Stochastic Gradient Descent (SGD) is a workhorse algorithm for continuous optimization. It is a folk belief among deep learning practitioners that learning rate decay in SGD substantially improves performance. However, it has remained unclear whether learning rate decay offers an asymptotic advantage for global nonconvex optimization. We answer this question in the affirmative by demonstrating a natural class of nonconvex functions for which, we prove that SGD with learning rate decay requires exponentially fewer gradient queries for global optimization than SGD with any fixed learning rate in the small-step size regime. To the best of our knowledge, this is the first exponential advantage that has been demonstrated for learning rate decay. The class of functions we consider includes many popular benchmark nonconvex functions. Our results provide a robust explanatory theory for the empirically observed benefits of learning rate decay for global optimization. We show that annealing performs the non-logconcave sampling task associated to the optimization problem in polynomial time, via a novel application of weak Poincar\'{e} inequalities.
Random tests of stateful code check laws, like reading back a written value, at states that random writes reach. The laws hold at every state exactly when the code behaves like a record with hidden data, so a pass is taken as evidence. This can fail: the laws can hold at every sample while the code is far from any record. The laws are local, but behaving like a record is global. Self-testing and stability theorems bridge this gap when the query distribution is controlled by the tester or preserved by the tested maps, but here it is generated by the tested writes, which need not be invertible. We ask how far a test must explore. An interface has $m$ binary variables, each with a read and two writes; in a record, reads return their bits and writes set only their bits. Samples are the ends of geometric runs of mean $\ell$ of uniformly random writes from a start distribution; a tester adds its own write chains of depth $d$. The distance is the least, over records with hidden data, of the largest probability that a read or write differs from it at a sample. From one start state, law checks at $O(m^2\log m/\epsilon)$ samples certify, at fixed confidence, any threshold $\epsilon$ after geometric runs of every positive mean, while runs of fixed positive length can pass at distance $1/2$. For the worst start distribution, take $\epsilon=m^{-\beta}$ with fixed $0<\beta\le1$. For $d\le cm$ with fixed $c<1/2$, a polynomial budget of samples and writes exists exactly when $\ell=\Omega(m)$, where law checks suffice. With no runs, the least depth for this budget is $(1/2+o(1))m$. Counting samples alone, in the worst case at $\epsilon=1/(32m)$ and $d\le m/16$, the critical mean for a polynomial budget is $\Theta(m\log(m/d)/\log m)$, by exhaustive search. So one start state needs only law checks, and the worst case at these thresholds needs exploration linear in $m$ if writes are counted.
For two $d$-dimensional point sets $A, B$ of size $n$, the Chamfer distance from $A$ to $B$ is defined as $\text{CH}(A,B)=\sum_{a \in A} \min_{b \in B} \mathsf{dist}(a, b)$, where $\mathsf{dist}$ is some underlying distance measure. We present efficient algorithms for approximating the Chamfer distance under the $\ell_p$ norm for $1 < p \le 2$. For the general case $1 < p < 2$, we utilize lopsided $\ell_p$ to $\ell_1$ embeddings with weak guarantees. We show that they are sufficient for preserving the Chamfer distance. In the high-dimensional regime, we use Fast Matrix Multiplication techniques to speed up the lopsided embeddings. These lead to $O(nd\cdot \mathrm{poly}(\log \log n, \log(1/\varepsilon))/\varepsilon^2)$ total time. For the specific Euclidean case ($p=2$), we leverage the Fast Johnson-Lindenstrauss Transform based on Toeplitz matrices and re-analyze the previous $\ell_1$-specific Chamfer algorithm in $\ell_2$. These achieve a runtime matching the recent state-of-the-art $\ell_1$ result of $O(nd(\log \log n + \log (1/\varepsilon))/\varepsilon^2)$, improving upon the previous $O(nd \log n/\varepsilon^2)$ bound for $\ell_2$. We also give additional results for the angular distance and upper and lower bounds in the streaming setting.
