arXiv subjects

Kai Hidajat

Publications and source records attributed to Kai Hidajat.

3 recordsLinked to original sources

arXiv2026

Priced Motion Through Optimal Faces: A Normal-Fan Geometry for Non-Stationary Adversarial MDPs

In a changing decision problem, standard dynamic-regret analyses have often equated the cost of non-stationarity to how far loss moves. However, it is simultaneously possible for a loss sequence to travel far and retain the same optimal policy, or for a small movement in loss to force the optimal policy to change completely. Thus, the size of the movement through loss variation, transition variation, or comparator path length describe the adversary's motion, but not the cost of that motion to the control problem. For a more faithful analytic interpretation, this paper develops a normal-fan geometry for finite-horizon adversarial MDPs with fixed transitions. Occupancy measures form a polytope, and each loss vector exposes an optimal face of that polytope. Non-stationarity in rewards is therefore a path through the normal fan, where motion inside one cone leaves the optimal face unchanged, while crossing a wall may carry regret. We pose the notion of a face-crossing price, which is the minimum regret incurred by remaining on the previous optimal face under the new loss. For any learner that tracks the previous face, dynamic regret decomposes exactly into intrinsic priced face motion plus within-face selection error. The resulting theory separates consequential from harmless non-stationarity, where loss variation can be arbitrarily large at zero price, and identical one-coordinate variation can hide horizon-scale differences in regret.

cs.LG↗

arXiv2026

Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization

Neural operators excel as deterministic surrogates, but inevitably collapse to the conditional mean when applied to stochastic PDEs, discarding the variance and tail structure upon which uncertainty quantification depends. Recovering this structure typically requires Monte Carlo rollouts or grafted generative models, both of which surrender the one-shot efficiency and resolution invariance that define the operator paradigm. To resolve this, we draw on the Doob-Meyer theorem, which establishes that any semimartingale fundamentally decomposes into a predictable drift and an unpredictable, zero-mean martingale. Translating this theorem into an architectural prior, we introduce the Martingale Neural Operator (MNO). MNO maps an initial condition directly to the conditional mean and covariance of the terminal law, parameterized by a drift-like mean and a low-rank factor $B_ϕ$ with $B_ϕ^\top B_ϕ$ positive semi-definite by construction. For our experiments, we use a Gaussian residual instantiation. Across 1D SPDEs, rough volatility, and 2D operator tasks, MNO reduces Wasserstein distance by up to $120\times$ on $ϕ^4$ field theory and $68\times$ on stochastic Burgers, evaluating $\sim 3\times$ faster than a conditional diffusion baseline at matched wall-clock training budgets. On 2D tasks, MNO is comparable to FNO on zero-shot resolution transfer and turbulent flow, while quasi-deterministic systems such as Gray-Scott remain a failure mode.

cs.LG↗

arXiv2026

Grokking as Structural Inference: Transformers Need Bayesian Lottery Tickets

Why does a Transformer that has memorized its training set wait thousands of steps before it generalizes? Existing accounts locate this delay in norm minimization, feature emergence, or the late discovery of sparse subnetworks. These explanations capture important parts of the transition, but ignore a constraint unique to attention-based models: if attention discards an informative token, no bounded downstream computation can recover it. We formalize attention as an implicit Bayesian posterior over the task dependency graph and prove that generalization requires two separable conditions: a familiar Goldilocks bound on MLP capacity, coinciding with norm-based theories of grokking, and a novel Bayesian structural condition requiring attention to place sufficient mass on every informative token. This decoupling explains delayed generalization as delayed structural inference. Early in training, the MLP memorizes through unaligned features, drives the cross-entropy loss near zero, and thereby starves attention of structural gradient. Weight decay must then erode memorization before the missing graph becomes learnable, yielding the known inverse-weight-decay delay, which we derive as a structural waiting time. We then prove that this explaining-away delay can be bypassed by a KL-based structural intervention, yielding an inverse-intervention-strength scaling law for the grokking time. Experiments on algorithmic sequence tasks isolate structure from capacity and show that this Bayesian ticket matches or outperforms lottery-ticket transfer.

cs.LG↗