Despite this promising empirical performance, little is understood theoretically about the properties and dynamics of PCNs in this regime. In this paper, we provide a comprehensive theoretical analysis of the properties of PCNs trained with prospective configuration. We first derive analytical results concerning the inference equilibrium for PCNs and a previously unknown close connection relationship to target propagation (TP). Secondly, we provide a theoretical analysis of learning in PCNs as a variant of generalized expectation-maximization and use that to prove the convergence of PCNs to critical points of the BP loss function, thus showing that deep PCNs can, in theory, achieve the same generalization performance as BP, while maintaining their unique advantages.

Nick Alonso, Beren Millidge, Jeff Krichmar et al.

Backpropagation (BP) is the most successful and widely used algorithm in deep learning. However, the computations required by BP are challenging to reconcile with known neurobiology. This difficulty has stimulated interest in more biologically plausible alternatives to BP. One such algorithm is the inference learning algorithm (IL). IL has close connections to neurobiological models of cortical function and has achieved equal performance to BP on supervised learning and auto-associative tasks. In contrast to BP, however, the mathematical foundations of IL are not well-understood. Here, we develop a novel theoretical framework for IL. Our main result is that IL closely approximates an optimization method known as implicit stochastic gradient descent (implicit SGD), which is distinct from the explicit SGD implemented by BP. Our results further show how the standard implementation of IL can be altered to better approximate implicit SGD. Our novel implementation considerably improves the stability of IL across learning rates, which is consistent with our theory, as a key property of implicit SGD is its stability. We provide extensive simulation results that further support our theoretical interpretations and also demonstrate IL achieves quicker convergence when trained with small mini-batches while matching the performance of BP for large mini-batches.

Tommaso Salvatori, Luca Pinchetti, Amine M'Charrak et al.

Recently, there has been extensive research on the capabilities of biologically plausible algorithms. In this work, we show how one of such algorithms, called predictive coding, is able to perform causal inference tasks. First, we show how a simple change in the inference process of predictive coding enables to compute interventions without the need to mutilate or redefine a causal graph. Then, we explore applications in cases where the graph is unknown, and has to be inferred from observational data. Empirically, we show how such findings can be used to improve the performance of predictive coding in image classification tasks, and conclude that such models are able to perform simple end-to-end causal inference tasks.

Sid Black, Lee Sharkey, Leo Grinsztajn et al.

Mechanistic interpretability aims to explain what a neural network has learned at a nuts-and-bolts level. What are the fundamental primitives of neural network representations? Previous mechanistic descriptions have used individual neurons or their linear combinations to understand the representations a network has learned. But there are clues that neurons and their linear combinations are not the correct fundamental units of description: directions cannot describe how neural networks use nonlinearities to structure their representations. Moreover, many instances of individual neurons and their combinations are polysemantic (i.e. they have multiple unrelated meanings). Polysemanticity makes interpreting the network in terms of neurons or directions challenging since we can no longer assign a specific feature to a neural unit. In order to find a basic unit of description that does not suffer from these problems, we zoom in beyond just directions to study the way that piecewise linear activation functions (such as ReLU) partition the activation space into numerous discrete polytopes. We call this perspective the polytope lens. The polytope lens makes concrete predictions about the behavior of neural networks, which we evaluate through experiments on both convolutional image classifiers and language models. Specifically, we show that polytopes can be used to identify monosemantic regions of activation space (while directions are not in general monosemantic) and that the density of polytope boundaries reflect semantic boundaries. We also outline a vision for what mechanistic interpretability might look like through the polytope lens.

Paul F Kinghorn, Beren Millidge, Christopher L Buckley

Predictive Coding Networks (PCNs) aim to learn a generative model of the world. Given observations, this generative model can then be inverted to infer the causes of those observations. However, when training PCNs, a noticeable pathology is often observed where inference accuracy peaks and then declines with further training. This cannot be explained by overfitting since both training and test accuracy decrease simultaneously. Here we provide a thorough investigation of this phenomenon and show that it is caused by an imbalance between the speeds at which the various layers of the PCN converge. We demonstrate that this can be prevented by regularising the weight matrices at each layer: by restricting the relative size of matrix singular values, we allow the weight matrix to change but restrict the overall impact which a layer can have on its neighbours. We also demonstrate that a similar effect can be achieved through a more biologically plausible and simple scheme of just capping the weights.

Beren Millidge, Christopher L Buckley

Recent work has uncovered close links between between classical reinforcement learning algorithms, Bayesian filtering, and Active Inference which lets us understand value functions in terms of Bayesian posteriors. An alternative, but less explored, model-free RL algorithm is the successor representation, which expresses the value function in terms of a successor matrix of expected future state occupancies. In this paper, we derive the probabilistic interpretation of the successor representation in terms of Bayesian filtering and thus design a novel active inference agent architecture utilizing successor representations instead of model-based planning. We demonstrate that active inference successor representations have significant advantages over current active inference agents in terms of planning horizon and computational cost. Moreover, we demonstrate how the successor representation agent can generalize to changing reward functions such as variants of the expected free energy.

Hassan Shapourian, Kasra Hejazi, Olabode M. Sule et al.