Elements arrive one at a time, and after each arrival an algorithm holds a set of at most $k$ arrived elements. Its goal is a held set whose value under a monotone submodular function is close to the best value that $k$ arrived elements can achieve. The recourse of an update is the number of elements inserted into or removed from the held set, and we require it to be bounded independently of $k$ and of the number $n$ of arrivals. With evaluations only on sets of arrived elements and polynomially many of them, the best ratio a randomized algorithm can achieve is $2-\sqrt{2}\approx0.5858$. We show that seeing future arrivals raises the best ratio for polynomially many evaluations to $1-1/e\approx0.6321$, for deterministic and for randomized algorithms; no algorithm with polynomially many evaluations exceeds $1-1/e$, even when all elements are available at once. For every $\varepsilon>0$, a deterministic algorithm that sees the next $\lceil\varepsilon k/3\rceil$ arrivals keeps a set of value at least $1-1/e-\varepsilon$ times the optimum at every time, changes $O(\varepsilon^{-3})$ elements per arrival and makes $(n+1)^{O(\varepsilon^{-2})}$ evaluations. With the whole arrival sequence known in advance, a deterministic algorithm attains the same ratio with $O(\varepsilon^{-2})$ changes per arrival. A randomized algorithm with the same short lookahead makes $O(n^2k)$ evaluations, changes $O(\varepsilon^{-2})$ elements per arrival, and attains the same ratio in expectation at each fixed time. The main tool is a greedy algorithm with an explicit guarantee for each of its partial solutions combined with each initial segment of the known future arrivals. A second tool moves between two such solutions one element at a time while keeping the value high, and the deterministic algorithm times each move using upper bounds on the optimum from failed searches.
Quantified Boolean formula (QBF), the quantified version of Boolean satisfiability (SAT), is a fundamental problem in constraint solving. Due to the $\mathsf{PSPACE}$-complete nature of QBF, designing theoretically efficient QBF algorithms is much harder and receives less attention in academia. In this work, we present a new algorithm based on memoized search, which solves QBF with $n$ variables and $m$ clauses in $1.685^m\cdot\text{poly}(n)$ time and space. This is the first improvement over the $1.709^m\cdot\text{poly}(n)$ bound by Williams in over 20 years. Our technique is based on the minimum vertex cover of the conflict graphs, which was proposed to study the parameterized complexity of SAT. We extend this idea to QBF and further show that QBF is slicewise polynomial-time solvable using this parameter.
Pairwise probabilities arise in rank aggregation, recommendation, and human or LLM evaluation, yet converting them into a maximum-likelihood list is a weighted Kemeny problem and is NP-hard in general. Known exact algorithms for rank-two tournaments use only comparison signs and do not optimize this likelihood. We show that the weighted problem is exactly solvable when the skew-symmetric log-odds matrix $W_{ij}=\log(p_{ij}/p_{ji})$ has rank two, including when the induced preferences contain cycles. Writing $W_{ij}=\det(h_i,h_j)$ and adding the closing vector $h_0=-\sum_i h_i$ turns the objective into twice the signed area of a closed polygon. This geometry yields CloseSort, which polar-sorts the $n+1$ edge vectors, cuts after $h_0$, and returns a global maximum-likelihood order in $O(n\log n)$ time from the factors. The same geometry identifies the runner-up and gives a stability certificate beyond exact rank two. When the smallest determinant $\delta$ of two consecutive edges in the closed angular cycle is positive, it is exactly the best-to-runner-up log-likelihood gap. For $W=W_0+E$, the CloseSort order of $W_0$ has regret at most $\sum_{i<j}|E_{ij}|$ on $W$ and remains optimal whenever this sum is below $\delta$, a worst-case tight threshold. In 5,120 determinant-logit simulations, increasing the observations per pair from 4 to 64 raises the fraction on which rank-two projection outperforms the empirical plug-in optimum from 68.6% to 85.7%. On 3,600 Arena-Hard LLM-judge profiles drawn from 63 systems and ten judges, the leading rank-two component captures 89.2% of the log-odds energy. On profiles containing a directed majority triangle, CloseSort reduces normalized regret by 44% relative to Borda and improves the mean normalized objective over each of eight baselines. With factors supplied, it is 8,407 times faster than exact dynamic programming on 18-item rankings.
Knowledge of job sizes is crucial for optimizing the mean response time in single-server scheduling. However, predictions of job sizes are typically inexact and may be costly to obtain. Thus, a long-standing research question is: what is the minimal information about job sizes required to obtain near-optimal mean response time? We introduce a novel policy, \emph{Sparse Predict and Demote (SPD)}, and prove that it achieves order-optimal mean response time in heavy traffic while requiring only one-bit job-size predictions for a vanishing fraction of arrivals. SPD achieves heavy-traffic order-optimality even in situations when predictions are (i) inexact up to any fixed multiplicative error bound, and (ii) costly, i.e., the server is used to compute predictions and is thus unavailable to serve jobs while processing prediction queries. Finally, we use numerical experiments to illustrate that SPD maintains near-optimal performance beyond the heavy-traffic regime.
