arXiv subjects

Alex Infanger

Publications and source records attributed to Alex Infanger.

10 recordsLinked to original sources

arXiv2026

Consequentialist Objectives and Catastrophe

Because human preferences are too complex to codify, AIs operate with misspecified objectives. Optimizing such objectives often produces undesirable outcomes; this phenomenon is known as reward hacking. Such outcomes are not necessarily catastrophic. Indeed, most examples of reward hacking in previous literature are benign. And typically, objectives can be modified to resolve the issue. We study the prospect of catastrophic outcomes induced by AIs operating in complex environments. We argue that, when capabilities are sufficiently advanced, pursuing a fixed consequentialist objective tends to result in catastrophic outcomes. We formalize this by establishing conditions that provably lead to such outcomes. Under these conditions, simple or random behavior is safe. Catastrophic risk arises due to extraordinary competence rather than incompetence. With a fixed consequentialist objective, avoiding catastrophe requires constraining AI capabilities. In fact, constraining capabilities the right amount not only averts catastrophe but yields valuable outcomes. Our results apply to any objective produced by modern industrial AI development pipelines.

cs.AI↗

arXiv2025

Misalignment from Treating Means as Ends

Reward functions, learned or manually specified, are rarely perfect. Instead of accurately expressing human goals, these reward functions are often distorted by human beliefs about how best to achieve those goals. Specifically, these reward functions often express a combination of the human's terminal goals -- those which are ends in themselves -- and the human's instrumental goals -- those which are means to an end. We formulate a simple example in which even slight conflation of instrumental and terminal goals results in severe misalignment: optimizing the misspecified reward function results in poor performance when measured by the true reward function. This example distills the essential properties of environments that make reinforcement learning highly sensitive to conflation of instrumental and terminal goals. We discuss how this issue can arise with a common approach to reward learning and how it can manifest in real environments.

cs.LG↗

arXiv2025

Distillation Robustifies Unlearning

Current LLM unlearning methods are not robust. A few steps of finetuning can revert their effects. We begin by showing that this is true even for an idealized form of unlearning: training to imitate a model that was never trained on unwanted information. This shows that training a model can drastically modify its input-output behavior while leaving its underlying capabilities intact. In light of this dynamic, we show our main result. Training a randomly initialized student on the outputs of an unlearned model transfers behaviors while leaving latent capabilities behind. In short, distillation robustifies unlearning. Based on this result, we propose Unlearn-Noise-Distill-on-Outputs (UNDO), a scalable method that distills an unlearned model into a noised copy of itself. UNDO introduces a tunable tradeoff between compute cost and robustness, establishing a new Pareto frontier on synthetic language and arithmetic tasks. At its strongest setting, UNDO matches the robustness of a model retrained from scratch with perfect data filtering while using only 60-80% of the compute and requiring only 0.01% of the pretraining data to be labeled. We also show that UNDO robustifies unlearning on the more realistic Weapons of Mass Destruction Proxy (WMDP) benchmark. Since distillation is widely used in practice, incorporating an unlearning step beforehand offers a convenient path to robust capability removal.

cs.LG↗

arXiv2024

The Persian Rug: solving toy models of superposition using large-scale symmetries

We present a complete mechanistic description of the algorithm learned by a minimal non-linear sparse data autoencoder in the limit of large input dimension. The model, originally presented in arXiv:2209.10652, compresses sparse data vectors through a linear layer and decompresses using another linear layer followed by a ReLU activation. We notice that when the data is permutation symmetric (no input feature is privileged) large models reliably learn an algorithm that is sensitive to individual weights only through their large-scale statistics. For these models, the loss function becomes analytically tractable. Using this understanding, we give the explicit scalings of the loss at high sparsity, and show that the model is near-optimal among recently proposed architectures. In particular, changing or adding to the activation function any elementwise or filtering operation can at best improve the model's performance by a constant factor. Finally, we forward-engineer a model with the requisite symmetries and show that its loss precisely matches that of the trained models. Unlike the trained model weights, the low randomness in the artificial weights results in miraculous fractal structures resembling a Persian rug, to which the algorithm is oblivious. Our work contributes to neural network interpretability by introducing techniques for understanding the structure of autoencoders. Code to reproduce our results can be found at https://github.com/KfirD/PersianRug .

cs.LG↗

arXiv2022

A New Truncation Algorithm for Markov Chain Equilibrium Distributions with Computable Error Bounds