We present ZAYA1-VL-8B, a compact mixture-of-experts vision-language model built upon our in-house language model, ZAYA1-8B. Despite its compact size, ZAYA1-VL achieves performance competitive with leading base models such as Molmo2-4B and InternVL3.5-4B, while surpassing models including Qwen2.5-VL-3B, PLM-3B, and MolmoE-1B across a range of image understanding, reasoning, and counting benchmarks. The architecture incorporates two key innovations: (1) vision-specific LoRA adapters integrated into the LLM to increase modality-specific capacity without increasing the number of experts, and (2) bidirectional attention over image tokens within the LLM to enhance visual understanding. We detail the full training pipeline including data composition at each stage, sequence packing, and the attention masking scheme. The model comprises 9.2B total parameters, with 1.4B active parameters including the vision encoder, and is publicly available at https://huggingface.co/Zyphra/ZAYA1-VL.

Leon Lufkin, Tomás Figliolia, Beren Millidge et al.

Recurrent neural networks (RNNs) and self-attention are both widely used sequence-mixing layers that maintain an internal memory. However, this memory is constructed using two orthogonal mechanisms: RNNs compress the entire past into a fixed-size state, whereas self-attention's state stores every past time step growing its state (the KV cache) linearly with the sequence length. This results in orthogonal strengths and weaknesses. Self-attention layers excel at retrieving information in the context but have large memory and computational costs, while RNNs are more efficient but degrade over longer contexts and underperform for precise recall tasks. Prior work combining these mechanisms has focused primarily on naively interleaving them to reduce computational cost without regard to their complementary mechanisms. We propose the Hybrid Associative Memory (HAM) layer, which combines self-attention and RNNs while leveraging their individual strengths: the RNN compresses the entire sequence, while attention supplements it *only* with information that is difficult for the RNN to predict, which is hence the most valuable information to explicitly store. HAM layers enable data-dependent growth of the KV cache, which can be precisely controlled by the user with a single, continuous threshold. We find that this fine-grained control of the KV cache growth rate has a smooth trade-off with loss and performance. Empirically, we show that our hybrid architecture offers strong, competitive performance relative to RNNs and Transformers even at substantially lower KV-cache usage.

Miguel de Llanza Varona, Christopher L. Buckley, Beren Millidge

Organisms have to keep track of the information in the environment that is relevant for adaptive behaviour. Transmitting information in an economical and efficient way becomes crucial for limited-resourced agents living in high-dimensional environments. The efficient coding hypothesis claims that organisms seek to maximize the information about the sensory input in an efficient manner. Under Bayesian inference, this means that the role of the brain is to efficiently allocate resources in order to make predictions about the hidden states that cause sensory data. However, neither of those frameworks accounts for how that information is exploited downstream, leaving aside the action-oriented role of the perceptual system. Rate-distortion theory, which defines optimal lossy compression under constraints, has gained attention as a formal framework to explore goal-oriented efficient coding. In this work, we explore action-centric representations in the context of rate-distortion theory. We also provide a mathematical definition of abstractions and we argue that, as a summary of the relevant details, they can be used to fix the content of action-centric representations. We model action-centric representations using VAEs and we find that such representations i) are efficient lossy compressions of the data; ii) capture the task-dependent invariances necessary to achieve successful behaviour; and iii) are not in service of reconstructing the data. Thus, we conclude that full reconstruction of the data is rarely needed to achieve optimal behaviour, consistent with a teleological approach to perception.

Quentin Anthony, Yury Tokpanov, Paolo Glorioso et al.

State-space models (SSMs) have recently demonstrated competitive performance to transformers at large-scale language modeling benchmarks while achieving linear time and memory complexity as a function of sequence length. Mamba, a recently released SSM model, shows impressive performance in both language modeling and long sequence processing tasks. Simultaneously, mixture-of-expert (MoE) models have shown remarkable performance while significantly reducing the compute and latency costs of inference at the expense of a larger memory footprint. In this paper, we present BlackMamba, a novel architecture that combines the Mamba SSM with MoE to obtain the benefits of both. We demonstrate that BlackMamba performs competitively against both Mamba and transformer baselines, and outperforms in inference and training FLOPs. We fully train and open-source 340M/1.5B and 630M/2.8B BlackMamba models on 300B tokens of a custom dataset. We show that BlackMamba inherits and combines both of the benefits of SSM and MoE architectures, combining linear-complexity generation from SSM with cheap and fast inference from MoE. We release all weights, checkpoints, and inference code open-source. Inference code at: https://github.com/Zyphra/BlackMamba

Paolo Glorioso, Quentin Anthony, Yury Tokpanov et al.

In this technical report, we present the Zamba2 series -- a suite of 1.2B, 2.7B, and 7.4B parameter hybrid Mamba2-transformer models that achieve state of the art performance against the leading open-weights models of their class, while achieving substantial gains in inference latency, throughput, and memory efficiency. The Zamba2 series builds upon our initial work with Zamba1-7B, optimizing its architecture, training and annealing datasets, and training for up to three trillion tokens. We provide open-source weights for all models of the Zamba2 series as well as instruction-tuned variants that are strongly competitive against comparable instruct-tuned models of their class. We additionally open-source the pretraining dataset, which we call Zyda-2, used to train the Zamba2 series of models. The models and datasets used in this work are openly available at https://huggingface.co/Zyphra

Yury Tokpanov, Paolo Glorioso, Quentin Anthony et al.