Byte-Pair Encoding (BPE) builds most language-model tokenisers by repeatedly merging a most frequent adjacent pair of symbols. How much shortening can this greedy rule lose against the best $k$ merges? Kozma and Voderholzer proved that greedy achieves at least one third of the optimal shortening; the best known upper bound is $(1-e^{-2})/2\approx 0.432$. We show that one third is a barrier for packing-based analyses, which see the optimum only through a pair packing, and we prove that greedy achieves at least $0.391$ of the optimum on every input, with any tie-breaking. The key idea is rule determinism: all copies of a rule of the optimum spell the same letters, so when one greedy replacement destroys three of the optimum's merge points, half a unit can be refunded on each side. A weaker bound, $0.374$, has a short proof that needs no computer; $0.391$ adds a certificate in exact arithmetic. We also show that the worst-case ratio is exactly $3/4$ for two merges and at most $7/12$ for three.
Dantzig-Wolfe reformulation strengthens the linear relaxation of a mixed-integer program by convexifying a chosen set of constraint families, and every candidate costs a column generation run. The choice is hard when families strengthen the bound only together; the classical inequalities of lot sizing, network design and facility location rest on such zero-marginal groups. We prove that when each evaluation convexifies at most $s$ of $k$ families, any method that observes only bounds needs a covering number of evaluations, of order $k^w$, to find a hidden group of $w$ families. A prediction that misses $e$ members reduces the cost to $\Theta(k^e)$ by a staged search, so the evaluations a prediction saves measure what the predictor knows. We study two sources of predictions: a lookahead at an optimal LP dual, blind to groups whose families share no variable, and a language model that reads the model, which JointDW verifies with the staged search and interleaves with the lookahead at a bounded loss. Zero-marginal groups appear in three of ten MIPLIB-NL models with an improvable bound and in all 59 standard lot-sizing, network-design and facility-location instances, where GCG's detector convexifies a set with no gain on 38 of them. On MIPLIB-NL the strongest language models find the groups about as often as the lookahead, and on textbook models their predictions rest on the names. Given every coefficient of a larger model, one reasoning model names the classical group in all but one answer, also with the names permuted or removed, and needs fewer evaluations than the lookahead for pairs of families; a predictor trained on the same class matches this only after hundreds of labeled evaluations. On the benchmark instances the language models name the best pair more often than the lookahead and the detector, and the sets of GPT-6.1 Sol come closest to the branch-and-price time of the best set.
We study online interval selection with binary predictions of membership in a final optimal solution. Requests arrive in arbitrary order; accepted requests may be revoked, but rejected or revoked requests cannot be recovered. With rewards proportional to interval length, the best consistency under finite strict robustness is $d=(3+\sqrt5)/2$ when both guarantees hold almost surely on each fixed input, including for randomized algorithms. A continuous lower bound forces consistency strictly above $d$ at every robustness budget below $4\varphi+3\approx9.472$, whereas our deterministic algorithm attains $d$ with exact robustness $\varphi^5\approx11.090$, where $\varphi=(1+\sqrt5)/2$. Each held interval inherits a ledger recording positive prediction mass lost through irrevocable decisions. Tuning the replacement costs and both special thresholds gives a continuous family with exact consistency throughout and exact robustness on two parameter intervals, including $(C,R)=(25/9,10)$ and $(4,7)$. Randomization permits expected consistency approaching one with finite expected robustness, whereas almost-sure robustness forces expected consistency strictly above two at every finite budget.