This paper introduces a new algorithm for numerically computing equilibrium (i.e. stationary) distributions for Markov chains and Markov jump processes with either a very large finite state space or a countably infinite state space. The algorithm is based on a ratio representation for equilibrium expectations in which the numerator and denominator correspond to expectations defined over paths that start and end within a given return set $K$. When $K$ is a singleton, this representation is a well-known consequence of regenerative process theory. For computational tractability, we ignore contributions to the path expectations corresponding to excursions out of a given truncation set $A$. This yields a truncation algorithm that is provably convergent as $A$ gets large. Furthermore, in the presence of a suitable Lyapunov function, we can bound the path expectations, thereby providing computable and convergent error bounds for our numerical procedure. Our paper also provides a computational comparison with two other truncation methods that come with computable error bounds. The results are in alignment with the observation that our bounds have associated computational complexities that typically scale better as the truncation set gets bigger.

math.PR↗

arXiv2022

A Short Proof of a Convex Representation for Stationary Distributions of Markov Chains with an Application to State Space Truncation

In an influential paper, Courtois and Semal (1984) establish that when $G$ is an irreducible substochastic matrix for which $\sum_{n=0}^{\infty}G^n <\infty$, then the stationary distribution of any stochastic matrix $P\ge G$ can be expressed as a convex combination of the normalized rows of $(I-G)^{-1} = \sum_{n=0}^{\infty} G^n$. In this note, we give a short proof of this result that extends the theory to the countably infinite and continuous state space settings. This result plays an important role in obtaining error bounds in algorithms involving nearly decomposable Markov chains, and also in state truncations for Markov chains. We also use the representation to establish a new total variation distance error bound for truncated Markov chains.

math.PR↗

arXiv2022

On Convergence of a Truncation Scheme for Approximating Stationary Distributions of Continuous State Space Markov Chains and Processes

In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior of the truncated chain or process is close to that of the untruncated system. For example, such questions arise naturally when considering numerical methods for computing stationary distributions on unbounded state space. In this paper, we use the principle of "regeneration" to show that the stationary distributions of "fixed state" truncations converge in great generality (in total variation norm) to the stationary distribution of the untruncated limit, when the untruncated chain is positive Harris recurrent. Even in countable state space, our theory extends known results by showing that the augmentation can correspond to an $r$-regular measure. In addition, we extend our theory to cover an important subclass of Harris recurrent Markov processes that include non-explosive Markov jump processes on countable state space.

math.PR↗

arXiv2022

On Convergence of General Truncation-Augmentation Schemes for Approximating Stationary Distributions of Markov Chains

In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior of the truncated chain or process is close to that of the untruncated system. For example, such questions arise naturally when considering numerical methods for computing stationary distributions on unbounded state space. In this paper, we study general truncation-augmentation schemes, in which the substochastic truncated "northwest corner" of the transition matrix or kernel is stochasticized (or augmented) arbitrarily. In the presence of a Lyapunov condition involving a coercive function, we show that such schemes are generally convergent in countable state space, provided that the truncation is chosen as a sublevel set of the Lyapunov function. For stochastically monotone Markov chains on $\mathbb Z_+$, we prove that we can always choose the truncation sets to be of the form $\{0,1,...,n\}$. We then provide sufficient conditions for weakly continuous Markov chains under which general truncation-augmentation schemes converge weakly in continuous state space. Finally, we briefly discuss the extension of the theory to continuous time Markov jump processes.

math.PR↗

arXiv2022

Solving Poisson's Equation: Existence, Uniqueness, Martingale Structure, and CLT

The solution of Poisson's equation plays a key role in constructing the martingale through which sums of Markov correlated random variables can be analyzed. In this paper, we study two different representations for the solution in countable state space, one based on regenerative structure and the other based on an infinite sum of expectations. We also consider integrability and related uniqueness issues associated with solutions to Poisson's equation, and provide verifiable Lyapunov conditions to support our theory. Our key results include a central limit theorem and law of the iterated logarithm for Markov dependent sums, under Lyapunov conditions weaker than have previously appeared in the literature.

math.PR↗

arXiv2022

Solutions of Poisson's Equation for Stochastically Monotone Markov Chains

Stochastically monotone Markov chains arise in many applied domains, especially in the setting of queues and storage systems. Poisson's equation is a key tool for analyzing additive functionals of such models, such as cumulative sums of waiting times or sums of rewards. In this paper, we show that when the reward function for such a Markov chain is monotone, the solution of Poisson's equation is monotone. This implies that the value function associated with infinite horizon average reward is monotone in the state when the reward is monotone.

math.PR↗