In this technical report, we present Zyda-2: a five trillion token dataset for language model pretraining. Zyda-2 was used to train our Zamba2 series of models which are state-of-the-art for their weight class. We build Zyda-2 by collating high-quality open-source tokens such as FineWeb and DCLM, then distilling them to the highest-quality subset via cross-deduplication and model-based quality filtering. Zyda-2 is released under a permissive open license, and is available at https://huggingface.co/datasets/Zyphra/Zyda-2

Quentin Anthony, Yury Tokpanov, Skyler Szot et al.

We report on the first large-scale mixture-of-experts (MoE) pretraining study on pure AMD hardware, utilizing both MI300X GPUs and Pollara networking. We distill practical guidance for both systems and model design. On the systems side, we deliver a comprehensive cluster and networking characterization: microbenchmarks for all core collectives (all-reduce, reduce-scatter, all-gather, broadcast) across message sizes and GPU counts over Pollara. To our knowledge, this is the first at this scale. We further provide MI300X microbenchmarks on kernel sizing and memory bandwidth to inform model design. On the modeling side, we introduce and apply MI300X-aware transformer sizing rules for attention and MLP blocks and justify MoE widths that jointly optimize training throughput and inference latency. We describe our training stack in depth, including often-ignored utilities such as fault-tolerance and checkpoint-reshaping, as well as detailed information on our training recipe. We also provide a preview of our model architecture and base model - ZAYA1 (760M active, 8.3B total parameters MoE, available at https://huggingface.co/Zyphra/ZAYA1-base) - which will be further improved upon in forthcoming papers. ZAYA1-base achieves performance comparable to leading base models such as Qwen3-4B and Gemma3-12B at its scale and larger, and outperforms models including Llama-3-8B and OLMoE across reasoning, mathematics, and coding benchmarks. Together, these results demonstrate that the AMD hardware, network, and software stack are mature and optimized enough for competitive large-scale pretraining.

Yury Tokpanov, Beren Millidge, Paolo Glorioso et al.

The size of large language models (LLMs) has scaled dramatically in recent years and their computational and data requirements have surged correspondingly. State-of-the-art language models, even at relatively smaller sizes, typically require training on at least a trillion tokens. This rapid advancement has eclipsed the growth of open-source datasets available for large-scale LLM pretraining. In this paper, we introduce Zyda (Zyphra Dataset), a dataset under a permissive license comprising 1.3 trillion tokens, assembled by integrating several major respected open-source datasets into a single, high-quality corpus. We apply rigorous filtering and deduplication processes, both within and across datasets, to maintain and enhance the quality derived from the original datasets. Our evaluations show that Zyda not only competes favorably with other open datasets like Dolma, FineWeb, and RefinedWeb, but also substantially improves the performance of comparable models from the Pythia suite. Our rigorous data processing methods significantly enhance Zyda's effectiveness, outperforming even the best of its constituent datasets when used independently.

Conor Heins, Beren Millidge, Daphne Demekas et al.

Active inference is an account of cognition and behavior in complex systems which brings together action, perception, and learning under the theoretical mantle of Bayesian inference. Active inference has seen growing applications in academic research, especially in fields that seek to model human or animal behavior. While in recent years, some of the code arising from the active inference literature has been written in open source languages like Python and Julia, to-date, the most popular software for simulating active inference agents is the DEM toolbox of SPM, a MATLAB library originally developed for the statistical analysis and modelling of neuroimaging data. Increasing interest in active inference, manifested both in terms of sheer number as well as diversifying applications across scientific disciplines, has thus created a need for generic, widely-available, and user-friendly code for simulating active inference in open-source scientific computing languages like Python. The Python package we present here, pymdp (see https://github.com/infer-actively/pymdp), represents a significant step in this direction: namely, we provide the first open-source package for simulating active inference with partially-observable Markov Decision Processes or POMDPs. We review the package's structure and explain its advantages like modular design and customizability, while providing in-text code blocks along the way to demonstrate how it can be used to build and run active inference processes with ease. We developed pymdp to increase the accessibility and exposure of the active inference framework to researchers, engineers, and developers with diverse disciplinary backgrounds. In the spirit of open-source software, we also hope that it spurs new innovation, development, and collaboration in the growing active inference community.

Robert Washbourne, Rishi Iyer, Tomas Figliolia et al.

We present ZAYA1-8B, a reasoning-focused mixture-of-experts (MoE) model with 700M active and 8B total parameters, built on Zyphra's MoE++ architecture. ZAYA1-8B's core pretraining, midtraining, and supervised fine-tuning (SFT) were performed on a full-stack AMD compute, networking, and software platform. With under 1B active parameters, ZAYA1-8B matches or exceeds DeepSeek-R1-0528 on several challenging mathematics and coding benchmarks, and remains competitive with substantially larger open-weight reasoning models. ZAYA1-8B was trained from scratch for reasoning, with reasoning data included from pretraining onward using an answer-preserving trimming scheme. Post-training uses a four-stage RL cascade: reasoning warmup on math and puzzles; a 400-task RLVE-Gym curriculum; math and code RL with test-time compute traces and synthetic code environments built from competitive-programming references; and behavioral RL for chat and instruction following. We also introduce Markovian RSA, a test-time compute method that recursively aggregates parallel reasoning traces while carrying forward only bounded-length reasoning tails between rounds. In TTC evaluation, Markovian RSA raises ZAYA1-8B to 91.9\% on AIME'25 and 89.6\% on HMMT'25 while carrying forward only a 4K-token tail, narrowing the gap to much larger reasoning models including Gemini-2.5 Pro, DeepSeek-V3.2, and GPT-5-High.