Randomized sketch-and-solve algorithms accelerate overconstrained $\ell_2$ regression by replacing the input with a much smaller, randomly projected problem. While standard subspace embeddings nearly preserve the regression cost, coordinate-wise accuracy is more delicate: rather than the objective value, we want the solution vector itself to be close to the optimum in $\ell_\infty$ norm. Specifically, we seek $x'\in\mathbb{R}^d$ with $\|x'-x^\star\|_\infty\leq\frac{\epsilon}{\sqrt d}\|Ax^\star-b\|_2\|A^\dagger\|_{\mathrm{op}}$, where $A\in\mathbb{R}^{n\times d}$ is the design matrix, $b\in\mathbb{R}^n$ the label vector, $x^\star$ the optimal solution and $A^\dagger$ the pseudo-inverse of $A$. Price, Song and Woodruff introduced this problem and claimed that the subsampled randomized Hadamard transform (SRHT) with $O(\epsilon^{-2}d^{1+\Theta(\sqrt{\log\log n/\log d})})$ rows achieves this guarantee; we point out a gap in the proof. Song, Ye, Yin and Zhang later claimed $O(\epsilon^{-2}d\log^3 n)$ rows, but their proof relies on an independence assumption that does not hold in general, and we exhibit an explicit instance on which it fails. To achieve a truly nearly-linear-in-$d$ row count, we introduce a new fast, dense randomized transform combining a randomized Hadamard flattening, a random permutation, and balanced, disjoint Gaussian pooling. Conditioned on the Hadamard-and-permutation stage, the sketched problem becomes an exact Gaussian regression whose noise is independent of the entire sketched design, exactly what Song, Ye, Yin and Zhang's argument was missing. Our sketch yields the $\ell_\infty$ guarantee with $m=O(\epsilon^{-2}d\log d)$ rows, using one Hadamard pass with padded internal dimension $N=\widetilde{O}(n+\epsilon^{-2}d^3)$. It is also efficient to apply: $(SA,Sb)$ can be computed in $O(Nd\log N)=\widetilde{O}(nd+\epsilon^{-2}d^4)$ time.
We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.
We study the query complexity of sampling from high-dimensional Gaussian distributions using gradient information. In the standard oracle model, exact gradients expose only matrix-vector products with the precision matrix, leading to polynomial approximation barriers and a characteristic $\sqrt{\kappa}$ dependence on the known spectral range. We show that this barrier disappears when the sampler is allowed to query smoothed scores, namely gradients of the logarithms of the Gaussian-convolved densities. For a centered Gaussian target with precision matrix $\Lambda$, a smoothed-score query at noise level $\tau$ gives access to the resolvent $(\Lambda+\tau^{-1}I)^{-1}$. Combining geometrically spaced noise levels with sinc-quadrature rational approximation, we obtain a sampler with $q=O((\log\kappa+\log(e\sqrt d/\delta_{\rm TV}))\log(e\sqrt d/\delta_{\rm TV}))$ smoothed-score queries for total variation error $\delta_{\rm TV}$, improving the condition-number dependence from $\sqrt{\kappa}$ to logarithmic. We also study finite-bit gradient oracles. Using coordinatewise quantization of the transformed smoothed-score answers and a final dithering step, we obtain a sampling scheme whose total communicated gradient information is polylogarithmic in $\kappa$; in particular, for fixed dimension and accuracy, the bit complexity is $O(\log^2\kappa)$. To complement these upper bounds, we introduce a channel-synthesis, or reverse-Shannon, converse technique for sampling lower bounds. This converts total-variation simulation guarantees into communication requirements and yields an $\Omega(d\log\kappa)$ bit lower bound for $d\ge2$ and fixed total variation error below one, as $\kappa\to\infty$. Together, these results identify smoothed scores as a provably more informative oracle for sampling and give nearly matching upper and lower bounds for its finite-bit complexity.
Random forests are widely used in fields involving sensitive tabular data, but existing approaches to enforcing differential privacy (DP) typically degrade performance to the point of impracticality. In this paper, we introduce Lumberjack, a differentially private random forest algorithm that achieves substantially higher utility by constructing large random decision trees and then applying aggressive, privacy-preserving pruning to retain only sufficiently populated nodes. A key component of our approach is a novel $(\varepsilon,\delta)$-DP heavy hitter detection algorithm for hierarchical data, whose error is $O_{\varepsilon,\delta}(\sqrt{\log h})$ for trees of height $h$ and may be of independent interest. This favorable scaling enables the use of significantly deeper trees than in prior work, leading to improved expressiveness under privacy constraints. Our empirical evaluation on benchmark datasets shows that Lumberjack consistently outperforms prior DP random forest methods, establishing a new state of the art. In particular, our approach yields substantial improvements in the privacy-utility trade-off for practical privacy budgets. Our findings suggest that carefully designed DP random forests can close much of the utility gap, highlighting a promising and underexplored direction for future research.
