A two-term filter of the last 32 order arrivals can beat sequence networks at quoting, provided orders never move the midprice. Acero, Plank and Zhang make a convincing case within that simulator. The boundary of the result matters as much as the win.

The filter behind HSN

The setting is Avellaneda-Stoikov with clustered order flow. Independent Hawkes processes generate buy and sell market orders; each arrival raises the rate of future arrivals through an excitation kernel g. The market maker chooses bid and ask spreads in [0.005, 0.30]. An order arriving on either side fills with probability exp(-10δ). Terminal cash plus inventory valued at S_T - κQ_T forms the reward, less a running penalty γ times the integral of Q². The paper uses γ = 0.10 and κ = 0.001, with inventory capped at |Q| ≤ 5. A forecast of the next burst helps the maker choose when tighter quotes will earn enough fills to justify the inventory.

The Hawkes-Structured Network (HSN, the authors' architecture) compresses that history. On each side, it maintains p sums: α_j times Σ exp(-β_j(t - τ_i)), taken over the last 32 event timestamps. In the reward experiments p = 2. A single asset thus gets four values, with a faster and a slower decaying sum on each side. PPO learns α and β along with the rest of the policy. Price, inventory and time remaining join those sums as inputs to a two-hidden-layer MLP. Its output is the mean of a Gaussian over spreads.

There is a mathematical reason to use exponentials. A completely monotone kernel can be written as an integral of them; finite quadrature then makes the control problem Markov in the sums HSN carries. The power-law kernel is where this device matters most. The paper cites a tail index of about 0.15 for S&P futures.

Where does the reward lead hold?

HSN wins in the five configurations where intensity should affect quoting. In each, its strongest rival is an MLP fed a raw stack of the last 8 events:

The paper's two single-asset exponential controls end in ties. The frame-stack has the higher numerical reward at both branching ratios: 0.0184 vs 0.0171 at n = 0.75, and 0.0211 vs 0.0208 at n = 0.90. The authors expected this and retained the columns as a control. Their explanation is that the leading-order optimal spread changes little with intensity in this regime. On those terms, the filter adds nothing over the raw 8-event stack.

Yet memory pays in those same columns. HSN beats every other architecture there, including the no-memory Simple MLP, by 45 to 90%. At n = 0.75, the frame-stack roughly doubles the Simple MLP (0.0184 vs 0.0090). The Simple MLP trails HSN by 36 to 57% in the five intensity-sensitive columns, so the value of history is at least as large in the controls. A nearly intensity-independent optimal spread sits uneasily with a no-memory policy losing about half its reward. The paper leaves that tension open.

For ε = 0.15, the GRU earns 0.0148, the LSTM 0.0102 and the Transformer 0.0105; the 8-event stack earns 0.0163. All architectures share PPO settings and receive 100 updates of 1024 steps, roughly 102k environment steps. Some of HSN's advantage over the recurrent and attention networks could reflect training speed under one shared hyperparameter set and a short budget. Table 1 gives the frame-stack comparison in its bottom row: significant gaps of 5.3% to 32.5%, alongside two ties. The abstract's "outperforms general-purpose neural architectures" and the conclusion's "consistent improvements" reach beyond the two single-asset exponential ties. We did not find a multiple-comparison correction for the pairwise Welch tests. Two of the five wins have p < 0.01 rather than p < 0.001.

The regret bound and the trained policy

Theorem 1 concerns a policy optimal under a truncated kernel, run in the true model. Its regret bound scales with the square root of the kernel's L2 approximation error. Halving that error therefore tightens the bound by only about 29%; the constant C depends on model parameters. The paper extends the claim to HSN "up to RL training error". Section 4.2 argues that, given large p and a sufficiently expressive MLP, the HSN policy class can mimic greedy policies, so "the policy learned by model-free RL can achieve performance at least comparable to these greedy policies". This establishes a case about representation. The tables contain a PPO-trained policy, whose distance from the truncated-kernel optimum is never measured. The L = 32 limit on the sums also falls outside Theorem 1. The paper says truncation comes "without material loss of information" but supplies no bound for it.

The intensity results require a similarly careful reading. Using HSN's PPO-tuned amplitudes directly gives an intensity MSE of 0.20 on a single-exponential kernel, close to the LSTM probe's 0.21. On a three-mode kernel, HSN scores 0.53, behind the LSTM at 0.31, the Transformer at 0.40 and the frame-stack at 0.43. The authors say PPO tunes α for reward. With the same OLS probe given to every baseline, HSN's held-out MSE drops below 10^-6. That explanation works for the quoting policy: its MLP can rescale the inputs anyway. The conclusion calls the connection between task-oriented representation learning and statistical recovery of the Hawkes process an open question, even as the abstract presents intensity recovery as a finding. We could not square the p-sweep, in which the three-exponential kernel reaches its floor only at p ≥ 6, with the probe's 10^-6 fit at p = 3. Settings differ across those experiments (n = 0.95 against 0.85, and T = 10 in the sweep), which may account for the discrepancy.

Orders leave the mid untouched

The model includes a per-order price-impact term, ε±. The experiments set it to zero: "a pure diffusion midprice, so that executed trades do not directly move the midprice". Hawkes intensity changes fill rates and nothing else. A burst of buys brings more fills at the same mid. On a real book, such a burst can call for wider quotes. The toxicity a desk worries about is switched off.

HSN earns rewards from 0.0171 to 0.0561 (T = 0.5, or 1.0 in the regime-switching runs). The fill model has no fees, queue position or latency.

We could not test any of this ourselves. HSN requires side-labelled arrival timestamps and fills that respond to the quotes we post. One-minute OHLCV bars provide neither.

I would want a rerun with ε± above zero. The authors identify richer market microstructure models as future work, so they acknowledge that direction. A surviving gap over the frame-stack would make the filter a genuine inductive bias for quoting. A vanished gap would leave a fill-rate forecaster tested in a market without informed flow.