Alexander Ororbia, Ankur Mali, Adam Kohan et al.

One major criticism of deep learning centers around the biological implausibility of the credit assignment schema used for learning -- backpropagation of errors. This implausibility translates into practical limitations, spanning scientific fields, including incompatibility with hardware and non-differentiable implementations, thus leading to expensive energy requirements. In contrast, biologically plausible credit assignment is compatible with practically any learning condition and is energy-efficient. As a result, it accommodates hardware and scientific modeling, e.g. learning with physical systems and non-differentiable behavior. Furthermore, it can lead to the development of real-time, adaptive neuromorphic processing systems. In addressing this problem, an interdisciplinary branch of artificial intelligence research that lies at the intersection of neuroscience, cognitive science, and machine learning has emerged. In this paper, we survey several vital algorithms that model bio-plausible rules of credit assignment in artificial neural networks, discussing the solutions they provide for different scientific fields as well as their advantages on CPUs, GPUs, and novel implementations of neuromorphic hardware. We conclude by discussing the future challenges that will need to be addressed in order to make such algorithms more useful in practical applications.

Nicholas Alonso, Beren Millidge

Recent advances have extended the context window of frontier LLMs dramatically, from a few thousand tokens up to millions, enabling entire books and codebases to fit into context. However, the compute costs of inferencing long-context LLMs are massive and often prohibitive in practice. RAG offers an efficient and effective alternative: retrieve and process only the subset of the context most important for the current task. Although promising, recent work applying RAG to long-context tasks has two core limitations: 1) there has been little focus on making the RAG pipeline compute efficient, and 2) such works only test on simple QA tasks, and their performance on more challenging tasks is unclear. To address this, we develop an algorithm based on PageRank, a graph-based retrieval algorithm, which we call mixture-of-PageRanks (MixPR). MixPR uses a mixture of PageRank-based graph-retrieval algorithms implemented using sparse matrices for efficent, cheap retrieval that can deal with a variety of complex tasks. Our MixPR retriever achieves state-of-the-art results across a wide range of long-context benchmark tasks, outperforming both existing RAG methods, specialized retrieval architectures, and long-context LLMs despite being far more compute efficient. Due to using sparse embeddings, our retriever is extremely compute efficient, capable of embedding and retrieving millions of tokens within a few seconds and runs entirely on CPU.

Tommaso Salvatori, Beren Millidge, Yuhang Song et al.

An autoassociative memory model is a function that, given a set of data points, takes as input an arbitrary vector and outputs the most similar data point from the memorized set. However, popular memory models fail to retrieve images even when the corruption is mild and easy to detect for a human evaluator. This is because similarities are evaluated in the raw pixel space, which does not contain any semantic information about the images. This problem can be easily solved by computing \emph{similarities} in an embedding space instead of the pixel space. We show that an effective way of computing such embeddings is via a network pretrained with a contrastive loss. As the dimension of embedding spaces is often significantly smaller than the pixel space, we also have a faster computation of similarity scores. We test this method on complex datasets such as CIFAR10 and STL10. An additional drawback of current models is the need of storing the whole dataset in the pixel space, which is often extremely large. We relax this condition and propose a class of memory models that only stores low-dimensional semantic embeddings, and uses them to retrieve similar, but not identical, memories. We demonstrate a proof of concept of this method on a simple task on the MNIST dataset.

Beren Millidge

Recurrent networks are typically trained with backpropagation through time (BPTT). However, BPTT requires storing the history of all states in the network and then replaying them sequentially backwards in time. This computation appears extremely implausible for the brain to implement. Real Time Recurrent Learning (RTRL) proposes an mathematically equivalent alternative where gradient information is propagated forwards in time locally alongside the regular forward pass, however it has significantly greater computational complexity than BPTT which renders it impractical for large networks. E-prop proposes an approximation of RTRL which reduces its complexity to the level of BPTT while maintaining a purely online forward update which can be implemented by an eligibility trace at each synapse. However, works on RTRL and E-prop ubiquitously investigate learning in a single layer with recurrent dynamics. However, learning in the brain spans multiple layers and consists of both hierarchal dynamics in depth as well as time. In this mathematical note, we extend the E-prop framework to handle arbitrarily deep networks, deriving a novel recursion relationship across depth which extends the eligibility traces of E-prop to deeper layers. Our results thus demonstrate an online learning algorithm can perform accurate credit assignment across both time and depth simultaneously, allowing the training of deep recurrent networks without backpropagation through time.

Nick Alonso, Tomas Figliolia, Beren Millidge

Standard sequence mixing layers used in language models struggle to balance efficiency and performance. Self-attention performs well on long context tasks but has expensive quadratic compute and linear memory costs, while linear attention and SSMs use only linear compute and constant memory but struggle with long context processing. In this paper, we develop a sequence mixing layer that aims to find a better compromise between memory-compute costs and long-context processing, which we call online vector-quantized (OVQ) attention. OVQ-attention requires linear compute costs and constant memory, but, unlike linear attention and SSMs, it uses a sparse memory update that allows it to greatly increase the size of its memory state and, consequently, memory capacity. We develop a theoretical basis for OVQ-attention based on Gaussian mixture regression, and we test it on a variety of synthetic long context tasks and on long context language modeling. OVQ-attention shows significant improvements over linear attention baselines and the original VQ-attention, on which OVQ-attention was inspired. It demonstrates competitive, and sometimes identical, performance to strong self-attention baselines up 64k sequence length, despite using a small fraction of the memory of full self-attention.