Vector quantization via random projection followed by scalar quantization is a fundamental primitive in machine learning, with applications ranging from similarity search to federated learning and KV cache compression. While dense random rotations yield clean theoretical guarantees, they require $\Theta(d^2)$ time. The randomized Hadamard transform $HD$ reduces this cost to $O(d \log d)$, but its discrete structure complicates analysis and leads to weaker or purely empirical compression guarantees. In this work, we introduce DASHQuant (Dithered Accelerated Single-Hadamard Quantization), a provable variant of this approach. Specifically, the quantizer applies $HD$ to the input vector and adds a random dithering before quantizing, injecting additional randomness at negligible cost. We prove that this approach is unbiased and provides mean squared error bounds that asymptotically match those achievable with truly random rotation matrices. In particular, we prove that a dithered version of TurboQuant, and a special case of EDEN, achieves mean squared error $\bigl(\pi\sqrt{3}/2 + o(1)\bigr) \cdot 4^{-b}$ at $b$ bits per coordinate, where the $o(1)$ term vanishes uniformly over all unit vectors and all dimensions as the number of quantization levels grows.
The Sinkhorn--Knopp (SK) algorithm is a cornerstone method for matrix scaling and entropically regularized optimal transport (EOT). Despite its empirical efficiency, existing bounds for achieving marginal accuracy $\varepsilon$ deteriorate severely in the presence of extreme entries, bottlenecked either by the global maximum regularized cost $\eta\|C\|_\infty$ (where $\eta$ is the regularization parameter and $C$ the cost matrix) or the matrix's minimum-to-maximum entry ratio $\nu$. These bounds may be overly sensitive to rare extreme entries that do not dominate the overall matrix structure. We identify a broad structural regime in which the dependence on the magnitude of global extreme entries can be circumvented. For EOT, we introduce well-boundedness, a local mass property that captures the row- and column-wise bulk structure of the underlying matrix while maintaining robustness against scattered entrywise outliers. We prove that, governed by this structural condition, SK converges to the target transport plan for a problem of dimension $n$ in $O(\log n + \log(1/\varepsilon))$ iterations. This complexity bound is independent of the maximum regularized cost $\eta\|C\|_\infty$. A computationally negligible pre-scaling step eliminates this dimensional dependence, reducing the iteration complexity to a strictly dimension-free $O(\log(1/\varepsilon))$. Beyond EOT, we establish a sharp phase transition for general $(\boldsymbol{u},\boldsymbol{v})$-scaling governed by a critical matrix density threshold. We prove that when a matrix's density exceeds this threshold, the iteration complexity is strictly independent of $\nu$. Conversely, when the density falls below this threshold, the dependence on $\nu$ becomes unavoidable; in this sub-critical regime, we construct instances where SK requires $\Omega(n/\varepsilon)$ iterations.
Modern data workflows are inherently adaptive, repeatedly querying the same dataset to refine and validate sequential decisions, but such adaptivity can lead to overfitting and invalid statistical inference. Adaptive Data Analysis (ADA) mechanisms address this challenge; however, there is a fundamental tension between computational efficiency and sample complexity. For $T$ rounds of adaptive analysis, computationally efficient algorithms typically incur suboptimal $O(\sqrt{T})$ sample complexity, whereas for arbitrary distributions, algorithms with $O(\log T)$ sample complexity are computationally intractable under standard cryptographic assumptions. In this work, we shed light on this trade-off by identifying a natural class of data distributions under which both computational efficiency and logarithmic sample complexity in $T$ are achievable. We propose a computationally efficient ADA mechanism that attains $O(\log T)$ sample complexity, optimal in its dependence on $T$, when the data distribution is dense with respect to a known prior. This setting includes, in particular, feature-label data distributions arising in distribution-specific learning. As a consequence, our mechanism also yields a sample-efficient (i.e., $O(\log T)$ samples) statistical query oracle in the distribution-specific setting. Moreover, although our algorithm is not based on differential privacy, it satisfies a relaxed privacy notion known as Predicate Singling Out (PSO) security (Cohen and Nissim 2020) whenever the number of its possible update transcripts is polynomial in the sample size. Our results thus reveal an inherent connection between adaptive data analysis and privacy beyond differential privacy.