Tomas Figliolia, Nicholas Alonso, Rishi Iyer et al.

Multi-headed Attention's (MHA) quadratic compute and linearly growing KV-cache make long-context transformers expensive to train and serve. Prior works such as Grouped Query Attention (GQA) and Multi-Latent Attention (MLA) shrink the cache, speeding decode, but leave compute, which determines prefill and training speed, largely unchanged. We introduce Compressed Convolutional Attention (CCA), a novel attention method which down-projects queries, keys, and values and performs the entire attention operation inside the shared latent space. This simple design dramatically cuts parameters, KV-cache, and FLOPs all at once by the desired compression factor. Because CCA is orthogonal to head-sharing, we combine the two to form Compressed Convolutional Grouped Query Attention (CCGQA), which further tightens the compute-bandwidth Pareto frontier so that users can tune compression toward either FLOP or memory limits without sacrificing quality. Experiments show that CCGQA consistently outperforms both GQA and MLA at equal KV-cache compression on dense and MoE models. Additionally, we show that CCGQA outperforms all other attention methods on MoE models with half the KV-cache of GQA and MLA, achieving an 8x KV-cache compression with no drop in performance compared to standard MHA. CCA and CCGQA also dramatically reduce the FLOP cost of attention which leads to substantially faster training and prefill than existing methods. On H100 GPUs, our fused CCA/CCGQA kernel reduces prefill latency by about 1.7x at a sequence length of 16k relative to MHA, and accelerates backward by about 1.3x.

Beren Millidge, Tommaso Salvatori, Yuhang Song et al.

The backpropagation of error algorithm used to train deep neural networks has been fundamental to the successes of deep learning. However, it requires sequential backward updates and non-local computations, which make it challenging to parallelize at scale and is unlike how learning works in the brain. Neuroscience-inspired learning algorithms, however, such as \emph{predictive coding}, which utilize local learning, have the potential to overcome these limitations and advance beyond current deep learning technologies. While predictive coding originated in theoretical neuroscience as a model of information processing in the cortex, recent work has developed the idea into a general-purpose algorithm able to train neural networks using only local computations. In this survey, we review works that have contributed to this perspective and demonstrate the close theoretical connections between predictive coding and backpropagation, as well as works that highlight the multiple advantages of using predictive coding models over backpropagation-trained neural networks. Specifically, we show the substantially greater flexibility of predictive coding networks against equivalent deep neural networks, which can function as classifiers, generators, and associative memories simultaneously, and can be defined on arbitrary graph topologies. Finally, we review direct benchmarks of predictive coding networks on machine learning classification tasks, as well as its close connections to control theory and applications in robotics.

Beren Millidge, Tommaso Salvatori, Yuhang Song et al.

A large number of neural network models of associative memory have been proposed in the literature. These include the classical Hopfield networks (HNs), sparse distributed memories (SDMs), and more recently the modern continuous Hopfield networks (MCHNs), which possesses close links with self-attention in machine learning. In this paper, we propose a general framework for understanding the operation of such memory networks as a sequence of three operations: similarity, separation, and projection. We derive all these memory models as instances of our general framework with differing similarity and separation functions. We extend the mathematical framework of Krotov et al (2020) to express general associative memory models using neural network dynamics with only second-order interactions between neurons, and derive a general energy function that is a Lyapunov function of the dynamics. Finally, using our framework, we empirically investigate the capacity of using different similarity functions for these associative memory models, beyond the dot product similarity measure, and demonstrate empirically that Euclidean or Manhattan distance similarity metrics perform substantially better in practice on many tasks, enabling a more robust retrieval and higher memory capacity than existing models.

Tommaso Salvatori, Luca Pinchetti, Beren Millidge et al.

Training with backpropagation (BP) in standard deep learning consists of two main steps: a forward pass that maps a data point to its prediction, and a backward pass that propagates the error of this prediction back through the network. This process is highly effective when the goal is to minimize a specific objective function. However, it does not allow training on networks with cyclic or backward connections. This is an obstacle to reaching brain-like capabilities, as the highly complex heterarchical structure of the neural connections in the neocortex are potentially fundamental for its effectiveness. In this paper, we show how predictive coding (PC), a theory of information processing in the cortex, can be used to perform inference and learning on arbitrary graph topologies. We experimentally show how this formulation, called PC graphs, can be used to flexibly perform different tasks with the same network by simply stimulating specific neurons, and investigate how the topology of the graph influences the final performance. We conclude by comparing against simple baselines trained~with~BP.

Pablo Lanillos, Cristian Meo, Corrado Pezzato et al.

Active inference is a mathematical framework which originated in computational neuroscience as a theory of how the brain implements action, perception and learning. Recently, it has been shown to be a promising approach to the problems of state-estimation and control under uncertainty, as well as a foundation for the construction of goal-driven behaviours in robotics and artificial agents in general. Here, we review the state-of-the-art theory and implementations of active inference for state-estimation, control, planning and learning; describing current achievements with a particular focus on robotics. We showcase relevant experiments that illustrate its potential in terms of adaptation, generalization and robustness. Furthermore, we connect this approach with other frameworks and discuss its expected benefits and challenges: a unified framework with functional biological plausibility using variational Bayesian inference.

Paul F. Kinghorn, Beren Millidge, Christopher L. Buckley

In cognitive science, behaviour is often separated into two types. Reflexive control is habitual and immediate, whereas reflective is deliberative and time consuming. We examine the argument that Hierarchical Predictive Coding (HPC) can explain both types of behaviour as a continuum operating across a multi-layered network, removing the need for separate circuits in the brain. On this view, "fast" actions may be triggered using only the lower layers of the HPC schema, whereas more deliberative actions need higher layers. We demonstrate that HPC can distribute learning throughout its hierarchy, with higher layers called into use only as required.

Beren Millidge, Anil Seth, Christopher L Buckley

The Free-Energy-Principle (FEP) is an influential and controversial theory which postulates a deep and powerful connection between the stochastic thermodynamics of self-organization and learning through variational inference. Specifically, it claims that any self-organizing system which can be statistically separated from its environment, and which maintains itself at a non-equilibrium steady state, can be construed as minimizing an information-theoretic functional -- the variational free energy -- and thus performing variational Bayesian inference to infer the hidden state of its environment. This principle has also been applied extensively in neuroscience, and is beginning to make inroads in machine learning by spurring the construction of novel and powerful algorithms by which action, perception, and learning can all be unified under a single objective. While its expansive and often grandiose claims have spurred significant debates in both philosophy and theoretical neuroscience, the mathematical depth and lack of accessible introductions and tutorials for the core claims of the theory have often precluded a deep understanding within the literature. Here, we aim to provide a mathematically detailed, yet intuitive walk-through of the formulation and central claims of the FEP while also providing a discussion of the assumptions necessary and potential limitations of the theory. Additionally, since the FEP is a still a living theory, subject to internal controversy, change, and revision, we also present a detailed appendix highlighting and condensing current perspectives as well as controversies about the nature, applicability, and the mathematical assumptions and formalisms underlying the FEP.

Beren Millidge, Anil Seth, Christopher L Buckley

Predictive coding offers a potentially unifying account of cortical function -- postulating that the core function of the brain is to minimize prediction errors with respect to a generative model of the world. The theory is closely related to the Bayesian brain framework and, over the last two decades, has gained substantial influence in the fields of theoretical and cognitive neuroscience. A large body of research has arisen based on both empirically testing improved and extended theoretical and mathematical models of predictive coding, as well as in evaluating their potential biological plausibility for implementation in the brain and the concrete neurophysiological and psychological predictions made by the theory. Despite this enduring popularity, however, no comprehensive review of predictive coding theory, and especially of recent developments in this field, exists. Here, we provide a comprehensive review both of the core mathematical structure and logic of predictive coding, thus complementing recent tutorials in the literature. We also review a wide range of classic and recent work within the framework, ranging from the neurobiologically realistic microcircuits that could implement predictive coding, to the close relationship between predictive coding and the widely-used backpropagation of error algorithm, as well as surveying the close relationships between predictive coding and modern machine learning techniques.

Beren Millidge

In this PhD thesis, we explore and apply methods inspired by the free energy principle to two important areas in machine learning and neuroscience. The free energy principle is a general mathematical theory of the necessary information-theoretic behaviours of systems that maintain a separation from their environment. A core postulate of the theory is that complex systems can be seen as performing variational Bayesian inference and minimizing an information-theoretic quantity called the variational free energy. The thesis is structured into three independent sections. Firstly, we focus on predictive coding, a neurobiologically plausible process theory derived from the free energy principle which argues that the primary function of the brain is to minimize prediction errors, showing how predictive coding can be scaled up and extended to be more biologically plausible, and elucidating its close links with other methods such as Kalman Filtering. Secondly, we study active inference, a neurobiologically grounded account of action through variational message passing, and investigate how these methods can be scaled up to match the performance of deep reinforcement learning methods. We additionally provide a detailed mathematical understanding of the nature and origin of the information-theoretic objectives that underlie exploratory behaviour. Finally, we investigate biologically plausible methods of credit assignment in the brain. We first demonstrate a close link between predictive coding and the backpropagation of error algorithm. We go on to propose novel and simpler algorithms which allow for backprop to be implemented in purely local, biologically plausible computations.

Alejandro Daniel Noel, Charel van Hoof, Beren Millidge

Intelligent agents must pursue their goals in complex environments with partial information and often limited computational capacity. Reinforcement learning methods have achieved great success by creating agents that optimize engineered reward functions, but which often struggle to learn in sparse-reward environments, generally require many environmental interactions to perform well, and are typically computationally very expensive. Active inference is a model-based approach that directs agents to explore uncertain states while adhering to a prior model of their goal behaviour. This paper introduces an active inference agent which minimizes the novel free energy of the expected future. Our model is capable of solving sparse-reward problems with a very high sample efficiency due to its objective function, which encourages directed exploration of uncertain states. Moreover, our model is computationally very light and can operate in a fully online manner while achieving comparable performance to offline RL methods. We showcase the capabilities of our model by solving the mountain car problem, where we demonstrate its superior exploration properties and its robustness to observation noise, which in fact improves performance. We also introduce a novel method for approximating the prior model from the reward function, which simplifies the expression of complex objectives and improves performance over previous active inference approaches.