We present a unified quantum sensitivity sampling framework that streamlines coreset construction for many classical approximation problems, significantly speeding up clustering, regression, and low-rank approximation. We assume $n,d\geq\mathrm{poly}(k/\epsilon)$ and, for regression, $n\geq\mathrm{poly}(d)$: - $k$-median/$k$-means: For $n$ points in $\mathbb{R}^d$, we build an $\epsilon$-coreset of input points of size $\tilde O(k^{4/3}\epsilon^{-2})$ (resp. $\tilde O(k^{3/2}\epsilon^{-2})$) in time $\tilde O(\epsilon^{-1}n^{0.5}dk^{1.67})$ (resp. $\tilde O(\epsilon^{-1}n^{0.5}dk^{1.75})$). Our better $d$- and linear $1/\epsilon$-dependence improve on [Xue, Chen, Li and Jiang, ICML'23] for $d\geq\Omega(k^{7/3})$ (resp. $\Omega(k^{2.5})$). - $\ell_p$ regression: For $\min_x\|Ax-b\|_p$, $A\in\mathbb{R}^{n\times d}$, we build an $\epsilon$-coreset of size $\tilde O_p(d^{\max\{1,p/2\}}\epsilon^{-2})$ in time $\tilde O_p(n^{0.5}d^{1.5}\epsilon^{-1})$ for $p\leq 2$ and $\tilde O_p(n^{0.5}d^{p/4+1}(\epsilon^{-1}+d^{0.5}))$ for $p>2$, matching [Liu and Ji, ICML'26] for $p\leq 2$ and improving on the prior best quantum sampling approach [Apers and Gribling, SICOMP'26] for all $p>2$. - Low-rank approximation: We give the first quantum sublinear-time algorithm for Frobenius-norm rank-$k$ approximation of $A\in\mathbb{R}^{d\times n}$ free of data-dependent parameters, running in $\tilde O(n^{0.5}dk^{0.5}\epsilon^{-0.5})$ time with factored output (plus $\tilde O(nk^2\epsilon^{-2})$ to materialize them). We also give sublinear algorithms for kernel and tensor variants. - Lower bound: For generic sensitivity sampling with total sensitivity $S$, any quantum algorithm outputting an $\epsilon$-coreset of size $o(\sqrt{nS})$ needs $\Omega(\sqrt{nS})$ queries, so our query complexity is optimal up to $\mathrm{poly}(\epsilon^{-1},\log n)$ and VC-dimension factors.
We study the problem of differentially private continual counting in the unbounded setting where the input size $n$ is not known in advance. Current state-of-the-art algorithms based on optimal instantiations of the matrix mechanism cannot be directly applied here because their privacy guarantees only hold when key parameters are tuned to $n$. Using the common `doubling trick' avoids knowledge of $n$ but leads to suboptimal and non-smooth error. We solve this problem by introducing novel matrix factorizations based on logarithmic perturbations of the function $\frac{1}{\sqrt{1-z}}$ studied in prior works, which may be of independent interest. The resulting algorithm has smooth error, and for any $\alpha > 0$ and $t\leq n$, it is able to privately estimate the sum of the first $t$ data points with $O(\log^{2}(t)(\log\log t)^{1+2\alpha})$ variance. It requires $O(t)$ space and amortized $O(\log t)$ time per round, compared to $O(\log(n)\log(t))$ variance, $O(n)$ space and $O(n \log n)$ pre-processing time for the nearly-optimal bounded-input algorithm of Henzinger et al. Empirically, we find that our algorithm's performance is also comparable to theirs in absolute terms: with optimized parameters, our variance is less than $2\times$ theirs for $t$ as large as $2^{32}$.
We study planning for finite-horizon partially observable Markov decision processes (POMDPs) with finite state, action, and observation spaces. Both exact and approximate planning are known to be computationally hard in this setting, so prior work has explored natural assumptions that make planning easier. Recent work [Golowich et al., 2022] gave quasipolynomial-time approximate planning algorithms for observable POMDPs. These are POMDPs whose observation channels satisfy an $\ell_1$-observability condition, but notably, no assumptions are required on the transition dynamics. In a similar vein, for every fixed $\rho>0$, we give quasipolynomial-time approximate planning algorithms for $\rho$-decodable POMDPs. We show that decodability is implied by both $\ell_1$- and $\ell_2$-observability and is strictly weaker than observability. Moreover, we show that the optimal decodability constant can be computed in polynomial time, whereas computing the optimal $\ell_1$-observability constant is NP-hard in the worst case. We also show that the fast filter-forgetting guarantee underlying the approach of [Golowich et al., 2022] does not extend directly from $\ell_1$-observability to constant $\ell_2$-observability. Thus, decodability is an easy-to-check, unifying assumption under which quasipolynomial-time approximate planning for POMDPs is possible.