Beren Millidge

While the utility of well-chosen abstractions for understanding and predicting the behaviour of complex systems is well appreciated, precisely what an abstraction $\textit{is}$ has so far has largely eluded mathematical formalization. In this paper, we aim to set out a mathematical theory of abstraction. We provide a precise characterisation of what an abstraction is and, perhaps more importantly, suggest how abstractions can be learnt directly from data both for static datasets and for dynamical systems. We define an abstraction to be a small set of `summaries' of a system which can be used to answer a set of queries about the system or its behaviour. The difference between the ground truth behaviour of the system on the queries and the behaviour of the system predicted only by the abstraction provides a measure of the `leakiness' of the abstraction which can be used as a loss function to directly learn abstractions from data. Our approach can be considered a generalization of classical statistics where we are not interested in reconstructing `the data' in full, but are instead only concerned with answering a set of arbitrary queries about the data. While highly theoretical, our results have deep implications for statistical inference and machine learning and could be used to develop explicit methods for learning precise kinds of abstractions directly from data.

Beren Millidge, Anil Seth, Christopher Buckley

The exploration-exploitation trade-off is central to the description of adaptive behaviour in fields ranging from machine learning, to biology, to economics. While many approaches have been taken, one approach to solving this trade-off has been to equip or propose that agents possess an intrinsic 'exploratory drive' which is often implemented in terms of maximizing the agents information gain about the world -- an approach which has been widely studied in machine learning and cognitive science. In this paper we mathematically investigate the nature and meaning of such approaches and demonstrate that this combination of utility maximizing and information-seeking behaviour arises from the minimization of an entirely difference class of objectives we call divergence objectives. We propose a dichotomy in the objective functions underlying adaptive behaviour between \emph{evidence} objectives, which correspond to well-known reward or utility maximizing objectives in the literature, and \emph{divergence} objectives which instead seek to minimize the divergence between the agent's expected and desired futures, and argue that this new class of divergence objectives could form the mathematical foundation for a much richer understanding of the exploratory components of adaptive and intelligent action, beyond simply greedy utility maximization.

Beren Millidge, Alexander Tschantz, Anil Seth et al.

The Kalman filter is a fundamental filtering algorithm that fuses noisy sensory data, a previous state estimate, and a dynamics model to produce a principled estimate of the current state. It assumes, and is optimal for, linear models and white Gaussian noise. Due to its relative simplicity and general effectiveness, the Kalman filter is widely used in engineering applications. Since many sensory problems the brain faces are, at their core, filtering problems, it is possible that the brain possesses neural circuitry that implements equivalent computations to the Kalman filter. The standard approach to Kalman filtering requires complex matrix computations that are unlikely to be directly implementable in neural circuits. In this paper, we show that a gradient-descent approximation to the Kalman filter requires only local computations with variance weighted prediction errors. Moreover, we show that it is possible under the same scheme to adaptively learn the dynamics model with a learning rule that corresponds directly to Hebbian plasticity. We demonstrate the performance of our method on a simple Kalman filtering task, and propose a neural implementation of the required equations.

Beren Millidge, Alexander Tschantz, Anil Seth et al.

The recently proposed Activation Relaxation (AR) algorithm provides a simple and robust approach for approximating the backpropagation of error algorithm using only local learning rules. Unlike competing schemes, it converges to the exact backpropagation gradients, and utilises only a single type of computational unit and a single backwards relaxation phase. We have previously shown that the algorithm can be further simplified and made more biologically plausible by (i) introducing a learnable set of backwards weights, which overcomes the weight-transport problem, and (ii) avoiding the computation of nonlinear derivatives at each neuron. However, tthe efficacy of these simplifications has, so far, only been tested on simple multi-layer-perceptron (MLP) networks. Here, we show that these simplifications still maintain performance using more complex CNN architectures and challenging datasets, which have proven difficult for other biologically-plausible schemes to scale to. We also investigate whether another biologically implausible assumption of the original AR algorithm -- the frozen feedforward pass -- can be relaxed without damaging performance.

Beren Millidge, Alexander Tschantz, Anil Seth et al.

Predictive coding is an influential theory of cortical function which posits that the principal computation the brain performs, which underlies both perception and learning, is the minimization of prediction errors. While motivated by high-level notions of variational inference, detailed neurophysiological models of cortical microcircuits which can implements its computations have been developed. Moreover, under certain conditions, predictive coding has been shown to approximate the backpropagation of error algorithm, and thus provides a relatively biologically plausible credit-assignment mechanism for training deep networks. However, standard implementations of the algorithm still involve potentially neurally implausible features such as identical forward and backward weights, backward nonlinear derivatives, and 1-1 error unit connectivity. In this paper, we show that these features are not integral to the algorithm and can be removed either directly or through learning additional sets of parameters with Hebbian update rules without noticeable harm to learning performance. Our work thus relaxes current constraints on potential microcircuit designs and hopefully opens up new regions of the design-space for neuromorphic implementations of predictive coding.

Beren Millidge, Alexander Tschantz, Anil K Seth et al.

The backpropagation of error algorithm (backprop) has been instrumental in the recent success of deep learning. However, a key question remains as to whether backprop can be formulated in a manner suitable for implementation in neural circuitry. The primary challenge is to ensure that any candidate formulation uses only local information, rather than relying on global signals as in standard backprop. Recently several algorithms for approximating backprop using only local signals have been proposed. However, these algorithms typically impose other requirements which challenge biological plausibility: for example, requiring complex and precise connectivity schemes, or multiple sequential backwards phases with information being stored across phases. Here, we propose a novel algorithm, Activation Relaxation (AR), which is motivated by constructing the backpropagation gradient as the equilibrium point of a dynamical system. Our algorithm converges rapidly and robustly to the correct backpropagation gradients, requires only a single type of computational unit, utilises only a single parallel backwards relaxation phase, and can operate on arbitrary computation graphs. We illustrate these properties by training deep neural networks on visual classification tasks, and describe simplifications to the algorithm which remove further obstacles to neurobiological implementation (for example, the weight-transport problem, and the use of nonlinear derivatives), while preserving performance.

Alexander Tschantz, Beren Millidge, Anil K. Seth et al.

The field of reinforcement learning can be split into model-based and model-free methods. Here, we unify these approaches by casting model-free policy optimisation as amortised variational inference, and model-based planning as iterative variational inference, within a `control as hybrid inference' (CHI) framework. We present an implementation of CHI which naturally mediates the balance between iterative and amortised inference. Using a didactic experiment, we demonstrate that the proposed algorithm operates in a model-based manner at the onset of learning, before converging to a model-free algorithm once sufficient data have been collected. We verify the scalability of our algorithm on a continuous control benchmark, demonstrating that it outperforms strong model-free and model-based baselines. CHI thus provides a principled framework for harnessing the sample efficiency of model-based planning while retaining the asymptotic performance of model-free policy optimisation.

Beren Millidge, Alexander Tschantz, Anil K Seth et al.

Active Inference (AIF) is an emerging framework in the brain sciences which suggests that biological agents act to minimise a variational bound on model evidence. Control-as-Inference (CAI) is a framework within reinforcement learning which casts decision making as a variational inference problem. While these frameworks both consider action selection through the lens of variational inference, their relationship remains unclear. Here, we provide a formal comparison between them and demonstrate that the primary difference arises from how value is incorporated into their respective generative models. In the context of this comparison, we highlight several ways in which these frameworks can inform one another.

Beren Millidge, Alexander Tschantz, Anil K Seth et al.

There are several ways to categorise reinforcement learning (RL) algorithms, such as either model-based or model-free, policy-based or planning-based, on-policy or off-policy, and online or offline. Broad classification schemes such as these help provide a unified perspective on disparate techniques and can contextualise and guide the development of new algorithms. In this paper, we utilise the control as inference framework to outline a novel classification scheme based on amortised and iterative inference. We demonstrate that a wide range of algorithms can be classified in this manner providing a fresh perspective and highlighting a range of existing similarities. Moreover, we show that taking this perspective allows us to identify parts of the algorithmic design space which have been relatively unexplored, suggesting new routes to innovative RL algorithms.

Beren Millidge, Alexander Tschantz, Christopher L. Buckley

Backpropagation of error (backprop) is a powerful algorithm for training machine learning architectures through end-to-end differentiation. However, backprop is often criticised for lacking biological plausibility. Recently, it has been shown that backprop in multilayer-perceptrons (MLPs) can be approximated using predictive coding, a biologically-plausible process theory of cortical computation which relies only on local and Hebbian updates. The power of backprop, however, lies not in its instantiation in MLPs, but rather in the concept of automatic differentiation which allows for the optimisation of any differentiable program expressed as a computation graph. Here, we demonstrate that predictive coding converges asymptotically (and in practice rapidly) to exact backprop gradients on arbitrary computation graphs using only local learning rules. We apply this result to develop a straightforward strategy to translate core machine learning architectures into their predictive coding equivalents. We construct predictive coding CNNs, RNNs, and the more complex LSTMs, which include a non-layer-like branching internal graph structure and multiplicative interactions. Our models perform equivalently to backprop on challenging machine learning benchmarks, while utilising only local and (mostly) Hebbian plasticity. Our method raises the potential that standard machine learning algorithms could in principle be directly implemented in neural circuitry, and may also contribute to the development of completely distributed neuromorphic architectures.

Beren Millidge, Alexander Tschantz, Christopher L Buckley

The Expected Free Energy (EFE) is a central quantity in the theory of active inference. It is the quantity that all active inference agents are mandated to minimize through action, and its decomposition into extrinsic and intrinsic value terms is key to the balance of exploration and exploitation that active inference agents evince. Despite its importance, the mathematical origins of this quantity and its relation to the Variational Free Energy (VFE) remain unclear. In this paper, we investigate the origins of the EFE in detail and show that it is not simply "the free energy in the future". We present a functional that we argue is the natural extension of the VFE, but which actively discourages exploratory behaviour, thus demonstrating that exploration does not directly follow from free energy minimization into the future. We then develop a novel objective, the Free-Energy of the Expected Future (FEEF), which possesses both the epistemic component of the EFE as well as an intuitive mathematical grounding as the divergence between predicted and desired futures.

Alexander Tschantz, Beren Millidge, Anil K. Seth et al.

The central tenet of reinforcement learning (RL) is that agents seek to maximize the sum of cumulative rewards. In contrast, active inference, an emerging framework within cognitive and computational neuroscience, proposes that agents act to maximize the evidence for a biased generative model. Here, we illustrate how ideas from active inference can augment traditional RL approaches by (i) furnishing an inherent balance of exploration and exploitation, and (ii) providing a more flexible conceptualization of reward. Inspired by active inference, we develop and implement a novel objective for decision making, which we term the free energy of the expected future. We demonstrate that the resulting algorithm successfully balances exploration and exploitation, simultaneously achieving robust performance on several challenging RL benchmarks with sparse, well-shaped, and no rewards.